Masonry Construction Structural Mechanics and Other Aspects -Jacques Heyman

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M SONRY CONSTRUCTION



M SONRY

CONSTRUCTION

Structural Mechanics and Other Aspects

Edited by

C. R. CALLADINE

Dept.

o

Engineering University o Cambridge u K

Reprinted from

Meccanica

Volume 27,

No.3

1992)

SPRINGER-SCIENCE BUSINESS

MEDIA, B.Y.



LIbrary

of

Congress

Cataloging In Publication

Data

Masonry

constructlon

structural mechanics and other aspects

ed1ted

by

C.R.

Calladine.

p.

em

ISBN 978-90-481-4172-2 ISBN 978-94-017-2188-2 eBook)

DOI 10.1007/978-94-017-2188-2

1. Masonry.

2. Structural analysis Engineering>

1. Calladine.

C. R.

TA67

.

M34

1992

624. 1 83--dc20

ISBN 978-90-481-4172-2

92-18221

Printed on acid free paper

All Rights Reserved

© 1992 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1992

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TABLE OF CONTENTS

Editorial vii

JACQUES HEYMAN / Leaning towers 53

R.K. LIVES LEY / A computational model for the limit analysis o three-dimensional masonry structures

6

SALVATORE DI PASQUALE/New trends in the analysis o masonry structures 73

MARlO COMO / Equilibrium and collapse analysis o masonry bodies 85

GIULIANO AUGUSTI and ANNA SINOPOLI / Modelling the dynamics o large block structures 95

ROBIN SPENCE and ANDREW COBURN / Strengthening buildings o stone masonry to resist earthquakes 213

FRITZ WENZEL and HELMUT MAUS / Repair

o

masonry structures 223

J.E. HARRIS / Weathering o rock corrosion o stone and rusting o iron 233



J CQUES HEYM N



This special issue of

Meccanica

contains eight papers

on

the common theme of

Masonry Construction: Structural

Mechanics and Other Aspects. Its appearance coincides

with the retirement of

Dr

Jacques Heyman from his

Professorship of Engineering in the University of Cam

bridge, and the Headship of the University's Engineering

Department.

t is,

of course, entirely appropriate to have a collection

of

papers in honour

of

Professor Heyman at this time; for

he has made signal contributions to our understanding of

masonry construction over the past thirty years or so. It is

no exaggeration to say that he has radically changed the

way in which Engineers think about masonry structures,

particularly in relation to old ecclesiastical buildings and

bridges; and indeed it

is

hard to imagine what this subject

would be like today in the absence of Professor Heyman's

seminal papers.

Heyman was a jun ior colleague of J. F. Baker, working

as a member of the Cambridge team which made

enormous contributions to the rational design of struc

tural steelwork; and he was in his early thirties when The

Steel Skeleton

Volume II:

Plastic Behaviour

(Baker, Horne

and Heyman) was published in 1956. He made many

contributions to that importan t work, of which some were

the fruits of his studies as a post-doctoral worker with

Professor Prager at Brown University. One of Prager's

achievements was to show how the three distinct sets of

relations in solid

mechanics the

equations of equilibrium

and compatibility, and the constitutive relationships

of

the

material - which are fused together into a single governing

equation in the classical theory of elasticity, combine in

quite other ways in the field of plasticity (mainly on

account

of the strong non-linearity and irreversibility of

the material). This fact requires us to adopt fresh and

different ways of thinking from those of classical elasticity

if we are to understand the behaviour of structures in the

plastic range.

Heyman saw that this new paradigm of structural

thinking provided a rational way of understanding mas

onry construction; and indeed that the two principal

theorems

of

plastic theory - appropriately adapted

provided useful tools for the analysis of masonry arches

and vaults.

Professor Heyman is also a considerable scholar, and he

has become a respected authority on mediaeval- and

earlier - writings on construction. His combination of

analytical power and scholarship, together with con

siderable practical engineering experience, has made him a

unique figure in relation to problems with old buildings.

Thus, he is much in demand by Deans and Chapters when

their Cathedrals are showing signs of distress; and indeed

he is a worthy successor to those upon whom Deans and

Chapters called in ages past.

Meccanica 27: vii-viii, 1992

©

1992

Kluwer Academic Publishers.

This, then,

is

the background to the present special

issue of M eccanica on Masonry Construction.

t

is

appropriate that the first chapter should be by

Professor Heyman himself. Here we have an illuminating

essay on the stability of leaning towers. This piece of work

had its origin in the analysis of a long mediaeval wall at

Peterhouse which was near the poin t of collapse in 1976.

I

can claim a little personal credit here, for one day as 1

walked past this wall 1 noticed a slight movement near the

ground. Closer examination revealed a field-mouse scurry

ing along a horizontal fissure

in

the wall at

~ r o u n

level;

and then 1 saw that the crack ran for many yards along the

wall. So 1 reported the matter to Professor Heyman.)

The next paper is by Dr Livesley, who has collaborated

with Professor Heyman at Cambridge for many years

on

computer algorithms for analysing the load-carrying

capacity of masonry arch bridges according to the well

known limit theorems . Here he moves from the con

ventional two-dimensional representation

of

an arch into

three dimensions; and in consequence he engages some

complex kinematic issues which involve not only the

formation of hinges

or

pivots between adjacent blocks, but

also rotational sliding of blocks over each other, which

demands consideration

of

frictional, dissipative effects.

The paper by Professor Di Pasquale, of Florence,

investigates in detail the general problem of calculation of

stresses and strains under plane-stress conditions for a

body

or

structure made from an elastic, no-tension

material. Three different kinds of sub-domain emerge in

the analysis, and the boundaries between them are not

known a priori. The paper includes some illustrative

examples, and some

of these are investigated by means of a

finite-element scheme.

Professor Como, of Rome, is also concerned with a

general analysis

of

structures made from no-tension

material. He studies the collapse state of a general body

under load, with the aim of clarifying the transposition of

the classical theorems for a plastic body to the new

situation of a masonry-like continuum. He succeeds

in

proving several theorems in relation to the collapse state

of

the body; and in particular he establishes kinematical

and

statical theorems for failure of a masonry structure.

Professor Augusti, also of Rome (who was formerly a

research student of Professor Heyman in Cambridge), has

collabora ted with Professor Sinopoli of Venice to produce

our

next paper,

on

the subject of blocks of masonry under

seismic loading. When

we

consider the behaviour

of

a

single block which stands on a shaking floor,

we

must

enter the world of dynamics, of course; and moreover this

is

a highly non-linear world, which involves impact and,

sometimes, bouncing. The paper includes a survey of much

recent work in this important area.

Drs Spence and Coburn, both of Cambridge, are also



Vlll

EDITORIAL

concerned with the behaviour

of

masonry structures in

earthquakes. They focus

on

the performance of a plain

masonry house, o f a kind widely used in Turkey, and on a

simple and inexpensive scheme for strengthening it. A

kinematic upper-bound analysis of one side of the house

under horizontal loading furnishes some clear design

guidelines.

The paper by Professor Wenzel, of Karlsruhe, brings us

to the specific field of res toration of old masonry buildings.

Among other things he is concerned with the practice of

stitching masonry blocks together by drilling, inserting

tie-bars and grouting. Old buildings have often stood for

centuries with their constituent stone blocks held together

mainly by gravity

and

friction. But uneven ground settle

ment over the years can lead to patterns of fissures

between the individual blocks; and the use of tie-bars and

grouting, by making the arrangement more monolithic,

can substantially improve the structural performance

of

the building.

The final chapter, by Professor Harris, broaches another

important topic in masonry construction: corrosion of the

stone by atmospheric effects. Here

we

are in the realm

of

the materials scientist; and we learn, for example, that the

corrosion and weathering

of

stone in buildings is on the

same pattern as the weathering

of

exposed rocks on the

geological timescale. Corrosion of metal inserts and ties is

also an important problem in masonry structures.

The obligation

of

the guest editor of a special issue of a

Journal is largely to provide a sense of cohesion between

the various individual contributions.

y

task is much

easier than tha t

of

Professor Wenzel, for I have no need to

insert artificial ties between the various parts, since the

coherence has already been provided by the occasion itself.

Each of the papers in this issue pays tribute, in its own

distinctive way, to the importance of Professor Heyman's

work in the field of Masonry Construction.

C R

CALLADINE

Guest Editor

University o Cambridge



LEANING TOWERS

JACQUES

HEYM N

University

of

Cambridge Department of Engineering

Trumpington Street Cambridge CB2 IPZ England

(Received: 1 Februar y 1992)

ABSTRACT, Masonry, as a unilateral material, can resist compressive stresses but has feeble tensile strength. A

masonry wall or tower, subjected to uneven foundation settlements, will crack,

and

these cracks may lead to overall

structural collapse. Of particu lar interest is the leaning tower, in which a fissure (of a shape t o be determined) develops

progressively up to the point of critical stability. A practical rule is developed for the maximum incl ination that may be

regarded as safe for a masonry tower.

SOMMARIO. La murat ura, materiale a compor tamen to unilaterale, puo soppor tare sforzi di compressione ma ha

llna modesta resistenza a trazione. na parete od una torre in muratura soggette a cedimenti fondali

si

fessurano e

tali lesioni possono condurre al collasso globale. Un caso di particolare interesse e quello di una torre inclinata nella

quale

una

lesione (la cui forma e da determinare) si sviluppa progressivamente fino al

punto

di stabilita critica. Nel

presente studio

i

sviluppa un criterio pratico di sicurezza per la massima inclinazione ammissibile per una torre in

muratura

KEY WORDS: Masonry, Unilateral material, Cracking, Settlement, Collapse, Mechanics of masonry.

1

INTRODUCTION

There is some fascination in the contemplation of the fall

of 13000 tonnes of masonry. Certainly the collapse of the

Campanile in Venice, on

14

July 1902, received extensive

contemporary discussion and analysis [1], particularly

since the final phase, from the instant

at which it was

known that the tower would collapse to the actual event,

lasted 3 days and

19

hours. The collapse was, indeed,

closely observed, but it was not apparently accompanied

by any tilt of the tower; rather, fissures were seen to widen,

and the final pile of rubble offered few clues as to the cause

of the defects.

t

is

equally fascinating to observe leaning towers which

are nevertheless stable; such towers are discussed below.

Perhaps the most famous is the campanile of Pisa, but

there are many other examples in Italy, particularly in

Venice and in the islands of the lagoon. A leaning tower

is

evidence of some geotechnical phenomenon; foundations

have given way to promote the tilt. However, towers may

be distressed without tilting, and there are many examples

of collapse having occurred within a few years of com

pletion

of

the work, for example

at

Winchester, Gloucester

and Worcester, and at Beauvais (twice). Occasionally

shores have been hastily inserted, as with the spectacular

strainer arches

at

Wells,

or

with the internal raking

buttresses at Gloucester.

The soil-mechanics timescale for consolidation of soil

within

an

area

10

or

15

m square

is

a decade

or

so. Those

towers tha t have survived this initial period may be seen to

have settled, by up to say 300 mm, with respect to the

surrounding masonry; the settlement was apparently uni

form during their first

20 years, and thereafter they were

reasonably assured of a stable existence, Some, however,

were not; the crossing tower at Ely collapsed in 1322 two

Meccanica

27: 153-159, 1992

1992


centuries after it had been built (and was at once replaced

by the present octagonal lantern), and the crossing tower

at Chichester collapsed in 1861 after seven centuries of

seemingly comfortable existence [2].

None of these towers, whether standing or fallen, is

reported as having tilted, t would seem that some other

defect

is

engendered by uneven settlement, and in this

respect the vertical fissures known to have been present in

the Campanile in Venice may provide some clues, Certain

ly, examination

of

the foundations after the fall of 1902

revealed that geotechnical failure was a most unlikely

cause for the collapse; although average bearing pressures

were high at about 600

k m

2

the piled clay had success

fully carried the load for several centuries, The Campanile

has been in existence for over a thousand years, but early

records refer to several fires and partial destruction, More

recently, the structure was struck by lightning in 1388, and

again in 1417 and in

1489; on this last occasion the

structure was virtually ruined [3]. Lightning again

damaged the tower severely in 1548, 1565 and

1653;

in

1745 it was almost destroyed, and 37 fissures had to be

repaired, Further damage was sustained in thunderstorms

in

1761

and

1762; in 1766, however, a Franklin lightning

rod was installed, and the Campanile had a more comfor

table existence until 1902,

(A

similar Franklin rod was

installed in Wren s St Paul s in 1769,)

A question of prime interest is why lightning should

distress a masonry tower, and perhaps cause collapse, The

answer

is

to be found in the consideration of masonry as a

unilateral materiaL The

full

theoretical consequences of

the unilateral properties of masonry will not be explored

here; considera tion is given to these in other contributions

to this journaL In broad, and not very rigorous, terms, it is

prudent and convenient to regard masonry as an as

semblage of dry stones (or bricks

or

other similar material),



154

JACQUES HEYMAN

some squared

and

fitted and some not, placed one on

another to form a stable structure. Mortar may have been

used to fill interstices, but this mortar will have been weak

initially, and will have decayed with time,

and

cannot be

assumed

to add

strength to the construction. Stability of

the whole is assured, in fact, by the compaction under

gravity of the various elements; a general state of com

pressive stress can exist,

but

only feeble tensions can be

resisted.

n accordance with this simple view of masonry

construction, it will be assumed that compressive stresses

are very low, so that there

is

no danger of crushing of the

material,

and that

tensile stresses cannot be developed.

(This

is

the unilateral model; the material can resist

compression, but has zero tensile strength.) t is evident

that this view of the material is imprecise.

For

example, it

is easy to envisage a dry stone wall in which the stones can

indeed be lifted away, but which, in the absence of

interference, will retain its structural shape. The stones

must, however, have a certain shape

and

be capable of

resisting lateral forces, by interlocking or by friction; an

attempt

to

build a vertically sided wall from small particles

(sand) will be unsuccessful. Coherence depends on compo

nents of the masonry structure having individual tensile

strength, even though the structure as a whole has none.

Medieval walls often have a thickness of

up

to

2

m or

more. Stone blocks used for building might have leading

dimensions of

about

300 mm, although occasional larger

through stones would be used for a wall say 500

mm

thick in order to provide the necessary coherence for the

structure as a whole. Stones larger

than

500 mm, a fraction

of the size of Greek monoliths, hardly exist in medieval

construction; the whole

art

of Gothic lies in the erection of

enormous structures using only small building blocks.

Thus the structure of a wall of thickness 1 m o r more

usually consists of oute r

and

inner skins of good coursed

masonry, with the central void filled with rough-cut

(rubble) masonry and mortar. Fissures can develop in this

centre

fill and

there

is

a tendency for the two skins of the

wall

to

drift apart. A square tower may be regarded as an

assemblage of four such walls; the internal skins are

constrained by each other to remain more

or

less in place,

but

there is nothing except internal tensile strength to

prevent the outer surfaces of the tower from moving. t is

for this reason that iron plates may be seen at various

levels at the corners of

many

existing towers; internal ties

connected to these plates restrain the outer faces at the

four corners

of

the tower.

The type of vertical cracking

just

discussed lies para llel

to the faces of the wall. Vertical fissures can also occur in a

perpendicular plane, through the thickness of the wall. A

material with no tensile strength has no shear strength,

and

such vertical cracking can be

promoted

by slightly

uneven foundation settlement taking place during

consolidation of the soil within the first decade of the

completed work. Figure 1 shows schematically the plan of

Fig. . Vertical cracking in the walls of a tower (schematic). The drift

apart

of the skins can lead to cracks in the rubble

fill;

cracks through the

thickness of the walls can lead to the isolation of a corner of the tower.

a corner of a tower, in which cracks in the rubble fill are

intersected by settlement cracks of the type envisaged.

t

will be seen that a corner of the construction has become

detached; should the corner be carrying a substantial load,

then there is

an

obvious opportunity for instability

to

occur.

Professor Willis s account [2] makes it clear that some

thing of the

sort

preceded the collapse of the crossing

tower

at

Chichester in 1861. Equally, a vertical crack had

existed at the corner of the Campanile for at least a

century,

and

it was the widening of this crack over nearly

four days

that

led to the final downfall.

t may be noted that there is another possibility for

promotion of vertical fissures in masonry. The well-known

cylinder test for concrete specimens makes use of the fact

that vertical compressive stresses applied to a block of

material can generate horizontal tensile stresses. Thus the

2

m thick walls of the Campanile are stressed

at

ground

level to, say, 1.5 N/mm2 in vertical compression,

and

corresponding tensile stresses would potentially be gen

erated. n fact, the existing crack which led

to

the collapse

connected a series of eight windows placed in the corner of

the tower, so that there was an easy path along which a

fissure could develop.

However vertical cracks are formed, whether induced by

settlement or by gravity-generated tensile strain, it seems

likely

that

they can lead

to

locally unstable elements

of

the

tower and can provoke overall collapse. Such collapse can

be prevented as, for example, has been noted

at

Wells,

where buttressing was applied in good time; in contrast,

the

153

m crossing tower

at

Beauvais, completed in 1569,

collapsed in 1573 some 13 days after work was put in hand

to save it. Fairly uniform settlement may not lead

to

dangerous cracking, and a tower may become relatively

safe after the initial consolidation period of a decade or so.

Alteration of the water table can, however, lead

to

a later

new high-risk period,

and

the collapses

at

Chichester

and

Ely may have been initiated by interference of this sort.

Vertical fissures will become wet in thunderstorms, and

will provide good conducting channels for the 30000

amperes associated with a lightning stroke. The corre

sponding rise in temperature can be 15000 DC and pres-



LEANING TOWERS

155

sures generated by the virtually instantaneous production

of steam can damage severely the overall fabric of a tower

[3]. (In the same way the sap-wood channel, acting as a

conductor, can lead to the explosion of a tree.)

In

the case of the Campanile in 1902, the immediate

cause of collapse was not a lightning stroke. Rather it

would seem that an existing fissure,

or

fissures, had

widened inexorably to the point where the tower became

unstable. An alternative reason for the collapse was put

forward by Alban Caroe [4J, although he gives no detailed

explanation to support his hypothesis. In a brief chapter

on belfries, bell-frames and bells in his book on old

churches, he makes the statement:

From

time to time a

request

is

put forward that a urinal shall be arranged in a

medieval church tower. Many of the dangers which must

be guarded against in any such provision are obvious, but

it

is

worth remembering that it was insanitary use of this

kind which caused the collapse of the great campanile of St

Mark at Venice. This paragraph, quoted in its entirety, is

not further expanded by Caroe.

2. LEANING TOWERS

A rectangular block, of height and width

b

may be tilted

on its base until the centre of gravity is vertically above one

corner; slight further movement will cause the block to

overturn. The cosine of the critical angle of tilt will be

approximately unity for a block whose height to width

ratio is say 4 or more, so th at the critical displacement, the

lean of the block,

is

equal to its width. The calculations

are

not

so simple for unilateral masonry.

Figure 2 reproduces the contractor s sketch of a

medieval boundary wall of Peter house, Cambridge. I t will

be seen that an inclined fissure has developed; because of

the unilateral nature of the material,

part

of the wall

remains attached to the base, and the fracture defines a

stress-free boundary whose shape

is

initially unknown.

Because of the fissure, the wall

is

in a potentially more

dangerous state than a solid block tilting about a corner.

The width of the wall

is

about 21 in, and the actual lean in

June

1981

was

14in

. The following analysis leads to an

equation for the profile of the fracture, and determines the

maximum inclination of the wall for stability; for the

Peterhouse wall this maximum lean

is

found to be

15

in.

Remedial buttressing was installed in time to prevent

collapse (and without waiting for the calculations to be

completed).

It

may be noted that the unsupported height

of the wall in Figure 2 is 104 in, so that the ratio H/b is

almost exactly 5.

Figure 3 shows a block of masonry of height

a

tilted to

such an angle IX that the support force acts just at the limit

of the middle third of the section. At this condition,

according to simple elastic theory, the left-hand bottom

corner will be just free of stress,

and

the block will be

supported by linearly increasing compressive forces along

/ -

.

\.

I .

~

'-8'

PETERHDUSE. CAMBRlDG L

e « < ~ thro u < i ~

~

\

w \ 0

P\lC• n 9ft

Fig. 2. A leaning masonry wall near to the point of collapse.

Fig. 3. A tilted block of masonry.

the bottom surface. Evidently

fa

tan

IX =ib, or

1 b

tan IX = .

a

(I)



156

JACQUES

HEYMAN

Fig.

4.

A tilted block for which the support force falls outside the middle

third of the original width of the wall. A fracture has developed cf. Figure

2) whose shape is to be determined.

Figure 4 shows part of a taller wall at the same inclination

ex,

for which a fracture has developed

at

the stress-free

surface,

and

some masonry has fallen away. At the general

section, distant

X

for the origin, the total weight

W

of the

masonry

is

supported by a force again acting

at

the limit of

the middle third of a base of reduced dimension

Y

Figure 5

shows

an

elemental slice of the wall of width

Y and

thickness

dX;

without loss of generality, the calculations

may be made for unit weight of material,

and

the weight

d

W

of the slice is given by d

W

=

Y

dX, or

f

moments are taken

about

for this slice, then

dW(tY cos ex-tdX sin

ex

+ W(tY cos ex-dX sin

ex

= (W +dWWY +dY)cosex,

so that, neglecting products of infinitesimal quantities,

2)

1 dW 1 dY

-

Wtanex

W . 3)

dX 3 dX

But, from Equation 2),

dY dY dW dY

d X = d W d X

=Y

dW

Fig. 5. An elemental slice of masonry.

so

that

Equation

3)

becomes

y2 _

W

tan ex =

WY

d

Y

6 3 dW

The non-dimensional variables

4)

may be introduced, and, using Equation 1), Equation 4)

becomes

y2 = 2w 1+ y ) .

5)

Equation

5)

may be simplified by making the substitution

so

that

dz

z

=

2w+w dw

or

dz 1

Z=

-2 .

dw w

This has the solution

z =

w(C-2log

w ,

or

y2 = w(C-2 log w ,

6)

7)

8)

9)

where C is a constant of integration. From Figure 4 it may

be noted

that

the crack starts

at

X

=

a,

Y

=

b,

for which

condition

W

=

ab;

that is,

Equation 9) must satisfy the

condition x =

1,

y =

1,

w =

1.

Thus C = 1, and

y2 = w(I-2log w .

10)

Now, from Equation

2),

y = dw/dx, so

that

Equation

10)

becomes

dw

dx = jw(I-2logw),

11)

that

is,

x = D +

f

w dw

1

jw(1-2 log w)

12)

where

D

is a second cons tan t of integrat ion, whose value is

determined as unity from the condition x = 1, w = 1.

Equation

12)

may be solved in terms of a parameter t,

where

1 - 2 log w = 4t

2

;

13)

making this substitution, Equation 12) becomes

r

/2

x=I+2e

l

Jt e-t dt.

14)

The integral

is

with the factor

2/Jn) that

of the error

function

erft,

so

that

Equations 13),

14) and 10)

lead to



LEANING

TOWERS

157

the final results

(15)

These expressions are valid for the parameter

t

in the range

O:;;;t:;;;t.

The shape of the fissure, corresponding to Equations

(15),

is plotted in Figure 6; this non-dimensional sketch for

the parameters x and

y

is plotted for

a

= b, and may be

stretched linearly to correspond to the parameters X and

Y for the real tower (cf. Figure 12(a»; it

is

thus valid for all

ratios a/b.

The maximum height of the tower (for the given lean IX

is

given by the condition

that

the crack penetrates through

the whole width of the tower,

that is,

by the condition

y = 0, for which the parameter t is also zero. Thus from the

second of Equations

(15),

the dimensionless height h of a

tower

that

is just becoming unstable

at an

inclination IX

is

given by

h

= [ x ] t ~ =

1

+ In

e

1

4(erf1 ,

and

since erf1= 0.5205,

h = 1 + 1.1846 = 2.1846. (16)

Thus, for the actual tower the height

H

is related

to

the

width b by

H h

1

)

=

b

= 3 cot X (2.1846) = 0.7282 cot IX,

that

is,

0.7282

tan X =

H/b) .

(17)

At this condition, the out-of-plumbness of the top of the

tower with respect to its base, i.e. its lean ,

is

H sin

IX, and

since sin IX tan IX for small angles, the maximum lean is

0.728b (compared with

b had

the tower been a solid block

Fig. 6. An accurate plot (non-dimensional) of the shape of fracture of a

wall on the point of collapse, drawn for a = b (see Figure

4).

The figure

may be stretched to give the shape for any ratio a/b.

Fig. 7. Development of fracture as a wall

is

tilted progressively

(height/width ratio of

5).

turning about one corner). As an example, for

H/b

= 5 (the

Peterhouse wall), the value of tan

X

is 0.146 (sin

X

=0.144),

and

IX = 8.3°.

Figure 7 shows how cracking develops as an inclination

is

progressively imposed on a block of masonry. The

sketches are for the same value

H/b

= 5, and at an

inclination of 3.8

0

tension is

just

reached in the outer

surface. Further tilting causes a fissure to develop, the

shape o f which

is

a portion of

that

of Figure 6,

and

this

fissure moves progressively through the fabric until the

block overturns. The last sketch in Figure

7

is identical

with Figure 12(a), which

is

in

turn

a stretched form of

Figure 6. Figure 8 shows schematically the fissure at some

intermediate state; a slight further inclination will transfer

some of the bricks

just

above the fissure

to

the passive pile

just

below, where they can play

no

further part in con

tributing to the stability of the masonry.

3.

PPROXIM TE SOLUTIONS

The fracture curve of Figure 6 is fairly straight,

and

suggests

that

an approximate analysis could be made on

the assumption that the crack is linear; such an analysis

would be safe, since the actual fracture is convex down

wards. Figure 9 shows a solid wall (or tower) of thickness b,

as before, inclined

at

an angle

IX; just

as before, the crack

starts at a distance a from the top of the tower, where

a

=

tb

cot

IX.

(1

bis)

The crack

is

assumed to extend linearly to a poin t

P,

which

is at a further distance s down the wall. The various

quantities in Figure 9 must be related at the condition at

which the wall

just

becomes unstable.

I I I

1.

\

L

...

L

L

I

l..

L

'I

I

I

,

,

I

I

j

I

Fig.

8.

Schematic illustration

of

an actual fracture.



158

JACQUES

HEYMAN

Fig.

9.

Approximate analysis of a wall assuming linear cracking.

The relationship can be found simply by taking mo

ments

about

P.

For

equilibrium (and using the relation

H=s+a , it is found that

tan

2

0:

~ ~ tan

0:

_ 2 ~ ) 2

=

0

6 H 18 H

from which

b

tan

0:

=

0.7125

Ii

18)

19)

The coefficient in Equation 19) may be compared with the

exact value 0.7282 of Equ ation 17).

A real tower consists, of course, of four walls sur

rounding a central hollow core; moreover, the wall thick

nesses usually diminish towards the top. t is not possible,

therefore, to derive a general expression corresponding to

Equations 18) or

19)

for the solid wall. However, it is of

interest to examine the stability of a tower of uniform thin

walled section. Figure 10 shows a square cross-section of

external dimension

d and

internal dimension c. The area is

d

2

- e

2

, and the corresponding section modulus is

d

4

- e

4

)/6d. Thus, using the same elastic approach as

before, cracking will start when the loading becomes

eccentric

to

the centre-line by

an amount d

2

e

2

j6d, and

for C d this has value d/3. The middle third rule for the

g

d .1

Fig.

10.

Cross-section of a hollow tower.

Fig.

11.

Approximate analysis of a hollow tower

cf.

Figure

9).

solid section is replaced by the middle two-thirds rule for

the thin-walled square box. (For a tower of usual wall

thickness the eccentric factor might be 0.29 or 0.30 rather

than 1/3.)

Thus the thin-walled tower, shown at its critical in

clination in Figure

11,

will start to crack when

a

=

ib cot 0:.

20)

The fissure is again approxi mated by a straight line in Fig.

11, and a simple analysis as before leads to

2 2 b 80 b)2

tan 0: 3

I i

tan 0: - 63

I i =

0

21)

from which

tan

0:

= 0.8418

~ ) .

22)

Figure 12 sketches the results corresponding

to

Equations

8.3

0

8.10

a)

(b)

c)

Fig. 12. Comparative solutions for the overturn of a masonry tower: a)

solid, exact; b) solid, approximate; c) hollow, approximate.



LEANING TOWERS

159

Fig. 13. Critical angles of inclination for towers of various ratios of width

to

height. A safe working rule is given by the line a= 24 bIH).

17),

19)

and

22) for a tower having

H/b

=

5. Figure 12(a)

shows the exact solution at the point where the solid tower

(or wall)

is just

overturning

at

an angle of

8.3°;

the

corresponding angles for the approximat e solutions for the

solid tower, b),

and

the thin-walled tower, c), are 8.1 °

and

9.6°. It may be noted that the overturning angle for a

cohesive (non-unilateral) tower, hollow or solid,

is 11.3

0

for

H/b= 5.

4

ON LUSION

Robert Willis gave, in 1835, a table of dimensions of the

principal Italian campaniles [5]. The ratios of height

to

base

H/b)

range from 3 (Pisa, which

is,

of course, circular)

to 12 (Torre Asinelli, Bologna). Table I gives values of

X

for

various values of H/b from Equations

19) and 22)

for the

solid

and

thin-walled towers respectively; the second two

lines give values of tan 1

tb H and

tan - 1 ~ b / H that

is,

the

angles at which the solid and hollow towers first develop

fissures.

The values of inclination in the table are very nearly linear

with

b/H,

Figure

13.

It will be seen

that

the overturning

TABLE

T

Values

of

inclination

of

tower.

a

Hlb

4

6

10

12

Overturn:

Solid.

13.4 10.1 8.1

6.8 5.1 4.1

3.4

Equation (19)

Hollow,

15.7 11.9 9.6 8.0 6.0 4.8 4.0

Equation

(22)

First crack:

Solid 6.3 4.8

3.8

3.2 2.4 1.9

1.6

Holl ow 12.5 9.5 7.6

6.3

4.8

3.8

3.2

angles for the solid and for the thin-walled towers are fairly

close; further, the finite wall dimensions of a real tower will

reduce the values of overturning angle from the values of

Equation 22). (Similarly, a hollow tower will first crack at

angles smaller than those given by the last line of the table.)

In

any case, Equation

22)

would seem

to

define a

dangerous limiting inclination; in degrees, the equation

may be approximated by

23)

An angle of

about

half this, say 24 b/H), might be such as

to cause concern if it developed in practice; at that

inclination a solid tower should exhibit some cracking,

whereas fissures in a real hollow tower might be slight.

This line

is

plotted in Figure

13.

According

to

this rule, the Campanile

at

Burano (on an

island in the Venice lagoon), which has a ratio

H/b

of

about

10, should cause concern. The inclination for

overturning

is about

4.8° from Table

I;

the angle of

concern would be 2.4°. At a plumbing height of 31.15 m,

the lean was measured

at about 1.96

m, corresponding

to

an angle of inclination of 3.6°. The tower has been

stabilized above ground by drilling and stitching, and

below ground by

root

piles

pa/i radice).

REFEREN ES

1. Beltrami. L.. Fall of the Campanile of St Mark s. Venice , Journ.

RIBA Third Series,

9(17). 26 July 1902;

ibid.,

27 September 1902.

2.

Willis, R., The Architectural History of Chichester Cathedral, Chiches

ter. 1861.

3. Schonlan d, Sir Basil, The Flight of Thunderbolts (2nd edn), Clarendon

Press. Oxford, 1964.

4.

Caroe, Alban. Old Churches and Modern Craftmanship. Oxford. 1949.

5. Willis. R

•

Remarks on the Architecture of he Middle Ages. Cambridge.

1835.

6.

Marra

M • II rafforzamento stat ico del Campa nile di BUfano .

Rivista

Italiana di Geotechnica. 5 (1971) 255-262



A COMPUTATIONAL MODEL FOR THE LIMIT ANALYSIS

OF THREE DIMENSIONAL MASONRY STRUCTURES

R. K. LIVESLEY

University of Cambridge Department ofEngineering

Trumping/on Street Cambridge CB2

lPZ,

England

(Received:

31

January 1992)

ABSTRACT. This paper extends previous work on the limit analysis of ductile frames and plane masonry arches to

the limit analysis of three-dimensional masonry structures. A lower-bound approach is developed which can handle

three-dimensional collapse mechanisms involving any combination of sliding. twisting and hingeingt

at

the block

interfaces. A compute r program for determining the collapse load of such structures

is

used to study a) the equilibrium

limits of a block with four contact points resting on an inclined plane and b) the collapse ofa semicircular arch of four

blocks. The paper also describes experimental and computational work on a radially symmetric model dome o f 380

blocks subject to foundation settlement.

SOMMARIO. II presentre contributo estende al campo delle stTUtture tridimensionali in muratura un precedente

lavoro sull analisi limite

di

telai duttili

ed

archi in muratura piani.

Si

e sviluppato un approccio statico che analizza

meccanismi

di

collasso tridimensionale

ottenuti

per combinazione dei meccanismi semplici di scorrimento e rotazione

nel piano e fuori dal piano delle supertici

di

interfaccia tra i blocchi.

Si

descrivono

a)

i limiti di equilibrio

di

un blocco

con 4 punti di contatto su base inclinata, b)

Ie

condizioni di collasso

di

un arco semicircolare costituito da quattro

blocchi. applicando un p rogramma di cal colo redatto per I analisi e la detinizione del carico

di

collasso

di

tali stTUtture.

La terza parte dell articolo presenta illavoro sperimentale e di calcolo sviluppato su un modello

di

cupola a simmetria

radiale costituita da 380 blocchi soggetta a cedimenti fondali.

KEY WORDS: Masonry, Collapse, Mathematical modelling, Static friction, Mechanics of masonry.

1 INTRODUCTION

The equilibrium and mechanism methods for determin

ing the collapse load of ductile framed structures (some

times known collectively as the plastic theory ) were

developed before the use of computers became widespread.

However, in 1951

Charnes and Greenberg [1J showed that

these two methods were dual linear-programming pro

blems: this discovery soon led to the development of

formal algorithms (and computer programs) for determin

ing the collapse load of plane frames and the closely

related problem of plastic minimum-weight design see, for

example, Heyman [2J and Livesley [3J).

computer programs) to problems of three-dimensional

collapse. The present paper describes this extension and

discusses some of its limitations.

In 1966 Heyman [4,5J showed that the collapse load of

a masonry structure in which the blocks were assumed to

be rigid

and

the joints incapable of carrying tension could

be determined by either an equilibrium or a mechanism

approach. Following Heyman s work the present

author

adapted an existing computer program for the collapse

analysis of plane rigid-jointed frames to the analysis of

plane single-span arches with in-plane loading [6]. This

program, which is based on

an

equilibrium approach, has

recently been extended [7J to deal with masonry bridges

having several spans.

The success

of

the work on plane arches naturally

suggested the possibility of extending the analysis (and the

The

term twisting implies relative rota tion of two blocks about an axis

normal to the plane of their interface. The term hingeing implies relative

rotation

of

the blocks about an axis lying in that plane.

Meccanica

27:

161-172, 1992

©

1992


2. THE IN PLANE COLLAPSE OF A PLANE ARCH

This section contains a brief summary of the equilibrium

approa ch as it appears when applied to a plane single-span

arch with in-plane loading.

Figure 1 shows a plane arch in which the block

interfaces are assumed to be planes normal to the centre

plane of the arch. The distributed traction which acts on

block

k

across interface

i is

represented by three forces

qi ,

Si

and t

i

as shown in the figure. These forces are assumed to

satisfy the constraints

1)

These constraints imply

that

the interface cannot transmit

tensile stress. [For the moment the limit tmaxli may be

, .Ai

l ~ \ {

Inle faC:),,,,--s,

Fig.

1.

Notation for the analysis

of

a plane single-span arch.



162

R. K. LIVES LEY

regarded as fixed: the case of Coulomb friction, where the

limit

is

a linear function of the forces

q i

and

Si is

discussed

later in the paper.J

It

is

straightforward

to

write down the three equations

of equilibrium for block k in terms of the external loads

and

interface forces acting on the block. These equations

have the general form

w

k

+ APk = [ c o e ~ c ; e t : ; : : ~ : ; i ~ : d

bY] [ :]

+ ...

the geometry of the block

t

2)

there being a term on the right-hand side similar to the one

shown for each of the interfaces associated with the block.

In these equations

Wk

is the contribution from the weight

of the block

and

) Pk is the contribution from the factored

live load.

Combining the equilibrium equations for all the blocks

gives the equilibrium equations for the whole arch. These

equations can be written in the form

w+Ap=Hr .

3)

For a single-span arch of n blocks each of the vectors W

and icp has 3n components, while the vector r, which

represents the complete set of interface forces, has

3 n +

1)

components. This vector satisfies the set of linear

constraints

4)

formed by combining the constraints

1)

associated with

the individual components.

The matrix H has

3n

rows

and

3(n+

1)

columns: for the

arch shown in Figure 1, n

is

equal to

16,

so that in this case

H has 48 rows

and 51

columns. An interface between two

blocks contributes two 3 x 3 submatrices to H - one for

each block, while an interface with a rigid foundation only

contributes one submatrix. Thus H consists of two dia

gonals of 3 x 3 submatrices.

[The representation of the normal component of the

stress-resultant across the interface by the two forces

q i

and

Si might appear

to

imply that the two surfaces in

contact are slightly concave. It is true that if they are

slightly convex then there will be a single normal compo

nent of force

at

an (unknown) contact point. However, it is

still legitimate to represent this single force by the two

forces

qi

and Si An arch with convex interfaces will rock

slightly when a live load is applied

and

the small change of

geometry will produce small changes in the coefficients in

the equilibrium equations. However, the constraints

4)

will not be affected.J

The collapse load factor, A

c

is

the maximum value of A

for which the equilibrium equations

3)

have a solution

satisfying the constraints 4). The probl em of determining

Ac can easily be put into standard linear-programming

format and solved using a general-purpose library routine.

However, a conversion

to

linear-programming format

involves the introduction of additional (slack) variables, so

that

the final size of the coefficient matrix is considerably

larger than that of the matrix H In the progr am described

in this paper the maximization of

A s

carried out using an

algorithm previously developed by the

author

[8J for the

limit analysis of frames, which takes advantage of the

simple form of the constraints, 4), and does not require the

augmentation of H.

The algorithm begins with a sequence of Gauss-Jordan

transformations which converts the equilibrium equations

3)

into

W

+ AP* = H*r.

5)

In this set of equations, 3n of the 3(n+ 1) columns ofH are

reduced ,

i.e.

have coefficients that are zero except for a

single 1. Using standard linear-programming terminology

a variable r associated with a reduced column is called a

basic variable and the row containing the 1

is

called the

associated pivotal row.

The remaining three components of

r, which are associated with columns of H*

that

have no

special arrangement, are called non-basic variables. [These

non-basic variables correspond to the redundancies in the

structure,

and

indeed this part of the algorithm is simply

the algebraic process for selecting a set of redundant

variables that

is

commonly used in the force method of

analysing elastic structures.J

The next phase of the algorithm finds a solution of the

equilibrium equations 5) which satisfies the constraint s

4)

for the case ), = 0, i.e. for the dead

load

alone. This solution

(which is not normally unique) is, in linear-programming

terminology, a basic feasible solution : its existence implies

that

the arch can carry its own weight.

The remainder o f the algorithm consists of the following

sequence of three steps, which

is

repeated until the max

imum value of A

is

reached.

a)

With all the current non-basic variables kept constant,

).

is

increased and the basic variables changed in

accordance with Equation 5) until one of the basic

variables, say r

j

reaches a limiting value.

b) A non-basic variable r

k

is

found such that

A

can be

increased further by keeping

r

constant and allowing

r

k

to vary.

f

r

k

is

at

one or other of its limits, then this

variation must be in a sense that does not violate the

constraints 4).

c)

A

Gauss-Jordan

transformation based

on

the pivotal

row

ofH

associated with

r is

used to make column k

a reduced column in place of columnj: this makes r

k

a

basic

and

r a non-basic variable, in preparation for a

return to step a).

The algorithm terminates when a non-basic variable can

no longer be found which satisfies the conditions set

out

in

step

b):

A is then equal to Ac.

When the algorithm terminates, a basic variable

r

and

associated pivotal row of H* will have been selected by

the last execution of step

a).

The variable

r

and the three

current non-basic variables all have limiting values: the



THREE-DIMENSIONAL MASONRY 163

relative displacements associated with these four variables

(which are the only non-zero relative displacements) define

the collapse mechanism. This mechanism can in fact be

obtained from row

i

of the final matrix H*. The reasoning

is

as follows.

Let d be the vector

of

displacements which corresponds

to the load vector f = w+

} cp

in a virtual work sense and

let e be the vector of relative displacements which corre

sponds to the vector of internal stress-resultants r. The

equilibrium equations 3) can be writ ten as f = Hr: it

follows from a standard virtual work argument that at

collapse the vectors e and d satisfy the equation

e=Htd

6)

The transformation of Equation 3) into Equation 5) can

be regarded as a pre-multiplication of both sides of

Equation 3) by a square non-singular matrix U, where

f* = Uf and H* = UH. f a set of generalized displace

ments d * is defined by the relationship

d =

Utd*

then Equation 6) becomes

e = HtUtd* = (H*)td*.

7)

8)

The vector d*, which defines the collapse mechanism,

must be such that the only non-zero components in the

associated vector of relative displacements e are those

which correspond to components of r that have reached

limiting values, i.e. the current non-basic variables plus the

basic variable r

j

•

This implies that d * must be a null vector

except for the it h component: it

is

convenient to make this

component equal to 1. Since d = Utd* this makes d equal

to the

ith

column

of U

t

i.e.

the

ith

row

ofU) and

e equal to

the ith column of (H*)t i.e. the ith row of H*). Note that

the vector e has j = 1 and (in general) non-zero elements

in the positions corresponding to the three current non

basic variables. Thus there are four non-zero relative

displacements associated with the collapse mechanism, as

expected.

In

this mechanism interfaces

at

which the vari

ables q; S; or

t

attain their limiting values either open out

to form hinges (in a hinge either q or S; is zero) or fail by

sliding (in a sliding failure

t

is

at

one

or

other of its limits).

[The fact that the mechanism

is

scaled so that

e

j

= 1

is

a

consequence of the non-zero component of d being

assigned the value

1.

The computed mechanism

is

actually

a small-displacement one, since

it

is obtained from the

equilibrium equations associated with the undeformed

state of the arch.J

The method can be used in cases where the limits on the

shear force t in Equation 1) are those associated with

Coulomb friction. However, it should be noted that

a) the value of

c

determined by the method may be an

over-estimate of the collapse load, even though the

method is essentially a lower-bound procedure;

b) the mechanism derived from the

ith

row

of

H* is

incorrect in any case where relative sliding occurs at an

interface, the sliding being accompanied by a sep

aration of the two surfaces.

Both these phenomena are due to the fact that the proof of

the lower-bound theorem assumes the normality rule: it is

well known

that

when Coulomb friction

is

present the

interface forces and the associated relative displacements

do not obey that rule. Reference [6J discusses the validity

of the computed value of

c

in the context of in-plane

collapse analysis and describes a technique for correcting

the collapse mechanism. The application o f this technique

to three-dimensional collapse mechanisms is described in

Section 5 of this paper.

3.

SOME

GENERAL FEATURES OF THREE-

DIMENSIONAL

MASONRY COLLAPSE

While three-dimensional masonry structures can collapse

in a variety of complex ways, there are a number of

impor tant types of structure where the significant collapse

mechanism

is

effectively a two-dimensional one.

In

mas

onry domes and vaults, for example, cracks can often be

observed

that

are parallel to the direction of thrust in the

material. In extreme cases these cracks divide the structu re

into a series of independent ribs, each of which is es

sentially a two-dimensional arch.

For

example, Figure 2

shows a dome in which a uniform outward movement of

the supporting ring has divided the dome into a series of

orange slices .

If

each slice

is

able to carry its portion

of

the

applied load as a two-dimensional structure then the

complete dome will not collapse.

t is interesting to note

that this fact was appreciated by Poleni as early as 1748

see reference [5J). Some experimental and computational

work on a model dome is described in Section 8

of

this

paper.

Another essentially two-dimensional problem is an arch

in which all the potential hinge lines, instead of being

parallel, pass through a point. In such cases a four-hinge

collapse mechanism is possible, with segments of the arch

Fig. 2. Meridional cracks in a masonry dome due to outward movement

of part of the supporting structure.



164

R

K.

LIVESLEY

forming elements of a spheric four-bar chain.

It

is also

possible to devise assemblies of rigid blocks with more

complex three-dimensional hingeing mechanisms: for

example, a ring of six similar semi-regular tetrahedra

connected by hinges

on

pairs of opposite edges forms a

mechanism capable of continuous rotation about the

circumference of the circle passing through the centroids of

the tetrahedra.

An extension of the analysis of Section 2 to three

dimensional structures in which collapse is associated

solely with hingeing about edges is straightforward. The

analysis of structures in which a combination of hingeing,

sliding and twisting may occur

at

any interface is much

more difficult.

There are two considerations governing the mathemat

ical modelling of such structures. First, the interface model

must reproduce, as far as possible, the observed relation

ship between the surface tractions and the associated

relative displacements for all possible combinations

of

hingeing, sliding and twisting. Second, the overall model

ling process must generate a tractable computational

problem when applied to a three-dimensional assembly of

reasonable complexity. In the context of the current

investigation this means that it must generate the equilib

rium equations for the blocks and the limiting constraints

on the interface stress-resultants in the simple linear form

Equations 3) and 4)) required by the limit-analysis

algorithm described in Section 2.

Figure 3 shows three possible ways of representing the

stress-resultant acting across a general plane interface. In

Figure

3 a)

the two surfaces are assumed to be slightly

convex. The stress-resultant acting across the interface

consists

of

a normal force and two components of shear

force at an unknown) contact point: there is no moment

resisting relative rotation

about

the normal at the contact

point.

In

Figure

3 b)

the normal traction is assumed to

vary in a prescribed way over the interface - a uniform

or

a

Hertzian distribution are obvious possibilities. The shear

traction is represented by two force components in the

plane of the interface acting

at

the centroid of the interface,

0

b)

c)

Fig. 3. Three possible ways of representing the traction acting across a

block interface.

Fig. 4. A circular arch of four blocks with an

otTset

load.

plus a moment about the surface normal.

In

Figure

3 c)

the

two surfaces are assumed to be concave, with three contact

points. The stress-resultant across the interface consists of

the three normal forces

at

the contact points plus the

tangential forces

at

those points: the latter forces can be

represented by three forces in the plane of the surface

acting along lines joining the contact points.

In problems of two-dimensional collapse the value of

Ae

is largely independent of whether the interfaces are con

cave

or

convex. In problems of three-dimensional collapse,

however, assumptions about the nature of the interfaces,

and the distribution of the normal traction across them,

can have a considerable effect on the value of

Ae

Consider,

for example, the behaviour of the four-block arch shown in

Figure 4 when the interfaces are convex, as in Figure 3 a).

If displacements are restricted to the plane of the arch then

the convex nature of the interfaces will cause the arch to

rock slightly when a live load is applied, but the value

of

Ae

and the form of the collapse mechanism will be virtually

the same as if the interfaces were concave. However, if

displacements out of the plane of the arch are allowed then

several mechanisms can be constructed for which Ae = 0,

due to the absence of torsional restraint

at

the interfaces.

The existence of zero-load mechanisms makes the repre

sentation shown in Figure 3 a) unsatisfactory as an inter

face model for general

use.

The representation shown in Figure 3 b) is general

enough to model the behaviour of any three-dimensional

interface. However, it is difficult to convert the const raints

associated with this representation into the simple linear

forms required by the limit-analysis algorithm.

The representation shown in Figure

3 c)

is clearly a

considerable simplification of a general interface, requiring

assumptions to be made

about

the positions of the three

contact points. However, it has the advantage

that

the

associated equilibrium equations and constraints are

much easier to linearize. The next section describes an

interface model which is a slightly generalized version of

the one shown in Figure

3 c).

4.

A

MO EL FOR THREE DIMENSIONAL

INTERFACES

The interface shown in Figure

3 c)

is a natural extension to

three dimensions of the interface shown in Figure

1.



THREE-DIMENSIONAL MASONRY

165

However, greate r generality can be achieved by regarding

the single-point contact shown in Figure 5 as the basic

interface element. This simplifies the algebra and permits

the modelling of

a)

interfaces with more than three

contact points and

b)

interfaces in which the normals

at

the contact points are not all in the same direction.

Each block in a three-dimensional structure has six

equilibrium equations.

I t

is straightforward to derive the

contributions which the triad of forces q

s, t

shown in

Figure 5 makes to the equilibrium equations of the block

or

blocks on which the forces act. If relative displacement

at the interface is governed by Coulomb friction (with

possibly some cohesion), then the constraints on q sand t

are

o q

9)

These constraints generate the conical yield surface in

q

s, t

space shown in Figure

6 a).

To

bring the non-linear constraints on

sand t

in

9)

within the scope of the algorithm described in Section 2,

the cone in Figure 6 a) is replaced by the octagonal

pyramid shown in Figure

6 b).

This change is analogous to

the change from a Von Mises to a Tresca yield criterion in

conventional elasticity theory.

[It

is advantageous, from a

computational point of view, for r to have a small non

zero value, even if there is no physical cohesion at the

interface. This ensures that the pyramid in Figure 6 b) has

a blunt tip, so

that

all its vertices are simple ,

i.e.

are points

q

5

Fig. 5. The three components of force at a point contact.

q

q

0)

b)

Fig.

6.

a) The constraint surface defined by the non-linear inequalities 9);

b) the constraint surface defined by the linearized inequalities

12).

formed by the intersection of exactly three constraint

planes.]

The change from the const raint surface in Figure 6 a) to

that in Figure 6 b)

is

achieved by defining two

additional

variables

u = s+t)/j2,

v = (s- t ) / j2

and eight auxiliary variables

W3

= lq

t

W4

= lq - t

Ws

=

lq +

s

+ t) / j2

W6 = M-(S+t ) / j2

W

7

= M+(s- t ) / j2

Wg

= M - (s -

t j

j2.

(10)

II)

The yield surface in Figure 6 b) can now be represented by

the simple linear constraints

}

s,

t

u v unconstrained

i W i ' s

12)

which have the form required by the algorithm.

The equilibrium equations for a three-dimensional

structure are assembled in much the same way as for a

plane arch. Equations 10) and 11), defining the additional

and auxiliary variables for each contact point, are added to

the equilibrium equations, the total number of equations

being

6nb

On

p

, where nb is the number of blocks and

np

is

p

r-:::::7I indicates coefficients depending

on geometry of structure

o

.. ....

not stored

Fig.

7.

The contribution of a single point contact to the vectors w, p and

to the matrix H.



166 R

K LIVESLEY

the total number of contact points. The complete set of

equations is written in the same form

w+Ap=Hr

13)

as before, the contribu tion of a single point co ntact to the

matrix H being shown in Figure 7.

Note that

in the

implementation of the a lgorithm described in Section 2 the

columns of H containing only a single 1 (the reduced

columns) are

not

actually stored. (The space freed during

the initial transformation of H into H *

is

used

to

store the

matrix U, from which the collapse mechanism is eventu

ally derived.) Thus the total number of columns of H

requiring storage is only 3n

p

• The sets of constraints 12)

for all the contact p oints are also written in the same form

as before,

14)

The algorithm described in Section 2 can now be

applied to the set of Equations

13) and

the constraints

14).

As mentioned in Section 2, if collapse involves sliding

at some of the interface points and of 0, then the

mechanism generated by the algorithm will be incorrect,

with separation occurring

at

those points where sliding

occurs. The next section shows how this separation can be

eliminated.

5. CORRECTING THE

COLLAPSE

MECHANISM

In any collapse mechanism that involves sliding at a

contact point either one or two of the auxiliary variables

W

b

Wg

associated with the point will have reached

their limiting values.

One

variable at its limit corresponds

to a po int on one o f the

faces

of the yield surface shown in

Figure

6 b).

Two variables

at

their limits corresponds to a

point on one o f the edges.) The associated relative displace

ment e

w

will be normal to the yield surface, and will

therefore have a component in the direction of

e

q

, as

shown in Figure 8 a). t

is

this component that causes

separation of the surfaces in the mechanism defined by

Equations 7)

and 8).

This mechanism can be corrected by replacing the

pyramidal yield surface in Figure 8 a) by the prismatic

yield surface in Figure

8 b).

Each active constraint

on an

auxiliary variable

W

1

,

•

,

Wg

is replaced by

an

equivalent

constraint on the corresponding variable

s

t or

v

Thus

the active constraint

1 :

W3 shown in Figure 8(a) is

replaced by the constraint t ; , tc shown in Figure 8(b),

where

tc is

the value of

t

in the computed collapse state.

The normal

e,

in Figure 8 b) clearly has no component in

the direction of

e

q

•

Each change of constraint implies a a u s s ~ J o r d a n

transformation of the matrix H in which a non-basic

auxiliary variable becomes basic

and

the corresponding

basic variable s t, or v becomes non-basic. When all these

transformations have been carried out the corrected mech-

q

t

-------

5

Fig. 8.

a)

Pyram idal yield surface showing relative displacement e

w

with a

component in the direction of e

q

;

b) prismatic yield surface showing

relative displacement

e

with no component in the direction of

e

q

•

anism can be obtained from the final version of U or H ,

as described in Section 2. The value of c

and

the values

of

the components of the internal force vector r are

not

altered by this procedure.

The separation of surfaces that should remain in contact

is

a relatively minor consequence

of

the fact

that

the

relative displacements associated with Coulomb friction

do not obey the normality rule. More serious is the

possibility, mentioned in Section 2, that the computed

value of c may be greater than the true collapse load

factor. The following section shows how such

an

over

estimate can arise.

6. EXAMPLE 1: A BLOCK WITH 4-POINT

CONTACT ON A ROUGH INCLINED PLANE

A program based on the modelling procedure described in

Section 4 has been used to study the equilibr ium of a single

rectangular block resting on a rough plane inclined

at

20°

to the horizontal, as shown in Figure 9. The block is

assumed to rest on four contact points at the corners of its

base.

The modelling procedure generates 12 primary

variables qi

Si t

i

,

8 additional variables

u; Vi and 32

auxiliary variables w

1

,w

s

);, where

i =

1, ... ,4 ranges

over the four contact points. There are 6 equilibrium

equations

and

40 equations similar

to 10) and

11). Thus

the matrix H in Equation 13) has 46 rows and 52 columns.

[Since columns which contain only a single 1 are not

stored, only 12 columns of H actually require storage.]

The live load consists of a single force applied

to

a point

on the top surface of the block, as shown in the figure. The

variation of

c

with for values of the coefficient of friction

in the range 0.364 < 0.5

is

shown in Figure 10. For

values of < 0.364

i.e.

tan - 1 20°) the block slides down

the plane under its own weight. For values of > 0.5 the

block tilts

about an

edge. There are four distinct failure




167

Fig. 9. A block resting on a rough inclined plane.

mechanisms A, B, C

and

D. The mechanisms A, Band C

all involve sliding

at

the contact points and therefore

require correction in accordance with the procedure de

scribed in Section 5. Figures 11 a)

and

b) show, re

spectively, mechanism A before and after correction.

The fact that the computer model only generates four

distinct failure mechanisms

is

a consequence of the re

placement

of

the conical yield surface in Figure 6 a) by the

pyramidal yield surface in Figure 6 b).

n

mechanism A, for

example, the variables

that

attain limiting values after the

0 1

0 09

0.08

transition

Q1 Q2;t. 0

correction procedure) are

q1

S[

v

1

t

2

V2

V3

and

S4 there

are actually no relative displacements associated with the

forces q1

and

VI) At each of the points 3

and

4 only one of

the tangential forces takes its limiting value - the force

V3

at point 3

and

the force S4 at point

4.

The directions of

relative displacement

at

these points are therefore defined:

these directions are sufficient to fix the instantaneous

centre of rotation, as shown in Figure 12,

and

hence the

failure mechanism. At point 2

both

tangential forces take

their limiting values: the corresponding point on the yield

pyramid

is

therefore an edge point. At such a point the

only constraint on the direction of the associated relative

displacement is that it must lie within the angle between

the normals

to

the faces that meet at the edge: this

condition is satisfied by the mechanism shown in Figure

12. Similarly,

at

point 1 all three forces take their limiting

values: the corresponding point on the yield pyramid

is

therefore a vertex. At such a point the only constraint on

the direction of the associated relative displacement is that

it must lie within the solid angle formed by the normals to

the faces which meet

at

the vertex: again, this condition is

satisfied by the mechanism shown in Figure

12.

t

should be possible to develop an iterative version of

the algorithm described in Section 2

that

would generate

solutions satisfying the non-linear yield constraint 9). A

program based on such an algorithm would produce a

more continuous variation of

A,

with

f than

the one shown

in Figure

10,

with a corresponding continuous variation in

1 2

o

0 . 0 5 - - - - . . . J - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

036

0.38 0.4 0.42

0.44

0.46

0.48

0.5 0.52

jJ.

Fig.

10.

Variation o f collapse load factor

20

with coefficient

of

friction J

for a block on an inclined plane. no additi onal const raint

imposed _ . _ . constraint q imposed. constraint q

imposed.



168

R.

K.

LIVESLEY

a)

b)

Fig. 11. A mechanism involving both sliding and twisting: a) uncorrected;

b) corrected.

Fig. 12. A mechanism involving both sliding and twisting: the instanta

neous centre of rotation is defined by the directions of the forces S and

V

-+

displacements of contact points.

t

tangential forces having limiting

values.

the form of the collapse mechanism. However, there is

another feature of the limit-analysis approach that war

rants investigation.

The block shown in Figure 9 has four contact points

with normal components of force

ql

q2 q3 and

q4

Since

these forces only appear in three of the equations of

equilibrium of the block, one can be chosen arbitrarily,

subject to the constraint

qi ;

O.

The limit-analysis al

gorithm always finds the set of interface forces which

maximizes

A

subject only to the equilibrium equations 13)

and the constraints

14). For

mechanisms A and B the

optimum set is one in which

q 1 = 0,

while for mechanism C

it is one in which q2 = O. Between mechanisms Band C

there is a transition region in which both ql and q2 are

non-zero.

In

this region the mechanism is indeterminate:

the instantaneous centre of rotation may lie anywhere on

the line joining points 1 and 4. [The indeterminacy is due

to the fact that

at

each of the points 2 and 3 two of the

tangential force variables s t u and v have limiting values,

so that the direction of relative displacement at these

points is not completely defined.]

In practice, of course, the distribution of normal trac tion

across the interface depends on both the initial flatness of

the surfaces in contact and the elastic properties of the

material. For example, if the material

is

rigid and the

interface is such that the block can rock slightly about the

line joining the points 1 and 3 then the block

will

initially

rest on the points

1,

3 and

4:

the force

q2

will remain equal

to zero as the load factor increases from 0 to

Ae.

Figure 10

shows the result of running the program with the ad

ditional constraint q2 = O.

It

will be seen that the im

position of this constraint produces a reduction of over 20

per cent in the value of

Ae

for some values of /1.

Alternatively, if the block can rock slightly about the

line joining points 2 and 4 it will initially rest

on

the points

1,

2 and 4. However, application of a small live load

will

cause the block to tip over

onto

point

3:

subsequently the

force ql will remain zero as the load factor increases to Ae.

Figure 10 also shows the results of running the program

with the additional constraint ql =

O.

Note that the graph

of

Ac

with

/1

for the 4-point contact problem is a combina

tion (in fact it is the convex hull) of the graphs for the two

3-point contact problems.

This example shows that in a collapse involving twisting

at one or more interfaces the value of the collapse load

depends on the assumptions made about the distribution

of normal traction across the interfaces. f more than three

contact point are specified at an interface then at least one

of the normal forces can be chosen arbitrarily (within

limits).

In

such cases the limit-analysis algorithm will

always seek out the distribution of interface forces that

gives the greatest collapse load factor.

7. EXAMPLE 2:

AN ARCH

OF FOUR

BLOCKS,

WITH FOUR POINT CONTACT

AT EACH

INTERFACE

The second example is the arch

of

four identical blocks

shown in Figure 4. The centre-line of the arch is an arc of a




169

circle with a semi-angle of 60°

and

there are contact points

at

the four corners of each interface,

i.e.

20 contact points

in all. The modelling procedure described in Section 4

generates a matrix H having 224 rows and 260 columns:

only 60 of these columns need to be stored. The live load

consists of a concentrated vertical force applied to the

upper surface of the arch, as shown in the figure.

The computer program was used to study the way in

which the collapse load factor and the collapse mechanism

change as the distance of the live load from the centre

plane of the arch increases. When the load lies in the

centre-plane collapse occurs in the mechanism with four

hinges shown in Figure 13 a) (the coefficient of friction was

made just large enough to prevent sliding at any interface):

this calculation involves

56 Gauss Jordan

transforma

tions

of

the matrix

H.

Collapse still occurs in the mechan

ism 13 a) (at the same load factor) for moderate move

ments of the load normal to the centre-plane. However, as

the load approaches the edge of the arch the mechanism

changes to the three-dimensional one shown in Figure

13 b).

In

this mechanism the collapse load decreases

linearly with the distance of the load from the centre-plane,

as shown in Figure 14: the ca lculation involves 104 Gauss-

Jordan

transformations of H. Further movement of the

load produces a transition to another three-dimensional

mechanism with a more rapid decrease in A as shown in

the figure.

The example in Section 6 shows that

at

any interface

where twisting failure occurs the algorithm always dis

tributes the normal traction among the contact points in a

way

that

maximizes A

and

that

the imposition of re

strictions on this distribution will, in general, result in a

lower value

of

A

even when the mechanism remains

unchanged.

In

the case of the four-block arch the im

position of the constraint = 0 at one of the contact

points

t

on each interface where twisting occurs produces

the reduction shown in Figure 14.

It

will be observed that

the magnitude of this reduction is considerably less than

that shown in Figure 10.

t At each interface the point chosen is the one at which is zero in the

hingeing mechanism (Figure l3 a)} but non-zero in the twisting mechan

ism (Figure l3(b)).

1 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

10.8

10.6

10.4

Ac

102

10

9.8

9.6

Ed:e of a rch I

0 ,

0 ,

\

\

I

I

3.6

3.7

3.8

4.1

4.2

4.3

Distance

of

load

from centre-plane of arch

Fig.

14.

Variation of collapse load factor A with the displacement of live

load from center-plane. no additional constraint s imposed.

- - - - additional constraints imposed on normal component of interface

forces.

\

\

\

\

\

I

4.4



170

R K. LIVES

LEY

Fig. 15. Experimental rig for testing a model masonry dome.

8

EXAMPLE

3:

A RADIALLY SYMMETRIC OME

OF 38 BLOCKSt

The approach developed in earlier sections of this paper

can also be used

to

predict the pattern of cracks

and

the

associated interface forces produced in a masonry

structure by foundation movements. This section describes

an experimental

and

computational investigation of the

effect of support movements on a model dome. This dome,

shown in Figure 15, has a base radius of 490 mm, a height

of 650

mm and

a thickness varying from 50 mm

at

the base

to

25 mm at

the top.

t is not

a precise scale model of any

existing structure, but many of the domes built in the

sixteenth

and

seventeenth centuries have similar

thickness/radius

and

height/radius ratios.

The dome is built from 380 concrete blocks arranged in

13

layers, all the blocks in

anyone

layer being of similar

shape. The number of blocks in a layer varies from 32

at

the base to 24 at the top. The dimensions of the blocks

were chosen in such a way that the vertical joints between

blocks in adjacent layers are separated by

at

least a quart er

of the meridional dimension of the blocks. Each block was

cast in situ to give as uniform a distribution of interface

traction as possible,

and

after casting the block interfaces

were faced with sandpaper

to

give a reasonably uniform

coefficient of friction. The top layer of the model dome

supports a rigid ring cast in one piece. In real structures

built in the sixteenth and seventeenth centuries this ring

usually supports a lantern: the model testing rig has

The

work described in this section was carried out by Dolt.ssa lng.

Dina

D Ayala of the University of Rome while visiting the Cam bridg e

University Engineering Department.

facilities for applying

an

additional dead load

to

the ring to

simulate the weight of the lantern.

The dome is supported on a base ring consisting of 16

concrete blocks whose linear dimensions are twice those of

the blocks in the lowest layer ofthe dome itself. Each of the

blocks in the base ring is supported on a separate steel

bearing plate which can be given independent vertical

and

radial displacements

and

can be rotated

about

axes

normal and tangential to the middle-surface of the dome.

Up to four of the bearing plates can be moved simul

taneously. These arrangements make it possible to

simulate virtually any of the settlement patterns

that

have

been observed in the supporting structures of real masonry

domes.

To date, tests have been carried

out

on the model for the

following pa tterns of support movement:

a) Outward

radial movement of two adjacent bearing

plates,

i,e. an

outward movement of a 45° sector of the

supporting ring.

b) Rotation of each block of the supporting ring about a

horizontal axis tangential

to

the middle-surface of the

dome. Alternate blocks are rotated in opposite senses

by the same amount.

c) Rotatio n of adjacent bearing plates

about

radial horiz

ontal axes to give a vertical settlement of the support

ing ring varying linearly from zero to a maximum and

back again to zero over a 45° or 90° sector.

In each test the change in shape of the dome and the

relative movement of the blocks adjacent to the cracking

were recorded. The results of these tests will be repor ted

at

a later date.

Further

tests are planned using more complex

settlement patterns.




171

The computer program used in the analyses described in

Sections 6 and 7 has been modified to analyse simple

domes with up

to 48

blocks. The program is

to

some extent

a special-purpose one, in

that

it places certain restrictions

on the types of failure mode allowed: this reduces the total

number

of independent force variables required. The

program has been used to analyse (a) a dome of three

layers having 16 blocks in each layer and

b)

a quadrant of

a dome of eight layers having six blocks in each layer. A

plot produced in connection with the latter analysis is

shown in Figure

16.

t is

apparent

that

a general three-dimensional collapse

analysis of the complete model dome of 380 blocks in

accordance with the procedure described in Section 4

is

a

computational problem at least an order of magnitude

greater

than

those discussed so far in this paper. The

interlocking arrangement of the blocks means that there

are, in effect, six interfaces associated with each block. If·

four contact points are assigned to each interface, then the

total number of contact points

is

380 x 12 = 4560 points,

which implies a mat rix H having 6 x 380

10

x 4560=

47880 rows and 3 x 4560 = 13 680 columns. While the

assembly and manipulation of a computational model of

this complexity requires considerable technical expertise, it

is

well within the limits of what

is

currently possible.

Indeed, many commercial finite-element and linear

programming packages can handle matrices of this size

and larger. Whether the algorithm described in Section 2

(and the present auth or s coding of it)

is

sufficiently robust

to be appropriate in such circumstances is an open

question: from a practical po int of view it might be safer to

reformulate the problem in conventional linear

programming format and use a standard package.

However, there is no doubt that a general three

dimensional collapse analysis of a masonry structure of

this complexity

is

feasible.

Fig. 16. Computer model of a segment of a dome.

9.

CONCLUSION

The work described in this paper illustrates two common

features

of

computer modelling. First, it demonstrates the

way in which an extremum problem involving non-linear

constraints can be linearized, and the penalty (in the form

of a large increase in the number of variables)

that

such

linearization incurs. Second, it illustrates the way in which

the development of a computational model

is

often in

fluenced by the availability of a program written for a

related problem.

t seems unlikely that the replacement of the conical

yield surface in Figure 6(a) by the pyramidal surface in

Figure 6(b) has a significant effect on the computed

collapse load. A more important source of error, as has

already been stated, is the use of a lower-bound approach

in problems where the collapse load involves limiting

friction. The assumption of Coulomb friction (or indeed

the more general assumption

that

the limit on a tangential

force depends on the associated normal force) implies that

internal forces which are not at a limit (and are therefore

indeterminate w ithout some form of elastoplastic analysis)

have an effect on the

amount

of work done in the collapse

mechanism and therefore on the value of the collapse load.

This is in contrast to problems involving ideal rigid-plastic

behaviour, where forces which are not at a limit have no

influence on the work equation. The examples in Sections

6 and 7 illustrate the way in which the lower-bound

procedure endeavours

to

increase the collapse

load

factor

by making adjustments to the normal interface forces -

adjustments

that

have no relation to the way in which the

collapse mechanism actually develops in a real structure.

Computer modelling can be a valuable tool in the

development of an intuitive understanding of the

behaviour of masonry structures. Three-dimensional col

lapse mechanisms are very difficult to visualize, even in

structures having only a few blocks. However, the work

described in this paper shows that in any mechanism

involving relative twisting at a block interface the value of

the collapse load factor

is

very sensitive to the distribution

of normal trac tion and, of course, to the assumed value of

the coefficient of friction. In any practical problem neither

of these can be determined with any degree of precision. t

follows

that

results obtained from the compu ter modelling

of such mechanisms are inherently less reliable than those

obtained from mechanisms where failure is due solely to

hingeing, since in the latter case the only assumption

is

that the interfaces cannot sustain tensile stresses.

ACKNOWLEDGEMENT

The author is grateful to Dr P. C.

Dhanasekaran

for the

hidden-line removal progra m used in the productio n of the

computer plots in this paper.



72

R. K. LIVES LEY

REFERENCES

1. Charnes, A. and Greenberg, H. J., Plastic collapse

and

linear pro

gramming , Summer meeting Amer. Math. Soc., 1951.

2.

Heyman,

J. and

Prager, W., Autom atic minimum -weight design of

steel frames , J Franklin Inst.,

266

(1958) 339-364 .

3.

Livesley, R. K., The automatic design of structural frames , Quart. J

Mech. Appl. Math., 9 (1956) 257-278.

4. Heyman,

J.

The stone skeleton , Internat. J Solids Struct., 2 (1966)

249-279.

5. Heyman, J., On shell solutions for masonry domes , Internat.

J

Solids

Struct.,

3 (1967) 227-241.

6. Livesley, R.

K.

Limit analysis of structures formed from rigid blocks ,

Internat. J Numer. Methods Engng., 2 (1978) 1853-1871 .

7. Livesley, R. K., The collapse analysis of masonry arch bridges , Proc.

Co, Applied oUd Mechanics 4 Elsevier, 1992, pp. 261-274.

8. Livesley,

R.

K.,

Matrix methods

o

structural analysis

(2nd edn),

Pergamon, 1974, pp. 138-141.



NEW TRENDS IN THE ANALYSIS OF MASONRY STRUCTURES

SALVATORE

DI

PASQUALE

Universitii di Firenze, Di partimento di Costruzioni,

Piazza Brunelleschi

6

50122 Firenze, Italy.

(Received: 2 March 1992)

ABSTRACT. The modern theory of masonry structures has been set up on the hypothesis of no-tension behaviour,

with the aim of offering a reference model, independent of materials and building techniques employed. This

hypothesis gives rise to inequalities which have to be satisfied by the stress tensor components and, as a dual aspect, to

the kinematic behaviour characteristics of media which can be classified as lying between solids and fluids: the

structure of the masonry material consists of particles reacting elastically only when in contact. An examinat ion of the

plane-stress problem leads us to define, within the prescribed domain under admissible loads, three different

subdomains with null, 'regular', or 'non-regular' principal stress tensors, respectively.

As

the boundaries of such

subdomains are not known

a priori,

the problem can be classified as a free boundary value problem. The analysis

concerns mainly the subdomains where the stress tensor is 'non-regular'; and a 'non-regularity' condition det

=

0 is

added to the equilibrium equations. This condition makes the stress problem 'isostatic' and leads to a violation of

Saint-Venant's compliance conditions on strains. Hence there is a need to introduce a strain tensor, not related to the

stress tensor, which can be decomposed into an extensional component and a shearing component;

we

prove that such

strains, of the class

y e,

are similar to those of the theory of plastic flow. From the point of view of computational

analysis the anelastic strains are considered as given distortions; they are computed by means

of

the Haar-Karman

principle, modified for computational purposes by an idea of Prager and Hodge.

SOMMARIO. La moderna teoria delle strutture murarie, fondata sulla rigorosa non reagenza a trazione del

materiale, ha 10 scopo di fornire un modello di riferimento indipendente sia dalle caratteristiche del materiale sia dalle

techniche costruttive impiegate. L'ipotesi di non reagenza a trazione

si

traduce in disuguaglianze che

Ie

componenti

del tensore di stress devono verificare; dualmente

il

comportamento caratteristico cinematico puo esser classificato di

confine, come del resto la stessa statica, tra solidi e fluidi: la struttura ipotizzata del materiale muratura consiste di

particelle che reagiscono solo se sono in contatto. L'esame del problema piano

porta

a definire all'interno del dominio

di definizione tre differenti tipi di sub-regioni in cui 10 stress enullo, canonico, 0 singolare. Poiche Ie frontiere di queste

sub-regioni non sono note a priori

il

problema puo anche essere c\assificato di frontiera libera. L'analisi concerne

fondamentalmente la sub-regione in cui il tensore enon regolare, perche deve verificare anche la condizione det

= O Cia

rende 'isostatico'

il

problema e conduce anche alla violazione della condizione di integrabilita delle

deformazioni. Questo passaggio puo essere superato introducendo un tensore di deformazioni a tensioni nulle che

si

puo decomporre in una componente estensiona le ed in una componente di scorrimento si dimostra che queste

deformazioni sono equivalenti a quelle che intervengono nella Teoria del flusso plastico. Dal punto di vista

computazionale

e

deformazioni anelastiche

sanD

considerate come distorsioni impresse determinate attraverso il

principio di Ha ar-Karman modificato, per Ie techniche computazionali, su idee di Prager e Hodge.

KEY WORDS: No-tension materials, Masonry, Historical Monuments, Constrained constitutive equations, Mechanics

of

masonry.

INTRODUCTION

We shall discuss an ideal material, which reacts elastically

to arbitrary pressure,

but

cannot withstand the slightest

tensile stress.

To give form to that idea, we can imagine a plane region,

R, delimited by a closed, regular boundary and completely

filled with the material indicated above; a generic section

or

split divides R into the parts R' and R .

The fundamental equations of statics are necessary

but

not sufficient for the equilibrium of this material; if

external forces are applied in an at tempt to separa te R' and

R

of R, they meet no resistance. Obviously this intrinsic

characteristic is not determined by the dimensions of

R,

because this characteristic is also valid for two infini

tesimal particles.

n

other words, the microscopic model of

such an ideal material is similar to a set of particles that

react only through contact, and, like the two particles,

when separated, show no mutual influences of any type.

The material considered here is granular, incoherent and

incohesive.

n the experimental stage, it is appropriate to imagine

Meeeaniea

27: 173-184, 1992

©

1992 Kluwer Academic Publishers.

that an ideal test consists of impressing a homogeneous

and uniform deformation defined by a shearing strain y in

the area R,

that

no longer has any external constraints. f

the material displays standard characteristics, for example

isotropic, in order to maintain this deformation, it would

be necessary to apply a system of boundary tensions

equivalent to a tangential stress

T

=

Gy

or, in other words,

two principal stresses, (T 1 2

=

±T. Such stresses are incom

patible with the nature of the material considered; the

material has to sustain the strains y wihout any stress

reaction; so that in this case its behaviour is characteristic

of ideal fluids, while it can behave also as a solid if it is

subject to compression.

Such materials were investigated in the eighteenth

century,

and

they were defined as being half-fluid

in

order

to characterize their capability

of

performing in a solid

like,

or

fluid-like mechanical behaviour, under the effect of

external forces [1].

Therefore they belong to a particular class of materials

whose behaviour must be described by means of appro

priate constitutive equations.

Before considering a mathematical formulation for the



174

SALVATORE DI PASQUALE

problem, it is appropriate to describe some qualitative and

quantitative aspects

that

can be deduced very easily, and

that will be very useful.

t is

helpful if

we

make

an

obvious

analogy with non-reacting-to-compression materials that

are commonly available and tested: for example wires

and

cloth are excellent specimens of such material.

The origin for this suggestin can be found in

J.

Heyman s research on the History of Mechanics; in par

ticular, in the study of Poleni s problem regarding the

stability of the St Peter s dome in Rome ([2J, [3J).

One of

the classic problems of the theory of elasticity

consists of a circular disk subjected to the action of two

equal and opposi te forces, acting on the boundary, along a

diameter o f the circle. The solution, provided by H. Hertz,

has been validated in

an

excellent way through experi

ments carried

out

with polarized light, and this demon

strates the capacity of the elastic material to distribute the

stress state inside, due to external actions.

From

a math

ematical point of view, this aspect can be attributed to the

ellipticity of the differential equations of the elastic

equilibrium.

f we

apply two tensile forces to a piece of woven cloth,

for example, when opposite edges of a handkerchief are

pulled along a line of fibres, we can observe a completely

unusual phenomenon: the only parts

that

react are those

constituted by the fibers of the cloth to which the external

forces are applied; there is no stress in the remainder

ofthe

cloth.

Other, more complex experiments, can be executed to

demonstrate a fundamental principle, which states that not

all the area

R

is in a state of stress, because

some

of its

zones are

not

under stress.

If, speaking about masonry,

we

define as construction

the region R and as structure that part of it under stress,

we

can see

that

these do

not

coincide in general;

we

may

also agree

that

a change in the external actions

on

the

same construction will bring about also changes in the

reacting structure .

This is

not

a surprising fact;

it

is well known in the

statics of masonry columns subjected to eccentric com

pression: the reacting area,

that

is the structure , does not

coincide with the whole section of the column, if the

pressure centre is external to a central core, but it is defined

by a width

that

depends

on

the ratio between the bending

moment M and the normal force N Some fractures occur

in the non-reacting part; the fact that is surprising is that

these fractures are disposed almost parallel to the axis of

the column, contrary to what one might expect; so that the

remedy consists

of

the installation of metal rings.

THE COLUMN P R DOX

In order to explain the main point

of

our problem, let us

consider a two-dimensional column

that

has been sub

jected to two concentrated and opposite forces

F

(Figure

a

a

F

Fig.

1.

The column paradox.

1 ,

generating in the section

a a

normal force

N

shear

T

and bending moment M.

Nand

M generate normal stress J

that we

suppose to be

compressive

and

trapezoidal over

S,

the shear force

T

generates tangential

stress

over the section S, so that we

have the principal stresses

J J2

)1/2

J1 .2

=

2 ± 4

2

Throughout this paper

we

shall regard tensile stress as

positive. Here, since J < 0, we have

J 1

> 0,

J 2

< 0.

This result is incompatible with the nature of our

hypothetical material because

it

is incapable of withstand

ing

t n s i l ~

\positive) stress.

No

one, however, can deny tha t

the structure is in equilibrium. Therefore we have a

paradox that requires

an

explanation.

In

fact, our reasoning is based

on

the accepted idea that

a masonry column is a struc ture whose axis is determined

by its geometric form; this is true for standard materials,

but

it is completely wrong for masonry materials.

The paradox disappears if

we

consider the action line of

force F to be the axis

of

the structure hidden in the

column,

and we

take the section S as being normal to this

axis: the tangential stress disappears and only normal

stresses are generated by the external forces.

This elementary paradox can explain that the funda

mental unknown in the statics of masonry

is

the resistant

structure that does not coincide, in general, with the

construction .

THE M SONRY PROBLEM

Now, let us consider a two-dimensional problem of stress,

defined by a flat disk

or

wall, subjected to force systems

on

the edges, and, if necessary, to body forces: it is kept in

equilibrium in order to avoid, for the moment, any



176

SALVATORE I PASQUALE

and, consequently,

in

Ro: trO =

0 deto

=

0

in

R

j

:

trO < 0

deto

= 0

in R

2

:

trO <

0

detO >

O.

Although it is

not

possible to give the complete list of the

conditions that must be fulfilled, the following example

(Figures

3

and

4)

has been set

out

to explain other

singularities of this problem.

Let us consider a rectangular plate; external pressures

p x)

and

q x)

satisfy the equilibrium equations. Let us

begin to demonstrate

that

the sub-regions

a b e f

and c-

d h g

are

Ro. In

fact, let

us

sketch the cross section

i j

and consider the equilibrium of the left part.

For equilibrium to occur in the

x

direction, there must

be

J:

O xdy=O.

However, given

O x

<:;

0,

we

must have, on the entire

section i - j

o x

=

O.

From

the condition det

0 ; ; . 0,

it follows

that x y = o.

I t

P

p xl

d

b

c

I

,

I

I

I

0

I

1

x

I

I

e

h

v

Fig. 3. Equilibrium

of

the strip

a e i j:

the stress tensor

is

null.

k

o

x

-

y

Figure 4. Equilibrium

of

the strip

a e i j:

the stress tensor is null.

pIx)

b

y

,,---

~

i

J:: x

I

0

I

I

I

r

\1/ V

'-'

..

X

g

q(X)

Fig. 5. Equilibrium

of

the loaded strip.

may easily be proved that

O y =

0 with the aid of section

k l.

Next, we must discuss the equilibrium of the strip b c-

f g

(Figure

5).

The cross section

o r is

close to

b f:

for equilibrium in

the x direction

we

have, as in the preceding analysis,

o x =

0

and

consequently

x y = O.

This shows that equilibrium in the y direction can be

satisfied

if,

and

only

if,

the distribution of pressure

is

such

that

p x)

=

q x)

on the two opposite sides.

This problem suggests further remarks about the regu

larity of the boundary data. The pressures

p x)

and

q x)

may be described by functions with concentrated or

distributed discontinuities, as Figure 6 shows, provided

that

equilibrium

is

verified within each strip of material;

the stresses are: o x = Xy =

O,O y

=

p x)

=

q x).

In fact, such

a solution satisfies the differential equations of equilib

rium, the boundary conditions

and

the condition

V

x

Fig.

6.

Equilibrium of a panel with discontinuous symmetrically distri

buted loads.



NEW

TRENDS

IN

MASONRY ANALYSIS

177

pIx

R

q(X)

Fig.

7.

Equilibrium of a panel with discontinuous symmetrically distri

buted loads.

det

J

=

O.

Obviously, the compatibility equations in term

of stresses, A« J x + Jy) = 0 for isotropic materials, is

violated.

They are verified only if Jy -

and

also p x), q x)

-

are

linear functions of x: that is the case shown in Figure 7. We

shall deal later with such problems, in order to discuss

their kinematical aspects.

Similar remarks are valid for load conditions for the

problem of a concentrated force acting at the tip of the

wedge. t is important to emphasize

that

in such cases the

lines, along which the external thrusts act, are the trajec

tories of the principal stresses and potential fracture lines.

Such simple remarks can highlight a fundamental aspect

of the problem: the failure of De Saint-Venant s postula te

and the inability of the medium to spread internally the

stresses due to the external actions.

From

a kinematical point of view we may develop some

preliminary remarks

about the strains within the sub

regions of R (for sake of simplicity, a plane problem will be

considered). f a region R is made up by elastic standard

material, the constraints preventing rigid-body motions

are sufficient to allow a determination o f the displacement

functions.

In our problem, on the other hand, the non-solid

material does

not prevent the points of the unconstrained

region R from moving, with the only condition that any

two particles cannot move to the same posi tion in the final

configuration.

In

other words, in the natural state all

displacements which cause a separation or a slip between

particles are possible.

But such displacements, from a physical point of view,

cannot exist without an external cause. Therefore, the

positions of the material particles are here uniquely de

fined, in the natural state, by the corresponding points in

the space, as if the material were solid. Every possible

compatible displacement is assumed to be zero in the

natural configuration; without such an assumption it is

clearly impossible to define the region R as an object of

investigation.

However, we have already stated th at R is divided into

sub-regions, as a consequence

of

the stresses generated by

the external forces. A so-called regular behaviour is

defined in

Rl

when, given the stress tensor

J,

the corre

sponding displacements can be uniquely determined with

out zero-stress strains

15.

In

a sub-region

Ro

with null

stress everywhere, the de termination of the displacements

has little meaning, given the previously stated assumption

in the natural configuration.

On the other hand, in the sub-regions R

1

where the

stress is determined by the further condition det J

=

0 the

evaluation

of

displacements is particularly significant,

since the strain compatibility condition

f ex f e

y

8

1

y

y

= ~

8y2

8Xl 8xy

is generally not satisfied.

In

the previous examples, where

Jy

=

j x), Jx

=

" "xy

= 0

the equation is satisfied in

RI

if and

only

ifj x)

is a linear function.

That

means

that

there are

no given displacements u, v, such that the differential

system

8u 8u

8v 8v

8x = ex: 8y + 8x = YXy; 8y = e

y

can be integrated. We could therefore introduce in the sub

regions

RI

the so-called zero-stress strains

15

and rewrite the preceding system in the form

The components of

15 depend on the stress J, since they

must satisfy the fundamental orthogonality condition

That is the constrained energy density must be zero.

Nevertheless, such a condition is not sufficient for the

evaluation

of

the displacements

u

and

v,

since there are

now four equations and five unknowns. Such kinematical

indeterminacy, which is an intrinsic property of the as

sumed material, may be explained as follows: since there is

one principal stress in R

1

two different anelastic strains

are possible, namely a positive strain along the zero-stress

direction and then a dislocation (slip) among the principal

directions. Since the class of positive strains has already

been investigated ([4J, [5J, [6J), we shall deal here mainly

with the slip strains (dislocations).

THE

TRAJECTORIES OF

THE PRINCIPAL

STRESS

IN Rl

Other information can be obtained for sub-region R

1

: for

this sub-region the equilibrium equations must be added



178

SALVATORE

DI

PASQUALE

to the condition det J = 0:

O Jx

O Xy _ o O Xy o(Jy - 0

ox oy px - ox oy py -

Thus, the problem of the stress analysis

is

statically

determinate according to H. Geiringer [7] because

we

have three equations and three unknowns; but we do not

know the boundary of RI and the corresponding con

ditions; However, it is possible to obtain some general

information that can justify the previous results. The

mechanical characteristics of the material are not neces

sary; that is the elasto-kinematics equations can be used

only if we want to know the displacements.

The fundamental aspect of this problem concerns the

trajectories of the principal stresses whose equations can

be obtained using the polar representation of stress tensor:

xy = - J

sin ip cos ip;

m which J = J2 and ip

is

the angle between the first

principal direction and the x axis.

The condition det

J =

0

is

automatically satisfied; the

equilibrium equations become

O J . O J . ( . Oip Oip)

- sm

2

ip

- -

sm ipcos ip

+ J

sm

2ip -

cos

2ip +

+

px

= 0

o(J o(J

- - sin ip cos ip

-;-

cos

2

ip -

ox

uy

- J cos 2ip - sin 2ip ) Py =

o

By

a simple linear combination, we have

[

Oip Oip

] .

J - sin ip - - cos ip = - px cos ip - Py sm ip.

ox oy

Ifwe introduce ljJ that

is

the angle between the direction

of principal stress

J

and the x axis, and tg ljJ =

z,

we

will

have

1 Z2)[ZPx -

p

y

]

J

=

(oz/ox) z(oz/oy) .

Ifwe consider Px = P

y

= 0, the solutions of the equation

{

z

=

const

z

= y x

justify the elementary results obtained previously, con

cerning the equilibrium of masonry walls loaded on the

boundary

and

without gravity forces.

The solution z

=

const represents the case of parallel

trajectories of the principal stresses; the solution z = y x

represents the case of radial trajectories.

We can note the analogy between the previous

equations and those regarding the statics of granular

media [8].

SUB REGIONS

ND

CONSTITUTIVE

EQUATIONS

The state

of

stress in the sub-regions of the domain

R

have

a corresponding state of strain; these are completely elastic

in Rb elastic and anelastic in R

I

, anelastic in R

o

.

The stre ss-st rain relations are:

e(u) =

K(J

15

in which

e(u)

are strains determined by displacement u;

K

is

the constitutive matrix of the material and the term

15

represents a tensor of anelastic deformations necessary for

the integrability of the elasto-kinematic equations in sub

regions RI and Ro, that

is

for the determination of the

displacement components.

The orthogonality condition

J15 = 0 may now be stated,

in the principal representation of J, as

The condition

Ci = Ci. x, y) 0

assures posItive strains along the directions of tensile

stresses; the function

y

=

y x, y

does

not

have, in general,

any conditions

on

its sign; in the sub-regions Ro the strain

is

completely anelastic with the same limitation for the

scalar function Ci

To summarize, these three cases can be described by the

first and second invariants of

J

and

15:

R

.

{tr J <0

{trl5

= 0

2· .....

det

J

> 0 det

15 =

0

R

. {tr J <

{trl5 >0

I· .....

det(J = 0 detl5 < 0

{

tr J =

0 {trl5 0

Ro: ..... .

det

J

=

0 det

15

0

The orthogonality condition J15 = 0 in the sub-region

RI

may be expressed, in coordinate system

0; x, y

by

keeping the separation between the normal and shearing

components, through the same components of the

J

tensor:

15 =Ci.[ Jy

y



NEW TRENDS IN MASONRY ANALYSIS

179

The tensors (J A

and

(J c are obtained by linear trans

formations

of (J :

n some cases it may be convenient to express the tensor

y(J c

by the angle

2cp

through the well-known relations

C

[-sin2CP

Y(J =

Y

cos2cp

COS2CP]

sin2cp .

From

the previous expression

we

can find

an

evident

analogy between the statics of masonry and the theory of

plastic

flow;

the difference between the two problems lies in

the fact

that

in our case the shearing (anelastic) strain takes

b

o

x

b

a

a

Fig. 8. Rectangular panel with self-equilibrated parabolic load; the

dashed line represents the deformation of the wall.

place

on

the principal directions of stress without

an

energy variation.

The two tensors

(J A

and

(J c

are orthogonal to each other

and to the stress tensor

(J ,

which lies on the cone

of

equation det (J = 0 in the stress space.

The strain tensor

O((J A

lies

on

the normal to the same

cone; since

(J c

is

orthogonal to

(J A, we

can state that (J c

must

lie

in a plane tangential to the cone; its direction is

also determined by its own orthogonality to

(J .

Finally the stress-strain relations are

in

R

2

: e(u)

=

K(J

in R

: e(u) = K(J +

Il((J A

+ y(J c; 0( :; O.

in

Ro: e(u)

=

£5

Let us suppose that, for an assigned problem, we have

determined the stress tensor; thus, the elastic-kinematic

equations are

[ ] ([kU

k12

k ] [0

0

:]

,y +

V,x

=

k12 k22 k23

+ 0

-20(

V,y k13

k23

k33

0

0

G

2y

+

[ ~ l

2y

in which the scalar functions

0

and

y

allow the integration

of the system. The integrability condition

is

expressed by

the following equation (note that the comma

is

a dif

ferential operator), in which

[

are the rows of the elastic

constitutive matrix and

(J

is

the stress tensor denoted as a

vector:

(ky(J

+

ll((J y

- 2y,xy),yy + (kj(J +

O((J x

- 2Y'x

y

Lx

=(ki(J

-

20(,xy +

2Y«(J x - (J y)),XY'

This equation

is

necessary but not sufficient to determine

the displacement components

u

and

v,

because of the

intrinsic lability of the assumed material.

We can separate the ane1astic strains

O((J A

from the y(J c

ones; it is also obviously possible to transfer the de

formation of one to the other without changing the state of

stress because of the internal lability of the material; in

several problems, in order to obtain a solution in terms

of

displacements, it

is

necessary to introduce a principle

that

allows one to obtain a solution that minimizes, for

example,

an

integral quadratic form of the displacements.

t

is

necessary to remark that non-regular load con

ditions at the boundary, i.e. discontinuity on the pressure

function, lead to discontinuous displacement functions.

NUMERIC L EX MPLES

(1)

The rectangular masonry wall shown in Figure 8 has

constant thickness

s

= 1; the material

is

isotropic with



180

SALVATORE I PASQUALE

E = 1 and v = O The boundary conditions are

{

y = ±b

Xy

=

0

J = - L

a

2

-

x

2

)

y a

2

{

X

=

+a

Jx

= x y = 0

The solution in the whole region

R,

in this case

R

1

,

is:

The trajectories of the principal stress are parallel to the

y direction; the kinematical analysis

is

defined by the

differential equations :

au

=

o

au

ov

_ O

ov

P 2 2

OX

oy

OX - , oy = - a2 a - X

).

These are not integrable; therefore we can take into

account the inelastic deformations that are compatible

with the state of stress

and

permitted by the material; here,

we consider anelastic deformations that correspond to

half-fluid behaviour:

The elasto-kinematic equations therefore become

au = 0; au

ov

= y L

a

2

_ x2);

ox oy ox a

2

ov p 2 2

- =

--(a

-x).

oy a

2

The scalar function y = y x, y) can be determined by the

condition of integrability; an elementary solution

that

eliminates the intrinsic indeterminacy in terms of displace

ments may be obtained with the condition

x=O

u=O

y=O

v=O

Then, the solution of the problem is

characterized by the anelastic shearing strain

xy

2b

xy

= 2p 2

a

Let us consider again the elastic-kinematic equations,

where we set:

The integrability conditions gives the equation

p

r

XY

-

2

=0

a

whose solution is

p

r

=

2 xy x) g y)

a

and then

a

2

[p ]

y

= 2p a2 _

x

2

) a

2

xy

x) g y) .

The functions f x)

and g y)

cannot be determined

without a further condition.

However, we have

to

notice that in the deformed state,

characterized by the elastic

and

anelastic components,

further zero-stress strains are admissible, complying with

the assigned displacements on the axes X and y, and

satisfying the system

au

au

ov ov

ox

=

b

x

0;

oy ox

=

2b

xy

;

oy

=

0

Such strains, and the corresponding displacements,

cannot be removed, except by the following assumption:

we shall consider only the zero-stress strains which are

necessary for the compatibility of the system and

we

shall

neglect all the strains which are already possible in the

natural state. Such an assumption may be carried out, e.g.,

by minimizing the euclidean

norm

of the displacement

vector.

2)

Rectangular panel with dead load, supported at the

base (Figure 9).

t is easy

to

prove

that

the whole region is

R l Let us denote by p the unit weight of the material; the

static solution is

Jx

=

xy

=

0

Jy =

-p h

-

y).

I f we

assume

E = 1,

then the elastic-kinematic equations

are

u,x = vp h

-

y)

V,y =

-p h -

y).

Ii

V

I

\

I

\

I

\

/

0

\

II

l

h

-

,

X

Fig.

9.

Plane problem of a rectangular wall; the dashed line represents

the deformation of the wall.



182

SALVATORE DI PASQUALE

Therefore, the distorsions

OS

produce displacements

s

and zero stresses. We shall explicitly remark that

0-0

is the

solution of the homogeneous equilibrium equations and

may be used to solve the unilateral constraint problem.

This can be accomplished, for example, starting from the

solution

o-e

standard material) and then correcting those

values by superimposing suitable distortions.

Let

us notice also that the Haar-Karmfm solution

implicitly uses the distortions 0°; the solution, if it exists, is

unique

and

satisfies the equilibrium equations and the

stress conditions. On the other hand, the kinematical

problem - i.e. the determination of strains and fractures-

is

still undetermined, due to the intrinsic kinematical

undetermination of the material.

From

such point of view,

it seems reasonable to eliminate from the set of possible

displacements all those which might occur in the natural

configuration. The same may be accomplished by mini

mizing a suitable norm of the displacement vector.

NUMERIC L EX MPLES

The first example tha t we shall present is a rough idealiza

tion of the problem of a square wall, subject to a concen

trated load, as in Figure 11. It will be solved by FEM with

only two triangular elements.

The complete discussion of this example

is

useful for the

understanding of the following result, obtained by a

refined discretization.

The equilibrium equations of nodes give for typogra

phical reasons we have posed

(o-x)i

=

Xi;

o-Y)i

=

Y ;

( xy)i =

Z,):

Xl =Z2; Y

l

=Z2

-2F;

Zl

=

-Z2;

X

2

=

-Z2; Y

2

=

-Z2;

Z2 =

Z2

Hence

we

obtain the complementary energy with E

=

1,

v

=

0;

F

y

J

Fig.

11. FEM

model for a square wall.

Its minimum

is

attained

by: Z2 =

0.25F.

From

this, in

the two elements, the stress tensors resulting for the

standard case are

l )=F[ 0.25

0- -0.25

0.25J

1.75

2)

=

F

[-0.25

0- 0.25

0:25J

0.25

This state of stress is

not

admissible in element 1,

because it gives a positive principal stress note that

det 0- 0). Therefore,

it is

necessary to introduce the

conditions

on

the unilaterality

of

the material; for the first

element, we have:

deto- 1) =

ZiZ2

- 2F) - 0

tro- 1) = Z2 - 2F ;:;;

0

while for the second, all conditions are satisfied if Z2

O.

But the condition det

0-

0 in the first element requires

Z2

;:;;

0;

therefore it has to be

Z2

=

0,

and then

we

have the

stress tensors

Next, let us determine the displacements. The elasto

kinematic equations, written by the obtained stresses,

would be E = 1; v = 0; F = 1):

It is easy to show that the system

is

incompatible: note

that it has

five

unknowns

and

six equations. This indicates

that

the static conditioned solutions cannot be derived by

a displacement field; so, it is necessary to add a system of

zero-stress strains.

We will solve the prob lem by extensional deformations

of the class ) o-A and by shearing strains of the class yo-c.

In

the first case Figure

12),

we have:

F

1

Fig. 12. Kinematical solution by extensional deformations and by

shearing deformations.



NEW TRENDS IN MASONRY ANALYSIS

183

F

r

J

Figure

13.

Kinematical solution by extensional deformations and by

shearing deformations.

The parameters

IX

f3,

j,

.:1. must verify the so-called

Lanczos identity:

-2.:1.

-

IX 2f3

-

j

-

2

=

0

with

IX{j - f32

;;;: 0;

X

j

;;;:

0;

.:1.:;;;

0

But this condition cannot uniquely determine the

displacements and v so we will introduce as a further

condition the minimization of the norm

IDI

of the

displacement vector:

n

min

IDI =

I

uf vf)·

i

1

In

this way we will obtain the solution:

In the second case of deformation Figure 13 , we will

F

Fig. 14. FEM model for a square wall vertically loaded

at

the upper

midpoint.

have the system:

U2=0; u

3

=4y; v3= 2 ; -U3 V4=-f3;

u

2 -

V3

u

4 V4

=

2 j;

V4

=

f3

to which we have to add the Lanczos conditions

4y

- 2

f3 2 j

-

f3 =

9,

The minimum

of IDI

gives the solution:

We can note

that

in element 2 we have a pure shear

strain without tension, while in the first case we

had

extensional strains in the element

1.

These last may be also

regarded as continuous fracture distributions.

Here it is

not

possible

to

give

an

exhaustive description

of the method of numerical resolution, which must be used

for the solution

of

problems that are characterized by a

great number

of

unknowns.

In general, however, the method of investigation is

based on imposed distortions with a numerical step-by

step procedure. I will present the solution of the problem

of a square wall, simply supported

at

the base and

vertically loaded

at

the upper midpoint Figure

14 ,

which

has been obtained with this procedure.

I I

I

I

I

Fig. 15. The

uomo

fayade

at

Siena: in the standard solution the

maximum tensile stress is 1.1428 N/mm2; in the masonry solution the

maximum residual tensile stress

is

0.009 N/mm2; for the kinematical

indetermination we chose not to represent the deformate structure.



184

SALVATORE

I

PASQUALE

t

is

important to note th at the solution of the problem

is easily understandable and that this solution is more

easily obtained by rigorous formulation than by numerical

procedure. The last example deals with the

uomo

far;ade

at Siena, erected in the fourteenth century. t is inter

esting to remark

that

in the elastic solution there are

positive stress of a neglectable order of magnitude, if

compared with the smaller negative stresses.That explains,

perhaps, many of those intuitive solutions founded by

ancient builders. The results of the

FEM

computations are

shown in Figure

15

for standard material (left) and for

masonry material (right).

ACKNOWLEDGEMENTS

This research is supported by a MURST grant. The

computer code has been worked out by Pierre Smars,

during a scholarship at the Dipartimento di Costruzioni

(Universita di Firenze), funded by the Belgian government;

the numerical example has been worked

out

by Cristiana

Pesciullesi, during her PhD.

REFERENCES

1 Delanges, P., Statica e Meccanica de semifluidi , Atti Soc. It. Scienze

Modena, V (1786).

2. Heyman, J., Coulomb s Memoires

on

Statics, Cambridge, 1972.

3.

Heyman, J., Poleni s problem , Proc. Inst. Civ. Engrs, 8 1988).

4.

Di Pasquale,

S., Statica

dei

solidi murari. Teoria

ed

esperienze ,

Preprint Dip. Costruzioni, Firenze, 1984.

5.

Giaquinta, M. and Giusti, E., Researches on the equilibrium of

masonry structures ,

Arch. Rat. Mech. Anal.

88 (1988).

6.

Del Piero, G., Cons titutive equation

and

compatibility of the external

loads for linear elastic masonry-like materials ,

eccanica,

24 3)

1989).

7.

Freudenthal,

A.

M. and Geiringer, H., The mathematical theories

of

the inelastic continuum ,

Handbook

o

Physics,

Vol. 4 (1953).

8.

Sokolovskii,

V.

V.,

Statics

o

Granular Media, Pergamon Press,

Oxford, 1965.

9. Prager, W. and Hodge, P., Theory o Perfectly Plastic Solids, New

York, 1961.



EQUILIBRIUM AND COLLAPSE ANALYSIS OF MASONRY BODIES

MARIO COMO

Universita

i

Roma Tor Vergata , Dipartimento

i

Ingegneria Civile,

Via della Ricerca Scientifica, 00173 Rama, Italy.

(Received: 10 February 1992)

ABSTRACT, This paper gives a general formulation of the statics of the masonry contin uum within the conceptual

framework set up by J. Heyman in his fundamental and pioneering studies of masonry arches

and

vaults, Here the

masonry body will be represented by an assemblage of rigid particles of stones. held together only by compressive

forces,

and liable to crac k as soo n as tensile stresses begin to develop. The very small size of the stones, compa red to the

overall dimensions of the body, permits a treatment in terms of a continuum,

The admissible mechanism displacement and stress fields of the masonry body are analysed, and an appropriate

formulation of the virtual work equatio n

is

given. A variational inequality, involving the sign of the work of external

loads along the mechanisms - necessary and sufficient for the existence of the admissible equilibrium states - is then

proved. The collapse of the body

is then properly formulated and, finally, new versions of the kinematical and statical

theorems of failure are proved.

SOMMARIO. II

presente articolo fornisce una formulazione generale della statica del continuo murario, seguendo

l indirizzo concettuale espresso da Heyman nel suo fondamentale e pionieristico studio

Stl

archi e volte

n

muratura.

Nel presente lavoro

il

solido murario viene schematizzato come un assemblaggio di elementi rigidi di pietra, tenuti

insieme da forze di compressione e soggetti a frattura non appena

s

inneschino trazioni. Le dimensioni ridotte dei

conci, in relazione aile dimensioni globali del corpo, consentono di trattare il problema nel continuo.

Vengono quindi analizzati possibili meccanismi, ed campi di tensione e spostamenti associati, espressi secondo

una

appropriata

formulazione del Principio dei lavori virtuali. Viene poi

mostrato

come una opportuna diseguag

lianza variazionale - che coinvolge il segno dellavoro delle forze esterne lungo i meccanismi - costituisca condizione

necessaria e sufficiente per l esistenza degli stati ammissibili di equilibrio. Utilizzando tale condizione viene quindi

fornita una nuova versione dei teoremi statico e cinematico di collasso.

KEY WORDS: Masonry, Unilateral mechanics, Limit analysis, Collapse load, Mechanics of masonry.

l COMPRESSIONALLY RIGID NO-TENSION

MODEL OF THE MASONRY MATERIAL

The aim

of

this paper

is to

give a general formulation of the

statics of the masonry body by assuming a compression

ally rigid, no-tension model for the materiaL With this

constitutive assumption, first proposed by Heyman [ lJ-

[5J, the masonry continuum can be represented as an

assemblage of rigid particles of stone held together by

compressive forces, and liable to crack as soon as tensile

stresses begin to develop, The very small size of the stones

compared

to

the dimensions of the whole structure allows

us to consider a continuous body instead of a discrete

system composed of a large number of particles,

justifies the omission of elastic deformations, At the same

time, with the assumed compressional rigidity, many

difficulties arising from the interaction of elastic

and

fracture strains can be avoided [13].

To define, in

more

detail, the constitutive model for the

masonry material, it

is

useful to recall the key assumptions

introduced by Heyman in the analysis of the strength of

masonry structures [1]:

• sliding failure cannot occur

• the masonry has an infinite compressive strength

• the masonry is incapable of carrying tension

• elastic strains are negligible,

The assumption of an absence of sliding goes back to

Coulomb

[6].

The

fact, thoroughly discussed by Heyman

[1J that elastic calculations of stresses are not relevant to

an assessment of the stability of a masonry structure,

Meccanica

27:

185-194, 1992

©

1992 Kluwer Academic Publishers,

The above assumptions can be formulated for the

masonry continuum in a more general form by means of

suitable conditions imposed on the stress

and

fracture

tensors J and

s:

tensile stresses can never develop inside the

masonry mass, Consequently, the condition

[7J [9J

1 )

holds in the sense that,

at

any point of the body, the

maximum eigenvalue of the stress tensor

cannot

be

positive, Condition 1) defines the locus Y of the admissible

stress tensors, Consequently, if is an arbitrary point

inside the masonry body, 0

is

the unit

outward

normal

vector representing the orientation of

an

infinitesimal

surface element having as interior point

and

t(o)

is

the

associated stress vector, from 1) tensile stress interactions

are n ot admissible

and we

get (Figure

1)

t O)·o ,,;:

0,

(1 )

Strains

s,

which

do not

contain

an

elastic component, are

produced by the internal fracture of the materiaL Because

of the interlocking of the stones

and

the high frictional

strength, no internal sliding occurs, Thus, strains scan

never be contractions

and

they have

to

satisfy the



186

MARIO

COMO

Fig. 1. The masonry continuum as an assemblage of rigid particles of

stone held together by compressive forces.

condition

2)

in the sense above specified. Condition 2) defines the locus

Y of the admissible strains.

Deformations, on the o ther hand, can develop

at

a point

of the masonry mass

and

along a given direction only if the

compression, acting along the same direction

at

that point,

vanishes. Thus the following normality condition also

holds Figure 2)

3)

where Ja is the actual stress and

E

is the corresponding

fracture strain occurring

at that

point. A general admis-

sible stress state J and a general admissible fracture strain

E

are consequently linked by the following inequality:

(JOE ; 0

V(JE

Y,

VEE Y

4)

where

Y and

Y are the loci of the admissible stress

and

strain states. Consequently, for the assumed chosen con-

stitutive model there is no internal dissipation of energy.

This lack of internal dissipation marks the difference

between the masonry model and the plastic model. Thus,

for masonry, it will be possible to refer to the actual values

of the strains, in place of the strain increments. Moreover,

because of the normality condition 3), the masonry

material thus defined is stable, in the sense of Drucker

[12], as depicted in Figure 3.

In the context of the previous assumptions

Kooharian

Fig. 2. Admissible stresses and strains for the masonry material.

o

o 0

a

o

o

a

Fig. 3. The Drucker normality condition between stress and strain for

the masonry material.

[11] and Prager [12] first applied limit analysis to

evaluate the strength of voussoir arches. In particular,

Prager showed that masonry voussoirs may be treated as a

material to which the limit theorems, first used for analysis

of the plastic behaviour of steel frames, may be applied. It

is

necessary, however,

to

remark

at

this point that for a

general formulation of the collapse of masonry bodies the

definition of limit load has to be revised [13].

To focus on this last problem let us consider the

masonry arch of Figure 4 loaded by its own weight

and

by

a single increasing point load AF When this load reaches

the collapse value AoF the line of thrust in the arch is

represented by the curve a a of Figure 4. This curve

touches the intrados of the arch at points A and C and the

extrados at points

Band D.

The failure of the arch

is

defined by the four hinges mechanism

A, B,

C, D

As

in the

collapse of perfectly plastic solids, the load distribution

acting

on

the arch attains its limit value

if:

i) there exists an admissible stress field J, represented

by a compressive stress state J ; 0 in equilibrium

with the loads;

ii) there exists an admissible velocity field it

to

which

correspond kinematically admissible strain rates j

associated with the stress state J by the normality

rule JOE

= o

The stress state acting in the arch is represented by the

stress vector

J

having components (M,

N),

where M

is

the

bending moment and N

is

the axial load. The correspond-

ing strain rate vector j has components <ii, A), which are

the relative

rotation and

the central displacement rates.

Fig.

4.

The limit state for the loaded masonry arch.



EQUILIBRIUM AND COLLAPSE ANALYSIS

187

The stress vector

J

that satisfies condition

1)

corre

sponds to a funicular thrust line for the loads, which is

contained within the arch. Thus condition ii) is satisfied if

the four hinges of the mechanism are

just

localized

at

the

points A, B, C and D, where the thrust line touches the

intrados

or the extrados of the arch. The limit stress state is

then represented by the thrust line passing through the

hinges A, B, C and D and never outside the arch. At the

limit state and during the motion of the mechanism

ABCD, the active positive work of the live

load

AoF is

balanced by the resisting work of the dead load loads g.

For

any other mechanism the resisting work is larger than

the active work. On the other hand, when A >

Ao

the

positive work of the force

AoF

in the four-hinge mechan

ism ABCD, is the prevailing term. Thus, as is well known,

at

the limit state the arch passes from the existence to the

non-existence of

an

equilibrium state. The case of the

masonry panel loaded by its own weight and a gradually

increasing lateral force

is

similar (Figure 5).

Let us now consider the case of the masonry panel of

Figure 6 made of rigid tensionless material and loaded by

the uniaxial compression Ap. A uniform field of uniaxial

vertical compressive stress occurs in the panel. In the plane

of the principal stresses

J

1, J 2 of Figure 3 the uniaxial

compression is located

at

the boundary of the limit locus Y

of the material. Any lateral dilatation 8 of the panel is thus

possible. This deformation, normal to the uniaxial com

pression

J

1 = Ap represents a mechanism. In fact, also in

this case conditions i)

and ii)

are both satisfied. Conse

quently, the panel is at a limit state for any value of the

applied vertical pressure

Ap.

On

the other hand, for any

A

the equilibrium of the panel

is

secure. This limit state

cannot therefore be taken as the collapse state for the

panel.

t

is only for plastic bodies, for which the internal

dissipation is never zero, that the mechanism state

and

the

loss of equilibrium are simultaneously attained. In the case

of no tension materials, on the contrary, these states can be

decoupled. The traditional limit load, defined by the

previous conditions i) and ii), cannot therefore represent

a satisfactory general definition of the collapse load for no

tension bodies. This definition, in fact, has

to

be strictly

connected to the passage from the existence to the non-

existence o equilibrium in the body, as firstly pointed out

by

Como and

Grimaldi [14]. Research into the conditions

on the loads which imply the existence of equilibrium

Fig.

5.

The limit state for the laterally loaded masonry panel.

tL_fL---,t AP

Fig.

6.

The limit load for the axially loaded masonry panel.

represents therefore a crucial point for the development of

a general theory of the collapse of masonry bodies.

2.

DEFORM TIONS

According to the rigid no-tension model, deformations of

masonry bodies are defined by the mechanism displace

ments fields

u P),

PEn.

5)

Here 0. denotes the region occupied by the body, with

boundary on which is assumed sufficiently smooth.

Displacement fields

5),

which we assume to be very small

quantities with respect

to

the leading dimensions of the

body, will satisfy the given boundary conditions as the

internal constraint condition 2). Condition 2), in par

ticular, requires that the displacement functions u P)

cannot

produce any contraction

among

points connected

by segments entirely contained in the body. Thus, if

P

1

P

2

is

such a

pair

of points in 0. and

(Qj,Q2) is

the

corresponding pair after the transformation 5),

6)

where d

,

and

s ,

respectively denote the distance

and the set segment connecting the points. A relative

sliding of one part over

another

in a body made of rigid

no-tension material is therefore not admissible.

The displacement fields

u P)

will be functions of

bounded variation in

0..

Thus, along any line passing

through the body these functions will show countably

many discontinuities, which are the fractures

that

occur

in the body. Each region enclosed by a fracture line, or in

part, by the boundary of the body, will represent a

fragment. Thus the

body

will fracture, at most, into a

denumerable set of fragments. In the same fragments a

partial fracturing can also occur. In this case the cracks

will be

not

connected together (Figures 7

and

8).

At each point inside the fragments, where the dis

placement function

u P)

is smooth, the strains E will be

represented by

E =

Du P),

7)

where D

is

the operator that associates, with the displace-



188

MARIO

COMO

Fig. 7. Fracture deformation in masonry bodies.

ment u, the infinitesimal deformation field e with

components

(8)

with the usual meaning of symbols

and

with derivatives of

the function u defined in the ordinary sense. On the other

hand, among the fragments and in their interior, where

partial fracturing occurs, the strains will be represented by

distributions.

The fracture lines represent points

of

discontinuity of

the displacement function

u.

At each of these points,

indicated by P, let n - and n

+

be respectively the two

outward normal vectors to the two faces of the fracture.

We can consider the two neighbouring points

P-

and p+

where P+, passing through P, is

an

infinitesimal seg

ment of the normal n -. Thus at P and along the positive

direction of n - normal to the fracture line, is defined the

jump shown in Figure 10:

L 1(n-)u(p) = u P+) -

u r )

= {u P+)

- u r)}n - 9)

where

u P)

is the scalar value of

u P)

and n - is the unit

outward normal

at

P-.

We have to postulate, of course,

that

10)

Restrictions of a geometrical nature are also imposed on

the deformation of the body. These constraints require

that the mechanism displacements u have to satisfy

suitable boundary conditions. Let an

be the portion of

the surface of the body where these restraints are imposed

and let v be the outward normal at the generic poin t P over

an .

The restrictions imposed usually require that the

displacements of the points of the boundary

an

cannot

cross a surface that is in contact, in the initial state, with

the same boundary an , i.e.

u(P) V

0

I;fP

E

an .

(11)

The set of all admissible displacements of the body - the

Fig.

8.

Fracture deformation in masonry bodies.

so-called mechanisms

-

satisfying the given bounda ry con

ditions and the internal constraint condition (6),

is

denoted

by M. The set M is a subset of the space

BVoffunctions

of

bounded variation.

For any

u

E

M the set

r u)

of all jump points is

measurable and represents a new

part

of the boundary of

the body, created by the fractures associated to the field

u

[15]. For a given displacement u the fracture-free region

is

n u)

=

n\r u).

12)

In this region the displacement fields u are smooth func

tions, for instance, with their first derivatives continuous.

Generalizations of the previous assumptions can be made

[16], [17]).

3. THE DMISSIBLE EQUILIBRIUM STATE: THE

PROPER FORMUL TION OF THE VIRTU L

WOR EQU TION

The development of a global analysis of the admissible

equilibrium of masonry bodies could be a very difficult

task because of the strong discontinuities involved in the

corresponding displacement functions. The idea, first pro

posed in [13], to analyse the internal equilibrium

of the

various fragments into which the body splits and, sub

sequently, to examine the connection among them along

the fractures, seems, however, to be very profitable and

will

be pursued in this section.

Let the masonry body be loaded by mass and surface

loadings

p(n)

and

p(an)

Figure

9).

The loaded

part

of the

body surface is an'. The surface region an is subjected to

appropriate boundary conditions, represented by the uni

lateral condition (11). Let the body be

at an

admissible

equilibrium AE) state, i.e. the AE state defined by a stress

field J which satisfies condition

(1)

and is also in equilib

rium with the given loads.

For

the sake of simplicity

we

shall assume that J E C

1

(n)S, the set of all the symmetric

second-order tensor functions, continuous in their first

derivatives, in n. However, some generalizations

of

the

Gauss-Green

formula [16] allow us to extend

our

results

to more general stress fields.

Fig. 9. The masonry body

at

an admissible equilibrium state.



EQUILIBRIUM AND

COLLAPSE ANALYSIS

189

Let

b

E M be a mechanism displacement tha t represents

a virtual admissible deformation of the body.

As

a rule,

with the development of

bU

fractures will spread in the

body. The body will be subdivided into fragments and

cracks will appear between them. Each one of these

fragments will displace with respect to its neighbours and,

at

the same time, may itself exhibit

an

internal continuous

or discontinuous dilating deformation. Let

n(bu)

be the

fracture-free region generated by the displacement bu.

Taking into account condition

(4),

we thus have

<50 = Dbu(P), PEn(bu), bUEM

13)

At the same time,

at

a point

P

crossed by a fracture, where

the

jump (9)

occurs, according to inequality (1 ) we get

Figure 10)

where

,1(n-) b = {u P+)-u P-)}n

,1(n+)bU = {u P+)-u P-)}n+

(14)

(9 )

Here

ten)

is the stress vector acting over the surface element

of normal n along which the virtual crack opening

,1(0) b

occurs. Inequalities (13) and (14) define, together with the

equilibrium conditions on the loads, the AE state of the

body.

The AE state

is

governed by the principle of virtual

work.

For

bodies made of rigid no-tension material this

principle will take a particular form, which we shall now

seek.

Let us consider a generic fragment k) of the body,

corresponding to the assumed virtual displacement bu.

Let n(k) be the region occupied by the fragment

k),

n(bU)(k) be the fracture-free interior of n(k), and let

a,1 bU) k)

=

an k)

u

r(bU)(k) be the new boundary

of

the

fragment k created by the displacement function

u

and

where r(buyk) is the

part

of an(bU)(k)

on

which partial

fracturing occurs Figure

9).

At any internal material point of the fragment, belong

ing to the set n(bU)(kl, the stress field

(J

will satisfy

(n -

)

t::

u

(n+)

t

Fig. 10. The jump of the displacement function across the fracture.

Fig. 11. The partial fracture boundaries.

inequality (1) together with the internal equilibrium

equations, which may be written as

(Jjj,j + pj = o

(15)

Let d

V

be a generic volume element of the fragment

k.

The

virtual work done to displace this element is

(16)

This work is zero, according to the equilibrium equation

(15). Integration of (16) over the volume n(k) thus gives

r « Jjj,j+pJbujdV=O

jO(6u) k)

(17)

Now the

Gauss Green

theorem, together with some

tensor calculations and the previous specifications, enables

us

to obtain from

(17):

r (Jij<50ijdV

=

f bujt\n)dS + r pjbujdV.

jO(6u) <

oQ(6u)'

j

0(6u)'<'

18)

We start from the statement

19)

where r(bu)(k)

is

the

part

of the boundary an(bU)(k) sur

rounding the partial fractures, anlf

is

the

part

of the

boundary an(bU)(k) facing the neighbouring fragments,

an<;.) is the part of an(bU)(k) on where the reactions rare

applied and any;) is the

part of

an(bU)(k)

on'

on which

the surface loads p are applied.

Thus we can specify the stress vector dn acting over the

various parts of the boundary an(bU)(k) and write

r

(Jij<50ijdV

=

r

bujt\n)dS

+

f bujt\n)dS+

jO(6u) k)

jrc6u) k)

O O ~

f bujr\n)dS +f

bUjp\n)dS

+

o . Q ~ a . Q ~

+

r

pjbUj

d

V (20)

JQ k)

Using

r(bu)(k)

= r ~ k u

r ~ k ,

(21)

where the portions n

k

)

and n

k

)

of r(bu)(k), the two sides of



190

MARIO COMO

the partial fracture,

are

indicated in Figure 11, and

evaluating the integral along the fracture line r(bU)(k) we

get

r

bu;tln)dS =

r

bu;(p-)tln-)dS +

r

bu;(p+)t n+)dS

J

)u

k) J i

k

JL

k

22)

because bu;(P-) = _ ~ n - ) b u ; + bu;(P+) and tl

n

-) =

_tl

n

+)

Figure 11). Taking Equation 22) into

account

and sum

ming up over all the fragments from

condition

20) we get

23)

We observe now that in the sum

(24)

there are pairs of integrals that evaluate the virtual work

tIn) dS bu; over opposite faces of contiguous fragments.

Thus, for any pair, the first integral will show the work of

the stress vector

t f

for the virtual displacement bu;(P+),

and the second will show the work of the stress vector

tl

n

-) = - tl

n

+>, acting

on

the opposite side

of

the facing

fragment, for the corresponding displacement bu;(P-). I f

the displacements bu;(P+) and bu;(P-), at the same point

but

relative to the opposite faces

of

the contiguous

fragments were equal,

then

the virtual

work

24) would be

zero. However, these faces undergo a relative disp lacement

because between them a fracture opens.

The

virtual

displacement bu;, crossing the fractures along their

normal

n,

will in fact exhibit a

jump

~ n - ) b u = { b u P + )

- bu P-)}n- that represents the virtual opening of the

crack. Thus

25)

where

F

represents the surfaces,

counted

in a given order,

of all the first sides of the fractures

that

open among the

various fragments.

With the same procedure we can also write

I {

bu;tln)dS}

= { ~ n - ) b u ; t l n + ) d S }

k)

J i

k

J

*

26)

where F* represents the surfaces,

counted

in the same

order, of all the first sides of the partial fractures inside the

various fragments. Summing Equations 25) and 26) we

get

i

n - ) bu.t n+) dS

, ,

={t

n - ) b u ; t l n + ) d S +

t

n - ) b u ; t l n + ) d S }

with

F(bu) = F(bu) u F*(bu)

(27)

(28)

where F(

bu) is

the

net

of all the first sides of the fracture

surfaces associated with the virtual mechanism displace

ment

bu. Then, with the following definitions

{t(n+>, ~ n - ) b u } = tl

n

+) ~ n - ) b u ; d S

where

n

=

n(l) u

n(2)

u ... ; on'

= anlP

u on):) u ;

on" =

an p u anll) u

we get

29)

30)

<rJ, be) =

{t(n+>,

~ n - ) b u } + <r,

bu

+ <p,

bu

VbUEM

31)

Taking into account

(12),

13) and 14) we have also

<rJ, be) ,;; 0 W+> ~ n - ) bu} ;;, 0 <r, bu ;;,

O.

32)

Vice versa, going

back

from Equations 31) and 32) we

arrive at Equation 15) and finally at Equations

(14), (13),

12) and, with

(2),

to (1). Thus the conditions 31) and 32)

are necessary and sufficient for t he admissible equi librium

state and represent, in a suitable form, the principle of

virtual work for no-tension masonry bodies.

4. THE VARIATIONAL INEQUALITY

FOR

THE

EXISTENCE OF THE AE

STATE

We

are now seeking conditions, involving only known

quantities, which would enable us to predict if a given

body, made of rigid no-tension material, can withstand the

action

of

assigned loads p.

In

this section we shall prove

that

the variational inequality on the loads p,

<p, bu) ,;; 0, VbUEM

(33)



EQUILIBRIUM AND

COLLAPSE ANALYSIS

191

is

necessary and sufficient for the existence of the AE state

in the body. As we shall see, necessity follows immediately

from the virtual work equations

(31) and

(32). But it is

more

complicated to prove sufficiency.

In

the context of

the elastic no-tension model a proof of this condition, but

with some specific assumptions, has been given by

Romano

and

Romano [18],

and

Romano

and

Sacco [19].

A new proof, in the context of the rigid no-tension model

which uses the virtual work equation (31), was given by

Como [13].

The ma in lines of this last proof are as follows.

f the variational inequality (33) was only necessary but

not

sufficient, it could be also satisfied by loads p

that

cannot

be sustained by the

body at

the AE state. This last

situation is, however, impossible to meet. t will be in fact

shown

that

any load p

that

cannot be sustained by the

body in

an

AE state and, consequently, puts the

body

in

motion, does positive work for the displacement v along

which the

body

itself begins

to

move. This contradiction

with the assumption proves the statement.

Let us assume therefore ad absurdam, together with

condition

(38), that

the body, under the action of the loads

p,

is

not at an

AE state. Let us consider the motion, defined

by the velocity field yep, t), that starts just after the

application of the loads. One or more fragments of the

body will begin

to

move. At any inst ant of the motion, the

stress J will satisfy the internal constraints, i.e. condition

(1) and the normality rule (3). Thus

o (P, t)

:;:; 0

o (P, t)· i;(P, t) =

0,

P

E

0.,

It

O. (34)

Let us apply the virtual work equation taking as virtual

displacement

On

the effective displacement

that

occurs

along the motion of the body during the time interval

dt

On = yep, t) dt.

(35)

Thus, with OE = i;(P, t) dt, l'1(n)on = l'1(n)v(p, t) dt,

and

taking

also into account the inertial forces produced in the body

because of the accelerations

V

we get

<O',i;) = {t(n+),

l'1(n-)v} <r, v <p, v - <pv,

v)t

> O.

(36)

Also, during the motion we have

<r, v

= o.

(37)

The first of(37) follows from

(34).

For the second condition

we

C:ln

observe

that

when cracks begin

to

develop, then

along them

l'1(n-)v i=

0

and the stress interaction t(n+) there

vanishes. Likewise if, during the motion, the body

is

going

to

come away from the constraint boundary 00. , there

v

i=

0, r = 0

and

also the last condition (37) holds. Thus the

condition (36) becomes

<p, v -

<pv,

v = 0, t >

O.

We now take into account that

dT

<pv,

v

= Cit

(38)

(39)

where dT/dt is the rate of change of the kinetic energy

T

= <pv,

v /2 of the body during the motion. Thus Equa

tion (38) yields

dT

<p, v = - t > O.

dt

40)

Let us evaluate the sign of the power <p, v during the

initial motion v, when the particles of the body begin to

move under the action of the external loads.

At any instant t, subsequent to the initial time t =

0,

the

motion is defined by the velocity vector field

v.

At the time

t = 0 the velocity field is zero

at

any point of the body. The

starting

motion

can in any case be obtained by continuity

as the limit ofv in the subsequent times. The displacement

function

s(P,

t), from the initial position

s(P,O) and

in the

neighborhood of the initial state, can in fact be evaluated

as

1 ( 0)

2

9(

).

I 19 P, t 1 - 0

s(P, t) = zV P, t P, t .

1m - ,

,-0 t

(41)

where

v(P,O)

is the initial acceleration field. Thus the

velocity

and

acceleration fields of the starting

motion

respectively are

yep,

t)

= v(P, O)t ... ; v(P,

t)

= v(P, 0) ...

(42)

Here the power of the external forces can be expressed as

<p,

v =

t<pv(P,O), v(P, 0) e (t):

lim

e (t)

=

O

(43)

-0 t

Taking

t

sufficiently small it

is

thus possible to obtain

sgn<p,

v

=

sgn t<pv(P,

0),

v(P,

0 .

But

t<pv(P, 0), v(P, 0) > 0, t > O.

Consequently, with t sufficiently small,

<p, v

= t<p,

v(P,

0)

> 0

44)

(45)

(46)

Therefore, from Equations (41) and

(46),

in a neighbor

hood of the initial state

we

have

<

(

p t) = t

2

<p, v(P, 0)

0

p,

s , 2

>

(47)

Thus, if the loads are applied

and

they cannot be statically

equilibrated, the body starts its motion with displacement

functions given by (41). The work done by these loads

along these displacements

is

positive. But this results

contradicts assumption

(33).

Hence we conclude that if

<p,on):;:; 0, VouEM, the body is

at an

AE state.

5. THE MECHANISM STATE

5.1. Definitions

The mechanism state corresponds

to

the peculiar con

dition in which the body, made of compressionally rigid,



192

MARIO

COMO

no-tension masonry material and

at an

admissible equilib

rium under the loads p can become deformed.

Let the body, under the action of the loads p be

at

an

AE state. Then

at

least one admissible stress field

j

1

in

equilibrium with the loads p exists in n. Let us assume now

that there exists a mechanism displacement Ve such that the

stress field j

1 at

any point of

n(vJ, i.e.

inside the fragments,

is

orthogonal

to the strain field

Ee

= DVe. Further, let it also

be assumed that at any point on the boundary an the

corresponding constraint reactions r 1 are orthogonal to Ve

and that along the fractures IF vJ the interactions t r are

orthogonal

to the crack opening functions

L ~ , < o - V e .

Thus

These conditions define the occurrence of a

mechanism

state in the body. We can recognize

that

any other

admissible stress state (J z in equilibrium with the loads p

will be orthogonal to the mechanism Ve.

Another equivalent definition of the AE mechanism can

be given [13].

As we can show immediately, by using the

virtual work equation (31), the existence of the mechanism

state is equivalent to the existence,

at

the AE state, of a

mechanism Ve along which the external loads p do no

work,

i.e.

to the condition

(48 )

For

plastic bodies

at

the limit state both the mechanism

and

the loss of equilibrium states are simultaneously

attained while, in the case of no-tension materials, these

conditions can be fulfilled separately.

In

the next sections

we shall consider in which cases the presence of a mechan

ism state has to be excluded or, when the mechanism state

occurs, under which conditions it corresponds to the

collapse of the body.

5.2. Admissible and safe stress fields

Stress fields that produce compression on any plane and at

any point in the body are defined as admissible and safe.

Thus, the following statement can be easily established:

If

an admissible and safe stress field

j in

equilibrium under

the given loads

p -

exists

in

the body, the mechanism state

is

not attained.

To prove the statement let us suppose, on the

contrary, that the body attains the mechanism state

defined by the displacement Ve. Thus there exist in the

body admissible stresses and reactions

(jz,

r

z

which are in

equilibrium with the given loads and such that j

z •

Ee = 0

in n, t ~ - · ,1(0) bu = 0 along I and r

z

' Ve = 0 over an . But

any other admissible state in equilibrium with the loads

must also be orthogonal to the same mechanism

Ve·

Consequently,

j-

. Ee = 0 in n. The contradiction between

this last result and the assumption

that

j is safe proves

the statement.

The mechanism condition cannot therefore be attained.

We find

that

<p,

bu

<

0, I: du

E

M,

and

the body does

not

become deformed.

5.3. Collapse

Let the masonry body be subjected to the loading process

p = p(A),

A

o.

49)

The collapse state corresponds to that particular mechan

ism state for which there occurs the transition from the

existence to the non-existence of the

AE

state in the body.

Let Ao be the value of the load factor

A

at which the

collapse

is

attained. Thus,

at

A

= Ao:

(i)

the body

is at

a mechanism admissible equilibrium

state, defined by the displacement

Ve;

(ii)

equilibrium is lost along the mechanism Ve as soon as A

becomes larger than

Ao

Thus the following conditions define the collapse state:

0,;;;

A ;;;

Ao, <p(A), u ;;;

0,

l:/uEM

<p(Ao),

v

e

)

= 0

{dd

A

p(A), ve

) ~ } o >

0

I:/u

Ve:

<P(Ao), u = 0,

{ A

<p(A), u ~ } o ;;; O.

Conditions (50) and (51) respectively imply:

(50)

(51)

(52)

(i)

the existence of the admissible equilibrium states for

0,;;;

A ;;;

Ao;

(ii)

the occurrence, at A =

Ao,

of the loss of the equilibrium

along the mechanism Ve.

Condition (52), established for the sake of simplicity,

implies the uniqueness of the collapse mechanism

Ve.

Thus,

for any A <

Ao,

the body either remains undeformed or it

displaces along mechanisms, according to whether the

work <p(A),

u

is negative I:/u E M or zero along some

displacement u. At A = Ao the body fails along the mechan

ism

Ve.

Consider as

an

example the case of a block uniaxially

loaded by the pressure AP but free to expand sideways

(Figure

6).

Taking for Ve any lateral expansion of the block,

condition (50) is verified for any A; but condition (51) fails

because, for any A

{d/dA<p(A),

v

e

)}}, =

O.

As

a second

example consider the panel, of width

B

and height

H

loaded by a vertical dead load G and a horizontal force AF

applied

at

the top, and gradually increasing (Figure 5).

Here the mechanism state

is

attained when the force AF

reaches the turnover value

AoF

=

GB/2H.

Conditions (50),

(51) and (52) all hold in this case and at A = Ao the collapse

is

attained.



EQUILIBRIUM AND COLLAPSE ANALYSIS

193

6. THE RESPONSE

OF MASONRY BODIES

TO

CHANGES IN THEIR ENVIRONMENTS

In

the context of the analysis developed above, in this

section we shall consider the following statement, firstly

proved by Heyman

[ ]: I f

the foundations

of

a stone

structure are liable to small movements, such movements will

never, of themselves, promote collapse of the structure.

Let us assume, in fact, that small changes of the

boundary

conditions - for instance, small settlements

or

spreadings of the foundations - occur for the masonry

body in an AE state at the initial configuration Co under

the loads

p.

Therefore at

Co and

under these loads

<p, u) ,,;; 0, VUEM.

Now if small settlements occur, the masonry body will

become deformed,

and

it will move from

Co

to the

displaced configuration C. Thus a mechanism displace

ment occurs to shift the body from Co to C. Additional

forces are, as a rule, produced by the external environment

and within the body changes in the stress distribution will

occur. However, in spite of the occurrence of the changes

of the bound ary conditions, because of the smallness of the

settlements produced, the work of all the external forces p

along the various mechanisms will be still the same as at

Co·

Thus, in the displaced configuration C, for the purposes

of evaluating work, we can take configuration Co; thus we

still have

<p, u) ,,;; 0, Vu

E M. Therefore , the equilibrium is

still admissible at the shifted configuration C.

7. THE CASE

OF LINEARLY INCREASING

IMPOSED

LOADS

ACTING ON HEAVY

MASONRY STRUCTURES

The case of constant dead loads

g

acting on a masonry

body together with imposed loads, the latter increasing

with

load

factor A is particularly significant. The real

strength offered by a masonry struc ture

to

the action o f the

loads

Aq

is just centered in the interaction between the

fixed resisting dead loads g

and

the imposed live loads Aq.

The assumed loading process is therefore defined by

p(A) = g

Aq, 0,,;;

A.

(53)

Let us partition the set M of all the mechanism displace

ments as

(54)

MPq = {uEM:

<q,

u) >

OJ

q

= {uEM:

<q,

u) ,,;;

OJ.

(55)

Thus,

d

dA <g Aq,

u)

= <q,

u)

> 0, VUEM

pq

•

(56)

The work of the loads (53) is therefore linearly increasing

with A along any mechanism

uE M

pq

•

For

the resisting

dead loads g, however, we have established

that

<g,

u) ;;

0, VUEM; (57)

Hence, according

to

condition (33),

at

the initial state

= ° f the loading (53) the masonry structure

is

at an AE

state. Let us analyse now the possibility of a change in the

conditions for the existence of

an

AE state along the

loading path

(53). For

the sake of that let us show first that

if an AE exists at A = A

1

, AE states exist also for any Asuch

that °

;; A ,;; A

1

.

In fact, the existence of the AE at

A

= A

yields

<g + Aq,

u) ,,;;

<g + A1q,

u) ,,;;

0, VUEM

pq

;

<g

Aq,

u) ,,;; 0, VUEM

Nq

•

Thus,

<g Aq, u) ,,;; 0, VUEM

(58)

59)

and there

is

AE for 0,,;;

A

,;;

} 1

Of course, the previous

statement

is

equivalent to the statement that if AE does

not exist

at A

=

}'2'

it does

not

exist

at

all for any

A

>

A

2

.

Consequently, once the AE state has been lost, it cannot

ever be recovered along the loading path, no matter how

large

A

becomes. With the loading

(53),

the occurrence of

collapse at

A ,;;

Ao requires

<g Aq, u) ;; 0, VUEM

60)

61)

7.1 The Kinematical and Statical Theorems

of

Collapse

With these results we can state now the kinematical and the

statical theorems of the collapse of masonry structures. The

statements of these collapse theorems follow the same

argument of the proofs given in [14]. However, the

sufficiency of condition (33), proved at Section

4

(see also

[13J), has here allowed the completion of the proofs. Let us

analyse the change of the sign of the work of the loads (53)

along a mechanism u+

EM

pq

• The kinematical multiplier

of the loads q

A

+(u+)

= _

<g, u+)

<q, u+)

62)

that cancels the work of the loads (53) along the mechan

ism U+, marks the passage, from positive

to

negative

values, of the sign of the work <g Aq, u+).

It

is also

possible, on the other hand, that along another mechan

ism UE MP

q

the corresponding kinematical multiplier A U)

can be lower

than A

+(u+). Thus, the condition

<g A+q, u) ,,;; 0, VUEM, can fail. Hence

(63)



194

MARIO COMO

This inequality represents the kinematical theorem of the

collapse of masonry bodies. We shall now state the

corresponding statical theorem. Let A

«J-)

be an admis

sible stress distribution in equilibrium with the loads

g + A «J- q.

Thus

A

«J-) is

defined as a 'statical mul

tiplier' of the loads

q.

Thus we get

64)

In fact, according to the the previous results, along the

loading path 53) AE states exist for any A such that

o

A A «J-).

This statement of the statical theorem of

the collapse

is

weaker than the formulation, given at

Section 5.2, based on the existence of an admissible and

safe stress field

J

. They become coincident,

of

course, if

the presence of mechanism states preceding the collapse

has to be excluded.

In the framework

of

the Heyman approach many

applications of the limit analysis of masonry structures

have been performed, particularly in evaluations of the

strength of masonry buildings

and

monuments under

seismic loads ([20J-[22 J).

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1. Heyman, J.,

The

stone skeleton', Internal. 1. Solids Struct., 2 (1966).

2.

Heyman,

J.,

On shell solution for masonry domes', Internat.

1.

Solids

Struct.,

3

(1967).

3. Heyman, J., The safety of masonry arches',

Mech. Sci.,

11 (1969).

4.

Heyman, J. Equilibrium

of

Shell Structures, Clarendon Press, Oxford,

1977.

5.

Heyman,

J. The Masonry Arch,

Cambridge Press, Cambridge, 1982.

6.

Coulomb, C.

A.,

'Essai sur une application des regles de maximis

et

minimis a quelque probleme de statique, relatif a l'architecture',

Mem. Math. Physique, Acad. Roy. Sciences Savants,

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(1773).

7. Giusti,

E.

and Giaquinta, M., 'Researches on the equilibrium of

masonry structures',

Arch. Rat. Mech. Anal.,

88 (1985).

8. Di Pasquale,

S.,

'Questioni di Meccanica dei solidi non resistenti a

trazione' AIMETA

VI

Congr. Naz. Ie Genova, 1982.

9.

Di

Pasquale, S., 'Statica dei Solidi

Murari

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Universita di Firenze, Dipart. di Costruzioni, .27, 1984.

10.

Baratta,

A. and

Toscano,

R.,

Stati tensionali in pannelli di materiale

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1.

Amer. Concrete Inst.

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13. Como, M., On the equilibrium and collapse of masonry structures',

Rapporto n. 30 del Dip.to di Ingegn eria Civile Edile,

II

Universita di

Roma,1990.

14. Como, M. and Grimaldi,

A.,

'An unilateral model for the limit

analysis of masonry walls',

Interna t. Congr. on Unilat. Problems

in

Struc. Analysis , Ravello, 1983; CISM , Spri nger Verlag, 1985.

15. Vol'pert, A. I. and Hudjaev, S. I., Analysis in Classes of Discontinuous

Functions and Equations

of

Mathematical Physics, NijhotT, 1985.

16. Del Piero, G., 'A generalized Gauss-Green formula for the math

ematical theory of plasticity',

Proc. I cclem Conf, Chongqing, China,

1989.

17. Del Piero, G., 'Recent developments in the mechanics of materials

which do not

support

tension', Internal. Call. Free Boundary Prob-

lems, Irsee, Baviera, 1987; IMTA/056 1st. Mecc. Teor. App ., Univ.di

Udine, 1988.

18.

Romano, G.

and

Romano, M., 'Elastostatics of structures with

unilateral conditions on strains and displacements', Internat. Congr.

on Unilat. Problems in Struct. Analysis , Ravello, Sett.

22-24,

1983;

CISM, Springer Verlag, 1985.

19.

Romano, G.

and

Sacco, E., Sui calcolo di strutture murarie non

resistenti a trazione, Atti Istituto i Scienza delle Costruzioni,

Universita di Napoli, 1986.

20. Como, M.

and

Lanni, G., 'Sulla verifica aile azioni sismiche di

complessi monumentali in

muratura , JOCogr

Naz.

di

lngegneria

Sismica, Dipartim. Ingegneria Civile, II Roma, 1987.

21. Como, M., Grimaldi, A.

and

Lanni, G., 'New results

on

the horiz

ontal strength evaluation of masonry buildings and monuments',

9th

World

Conf

Earthquake Eng., Tokio, 1988.

22. Abruzzese, D., Como , M. and Lanni, G., On the horizontal strength

of the masonry cathedrals',

ECEE,

9,

Moscow, 1990.



MODELLING THE DYNAMICS OF LARGE BLOCK STRUCTURES

GIULIANO AUGUSTI

1

and ANNA SINOPOLl

2

lDiparti mento di Ingegneria Strutturale e Geotecnica, Universita di Roma La Sapienza ,

Via Eudossiana

18

00184 Ramo Italy.

2Dipartimento di Scienza

e

Tecnica del Restauro, lst tuto Universitario di Architettura di Venezia,

Tolentini 197, 30135 Venezia Italy.

(Received:

21

February 1992

ABSTRACT. This paper summarizes the main critical points

that

arise when the problem of modelling the dynamics

of block structures is tackled. In the first sections, a rigorous formul ation

of

dynamics and impact problem is presented

for a single rigid block freely supported on rigid ground, in order to illustrate the basic difficulties concerning the

modelling of more complicated structures. Then, a critical review

is

presented on the numerous researches performed

on this subject and the results achieved, and the problems still open, are put in evidence.

SOMMARIO

In questo lavoro,

si

illustrano i punti salienti e critici che devono essere affrontati nella modellazione

del comportamento dinamico di strutture costituite da grandi blocchi assemblati a secco. Nei primi paragrafi, viene

presentato e discusso

il

problema generale della dinamica e dell urto del blocco singolo semplicemente appoggiato su

suolo rigido: equesta la base necessaria per affrontare in modo rigoroso la modellazione di strutture pili complesse.

Viene quindi presentata una rassegna critica di vari modelli proposti in letteratura evidenziando problemi risolti e

quelli ancora aperti.

KEY WORDS. Blocks rocking,

Dry

friction, Impact, Structural dynamics, Mechanics of masonry.

1

INTRODUCTION

The conservation

and

maintenance of the monumental

patrimony from the past pose many difficult problems of

various natures, from philosophical

and

historical

to

architectural and structural. t is particularly important to

understan d the mechanical behaviour of old constructions

with respect to the most probable causes of damage

and

failure, including earthquakes, in order to check and

possibly improve their durability.

As

a

matter

of fact, a

common

feature of most con

structions that have survived through several centuries,

is

the low value of the static stresses compared with the

mechanical strength of the material. Therefore, their safety

and

reliability depend essentially on the long-term re

sistance of the material with respect to environmental

effects (which are outside the scope of this paper),

and on

the structural resistance to dynamic actions,

among

which

the seismic ones are the most important.

This

paper

deals with the mechanical behaviour of

structures made by blocks of very large dimensions,

usually not connected with each other, so that the most

probable cause of failure

is

the loss of equilibrium of the

whole structure or part of it.

t is to be noted that in many works on masonry

structures, following essentially the limit analysis metho

dology, a multiplier of some superimposed load is sought

that

corresponds to the formation of a failure mechanism

( collapse load factor ): this coincides with the stability

analysis of the initial equilibrium configuration

[IS]

Meccanica 27:

195-211, 1992

©


Often, the resistance

to

earthquak es is dealt with by means

of the same technique (,quasi-static approach ): the

superimposed load is the inertia force co rresponding to a

constant ground acceleration. This approach, however,

yelds a necessary but not sufficient condition for the

actual collapse of the structure, because it refers only to the

incipient stage of the motion; this means

that

the dynamic

collapse mechanism

and

load factor are not necessarily

coincident with the static ones. Therefore, in order

to

have

a complete picture of the structural response, it is neces

sary

to

follow the successive dynamic phases, taking

account not only of possible geometric non-linearities

(which may be

important

also in true static situations)

but

also

of

the variability of the external action(s) with time

and

of consequent modifications of the structural diagram

and response.

n the authors opinion, a complete understanding of the

behaviour

of

such structures requires

that

the dynamics,

the impact and the friction be rigorously modelled. There

fore, in the present paper they present a thorough dis

cussion of the single block problem, thinking

that

its

assemblage into multiblock structures may follow with

comparative ease. After recalling briefly early works on the

subject, the general dynamics

and

impact equations are

presented in Sections 3

and

4. Then, Sections 5

and

6 are

devoted to a critical review of recently published works,

focusing in particular on the stability

and

boundedness of

the resultant motion. Finally, the concluding section dis

cusses the problems that can be considered solved

and

those that are still open.



196

GIULIANO AUGUSTI AND ANNA SINOPOLI

2. PURPOSES

AND QUESTIONS

OF

EARLIER

WORKS

The papers by Milne

[lJ,

[3J

and

Perry [2J can be

considered the first modern works on dynamics of block

structures: their

main

aim was to estimate the peak

acceleration of an earthqua ke excitation, by observing the

overturning of tombstones or monumental columns, in

regions where no seismographic records were available.

Then, after the fundamental work by Housner [4J, the

last

few

years have seen a wealth of research papers on

block structures, mainly devoted to the dynamics of the

single block.

Most

of these papers, especially in Italy, are

aimed at the preservation of monumen tal structures, and

in particular of archaeological remnants (for a summary

review,

cf.

[28J); other papers deal with different,

but

analogous aims connected with seismic problems, e.g.:

a) evaluation of historical peak ground acceleration from

the effect induced on existing elements that can be

assimilated to free standing blocks (like isolated

columns, votive stones, etc.);

b) seismic stability of protection shields

and

other ele

ments of industrial machinery;

c) design of elevated tanks; etc.

Most of these researches, however, tackle the

mathematical-numerical problem of the identification

and stability of a given structural response to an assigned

dynamic input, and accept the model of the single block

motion proposed by Housner [4J which, as he himself

pointed out, holds only for slender monolithic structures.

Thus, not enough attention has been paid to the

mechanical modelling, and this

is

relevant in particular

when multiple-block structures are dealt with. In fact,

problems arise with reference

to

the treatment of impact,

to the role of friction, to the recognition of the most

relevant mechanisms, to the coupling of the degrees of

freedom

and

the consequent transition during the dynamic

evolution from a given mechanism

to another

one,

to

the

sensitivity of the system to the parameters of the external

action.

The most critical point appears

to

be the interplay

between dynamics and friction, particularly at the time of

impact

cf.

e.g. [24J, [37J, [46J). In fact, whatever the active

mechanisms, the impacts that happen every time the

structure passes through the static equilibrium configura

tion may-thanks to the lack of connections between

adjacent blocks

and

the role of friction-couple or de

couple degrees of freedom and set in motion different

mechanisms, each of which can lead to different modes of

collapse. Moreover, sliding displacements, almost always

present after an impact, may lead to loss of equilibrium

due to the variation in the geometrical configuration: this

phenomenon has also been so far rather neglected in the

literature.

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I

I

I

I

I

I

I

I

I

I

I

I

I

I

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I

I I

I

I

A B

A'

A

B ' A

B

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11111111111111111/11111 1111

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e)

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(b)

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Fig. 1. Rigid block at rest

and

five plane mechanisms.

3. DYNAMICS OF A SINGLE BLOCK

3.1. The Problem

Consider a rigid block of mass m simply supported on a

horizontal plane rigid ground in its static equilibrium

configuration (Figure l(a». Assume, for simplicity, that the

block is a parallelopiped (the generalization of the follow

ing treatment to other shapes

is

elementary). Let

hand

b

be, respectively, the height and the base width of the block;

Is

and fk the static and kinetic dry friction coefficientst for

the materials in contact. This system can be considered the

plane diagram of a monolithic stone pillar or column on its

support, in its equilibrium configuration.

The dynamics of a such apparently simple system are

actually very complicated.

tThe static and kinetic friction coefficient are actually limit values.

respectively, for vanishing and very large relative velocity: although most

treatments of structural problems unify the two coefficients (Coulomb

friction), we prefer to keep them conceptually distinct [40].



DYNAMICS

OF

BLOCK STRUCTURES

197

Let XG

YG and )

be the Lagrangian coordinates, describ

ing the degrees of freedom for plane motion of the block.

The dynamic equations, for positive values of the angle ),

are [24]:

1)

2)

3)

where Al and

A2

represent, respectively, the horizontal

reaction due

to

friction

and

the vertical reaction due

to

the

ground,

m

is

the mass of the block, g the acceleration of

gravity and IG the moment of inertia with respect to the

centre of mass.

The sta rt of a given mechanism or the transition during

the

motion to another

mechanism,

among

all the possible

ones, depends

on

the values of

Al and

A2 hence

on

the ratio

b/h,

on the friction coefficients,

J;

and

fk

but

mainly on the

effects of the impacts which occur every time some point of

the block comes suddenly into contact with the ground.

In

particular, if A2 0 always holds during the motion,

the block remains in contact with the ground at least in

one point no uplift). From now on, we shall work under

this assumption; therefore, the motion of the block has two

degrees of freedom: namely, the

rotation

rocking)

around

either corner edge

or

B and the translation slide) of the

contact point.

As

a consequence, there are five possible

mechanisms, depending on the coupling between the

degrees of freedom Figure

l b

- f)).

In

case of free dynamics, the simple sliding motion

Figure l b)) can be ignored in

our

discussion because it

requires a non-zero initial velocity which would become

zero in a very short time due

to

the action of kinetic

friction.

In order to understand the transition from one mechan

ism to another, let us assume

that

the block starts rocking,

from given initial conditions; in this case, the contact at

point without sliding is expressed by the constraint

equations:

4)

5)

The motio n is a rocking governed by Equat ions 1)- 5), if

and

until:

6)

where J;

is

the static friction coefficient. Inequality 6),

under the assu mption of small angles and for free motion

started at rest from a given angular displacement), corre

sponds Figure 2) to [24]:

3(b/h)

J ; ~ 4 b 2 / h 2 ·

7)

s =0 75

b h

10

9

8

5

4

3

2

o

Is

Fig. 2. Regions

of

rocking

or

slide-rocking shaded), as functions of hand

b/h [24].

f nequality 6) is not satisfied, the slide rock mo tion starts,

governed by Equations 1)- 3), 5) and by:

8)

which substitutes Equation

4),

where

fk is

the kinetic

friction coefficient.

t

is interesting to observe that, under the assumption of

small angles, as shown in Figure 2, for each value of the

ratio

b/h

a value of the static friction coefficient exists

above which only rocking motion is allowed until the

block comes back to the static equilibrium configuration.

3.2.

Free ro king motion

Let us focus on the case of simple rocking. The block can

move according

to

two possible mechanisms,

that

is the

rotation around point or B Figure 3 a), b)); correspond

ingly, the Lagrangian coordinate

)

is assumed

to

have

positive and, respectively, negative values.

Rearranging Equations 1)- 5), the dynamic equations

for free rocking are:

) -

0 2

[sin

)

- cos

)

]

= 0, )

> 0

9)

.. [

b

]

) -

0 2

sin ) +hcos ) = 0, ) < 0

10)

where b/h

is

the ratio between the base width

and

the



198

GIULIANO

AUGUSTI AND

ANNA SINOPOLI

a)

b)

Fig. 3. Mechanisms for positive a) or negative

b)

values of the angle 8.

height of the block; x

2

= 3gj2h)/ 1 b

2

h2); and g

is

the

acceleration due

to

gravity.

A special time instant, durin g the dynamic evolution,

is

when the system, coming from the motion governed by

Equation 9) or 10), reaches its static configuration and

hits the ground: an impact occurs and, as will be shown in

a later section, because a finite area

is

involved in the

impact, the subsequent motion can follow any of the

possible mechanisms, depending on the friction and on the

size of the block.

Under

the assumption, valid for very slender block, that

only rocking around point

A

or

B

is allowed [4], [24],

[29]), the motion continues to be governed alternatively

by either Equation

9) or 10), and

the angular velocity

after each impact can be expressed as a function of the

corresponding velocity before it:

11)

with 0 ;;;

f ;;;

1. Therefore, the dynamics exhibits as many

discontinuities as impacts

and

an oscillatory motion

around the equilibrium configuration results.

It

can be observed that in both Equations 9) and 10),

i.e. for both mechanisms A

and

B, the angle

e

has been

taken equal to zero in the static equilibrium configuration:

this assumption allows one

to

consider the motion as a

single oscillatory motion, even if it follows from the

matching of rotations according

to

two different mechan-

isms, which admit:

12)

as natural unstable equilibrium configurations. Each

mechanism

is

equivalent to an oblique inverted pendulum,

the dynamics of which are usually described by assuming a

zero value for the angle corresponding to the unstable

equilibrium configuration [43].

For

the block, because

ofthe

presence

of

the ground, the

configuration e= 0 is of stable equilibr ium with respect to

the gravitational field by which the dynamics

is

governed.

This feature can be confirmed by noting Figure 4)

that

the

diagram restoring gravity moment M e) versus rotation

angle eexhibits a discontinuity in the origin; then, a finite

value of external excitation

is

required to start the motion

from the static equilibrium configuration. The system

is,

for e= 0, in a potential well [36].

In

Figure 5 a), the trajectories integral curves) in the

phase plane <p,

til)

for free oscillations of a pendulum are

shown <p = 0 corresponds to the equilibrium configura-

tion of a direct pendulum, while <p =

±

T of an inverted

pendulum). The same curves can be referred to Equations

9) and 10): the two vertical lines crossing points A and B

represent the configuration corresponding to the zero

value

of

the angle

e

or either mechanism

of

the block [43].

But the arcs of the trajectories included between these

vertical lines are not allowed due to the presence of the

ground; therefore, the integral curves for the block are

obtained by cutting away the

A B

strip

and

matching the

remaining arcs Figure

5 b)).

When the system crosses

e = 0, the representative point is forced by the impact to

jump onto

a trajectory o f the other mechanism at a lower

energy level: therefore, the static equilibrium configuration

is orbitally stable.

It

can be also noted

that

the reference

to

the dynamics of

a pendulum allows a physical interpretation of

x

in

Equations 9) and 10), as the angular frequency for small

oscillations o f the direct pendul um around <p = o

3.3. orced rocking motion

Let us consider now the forced dynamics induced

by a horizontal harmonic ground acceleration: ) g =

Ksg sin wt

1» where

g is

the gravitationa l acceleration.

Let the

motion

of the block

start

with a positive angle

M e)

e

Fig. 4. Restoring gravity momen t M 8) versus rotation angle 8 [36].



DYNAMICS

O

BLOCK STRUCTURES

199

a)

B

A

J

b)

B=A

Fig.

5.

Integral curves for free oscillations of a pendulum a) and of the

block on rigid ground b) [43J.

Figure 3 a)); the equation describing the dynamics

is

[43]:

.. b J

) -

rx

2

1

+ KSh

sin(wt

+

» sin )

+ x 2 [ ~

- Kssin(wt

+

4»}OS }

= 0

13)

where > is the initial phase required to star t the motion at

t

= 0; it

is

such that

. b

Kg sm >?o h

An expression similar to inequality

14):

a

b

> -

g r h

14)

14a)

where

a

is the

peak

acceleration of the earthqua ke treated

as an instantaneous impulse or a constant horizontal

force), has been used in the so-called West formula [3J to

evaluate the ground acceleration responsible for overturn

ing rigid bodies. But, as first recognized by Housner [4J,

Equation 14a) gives only a necessary condition to start the

motion and, consequently, cannot be considered as a

criterion for overturning.

Equation 13)

is

not

linear.

Under

the assumption of

small angles, it becomes [43]:

15)

Let us change the time scale by putting wt

=

; assume

8 = -

K,(b/h)(rx

2

W

2

) and

j

= -

2

W

2

; Equation 15)

becomes [43]:

u + [ j +

8

in , +

4»JU

= j [ ~ - K sin , + 4»J

16)

where the variable ) has been substituted by u

to

take

account of the change in the time variable.

An equation similar to Equation

16)

can be obtained if

mechanism

B is

started Figure

3 b)):

u"+[(j-ssin( ,+ fr)Ju= - ( j [ ~ + K , S i n ( ' + f r J (17)

The

motion

is governed by Equations 16) or

17),

until

the block comes back to u = 0 and hits the ground.

Therefore, as in the free case, the dynamics exhibits as

many discontinuities as impacts and the motion is ob

tained by matching alternatively the solutions U

and U

B

of

Equations 16) and

17),

at each impact instant

,*.

The

matching conditions are

18)

and

u ~ , * ) = f 3 u ~ , * )

19a)

or

u ~ , * ) = f 3 u ~ , * ) .

19b)

The same equations 16)- 19) govern the motion of a

slender, but multiblock column, in its simplest mode

rocking motion):

that

is, the one where the relative

rotations

and

slide displacements between adjacent blocks

can be ignored. The behaviour of a monolithic column and

even the simplest mode of a multi block one are very

different

at

the instant of impact, particularly in the

amount of dissipated energy and, consequently, in the

value of f [29].

3.4. Free nd forced slide rock motion

The dynamic equations off ree plane slide-rock motion can

be obtained rearranging Equations 1)- 3),

5)

and

8). or



200


positive values of the angle

(J,

they are:

mXG

= - fkA2

sgn(x

A

)

b

h )

G J = -

fk

A

2

s

g

n

X

A

) :

sin

(J

+ : cos

(J

+

A2

Gin

(J

- cos

(J

)

where

20)

21)

22)

In presence of a given horizontal ground acceleration

x

g

Equation

20)

must be substituted

by:

23)

and the forced dynamics are described by Equations (21)

23).

A similar formulation can be obtained if the motion

occurs according to the other mechanism, where point

B

is

in contact with the ground (Figure 1

f)).

Obviously, during the continuous dynamic evolution,

the condition

x

A

0 or

XA = 0,

with

m xG - Xg) :s; hAz

decides, respectively, the permanence of the slide-rock

mode or the transition to a rocking mode.

4.

THE IMPACT PROBLEMt

As

has been already stated, the dynamic evolution of a

rigid block and, particularly, the persistence

of

a given

mechanism are strongly affected by the impacts; in fact,

they can modify, besides the velocities, the degrees

of

freedom, depending on the value of the friction coefficients

and on the sizes of the block.

Consider the rigid block (Figure

6 a)) at

the instant

when it hits the ground, coming from any plane motion,

characterized by the velocity Va of its centre of mass

G

and

by the angular velocity r .

The impact problem can be formulated as follows: What

will be the initial conditions for the post-impact motion

(Figure

6 b)),

namely

v;t

and

0+,

given the posit ion

and

the

pre-impact velocities? How do these quantities depend on

the sizes of the block

b,

h, and

on

the static and kinetic

friction coefficients?

When two bodies come suddenly into contact, transient

or

permanent deformations can occur, connected to

partial

or

total energy dissipation. Due to the fact that the

transient deformations disappear after the impact, while

the permanent ones are negligible with respect to the

variations of the position of the system, the typical

assumption for the impulsive motion

is that

the position of

tIn

this section, bold characters indicate vectors.

V G ~

,

,

,

,

A B

111111111111111111111111111/111111111[11111111111

,

.

)

,

,

,

,

c+

Fig.

6.

Rigid block hitting the ground

a)

and immediately after the

impact b). C is the instantaneous centre of rotation.

the system does not vary, while the velocities are subjected

to an instantaenous variation.

Then, the classical formulation of the impulsive motion

assumes that the bodies are rigid and the sudden dis

continuities in the velocities are due to the action of a force

F, applied

or

introduced to justify the dynamic effects,

which reaches a very high intensity during the infinitesimal

duration of the phenomenon, but whose impulse I is

defined and finite.

The classical equations of the impulsive motion for a

rigid body are:

1=

f

dt = L\Q = mL\vG = m v;t - va)

M(G)

=

f

P - G) x F]

dt

= L\K(G)

=O GL\O =

0 0<0+

-

0-

24)

25)

where I, Q VG and

m,

are, respectively, the impulse applied

in point

P

the momentum, the velocity

of

the centre of

mass G and the mass of the system; while

0 O G,

M(G) and

K(G)

are, respectively, the angular veloclty, the inertia

tensor, the moment

of

the impulse and the angular

momentum, evaluated with respect to

G.

Equation

25),

if written with respect to a generic point

0 becomes, with symbols of obvious meaning:

M O)

=

f

P - 0) x

F] dt

= L\K(O)

= m[ G - 0)

x

M

G

] + O GL\O.

26)

In general, Equations 24) and 25)

or

26) are not

sufficient to solve

an

impact problem, especially in our case

in which a finite area is involved in the contact, so that

point P, where the impulse is applied, is not

a priori

known.

Further

relationships are therefore required.

In

most papers where the problem has been tackled,

these relationships have been generally obtained from

some

a priori

assumptions either about the degrees of

freedom (for example: only rocking

both

before and after



DYNAMICS OF BLOCK STRUCTURES

201

the impact ([4J, [7J, [12J, [30J-[33J, [36J, [39J, [43J)),

or

about the point P of application of the impulse and its

velocity

and/or about

the dissipated energy by means of an

empirical parameter (restitution coefficient)

([9J, [38J,

[46J).

Other approaches refer to variational principles

concerning the dynamic evolution of the system

[llJ,

[24J, [29J, [37J, [42J).

The implications of these as

sumptions will be examined below.

Some researchers have altoghether avoided tackling the

impact question

and

given continuity

to

the phenomenon

by inserting an elastic layer between the rigid block

and

the rigid groun d ([8J, [13J, [15J, [21J, [22J among others):

this type of approach will

not

be discussed in this paper.

5.

ROCKING MODELS

5.1. Housner rocking and impact model

Housner first investigated systematically the dynamics of a

slender rigid body, simply supported on a rigid ground

[4]. In the Housner model, the block is allowed only to

rotate

around

the corner edges of the base and the static

friction coefficient is large enough to prevent any sliding

displacement of the contact point. As Housner himself

pointed out, such a model

is

valid only for blocks of

sufficient slenderness (Housner puts the limit at

b/h

0.35).

t

can be shown

that

Housner s equation s of motion

coincide with Equation s

9)

and

10), provided the exact

) 2

is substituted by the expression, valid for b/h

«

1:

27)

Under

the further assumption of small oscillations,

Housner s equations for free rocking are:

2 2

b

e

- pO = - p h

0>0

28)

b

p

2

0= p2

h

,

0<0.

29)

Let us recall the most important points in Housner s

paper. Referring to either Equati on

28) or 29),

and to the

initial conditions:

30)

the time duration

to

until the system hits the ground is

shown in Figure 7 as a function of the initial angle

0

0

,

With regard to the impact model, Housner assumes that

the impact is inelastic (no bouncing) ; nevertheless,

because of the rotational inertia of the block, the post

impact angular velocity can be different from zero. The

centre of

rotation

after each impact coincides with the edge

5

I

I

/

P to

4

/

:3

/

2

/

/

...-

/

0.2

0.4

0.6

O B

1 0

8 J r

Fig. 7. Relationship between the time duration to and the amplitude of an

oscillation [4].

B (or A opposite to the instantaneous centre of rotation

before impact, A (or B ; then, the impulse

is

applied in the

new centre of rotation. Consequently, the angular mo

mentum about the new centre B (or A does not vary

during the impact and the ratio between the post-impact

and

pre-impact angular velocity, from Equation

26),

is

{j+ 2 b

2

h2

f =

{

= 2 1 + b

2

/h2)

31)

Thus the value of f depends on the ratio b/h; in order to

maintain the sign of the angular velocity over the impact, it

is

essential

that

f

;; 0,

whence b/h j2, which is satisfied

by Housner s slenderness requirement. In free motion, the

energy of the system is progressively reduced together with

the

duration and

the amplitude of the oscillation (Figure

7).

Furthermore, it can be noted that, under the assumption

of simple rocking, the ratio 31) coincides with the ratio

between the velocities of the centre of mass, normal

to

the

surface o f contact,

and

with the square

root

of the ratio

between the post-impact and pre-impact kinetic energy:

f

{j+ ylt

=

{

= -

Y

. 32)

Nevertheless, the restitution coefficient f cannot be

identified with the restitution coefficient e of the experi

mental Newton s impact law, which gives a measure of the

relative energy dissipated in a collinear impact. In fact,

under the assumption of inelastic impact (e

= 0),

a value

different from zero of the post-impact angular velocity

might appear contradictory.

Housner analyses also the case

of

motion forced by a

half sine-wave and determines the minimum value of the

excitation amplitude required to overturn the block, ob

taining the curve shown in Figure 8.

Finally, in orde r to simulate the effects of an earthquake,

Housner utilizes an energetic approach

and

treats the

ground

motion

as a succession of

n

impulses inducing

n

velocity variations, randomly distributed in a given time



202

GIULIANO AUGUST

AND

ANNA SINOPOLI

2.2

2.0

K,

bib) 1.8

1.6

1 4

1.2

1.0

I

I

V

I VI

I

v

V

L

V

I

I

o 02 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

w p

Fig.

8.

Minimum amplitude of

a

half-sine wave required to overturn the

block [4].

interval. The most interesting indications obtained are, in

this case, a scale effect which, with two similar blocks,

makes the higher one more stable and, with two blocks of

the same height, makes the stockier one more stable.

5.2. Housner type models

In most papers produced after [4], a mechanical model of

the same type has been assumed; i.e. the motion of the

block has only one degree offreedom, the rota tion,

and

the

impact is governed by a restitution coefficient either as

defined in Equations

31)

and

32)

or treated as an

experimental parameter.

The attention of these researches has been mainly

devoted to investigate either the response of the block,

when excited by a given recorded earthquake, or the

experimental value of the ra tio

{J

=

0+/0-,

or

the stability

of the block motion, under a horizontal harmonic

excitation.

Aslam et al [6] performed an extensive investigation,

both

numerical

and

experimental, on slender concrete

blocks. The results obtained in free dynamics are com

pared in order to obtain the value of the ratio {J which

gives the best agreement.

By

means of a shaking table, the

analyses are then extended

to

the forced motion induced

either by a ha rmonic excitation (horizontal and vertical) or

by a recorded earthquake. The system is extremely sen

sitive to the initial conditions

and to

the details

of

the

exciting motion: in particular, the numerical results agree

closely with the experimental tests in the presence of large

amplitude and low frequency harmonic excitation; a para

metric numerical study shows a dependence of the max

imum amplitude response on the ratio b/h, on the size of

the block and on the value of {J.

On the contrary, in the presence of a simulated seismic

motion, the experimental results were not found

to

be

repeatable.

Similar results are obtained by Yim

et al

[7].

In

their

opinion, however, it is impossible

to

investigate determin

istically the stability of the response of the block to a

seismic excitation, as a function of features of the system

and

of the external excitation; for example, if overturning

occurs for a given earthquak e, overturning may not occur

in the presence of

an

earthquake with the same time

history, but with an intensity proportionally higher.

Systematic trends can be nevertheless identified if a numer

ical statistical analysis is performed; in this case, the

probability of overturning increases with the acceleration

peak and the slenderness, and decreases with the size of

geometrically similar blocks. Such a behaviour is analo

gous

to

the one obtained by Housner [4], exciting the

block by an impulsive acceleration.

The first investigation on the period and, the stability of

the responses of a block to a horizontal harmonic ground

motion was performed by Spanos

and

Koh [12]. The

periodic motions were numerically first identified by

matching approximate analytical solutions

at

the instants

of impact,

and

then verified by integrating the exact

equations (without the small angles assumption). Different

periodic motions were found; they were labelled by two

integer numbers m, n), which are, respectively, m the

number of impacts in a semi-period of the response, and n

the ratio between the periods of the response

and

of the

excitation.

The approximate solutions correspond to Equations

16) and

17),

with e =

0,

and C substituted by p (see

Equation 27)). Referring to Equation 16), if

at

each

impact r = 0 is assumed, the expression is [36]:

u r)

= -

uosin r

+ ¢ +

C

co s { ; r + D s i n h ~ ) r

where

K

C(2

U

o

= w

2

C(2;

c

= - G

+

Uo sin

¢;

w[u

o

cos

x + u (O)]

D = --- '-- ----... .:. . . .:.=.

33)

A similar expression can be obtained for the solution

according

to

the other mechanism (Equation 17), with

e = 0), which matches Equation

33) at

the instant of

impact. For these solutions,

an

orbital stability analysis

has been performed by Spanos and Koh [12], for given

kinds of response, by means of a perturbation method.

Stability

maps

are then introduced, as a function of the

amplitude

and

the frequency of the external excitation; as

an example, the map corresponding to the (1,3) mode is

shown in Figure

9.

Three regions can be identified where

a) the motion does not exist because the external excita

tion is always lower

than

the restoring gravity moment

(potential well);

b)

the given rocking mode

is

stable; c) the

given

motion

is unstable (overturning). This shows the

influence of the restitution coefficient

{J

treated as

an

experimental parameter, on the area of the stable motion

regions, which increase when {J decreases.



DYNAMICS OF BLOCK STRUCTURES

203

~

.

....

. .J

<

0

N

/

/ / . . . . . ~

.

,/

_ • • 0.12 )

?

J ~ ~ ~ ~ ~ ~ ~ ; : : ~ ~ = = = = : = = = = = = = ~ = = = = = : =

0 , __

....

0

0

.

0

1.5 3.0 4 5 6 0 7.5

EXCITATION FREQUENCY

W p

Fig. 9. Stability map for (1,3) mode rocking, for different values of the

restituti on coefficient f [12].

Many successive papers elaborate the above concepts,

gIVIng further

but

similar results either by analytical

approach

([16], [23], [26], [34]), or by numerical

and

experimental investigations ([32], [33]).

An important contribution is given by Hogan [31], who

starts from the model, the analysis

and

the response

classification of Spanos and Koh [12] and performs a

complete investigation on the existence and stability of

single-impact subharmonic responses

1,

n (with n ;: 1), as

a function of the restitution coefficient

p

Hogan verifies the existence of motions, characterized

by a period increasing with the amplitude of the harm onic

excitation, until the response becomes aperiodic or more

probably chaotic. n Figure 10, examples of stability

boundaries for symmetric 1,

n

orbits are shown, as a

function of the non-dimensionalized exciting frequency

w/IY and amplitude K, h/b).

Due to the high sensitivity of the response to the initial

conditions, Hogan determines the domains of attraction,

Ks

(hlb)

4

2

/

/

/1 3/

/

/

/

/

1 =5

/

/

./

=7

....

/ / /

/

....

1/ 5

; ./'

~ ~ ~ ~ ~ = ~ = ~ ~

()

4 8

w p

Fig. 10. Stability boundaries for (1,

n

symmetric orbits, for

b/h =

0.001

and f = 0.925 [31].

shown in Figure 11 for four subharmonic orbits

1, n :

he

finds that the behaviour of the system is in some ways

unpredictable, but fails to explain fully the reason for this.

Note, however,

that

Figures

10

and

11

correspond to

b/h

=

0.001, a value

that not

only is unrealistic, but makes

the block very much like a single inverted pendulum

The problem of identification of the responses and of

stability analysis of the motion

is

also tackled by Sinopoli

[36], [43]. First, she performs a systematic numerical

analysis [36] on the exact equations «16) and 17)) for

simple rocking un der harmonic excitation of a multi block

column, made

of

very stocky blocks, for which p of

Equation 11) is assumed equal

to

zero [29]. The para

meters of the analysis are the angular frequency

wand

the

acceleration amplitude Ks of the excitation. An interesting

result is shown in Figure 12, where the periods of the

responses have been represented, for 0 < Ks 1 and

w 6 rad/s. Let us emphasize that they have been ob

tained starting from a quiescent position

and

with an

initial phase corresponding to the lowest value required to

start the motion.

As

in [12], three regions can be recognized in the (K

w

plane. The lowest one is that fOf which the motion is not

possible; the highest corresponds to overturning of the

column. The region in the middle corresponds

to

bounded

periodic motions; its height increases with w These

motions generally are not symmetric and characterized by

different values of period and of number of impacts per

period. Nevertheless, some systematic trends can be ob

served. The number of impacts per period and the periods

generally increase with

K

while the maximum amplitude

of oscillation decreases rapidly with

w

and increases with

Ks. Many responses have intervals of motion and rest,

because the block impacts when the amplitude of the

excitation is lower than the value required to move the

system; the consequent motions are periodic

and

a transi

ent does not exist: these cases are labelled in Figure 12 by a

number indicating the period (or semi-period in case of

symmetric motions); examples of motion s with n = 1 are

shown in Figure

13.

n the other cases of periodic motions, both transient

and steady state are present: in Figure

12,

these motions

have been labelled by symbols. Among them, only motions

with period n = 1 are comparatively numerous: in Figure

12 the lower group (around Ks;::::; 0.3) corresponds to

symmetric motions, with two impacts per period; the

upper gro up to asymmetric motions with only one impact.

Figure

14 shows the values of the time interval between

two zeros versus the maximum amplitude of the steady

state responses, for a given w Ks increases along the

curve): apart from the jumps, which correspond to a

variation of the response period, this relationship is quite

similar to the one obtained by Housner for free oscillations

(Figure 7) and suggests the prevalent importance of the

natural component of the motion.

n order

to

investigate the stability of all these kinds of



204

GIULIANO

AUGUST AND ANNA

SINOPOLI

0

0

5

} o 8 / 8 r

. 1

- 0 5

Fig. 11. Domains of attraction for four subharmonic orbits, for:

b h =

0.001, f

=

0.925. w a

=

9. and

K h

b

=

2.5 [31].

response, Sinopoli [43] observes that the differential

equations

16)

and 17), the solutions

of

which must be

matched, are non-autonomous: the integral curves cannot

be obtained analytically

and

more than one trajectory can

pass through each point of the phase plane. More specifi

cally, Equations 16) and 17) are forced Mathieu

equations, a classical example of parametric resonance:

they do

not

admit general solutions in a closed form.

Nevertheless, it is possible to determine in the plane 8, e)

the regions corresponding to stable

or

unstable motions;

within these regions the motions are periodic and they

correspond to sub harmonic, super harmonic and super

subharmonic solutions i.e. expressing the period as ratio

of two integers p and

q:

p/q, its value corresponds re

spectively to q

= 1,

p

= 1 and

q

=f 1,

p

=f 1 and

q

=f 1 .

t is not correct to reduce the stability regions for the

motion to the ones given separately by Equation 16) or

17),

whatever the initial conditions are. Such a criterion, in

fact, gives only sufficient conditions for the stability of the

motion in regions coincident with those of

an

inverted

pendulum.

On

the contrary, it is expected that the motion

is characterized by more extended stable regions, because

of the presence of the potential

well

and the match

imposed by the impacts.

Let r j be the instant of the first impact and assume that

the motion comes from the former mechanism

A,

Equa

tion

16));

for r > ri, the rotation occurs around point B

and Equation 17) can be written as [43]:

+ [8 - esin(r +

rf

+ 4»]u

1

= - + K,sin(r +

rt

+ »J

34)

where, to emphasize the phase changes in the equations

describing the motion, the time variable has been assumed

to restart from zero after the impact.



DYNAMICS

OF

BLOCK STRUCTURES

205

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

UNSTABLE

MOTION

•

.5

2

2

0

1

1

7

1

2 1

0

1

1

I

I

1

1

1

11

1

I

1

I

1 1

1

1

1

1

I

1

I

0

1

0

1

0 0

I

1

0 0

1

0

0 0

1

0 0 0

0 0

0

0

1 1

1

1

1

1

1

,

1

1

,

,

1 I

,

1

1

,

5

5

3

7

7

•

t

Z

2

7,

}1

,5

7

2

•

2

2

t

0

0

\5

Q

1

2

,

2

,

2

,

2

,

0)

,

I

,

I

,

I

,

1

,

1 1

1

,

1

,

1

,

I 1

I

,

1

,

I

,

1

,

I

,

I

,

1

0

0

0

0

0

0 0

0 0

0

0

0

0

0

0

0

,

1

,

1

I

,

,

,

1

1

1

,

1

0

n=

1

'*

n=

6,.

n= 3

c

n= 4

5

n = 5

....

•

;::

1

f

•

3,/

2

17 5

2

•

2

a

2

2

5

2

~

z

3

•

2,

N

,5

2 B

•

•

3

•

3

z

2

•

•

2

Z

2

•

t

0

0

0

0

0

0

0

,

t

,

,

1

, ,

,

,

,

,

, ,

I

,

,

,

,

,

,

,

,

,

,

,

,

1

,

,

,

1

,

1

,

,

1

,

,

,

,

1

1

,

,

0

t

0

0

0

0 0

0 0

0 00 0

0

0

0

0

0

,

,

,

,

,

,

,

,

,

1

,

1

,

,

,

NO MOTION

0.1

2

3

4 5

w (rad/sec)

Fig.

12.

Regions of periodic or unbounded motions: b h = 0.2;

f

= 0;

a

= 1.19 rad/s [36].

Let

i

represent the counter of the impacts

and

t

the

time span between

an

impact

and

its predecessor; after the

kt impact, if

k

is even, the motion is an oscillation

according to the first mechanism

A);

its equation is

u + il + B siner +

r.

+ tfJ ]u

k

35)

where

r.

=

~ ~ l

r1-

After an impact o f odd order k + 1, on the contrary, the

motion follows mechanism

B

and

the equation is

uZ+

1

+ il -

B

sine +

rt+1

+ tfJ ]Uk+1

= - i l ~ + Kssin r + rt+1 + tfJ)].

36)

The resultant mot ion of the block is obtained by matching

alternatively the solutions of Equations 35) and 36),

increasing

k, at

the instants

r.;

the matching conditions

18) and

19) can be written:

Uk+

1 0) = Uk r:) = 0

ui+

1

0) = j3ui r:).

37)

38)



206


J CradY

o ~ .

__________________________

n = 1

Ks

= 0.24

.

..

0 ,

. '

-000.

O . T - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~

0_

n = 1 Ks = 0.27

0 ....

000 .

000

-000.

-o.OOl

-<lao.

_______________________

0.01

00

00 .

0 0 .

003

002

0.01

n

= Ks =

0.37

O ~ . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - _

007

a o

O O ~

0.04

O.Ol

n =

Ks

= 0.52

t sec)

Fig. 13. Examples of motion without transient: blh = 0.2; f3 = 0; a =

1.19 rad/s; 0 = 5.5 radls [36].

Due to the character of Equations (35)

and

(36) and the

impossibility of finding analytical expressions for the

corresponding solutions, Sinopoli tried to identify a single

differential equation governing the motion of the block

and satisfying the matching conditions (37) and 38).

To this aim, a motion characterized by a periodic

succession of impacts is considered:

k

r

K

= I

r t

=

2nn:

i= 1

39)

whatever the period

n

and the number k of impacts per

period. Thus, Equations (35) and (36) are substituted by

the single differential equation:

u

+

p(r)u

+

q(r)u = f r)

40)

where p(r), q(r) and f r) are periodic functions of period

2nn:

(n

= 1 2 3 ... . The function p(r) takes into account

' ::j

(a)

,

,1

, J

1

I

I t

f

0

0.15

t

sec)

8

r.:)

I

b)

5

t sec)

Fig. 14. Relationship time vs. maximum amplitude, for a) 0 = 2.5 and b)

w = 5.5 radls [36].

the energy dissipated

by

the impacts

at

the instants

ri

by

means of the relationship

k k

f3

-

1

p(r)u (r) = I M(r;) = I b*(r - r;)u (r)

i l i l

OJ

41)

where b*(r - r;) is the Dirac delta function.

Equation (40) is a forced damped Hill s equation, that is

a linear differential equation with periodic coefficients; it is

general and valid also for n approaching infinity, in which

case the motion is not periodic and probably chaotic.

It

can now be understood why the details of the

response motion of the system

cannot

be predicted. In fact,

criteria of existence

and

stability for the solutions of Hill s

equation like Equation (40) are provided by the well

known theory of Floquet. Namely, periodic motions of

period 2nn: exist only inside the parametric regions where

the solutions

of the homogeneous associated equation are

stable; such stable solutions are periodic orbits ranging

from subharmonic, to superharmonic, to supersubhar

monic.

Further

due to the linearity of the equation, the

stability of the general solution is determined by the

stability of the corresponding homogeneous solution, so

that conditions of existence and stability coincide. There

fore, any numerical investigation is sufficient to give the

regions of periodic and, consequently, stable motions,

without any further analysis.



DYNAMICS

OF BLOCK

STRUCTURES

207

This is confirmed by the qualitative coincidence between

the boundary lines of the stable symmetric subharmonic

1, n) orbits obtained by

Hogan

[31J for f3 = 0,

and

the

ones obtained numerically by Sinopoli [36J

and

shown in

Figure 12.

6. ALTERNATIVE IMPACT MODELS

Ishiyama [9J does not strictly follow Housner s model in

his numerical investigation of the block motion, induced

by a recorded earthquake.

He considers the possibility of

slide, rotation, slide rotation, translation jump, rotation

jump and the variations of mechanism,

and

gives criteria

for overturning.

A particular attention is devoted

to

the impact formu

lation, which refers to Equations 24)

and

25): the further

relationships necessary to solve the problem are obtained

by specifying the point P of application of the impulse

(impacting corner edge, in the case of

rotation

jump for

J P 0 (Figure 15 a)); or centroid of vertical impulses, in the

case of

rotation

or

rotation

jump for

J

= 0 (Figure 15(b))

and

introducing

both

a vertical

e

y

and

a horizontal ex

restitution coefficient, in order

to

take into account

both

the structural and frictional dissipation of energy:

42)

Y1

e

y

=-;-=-, 0

e

y

1

YA

43)

Such coefficients relate the post-impact and the pre

impact velocity of the impacting point; because the resul

tant

impulse does not cross the centre of the mass of the

block (not collinear impact), the impact can be considered

as that of a concentrated mass by means of the intro

duction of

an

equivalent mass.

Therefore, the dynamic evolution

and

the estimation of

~ o

8 0

Iy

Iy

a)

b)

Fig. 15.

Impact

from rotation jump: a) for 0 oF 0; b) for e= o

the probability of failure (overturning) depend on the

values of ex and e

y

for which, in any case, sufficient

experimental tests are

not

available.

As

a consequence, a compatibility condition must be

imposed for the post-impact vertical velocity of the im

pacting point (Figure 15 a)), that is: y; ;;>

0;

and, if it is

unsatisfied,

an

additional vertical impulse must be applied

to the system. Moreover, during free dynamics, rock and

slide exclude each other if the static friction coefficient is

respectively higher

or

lower

than

b/h.

Therefore, in our opinion, the formulation of Ishiyama

does not give an efficient new model. In fact, exper imental

results [44J have shown that after

an

impact, even if the

block approaches from simple rocking, a component of

slide is always present.

More recently, Lipscombe [38J studied the dynamics of

single block and multiple-block structures. With respect to

the single block analysis, he extends the Housner model to

the case where, besides rocking, the block can bounce, as a

consequence of a not completely inelastic impact. Like

Ishiyama, Lipscombe refers to the equations of the im

pulsive motions

24) and 25), and

to the vertical restitution

coefficient e

y

; but, in his model, only either corner edge

A

or B impacts on the ground; further, there is the possibility

of sliding.

Lipscombe evaluates the expression ofthe ratio between

post-

and

pre-impact angular velocity, as:

Ii+ 2 - 1 3e

y

b

2

/h2

f3 = Ii- = 2 1 b

2

/h2

44)

Equation 44), for

ey

=

0 (inelastic impact and, then,

no

bouncing), coincides with Equation

31)

of the Housner

model. Lipscombe introduced bouncing in the impact

formulation in order to explain the results of experiments

performed on freely rocking steel blocks of different sizes,

which showed a behaviour more conservative

than

the

prediction of Equation 31); however, the introduction of

bouncing was able to justify the experimental results only

for stocky blocks

b/h

= 0.5 and

b/h

= 1).

Shenton

and

Jones [46J formulate

an

impact model for

a block approaching from a slide-rock motion and concen

trate the impulses applied to the block in the base edges,

named respectively rotating

and

impacting corner. Besides

the equations of the impulsive motions 24) and 25),

Shenton and Jones introduce the vertical restitution coeffi

cient e

y

related

to

the impacting corner;

and

compatibility

conditions, on the horizontal

and

vertical velocities, re

spectively, for impacting and rotating corners. Such con

ditions say

that

the impacting corner will slide or not,

depending on the horizontal impulse required

to

static

friction; while, in the rotating corner edge an additional

vertical impulse must be applied, if its velocity does

not

respect the assumed inpenetrability law.

Shenton and Jones also propose a slide-rock model [47J

which accepts the above formulation of the impact. The

authors investigate the conditions of existence

and

sta-



208 GIULI NO

UGUSTI AND NN

SINOPOLI

bility of the steady-state slide-rock, characterized by a

harmonic behaviour for both rocking and sliding.

The alternative formulations of the impact discussed

until now always use the concept of a restitution coeffi

cient. A different mathematical formulation was proposed

by Moreau [5], [11], [19], [42] to simulate the dynamics

of elastic or rigid bodies with frictional contact

and

in the

presence of unilateral constraints (incompenetrability law).

The dynamic evolution is

formulated as a convex mecha

nics problem, in order

to

describe any kind of contact,

including frictional contacts

and

collisions. The presen

tation of such a rigorous mathematical formulation

is

beyond the scope of this paper. With respect

to

the use of

the restitution coefficient,

Moreau

[42] says

that

its

introduction in the treatment of collisions would require

data usually impossible to identify and to collect experi

mentally ; further, such a coefficient bears little consis

tency beyond the special case of the collision of two

otherwise unconstrained perfectly rigid bodies .

Also in the approach proposed by Sinopoli in [24], [29],

there

is

no reference to the restitution coefficient. Sinopoli

formulates the inelastic impact between a rigid block and a

rigid gro und as a p roblem of dynamic evolution governed

by the variational principle of Gauss; this principle says

that, during the dynamic evolution of any system subjected

to

any k ind of forces

and

constraints, the real motion is the

closest to the one characterizing the system if it could

become completely free. Gauss assumes as a measure of

distance between the real motion

and

all the other possible

ones the function R the expression of which for the

impUlsive plane

motion

of a rigid body is [25]:

45)

where the symbols are the same as in Equations

24) and

25). Therefore, the real evolution of any system corre

sponds to the stationarity of

R

and, particularly,

to

its

minimum value.

The advantage in using the variational principle of

Gauss

is

that it is also valid for unilateral constraints [5],

so

that

a weak formulation (valid only under the as

sumption of persistent constraints [25]) can be adopt ed for

the impulsive dynamics.

For

a rigid block hitting the

ground

and

coming from a rocking motion, this formula

tion has also been named the kinematic approach [29] in

the sense that the inelastic impact

is

studied as an

evolution problem, characterized by a sudden imposition

of persistent unilateral constraints (impenetrability con

dition for the surfaces in contact),

and

all the constraints

can be expressed in terms of distribution of velocities,

imposed

to

the

motion

of the system.

Consequently, the terms I and M in Equation 45)

represent external impulses applied to the system, contem-

y

_

\

\

\

\ \

\

\

-

\

\ G ~ \

\

\

\

\ \

\ \ J

_

A--=:t::q]]JJ )

Fig.

16.

Impact of a rocking rigid block [12].

poraneously to the shock. f they are absent, the problem

can be formulated as [29]:

minR = mint{m[xG - XG)2 YG - YG)2] ()G(8 - 1:i-)2}

46)

with the condition of impenetrability:

Y 0

47)

which must be satisfied by all points Qof the base coming

into contact. Equations

46) and

47) give exactly the

solution of an inelastic impact in the absence of friction;

furthermore, it can be interesting to observe that such a

formulation is completely equivalent to the one proposed

by

Moreau

by means of the convex analysis [11].

The most critical point, in the presence of friction,

is

the

evaluation

of

its role during impact. Following a weak

formulation, the friction performance must be expressed in

terms of restrainsts on the velocities, which remain con

stant d uring impact (persistent constraints); but, the char

acter of a frictional contact is

that

it generally varies

and

depends on the instantaneous redistribution of the veloc

ities. Then, a weak formulation can be

adopted

only by

means of some drastic assumptions

about

the role

of

friction.

In this respect, Sinopoli suggests two different formula

tions [24], [29], where respectively the friction perfor

mance is either roughly taken into account

or

neglected.

In

the first [24], noting that the pre-impact velocities distri

bution of points Q

s

vertical, due to the pre-impact simple

rocking, that is:

X =

48)

Equation 48) is assumed valid

both

during and after the

impact if the instantaneous velocities redistribution im

plies a rotation in the post-impact motion; otherwise, the

friction impulse is neglected. This means

that

the friction is

required to act at the beginning of impact (contemporarily

to

the starting of persistent constraint 47))

and

is either

high enough to prevent sliding, if rotation is possible,

or

otherwise negligible. Consequently, if xG YG

and

8- are

the velocities of the block before the impact, the corre-



DYNAMICS OF

BLOCK

STRUCTURES

209

sponding post-impact quantities depend on the size ratio

blh

[24]. In particular, for

blh

< j2:

· +

h

'+

XG

=

-- }

2

(49)

so that the post-impact motion is a rocking, characterized

by a reduced angular velocity. This result

is

the same as

Housner's; however, it must be noted that, referring to

Figure

2,

the relationship

blh

<

j

s

a necessary, but not

sufficient, condition in order

to

have simple rocking, which

depends also on the value of the friction coefficient.

On the contrary, if

blh

;;,

j2:

+ • _

h ._

xG = XG = - 2(}

Pt

= 0

e

= 0

and the post-impact motion

is

merely a translation.

SO)

In the second formulation [29J, Sinopoli assumes that,

due to the assumption o f rigidity, the time durat ion of the

impact is zero

and

therefore the friction performance,

which

is

a consequence of the velocity redistributions, can

start to act only after the impact; the friction impulse is

then always negligible during the shock and no restrictions

must be imposed

to

the tangential velocities of points

P.

The results obtained are that, if

blh < j212:

· + . _

h

(}._

XG = XG = - 2

· +

b

e

Y = -2

'+

1 - 2b

21

h

2

)

. _

(} = 1 + 4b

z

h

2

}

On the other hand, if

blh

;;, j212:

. + . -

h

e-

XG = XG = - 2

= 0

e = o.

SI)

(52)

It

is

relevant to observe, from Equations SI)

and

S2),

that a slide component is always present in the post

impact motion; particularly, the motion

is a simple slide

for stocky blocks and a slide-rock for the slender ones

(Figure 17). The Housner 's results can be obtained, in this

case, asymptotically for

blh

approaching zero.

Referring to the second formulation by Sinopoli, results

similar to those of Equations

SI)

and S2) have been

qualitatively obtained for the impact of a multi-block

y

( a

G< _.

\ ~ Y l i

\

A \ B

'l7.7 / / / / / / / //> : , / / ( ; / / / x

I \1

lc

I

i

y

( b)

A

7/.1.-://,, ,.//, ,./.7/-:-/:' :."

:/ : / . 7 / . 7 / ~

./fT7

I

cl

II

I

Fig.

17.

Velocities after the impact, respectively, for (a) slender

and

b)

stocky blocks [29].

column, rocking around its base corner edge (Figure 18).

In this case, the motion after the impact does not depend

on the geometry of the whole column,

but

on the size ratio

bJh,

of the single blocks [29].

In particular, if

b,jh, ;;, j212,

where i is the counter of

the blocks (assumed equal to each other) from the bottom,

the velocities after the impact (Figure 17) are:

x

G

= _

h, 2i -

1)

(}_

, 2

yt = 0

(}+ = O.

I _'...,

r \

\

\ nr\

\ _-1-1

r-A,;

I \

S3)

:

+

.Y. Gn

I

I

I

I

I

I

r

-,

I

I

I

-t

I

I

I

\ Ir

..-- _.-1--,

\ /'

\

\

/Ir-\

I

I

I

I

,

I

I

..J

I

I

I

IAi

1

__

-1

i '

/ \

\ ; ;S2 I

J ; 1 . c - ~

.'I' /

I

\./ij

G, \

IIi(". / ,

A, i:/ _ ---

I

1

1

1

+

1

I.Y.G2

I

1

I

1

1

~ ; ,

I

1

I

////////////

( a )

( b )

Fig. 18. The multi-block column before and after the impact [29].

I

...,.1

1

I

I

I

..J

J

I

1

1



210

GIULIANO

AUGUSTI

AND

ANNA

SINOPOLI

IS

SLIDE (mm)

Fig.

19.

Cumulative distribution of slide displacements as a function of

b/h [44].

After the impact, a multi-block column exhibits relative

sliding during free motion; relative sliding,

and

probably

rocking, in forced dynamics. Therefore, a new cause of

failure arises, in addition to overturning: one due to

excessive slidings in the contact surfaces.

The validity of formulations governed respectively by

Equations (49)-(50) or by Equations (51)-(52) depends

mainly on the assumptions concerning the starting instant

of the frictional behaviour and, then,

on

the impact

inelasticity, on the rigidity of both block

and

ground, and

on the duration of the impact.

For this reason, an experimental investigation has been

performed by Ageno and Sinopoli [35], [44] on marble

blocks of several sizes, impacting a marble ground. The

test results showed

that

the slide displacements suggested

by Equations

(51)

and

(52)

are always present; a

cumulative distribution of such displacements as a func

tion of the ratio b/h

is

shown in Figure

19.

With respect to

the time histories, for slender blocks, a good agreement has

been found with experimental results, lying between

Equations (49)

and 51);

on the other hand, for stocky

blocks, the results cannot be explained by the mechanical

adopted model.

A new theoretical model has been formulated recently

by Sinopoli [48], in order to take account of these

disagreements and to follow instant-by-instant the

dynamic evolution and the performance of friction.

7. CONCLUSIONS AND OPEN PROBLEMS

From the preceding review, it has been seen

that

in the

study of the dynamics of the single rigid block on rigid

ground, some problems have been solved satisfactorily,

but

some have not. Among the first kind, the simple

rocking motion of slender blocks under harmoni c shaking

has been investigated in detail. However, the ascertained

chaotic characte r of the response makes it necessary

to

investigate further the behaviour in

random

and stochastic

conditions, particularly relevant for seismic reliability.

Some tentative investigations

on

this subject have used the

statistical linearization technique,

but

without investiga

ting in depth its validity when the non-linearity

is

due

to

a

potent ial well : much furthe r research

is

therefore

necessary.

Many

possible alternative models

of

impact have also

been thoroughly investigated, as summarized in this paper.

However, the experimental evidence on the values of the

resti tution coefficient and its applicability to different

materials, contac t surfaces

and

dynamic conditions is as

yet insufficient.

The

main

point whose treatment is still unsatisfactory, is

the coupling between rocking

and

sliding. This point

is

particularly relevant for actual structures, made of many

blocks: slides in fact make loss of equilibrium possible, but

at the same time dissipate energy; the prevalence of either

aspect decides the safety

of

a structure.

Another question concerns the validity of the

Coulomb

friction hypothesis,

or

the respective values of static

and

kinetic friction coefficients: the first can be particularly

high when ancient constructions are concerned, in which

the surfaces may have remained in standing contact for

centuries.

Of course, the combination of the dynamics of single

blocks into that of more complex structures is a further

research task. Only a few tentative steps have been taken

in this direction.

ACKNOWLEDGEMENTS

The authors acknowledge the

support

received from re

search grants by the Italian Ministries of Universita e

Ricerca Scientifica e Tecnologica , and of Beni Culturali

ed Ambientali .

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in

preparation).



STRENGTHENING BUILDINGS OF STONE MASONRY TO RESIST

EARTHQUAKES

ROBIN SPENCE

and

ANDREW COBURN

2

1University

oj

Cambridge Department oj Architecture

Cambridge CB2 2EB U.K.

2Cambridge Architectural Research Ltd The Oast Hause

Malting Lane Cambridge CB3 iHF u K


ABSTRACT. Stone masonry buildings are common in many areas in the Alpine-Himalayan earthquake zone, and

their failure in recent earthquakes has been the cause

of

many deaths. Poverty and lack of alternatives prevent the

replacement

of

stone masonry with more ductile materials, but the brittleness of unreinforced stone masonry can be

considerably reduced

by

the incorporation of horizontal lacings of timber

or

reinforced mortar.

As

part

of a joint research project with the Turkish Earthquake Research Institute in Ankara to study low-cost

upgrading strategies for rural earthquake protection, full-scale wall elements were subjected to static lateral loading

test and dynamic tests on an impulse table. A simple yield-line analysis based on wall overturning was shown to be

able to give a useful prediction ofthe mode offailure and failure load in the static test, and give guidance on the relative

performance of unreinforced and reinforced walls in the dynamic test. The results of these tests were used to assess the

cost-effectiveness of a large-scale programme of upgrading rural buildings in eastern Turkey.

SOMMARIO. Gli edifici in muratura di pietra sono molto diffusi in ampie zone della regione sismica alpino

himalayana ed illoro crollo ha provocato la perdita di numerose vite umane durant e gli ultimi eventi sismici. Spessola

poverta e la mancanza

di

alternative impediscono la sostituzione della mura tura di pietra con materiali piu duttili:

ciononostante t comportamento di tale materiale puo essere sostanzialmente migliorato incorporando cuciture

orizzontali di legno od introducendo elementi metallici nei ricorsi di malta. II progetto di ricerca che si va sviluppando

congiuntamente con I Istituto Turco di Ricerca sui Terremoti di Ankara

si

propone di individuare una serie di

provvedimenti di basso costo per migliorare la qualit a antisimica dell edilizia rurale. In tale schema sono stati

condotti una serie di es perimenti su pannelli murari in scala

eale

soggetti a carichi statici laterali

ed

a prove

dinamiche su tavola vibrante. U na semplice analisi a rottu ra bas ata sui ribaltamento dell elemento murario si

e dimostrata in grado di fornire accurate previsioni sui meccanismi e sui carichi di rottura nelle prove statiche, e dare

utili indicazioni sui comport amento di pannelli semplici e rinforzati sottoposti a prove dinamiche.

I risultati di tali prove sono stati utilizzati per una analisi cost-benefici per la definizone di un programma su larga

scalia del miglioramento del comportamento sismica di edifici rurali nella Turchia Orientale.

KEY WORDS: Earthquakes, Stone masonry, Rural buildings, Turkey, Mechanics of masonry.

1 INTRODUCTION

Stone and adobe masonry houses are characteristic of the

rural areas throughout most of eastern Turkey. The

materials are freely available, the building skills are well

known, and the houses are

well

adapted to the climate,

with its extremes of temperature. However, these houses

are notoriously vulnerable to earthquakes. Well over

100000 houses have been destroyed by earthquakes in

eastern Turkey this century, killing over 50000 people.

There has been some movement away from the traditional

materials in recent years, towards lighter-weight pitched

roofs, but these require expensive modern materials, and it

seems inevitable that stone and adobe buildings will

predominate for some time to come.

During the summer of 1982 a combined research team

from three institutions (Cambridge University, the Turkish

Earthquake Research Institute

and

Middle East Technical

University, Ankara) carried out a field study of earthquake

vulnerability in the villages of Bingol Province, eastern

Turkey [1]. The study concentrated

on

examining the

housing, construction processes and local building indus-

Meccanica

27: 213-221, 1992

1992


try in an area of especially high seismiCity. One of the

major conclusions of the field study was that there was a

need for low-cost strengthening measures which could be

included in the construction of new houses of otherwise

traditional form and materials, so that the rural popula

tion could afford to build more safely. The extent to which

different methods of strengthening buildings of this type

could be carried out, their relative earthquake resistance

and effectiveness in reducing damage was the subject of the

second phase of the project.

During 1983 to 1986 a series of construction experi

ments was carried out at the laboratory of the Earthquake

Research Institute in Ankara to assess a range of different

strengthening methods for stone masonry. The assessment

involved observing traditional construction techniques in

operation, comparing the performance of un strengthened

and strengthened walls under static lateral loading, and

dynamic loading on an impulse table designed and built

specifically for this purpose. Other papers [2J have con

sidered the economics of alternative upgrading strategies.

This paper considers in more detail the assessment of the

strength of stone masonry buildings under seismic loading.



214

ROBIN SPENCE AND ANDREW COBURN

2

TR DITION L STONE M SONRY BUILDINGS

ND

THEIR SEISMIC RESIST NCE

The most commonly occurring house type in eastern

Turkey consists

of

a detached, single-storey random

rubble stone mason ry structure with a thick, flat

mud roof

on timber joists Figure 1). The house has a number of

rooms, added at different stages in its history,

and

often

has abutting stores

and

animal sheds. Variations on this

general pattern occur throughout eastern Turkey in a

number of architectural expressions

and

configurations.

Characteristic of most of the building traditions are a

common structural system of room size, wall lengths,

heights and roof span dimensions. The appearance and

layout of the traditional house

is

more standardized

than

are materials of construction

and

building techniques. The

major variation in the structural characteristics of houses

within villages

and

between villages

is

in the quality of

construction of the load-bearing maso nry walls. A number

of grades of stone masonry are found, from rounded,

riverbed stones set in thick

mud

mortar, through knapped,

angular rocks fitted with mortar infill, to dressed stone

facing blocks, scribed together in courses.

In order to estimate the strength of buildings of tradi

tional construction the mode of failure must be known,

and a theory developed which

is

able to predict the loading

which will cause failure in

that

mode.

For stiff masonry buildings with stiff diaphragm-like

floors and roofs effectively tied

to

the walls, the seismic

resistance is related to the in-plane shear strength of the

walls; by investigating this strength for all the walls a base

shear coefficient can be developed which

is

useful in

predicting performance under a particular loading. This

approach is only valid, however, as long as the roof and

floor construc tion are able to transfer the horizontal loads

in the floors

and

roofs into the walls in the form of in-plane

forces. Most traditional forms of stone masonry con

struction have floors

and

roofs which span only one way,

and

entirely lack in-plane shear strength. In such cases it

is

the out-of-plane strength of the walls which is critical to

earthquake performance.

Earthquake

damage surveys in

Italy [3], the Yemen [4]

and

Turkey [5] have convincingly

demonstrated

that

the primary cause of failure in such

buildings is the lack of out-of-plane strength of the walls,

coupled with the lack of continuity

at

corners

and

other

Fig.

1.

Appearance of typical stone masonry house in eastern Turkey.

wall-to-wall connections. The predominant modes of

failure in all these earthquakes are vertical corner cracks,

corner failures, skin-splitting and wall bulging Figure 2).

The characteristic shear cracks which are associated with

in-plane shear failures are relatively rare.

oburn

and Hughes [5] have charted the process of

structural collapse of typical houses in the 1983 Erzurum

earthquake by examining the characteristic modes of

failure in areas which can be assumed

to

have experienced

different degrees

and

periods of shaking. They have con

cluded

that

loss

of

strength

is

initiated by the reactivation

of existing weaknesses possibly even caused by earlier

earthquakes) in the wall construction; followed by the

separation of the structure into separate components

oscillating independently. Collapse of one

or

more wall

elements or the failure of the bearing of the roof on its

supporting wall then follows.

In

this process, the critical

elements are:

a)

the integrity of the wall construction;

b) the wall-to-wall connections;

c) the out-of-plane bending strength of the walls;

d) the wall-to-roof connection.

The wall-to-roof connection

is

sometimes cited as the

principal critical element, but is in fact best seen as one of

several critical elements,

and

indeed a secondary one since

its strength

is

likely

to

be tested only when the wall-to-wall

connections have failed. It

is

the out-of-plane wall strength

which needs to be considered first in estimating and

improving seismic resistance of stone masonry buildings.

3.

STRENGTHENING STONE M SONRY

BUILDINGS

Following the above analysis, the means

to

upgrade

traditional stone masonry so as to improve its perfor

mance in earthquakes are:

a)

to improve the integrity of the wall construction so as

to prevent separation of individual stones or failure of

sections of a wall independently of the whole wall;

b)

to provide continuity

at

wall-to-wall connections, with

sufficient strength to resist the tensile and shear forces

acting;

c) to increase the out-of-plane bending strength of the

walls, so that loads applied perpendicular

to

them can

be transferred to walls in the plane of these loads;

d) to attach the roof-members securely to the

top

of the

walls to prevent relative movement.

The use of two-way spanning roofs, such as reinforced

concrete, which can connect to the walls in both directions,

is

desirable structurally

but

too expensive for use in most

rural situations.

In some parts of eastern Turkey there

is

a tradition of

using horizontal timber courses or hat Is at approximately



STRENGTHENING STONE

MASONRY

215

Reactivation of Existing Weaknesses

Vertical Cracking at Corners

iagonal Cracking and Around

Openings

Skin

Splitting

• Often old movement or settlement cracks reactiv ated

• Existing masonry instabilities triggered

Structural Separation

End or Non Ioadbearing Wall Separation

Wedge shaped Corner Failure

Fig. 2. Main modes of failure observed in rural Turkish houses.

90cm intervals up the walls Figure 3 . Where these have

been continuous and well-joined they appear to have

reduced the level of earthquake damage significantly by

halting crack propagation, by providing continuity

at

the

critical wall-to-wall connections and by increasing the out

of-plane bending strength of the wall.

One

possible

method of upgrading traditional stone masonry buildings

would be

to

encourage the wider and more effective use of

these hauls An alternative method of achieving the same

effect would be

to

use concrete ring-beams at the ground

and eaves levels. n each case the

roof

joists would be

connected to the upper ring-beam to prevent slippage.

Additional improvements which could be used either in

conjunction with, or independently

of,

the above tech

niques are the use of

sand-cement

sand-lime-cement or

stabilized soil mortars,

and

the use of cut stone as opposed

to random-rubble masonry.

All

these modifications can be expected

to

have a direct

effect on the bending strengths of wall elements, and

to

test

this effect, two series of tests were designed.

n

the first

series, a set of stone masonry walls was constructed and

subjected

to

a static out-of-plane loading. This test series

is

summarized in this paper. n the second series, a set of

complete building mock-ups, of 4.0 x 4.0 m plan size, and

using full-sized wall construction, was tested under sinus

oidally oscillating lateral acceleration on an impulse table.

The results of these tests are reported elsewhere [6].

Both test series used test pieces constructed

at

full size,

by masons familiar with the traditional techniques of

construction, because it

is

believed that small-scale

1

Dressed coursed stone.

2

Knapped a ngular stone.

3. Round stones

n

deep mud mortar.

Fig.

3.

Timber

hatils

used in traditional stone masonry.



216


models, or test pieces constructed by technicians under

laboratory conditions, cannot adequately represent the

material and stress conditions found in prototype

structures. Accurate representation

of

the type of con

struction being investigated is thought

to

be of much

greater significance

than

the accurate representation of the

earthquake loading to which they may be subjected.

follows:

W1 A standard, unstrengthened, random-rubble wall

with mud mortar as built by large numbers of

villages in eastern Turkey today.

W2 A standard random-rubble wall with mud mortar,

reinforced with horizontal timber

ha ls

as described

above.

W3 A random-rubble wall built to Turkish

Standard

4.

STATIC TESTING OF

W LL

ASSEMBLIES 2510, Design

and

Construction Methods for

Description of Walls

In the surveys of traditional building stock

that

were

carried

out

during the field study in eastern Turkey, a

common characteristic of many of the building forms was

a structural unit of

around

3.3 m span, corresponding to

the average length of timbers used as

roof

joists. This

structural bay was taken for building portions of wall at

full size. The test elements were constructed as free

standing walls 4.5 m long with two side walls 1.5 m long

and

0.6 m thick giving the

standard

unsupported length of

3.3 m. The wall height,

2.8

m, corresponded to that of the

common single-storey houses surveyed. The walls carried

no roof load

and

in

that

way were analogous with the

nonload-bearing end wall of a rectangular room.

Four

walls were built. The dimensions are shown in

Figure

4,

and the form of construction of each wall was as

I . ~

,

JU

,

Fig.

4.

Set

up

of testing apparatus for static tests.

PLAN

ELEV TION

Masonry , using cement mortar and reinforced with

horizontal reinforced concrete beams. Such walls are

too expensive for rural housing

but

are used for

school and government construction in earthquake

areas.

W4 Cut and dressed stonework with cement mortar, a

masonry system commonly used for mosques

and

community buildings.

Testing Programme

The testing

apparatus

was not designed

to

simulate pre

cisely the loading which might be caused by an earth

quake,

but to

create a set of internal forces in the wall

similar to those which would be caused by the horizontal

out-of-plane component of earthquake loading. By this

means it is possible:

a)

to make effective comparisons between the strengths of

different types of masonry which would indicate their

relative performance in earthquakes;

b) to test the validity

of

a theoretical approach

to

the

estimation of the out-of-plane strength of a masonry

wall.

t was important also to devise a cheap and simple test

that

could be replicated for a wide variety of wall types,

and could be carried

out

without the resources of a

sophisticated structural testing laboratory. The testing

apparatus

adopted is shown in Figure 4.

To spread the load

and

prevent punching failure, it was

decided to use the distributed loading system shown. By

means of an H-shaped loading yoke and ball seatings, a

single concen trated load acting on a

30 mm

diameter bar

passing through the wall was distributed into four equal

loads on four spreader plates. Each of these plates was

40 cm square by

1.5

cm thick

and

load was transmitted

from them into the wall through a pad of gypsum plaster

to distribute the stress evenly. There was

no

evidence

of

punching failure in any of the tests.

Load was applied to the bar by means of a hollow

hydraulic ram coupled to a loading cylinder. The ram was

supported on a hollow steel beam which in turn was

supported from the ends of the side walls thro ugh a timber

spreading beam. When supports were removed this

created

an

internally consistent set of forces independent



STRENGTHENING

STONE MASONRY

217

1 0 0 ~ - - ~ - - - - . - - - - - - - - - - - - - r - - - - r - - - - r - - - - r - - - - r - - - - r - - - - - - - - - - - - - - - - ;

Load

KN)

50

Load: Deflection

for mid-point

of

wall

Wall 1

Deflection em)

Fig. 5. Load-displacement curves for the four tests walls.

of any external reactions. The load was measured by

means of a hollow load-cell (using electrical resistance

strain gauges) introduced between the

ram and

the steel

beam, so that an accurate reading of the force in the bar

was obtained.

Displacement was measured on the loaded face in three

ways. Small displacements were measured by a pattern of

dial guages supported on an independent frame. Larger

displacements were measured from a pair of string lines

attached

to

independent frames

at

each end of the wall.

Displacements were also monitored by means of measur

ing tapes fixed perpendicular to the wall, observed

at

each

load increment from a theodolite station established out

side the testing site. A photographic

and

cine-film record

of the progress of the tests was also maintained.

Initially the load was increased in equal increments, but

as displacement increased the

load

increments were re

duced to follow the displacement In tests W1 and W2 the

displacement exceeded the (90mm) displacement capacity

of the rams,

and

the additional displacement was followed

by tightening the nut holding the jack assembly in place.

Displacement measurement terminated in these two tests

at a mid-wall displacement of 130mm, but the loading was

continued until partial failure ofthe wall took place. In test

W3

the test was terminated

at

a load of 100 kN when the

capacity of the loading

apparatus

was exceeded. In test W4

the test was terminated at the end of the displacement

capacity of the rams.

Load displacement

curves for the mid-wall point of

load are shown graphically in Figure

5.

Figure 6(a) and b)

show, respectively, the conditions at failure of walls WI

and

W4_

5. THEORY:

LATERAL STRENGTH OF STONE

MASONRY

WALL

ELEMENTS

To investigate the structural action of a stone masonry

wall acted on by lateral and gravitational forces a simple

yield-line theory has been developed. The theory is based

on the following assumptions:

1.

The wall retains its overall integrity

i.e.

does not

disintegrate) under the applied loading.

2. The mortar used, whether of soil, cement or lime, is

assumed to have zero tensile strength, but high com

pressive strength.

3.

At failure, the wall separates into rigid blocks which

adjoin each other along hinge lines, i.e. a failure mech

anism has developed_

4. The hinge lines are assumed straight, and located in the

faces of the walls.

5.

Displacements at failure are considered too small to

affect the initial geometry significantly.

6. Failure occurs slowly

and

in equilibrium so

that

a work

balance between the work done by external load

and

that used internally is maintained.

7.

Timber or steel reinforcement members which cross

hinge lines are assumed to carry tensile or compressive

forces, but not shear force or bending moment



218


Fig

. 6 b). Conditions at railurc

ror

wall \ 4.

8.

The wall is made from a material with a uniform specific

weight p

and

is assumed subjected to a centrally located

patch load, directed inwards.

f he wall

is

free-standing and without wings, we can easily

see, Figure 7, by equating the work done by the load to

that

required

to

raise the centre of gravity of the wall, when

the wall pivots an angle ¢ about edge A B , that:

h¢ t)

·

T

= h tLp ) ·2 ¢

Therefore,

P = t

2

L.

1 )

This is of course the same as the value for P which would

be obtained by taking moments about A B .

A more complex situation occurs if there are rigid

vertical support s

at

B C

and

A D . In this case a triangular

block BCF

F

C B will form, which rotates about BF,

Figure 8.

I

Fig. 7. Collapse mechanism for a free-standing wall.

C

Fig. 8. Collapse mechanism for a wall with vertical end supports.

We will assume

that

the sloping edge of this block,

BF

F B , reaches the top surface at a distance yt from the

end of the wall. If, as before, the trapezoidal piece rotates

by ¢ about A B , its centre of gravity rises by ¢t/2. The

triangular piece rotates about FB in such a way that its

vertical faces remain vertical; so its centre of gravity lifts by

t¢, the amount by which every point on the loaded face of

the wall rises. Thus the work balance for half the wall

becomes:

Therefore,

2)

Note that Equations

1)

and

2)

are both independent of

the wall height h The effect of the vertical suppor ts

is

seen

to be equivalent

to

an additional length of free-standing

wall yt

If the vertical supports at A D and B C are replaced by

a pair of perpendicular wing walls, the mechanism de

scribed above

is

resisted, because it involves motion of

B C into the wall. This can be allowed for, however, by

assuming a mechanism in which a triangular part of the

wing wall (triangle II) separates and rotates

about

edge

LM, Figure 9.



STRENGTHENING STONE MASONRY

219

Triangle I

Triangle II

Fig.

9.

Assumed collapse mechanism for a wall with end wing walls.

In this case we have three rotating pieces. The analysis

is

faciliated by using e, the rota tion of the to p face o f triangle

I, as the rotational variable,

and

using pattern parameters

v

and

l defined as in Figure 10.

I is the instantaneous centre for rotation of triangle I.

Triangle

II

can be seen to rotate

ellJ

if the two triangles

remain in contact

at

point

e

(although this implies a

movement into the wing wall of all points on B C .

The work balance for half the wall, calculated as before,

then gives:

~ ~ v + ~

+ ~ l J

2 + ~ + ~

pt

3

t lJ v lJ

lJ

which has a minimum value:

P L

-=-

5.3

pt

3

t

(3)

(4)

when l = 2.51 and v = 2.25. Note

that

the effect of the

wing walls is equivalent to an increase of

5.3t

in the length

of a free-standing wall.

The effect of introducing tensile reinforcement into the

arrangement of Figure 9 can now be examined by con

sidering the additional work done in this reinforcement

I

I

,

I

-

T t

F

I

fa

I

I

I

ie

1 t - - _ u _ t

~ I ~ 1

M

I

I

I

I

I

I

ny

L

Fig.

10.

Definition of pattern parameters for mechanism of Figure

9.

across the various hinge lines. For a

bar

generating a yield

force H across the hinge B F

and

fJ across the vertical

hinge

Be,

and located in the wall face, an additional term

should be added

to

Equation

3):

- 4H 1

fJ

t

pt

3

-

pt

3

v

l .

5)

A minimum value for the overturning load mayor may

not now be possible depending on the value of

Hlpt

3

•

6. COMP RISON OF THEORY WITH

TEST

RESULTS

The failure mechanism proposed assumes the formation of

hinges in both faces of the walls at failure. In all the three

tests where failure was achieved major cracks in the

opposite faces of the walls were seen (Figure 6), corre

sponding to the hinges

A B ,

AE,

BF,

A D and

B

C. These

cracks are neither exactly straight

nor

in precisely the

place assumed, as would be expected in walls of non

homogeneous construction. But the general corre

spondence is good. There was also some evidence of

cracking in the side wall corresponding to hinge-line

LM,

but this was clear only in Wall 4.

The method of loading adopted allows the hinge BF

to

intercept the top of the wall a maximum distance 1.2 m

from the wall end e, which is smaller than the theoretical

optimum. Hence

Pc is

minimized with this constrained

value. In all three tests, the angle of the cracks in the rear

faces is quite close

to

this constrained value.

Using the constr ained crack location, failure loads were

calculated for all four walls using Equations

4) and

5). In

Wall 2, the force generated in the timber

hatlls

was

calculated by assuming that it depends entirely on the

shear strength of the connection with the cross-members.

Using data on nailed timber connections given by Ozelton

and Baird [7J, each joint was assumed to have a failure

load of

4.5

kN. This leads to failure loads for the haUls

at

cill

and

lintol level of 9 kN (at plane

BC) and

18 kN (at

plane BF). The

top

hatll was assumed ineffective since

there was no vertical load on it to prevent it from slipping.

In

Wall

4,

the reinforcing bars in the

top

ring-beam were

assumed

to

act at their yield strength at an assumed stress

of 250N/mm

2

, and the concrete was assumed cracked.

The

unmortared

random-rubble walls

1 and

2) were

assumed to have a specific weight of 20 kN/m

3

. The

cement mortared walls 3

and

4) were assumed to have a

specific weight

of22kN/m

3

.

All

dimensions were as shown

in Figure 4. The calculated collapse loads are shown in

Table I

In all cases the calculated collapse load was higher

than

that

actually achieved. This can be partly explained by the

use of the assumption that the hinge line was in the face of

the wall, clearly

not

valid, particularly in the case of Walls

1

and

2 with very soft mortars.



220

ROBIN SPENCE AND ANDREW

COBURN

TABLE 1.

Comparison of

calculated and observed collapse loads.

Wall

WI

W2

W3

W4

Failure

load

P

(kN)

19

95

>100

40

Assumed

Calculated

values

of

collapse

load

specific P, kN)

weight k m

3

20

47

20 155

22 212

22

51

P,IP

2.47

1.63

1.27

In both

ofthe

walls with mud mortar, the stiffness is very

low, and substan tial displacements have taken place before

the maximum load is reached. Also there is evidence of

very considerable distortion within the blocks at failure:

thus the assumptions of small displacements and rigid

blocks are

not

valid.

t

may be

that

due to internal

distortion

ofthe

blocks the side walls are unable to provide

the lateral restraint needed to mobilize the wing wall

failure. In these two cases the theoretical collapse load is

not achieved, and a collapse mechanism which allows for

internal distortion

and

large displacements may need to be

considered. Indeed, the actual collapse load for Wall I is

slightly below the theoretical collapse load for a free

standing wall of the same length.

The observed collapse load for wall W2, reinforced with

timber

hatlls

was more than four times that for the

unreinforced wall

WI

an increase which is reasonably well

in accordance with the prediction of the theory.

In the case of Wall 3, with cement mortar, the capacity

of the testing apparatus was exceeded before any signifi

cant cracking

had

taken place, and the validity of Equa

tion 5) was not tested.

In the case of Wall 4, with cut stone blocks in cement

mortar, the stiffness

of

the wall is initially much greater

than that of Walls I

and 2,

owing to the tensile strength of

the mortar, and a peak strength is achieved at small

displacement (Figure 5) which is well in excess of the

calculated load. However, as displacement increases, pro

gressive failure of the mortar leads to a reduction in

strength with the development of a cracking pattern

similar to those of Walls I and

2.

At the maximum

displacement achieved (100mm), the essential features of

the mechanism of Figure 10 were clearly seen, including

lateral tipping of the side walls. The load required to

maintain equilibrium

at

this stage is 27 greater than the

calculated collapse load, a difference which might partly be

accounted for by interlocking faces along separation

planes which are not allowed for in the theory.

7.

STATIC LOAD TESTING: CONCLUSIONS

Given

that

only one wall of each type was tested, the

validity of the test results numerically must be considered

uncertain. Nevertheless, in a general sense, the results are

useful. They indicate that:

1.

2.

Under static loading, even unreinforced random-rubble

masonry walls do not simply disintegrate,

but

deform

under gradually increasing load until a pronounced

failure mechanism has developed. Very considerable

displacement is possible before the wall topples.

The failure load of

an unreinforced random-rubble

masonry wall with mud mortar is rather low, and

depends primarily on its resistance to toppling, without

any contribution from the interlocking of stones

or

internal friction.

3. The failure load of a random-rubble masonry wall in

mud

mortar

with timber reinforcement is much higher,

on account of the contribut ion from the tensile strength

of the timber hat/Is. This tensile strength depends on the

strength of the nailed connections with the cross

members, which are therefore a crucial part of the

reinforcement.

4. The failure load of a random-rubble masonry wall in

cement

mortar with concrete ring-beams is substan

tially higher than either of the above wall types. t is

also very stiff under small levels of lateral load. The

mode of failure

of

this type of wall has not been

observed.

5. A cut stone masonry wall in cement mortar behaves in a

brittle-ductile manner. After an initial load peak at

small displacement (due perhaps to the tensile strength

of the mortar and also to restrictions imposed by the

block geometry on the formation of yield lines) the

strength drops sharply, and at large displacements the

strength is not much greater than that of an unrein

forced random-rubble wall, i.e.

the contribution from

interlocking and internal friction is not very great.

6.

In the three cases of walls which were failed, a simplified

yield line theory based on the overturning strength of

the wall under gravitational forces, and ignoring the

tensile strength in the mortar or internal friction, is able

to predict the mode

of

failure and give a reasonable

estimate of the failure load. The contribution of the

strength of timber reinforcement can be allowed for in

the theory.

8.

DYNAMIC TESTING AND APPLICATION

The results of the test described here were used to design a

series of test houses for dynamic testing using an impulse

table built for the purpose at the Department of Earth

quake Research in Ankara. The table

had

plan dimensions

of 5 m x 6 m, and was designed to impar t a sinusoidal

loading with a peak acceleration of g to a payload of up

to

50

tonnes. Three test houses, of dimensions 4 m x 4 m in

plan, with walls 2.2 m high, were tested. The first was of

unreinforced rubble masonry walls, like wall WI. The

other two had walls of rubble masonry reinforced with



STRENGTHENING STONE MASONRY

221

horizontal hauls

at

the levels of the cill, lintol

and

eaves.

One of these had timber hatlls, the other thin reinforced

mortar

bands.

Each of the three test houses was subjected to the same

set of gradually increasing increments of load until failure

took

place. As in the case of the static tests, the presence of

the hat Is

had

a considerable effect on the amount of load

which the test houses could carry, and also on the failure

mode.

In

the test house without reinforcement, failure

occurred by the wall overturning in the manner observed

in the static tests. In the test houses with horizontal

reinforcement, the out-of-plane failure mode did not occur.

The reinforcement was able to transmit the lateral forces

into in-plane forces in the adjacent walls, which failed by

shear at a higher load.

The results of these tests were used

to

assess the likely

effect of a general programme of upgrading of stone

masonry houses thro ughout the region of high seismicity

(the

13

provinces of eastern Anatolia where this form

of

construction predominates). t was estimated that over

25

years, such

an

upgrading programme could be expected

to

save over 70000 houses from destruction in future earth

quakes, and save

around

3000 lives. The cost of the

upgrading would be about £10 million over that period of

time,

but

the estimated saving just in terms of the avoided

replacement costs of the destroyed houses would be over

twice that sum. The programme, in other words, would

save money

and

lives.

Since the completion of this test programme, there has

not been a serious damaging earthqu ake in the region, but

it is understood

that

in future reconstruction programmes

there will be a greater emphasis on the strengthening of

buildings of traditional construction rather

than

their

replacement with modern houses, i.e. concrete block

houses with sheet metal roofs.

CKNOWLEDGEMENTS

The work described in this paper was

part

of a

joint

research project between the Directorate of

Earthquake

Research, in Ankara, Turkey, and the Martin Centre for

Architectural

and Urban

Studies. The authors acknow

ledge ample assistance from the Cambridge University

Engineering Department, in particular to Professor

C. R.

Calladine for

important

suggestions in the formula

tion of the theory, to

Arthur

Timbs for assistance in the

testing

apparatus

design, and to final-year students

Johnny Chiu and Mark Cowdrill for assistance in Turkey

with the testing work.

REFERENCES

1

Coburn, A.

W. ed.), Bingol Province Field Study, 2 24 August 1982,

Report to the Turkish Committee on Earthquake Engineering, The

Martin Centre for Architectural and Urban Studies, 1982.

2. Spence,

R.

J. S and Coburn, A.

W.,

Earthquake protection - an

international task for the 1990 s, The Structural Engineer, 65A (August

1987).

3. Spence, R. J. S, Hughes,

R.

E., Nash, D. F. T and Coburn, A

W.,

Damage assessment and ground motion in the Italian earthquake of

23.11.1980', Seventh European Con on Earthquake Engineering,

Athens, August 1982.

4. Coburn, A.

W.

and Hughes,

R. E., Dhamar Province Earthquake,

3

December 1983, Report to the Joint Relief Committee, Yemen Arab

Republic, The Martin Centre for Architectural and Urban Studies,

1984.

5. Coburn, A.

W.

and Hughes, R. E., Report on Damage to Rural Building

Types in the Erzurum-Kars Earthquake, 30 October

1983, Report to the

Turkish Committee on Earthquake Engineering, The Martin Centre

for Architectural and Urban Studies, 1984.

6. Spence,

R. J.

S and Coburn, A.

W., Reducing Earthquake Losses in

Rural Areas, Report to the Overseas Development Administration,

The Martin Centre for Architectural and Urban Studies, 1987.

7.

Ozelton, E.

C.

and Baird,

J. A.,

Timber Designer's Manual, Crosby

Lockwood Staples, London, 1976.



REPAIR OF MASONRY STRUCTURES

FRITZ WENZEL and HELMUT MAUS

Universitiit Karlsruhe Institut ur Tragkonstruktionen

Englerstr. 7 D-7500 Karlsruhe J Germany

(Received: 28 February 1992)

ABSTRACT. Besides the traditional rep air techniques of craftsmen for mason ry structures, engineering method s and

procedures such as grouting and reinforcing of old masonry are available. These technical measures can help to save

the monumental value of historically

important

buildings more effectively than the procedure of dismantling and

rebuilding; and, as a rule, they are distinctly less costly. Nevertheless, too mu ch technical aid can destroy w hat is meant

to be preserved.

For

that reason the investigations described in this

paper on

both improvement

and

development of

engineer-like repair techniques have been focused on the goal of minimizing interventions and modern additions as far

as possible.

SOMMARIO. In alternativa aile tecniche tradizionali usate dagli artigiani e dai capomastri per la riparazione delle

strutture murarie, sono oggi ampiamente sperimentate ed applicate alcune tecniche e metodi di ingegneria strutturale

quale Ie iniezioni e l inserimento di elementi resistenti a trazione. In terventi di questio tip o passano aiutare a

preservare il valore storico e monumen tale di edifici antichi, in maniera piu filologica rispetto al met odo dello

smantellamento e ricostruzione con nuovi elementi della stesso materiale; e di regoia, hanno costi piu con en uti.

Ciononostante, l uso indiscriminato di tali tecniche puo finire per distru ggere cio che si aveva intenzione di

conservare.

Per

tale motivo gli studi descritti nel presente contributo, sui miglioramento e sviluppo di tecniche di

intervento strutturale, applicano la filosofia del minimo disturbo possibile.

KEY WORDS: Retrofitting, Repair techniques, Mechanics of masonry.

INTRODUCTION

When dealing with old masonry the first question en

gineers are confronted with is whether the masonry needs

structural repair at all. If repair proves to be necessary a

second question arises: Is it possible to repair the masonry

conventionally

or

is it more appropri ate

to

apply engineer

ing methods

and

to use relatively modern techniques?

Conventional repair of masonry should be carried out

wherever possible and wherever compatible with the

monument s value. That is indisputable. Besides the tradi

tional repair techniques of craftsmen for masonry

structures, engineering methods and procedures such as

grout

injection and stitching, as well as the prestressing of

old masonry, have been practised for a long time.

The techniques of mortar injection, steel reinforcement

and prestressing have been used since the 1920s

to

strengthen old masonry. Due to follow-up examinations of

buildings repaired in such a manner, together with recent

research results achieved

at

the University of Karlsruhe,

rules for dimensioning

and

execution have been made

available for the structural repair of old masonry.

No universal standards can or should be established for

historical buildings, but rules and recommendations can

be given for application in practice which can be adapted

to

the special circumstances of each object.

GROUT INJECTION

OF MASONRY

Old masonry

is

grouted to increase its supporting capac

ity,

to close cracks and cavities, to strengthen loose

M eccanica 27:

223-232, 1992

if

1992


masonry and mortar, to replace missing mortar, to allow

the introduction of new, larger forces into the masonry at

local points, to involve the inner filling of multi-leaf walls

and pillars in the supporting structure, to link reinforce

ment bars and prestressed anchor ties to the masonry and

protect them against corrosion. Where these or similar

problems do

not

occur, grout injection does not need

to

be

performed.

Injection Material

All

types of cement customary in the trade, including those

with additives of trass, are suitable as injection materials.

Clay

and

expanding cement are not suitable. Very impor

tant are cements with a high sulphate resistance (HS

cement), which normally help to prevent damage of expan

sion in old mortar containing gypsum ([IJ-[4J). The

disadvantage of these cements is their dark colour; if they

leave the masonry they can easily cause stains

on the

surface which

is

why particular caution

is

advisable.

Although super-hydraulic limes may also be injected, these

do

not obtain sufficient strength in the masonry and tend

to expand because of their C

3

A content.

For

cement

injection, water/cement (w/c) ratios from 0.8

to

1.0

and

pressures up

to

6 bar are used.

Necessary Information Obtained from Preliminary

Investigations

Preliminary surveys must to some extent reveal the cav

ities and those areas in the masonry that are not strong



4

FRITZ WENZEL AND

HELMUT

MAUS

enough to transfer loads with sufficient safety. This

is

especially important where concentrated loads are or will

be imposed. These areas are decisive towards the arrange

ment of the drilling holes and the spread of suspension

within the masonry and their ascertained frequency can

help assess the

amount

of injection mortar needed.

Cavities that absorb injection mortar may consist of

areas neglected when applying

mortar

during construc

tion of cracks

and

gaps caused by deformation or of voids

in joints and foundations where mortar has disappeared

due to erosion. Unknown channels of circumferential

tendons

and

concealed holes formerly used to support

scaffolding may also be among them.

There are also considerable differences between solid

and multi-leaf masonry

that

are reflected in the injection

performance. In one-leaf and solid walls there are usually

fewer cavities the propor tion of stone

is

greater and the

mortar

application is more thorough. In the case of multi

leaf structures there is often less bonding through mortar

than in the outer skins. Cavities occur more often in

natural stone masonry made with lime

mortar

than in

masonry of bricks and gypsum mortar.

In

some cases large

stones may reduce the degree of cavities substantially.

Beyond the rising masonry the foundation and its

related areas are of interest. The proportion of cavities

may be so great due to eroded

mortar or

underground

channels that the injection mortar assessed for the whole

repair operation may be used almost entirely in the

foundation area.

A further point

of

concern

is

the ability of the injected

areas to absorb and store moisture. The mortar is pre

dominantly involved in this process but the stones can also

participate. This refers especially to bricks. Brick masonry

can influence the flow of suspension to a greater extent

than

stone masonry can. Pieces of brick may have been

used to fill the core of multi-leaf stone masonry which

shows the importance of preliminary surveys.

The main factor in

mortar

is

the content of binder.

Gypsum

mortar

generally contains only small amounts of

aggregate and therefore consists mainly of binding

material. Gypsum mortar

or

the contents

of

gypsum in

mortar call for special atten tion when selecting grout for

repair purposes. At times lime mortar may contain only

small amounts of aggregate

but

usually lime

or

lime

gypsum mortar

is

quite lean compared with gypsum

mortar. In some cases one can effortlessly scratch it away.

Occasionally

mortar

may even contain straw charcoal

or

clay material.

The composition of

mortar

has

an

effect

on

its pore

structure which in turn influences the ability to absorb

water. t

is

important therefore to know the moisture

content of the masonry and its distribution thr oughout the

areas in need

of

repair. f the values differ significantly as

say between the base and the eaves

or

if extremely high

or

low values occur this should be noted.

Special attention is required with damp masonry. The

cause of dampness should be determined and if possible

should be dealt with in advance. Salt efflorescence

at

the

surface

is

not

only a sign of increased moisture content

but

also

ofthe

salt load of

mortar and/or

masonry stones. This

requires additional examination to determine the type and

origin of the salt.

Additional Surveys During Drilling Operations

Sometimes it can be useful to determine on site whether

the proposed measures can be applied effectively and to

make adjustments. During the drilling

of

injection and

reinforcement holes valuable information can be gathered

and may lead to adjustments in the injection procedure.

The flushing of drilling materials regardless of whether

by air or water may give clues to the content of binding

material in old mortar. Should the content be extremely

low then parts of the

mortar

surrounding the drilling hole

may also be broken down and washed out. This creates

additional cavities up to several litres in volume.

If

the amount of drilling dust remains small

or

only a

little of the crushed drilling material is washed out the

mortar is

usually more dense and has a higher binder

content. This does

not

refer to structures that have very

little mortar in the first place. A good look into the drilling

hole will clarify this.

f drilling

is

done with water flushing the return of

drilling water may indicate the expanse of cavities. Inter

connected and larger channels allow the drilling water to

intrude more quickly. Usually the content of

mortar

in the

core masonry

is

fairly low. Similar but weaker effects can

be caused by porous

mortar

containing coarse sand and

little binding material which can absorb water like a

sponge.

Drilling progress may help to assess the content of stone

and mortar. Tedious drilling squeezing and damaged

segments of drilling bits usually result from a high content

of stone and a low proportion of mortar.

By

contrast

drilling steels driven into masonry with a large proportion

of solid

mortar

can be guided easily and do

not

vibrate as

much. The speed of drilling can often be increased.

ll

the phenomena described above may occur in

various combinations. Final certainty though is obtained

by a look into the drill-hole using a flashlight or for a

more precise result and for documentary purposes -

an

endoscope. Rough and uneven drill-hole walls as well as

mortar and stone caving point towards mortar with little

binding material in this area.

The Degree of Injection and its Conditions

The degree of injection refers to the percentage of suspen

sion in comparison with the entire volume

ofthe

masonry.

t depends

on

the specific components of the old masonry



REPAIR

O

MASONRY STRUCTURES

225

being repaired

or

strengthened

and on

its structure and

moisture content,

on

the composition of the suspension as

well

as

on

the procedure selected to prepare and carry out

the injection, including the applied pressure. A closer look

at

the individual parameters shows how they interdepend.

One example

is

the mixture recipe of the suspension which

depends on parameters such as the area to be injected as

well as the drill-hole intervals, the depth and diameters of

the holes, etc.

In

this respect it

is

only possible to determine

each factor precisely on a local basis. During injection the

parameters may change. This

is

one reason why prelimi

nary surveys should

not

be limited to one spot but should

be conducted at several points to guarantee a closer

coverage of the object.

The parameters of each of the three areas masonry ,

suspension and method have been compiled and

grouped according to the impact they have

on

the success

of injection.

igh Impact

Masonry: Cavity Content

The factor with the greatest influence

is

the structure of the

masonry and the proportion of cavities. The best injection

results are achieved in areas with a high proportion of

stone and a low content of mortar. In these circumstances

a continuous flow of grout travelling great distances

through connected cavities is possible. Stones and

mortar

are covered with suspension and cavities are filled. The

tighter the fabric

and

the greater the proportion of mortar,

the smaller is the opportunity for the suspension to make

its way through the masonry and enter the fabric. t may

spread within only a small radius

around

the drill-hole.

Suspension: The Water/Cement Ratio

To be able to reach distant cavities and perhaps stabilize

porous mortar, the flowability of the injection grout must

be sufficient. This depends primarily on a high content of

water (w/c 1 . This makes it unlikely that the absorption

of mixing water will cause the cement grout to thicken and

reduce the cross-sections of the holes. The cement particles

are practically swept along by the stream

of

water and

washed into the mortar. However, instead of using a high

proportion of water in the suspension, it may be better to

increase the number

of

injection holes. This refers especi

ally to masonry that hardly influences the water/cement

ratio of the suspension such as damp masonry or masonry

with mortar of high binder content. There is a danger of a

cement stone being created after hydration with a high

content of capillary pores. This

is

of no disadvantage to the

flow or force but if the cement happens to have a high

alkali content and contains elements that can be leached

out it can lead to efflorescence. Should the suspension have

been injected to protect steel reinforcements, corrosion

may occur due to introducing moisture.

Method: The Pattern o Drilling and Injection Holes

The most obvious way to obta in a high degree of injection

is

a dense drilling pattern. (This,

of

course, would

not

be

applicable in the cautious

and

substance-preserving repair

of historical buildings.) This not only refers to the drill

holes remaining visible after they have been sealed (which

of course is not the case when rendering

is

applied), it also

means that considerable quantities of both the ou ter leaves

and the core infilling material may be extracted in the

process of drilling

and

flushing.

In

view of the repair task a

decision must be made between obtaining totally pierced

masonry (with the highest degree of injection)

or

old

substance with restrictively applied drilling and mainly

local injecting.

Medium Impact

Masonry: Content o Binding Material n Mortar

A high content of binding material

not

only results in solid

but often in relatively dense old mortar. Therefore only

small amounts of water are absorbed from the suspension

in the immediate surroundings of the injection holes. The

suspension can flow to greater distances and there are

hardly any signs of thickening and stiffening in the areas of

contact with the old mortar.

With increasing porosity

or

decreasing content

of

bind

ing material this picture changes. Layers of thicker cement

grout appear

at

the areas

of

contact with the mortar. The

mortar absorbs water much like a filter thereby changing

the consistency of the injection grout closest to the mortar.

Cement particles seal the pores of the

mortar

preventing

suspension and water from entering further. In this phase a

type of tube forms, which still allows the suspension to

continue to flow.

Suspension: Micro-Cement Thixotropic Additives

The flow and penetration performance of the suspension

can be influenced positively with the help

of

micro-cements

and thixotropic additives. A suspension made of micro

cement has less tendency to clog and penetrates into even

the finest openings and cracks as opposed to suspension

made of normal cement. The grout consumption increases

enormously and therefore the degree of injection as well. t

must be decided in each case whether

or not

this condition

should be reached and how useful it actually is for the

repair task. Thixotropic liquids function as a kind of

grease for the suspension.

t

therefore flows further and

obtains similar characteristics to those of micro-cement

mortar. Such additives should generally be examined for

their suitability for use in old building substance.



226

FRITZ WENZEL AND HELMUT MAUS

Method: Wetting

Areas of masonry that are dry or have been dried by the

air-flushing system during drilling may absorb water from

injected suspension to a great extent, thus causing the

grout t o thicken and channels to become clogged. This can

be prevented by injecting water into the holes after drilling

except where other aspects, such as valuable wall paint

ings, must be considered). One advantage is that the stones

and

mortar

are moistened, which has a positive effect on

the flow performance of the suspension. Another is th at the

returning amounts of water can help to assess the porosity

of the masonry and how strongly the mortar can absorb

water.

Low Impact

Masonry: Moisture Content

The wetter the area being injected, the less it will influence

the water content of the suspension. For this reason the

water/cement ratio may be reduced in some areas of the

masonry. In practice, this will hardly be significant.

Not

only here, but especially in this case, the application of

spray plaster to the inside leaf of the wall is not re

commended, as this could stop vapour diffusion and the

masonry would

not

be able to breathe. The natural

moisture content thus captured would be increased by

surplus injection water.

Suspension: Cement Aggregate Ratio

Pure cement paste has a better flow performance than

cement grout containing aggregate. If material such as

sand

and

stone flour are added, these larger

and

heavier

particles tend to settle and segregate. They cannot be

recommended for long-range injections or for masonry

with

few and

narrow internal channels. They are more

suitable for filling large cavities

or core areas behind

detached outer leaves.

Method: Pressure o Injection

Precise information cannot be given on this subject. The

differences in masonry composition, in the forces within

the leaves of the wall and in the ingredients ofthe grout are

too great.

On

the one hand, the injecting pressure reaches

values up to 6 bar; on the other hand, experienced foremen

can assess the pressure situation within the masonry with

the loop

of

the hose and inject with just enough pressure to

give a light flow. Generally, a solution will be found

regarding the situation on site and the repair goal set.

There is a tendency, though, to fill up the masonry with

controlled pressure maintaining a minimal flow of grout

and reducing the pressure to be taken into consideration

to a horizontal component against the outer leaves.

Applying a constant pressure only monitored by the

manometer would endanger the employees as well as the

substance of the old building.

Method: Degree o Destruction Caused by the Drilling

Method

Rotary percussive drillings methods cause vibrations that

can create new cracks

and

loose areas in damaged mas

onry. These may connect existing cavities thereby increas

ing both the degree of injection and the amount of grout

necessary. Rotary drillings just cut through cavities but

usually

do not cause new discontinuities in the old mas

onry fabric.

EXPERIENCE OBT INED

FROM

PAST

REP IR

WORK

Drilling Patte rn

In two-thirds of all areas examined containing drill-holes

the density of the holes was less or equal to one per square

metre of the wall surface. Approximately one-fifth of the

areas had more than one drill-hole per square metre. The

upper limit for the common case was at about two and a

half. Only building members with increased structural

requirements had higher values.

Degree of Injection

In multi-leaf masonry the degree of injection reaches

values of 5 to 15 . Some brick walls and some of natural

stone which are obviously solidly bonded and dense have a

percentage

of

filled cavities distinctly less

than

5 . Much

higher degrees of injection are reached in the base and

foundation masonry. This makes it apparent that in these

regions additional, perhaps remote, cavitities have been

filled.

The Behaviour of Grout Within the Masonry

see Figures

1-6

The evaluation of repair documents, the computational

analysis of stress conditions in repaired masonry and the

knowledge obtained from interference into the substance

prove that there is an increase in load-bearing capacity

after injection. This is obviously caused by the reduction of

cavities and faulty areas in the old masonry. Loads can

then be transferred directly and peaks of strain can be

reduced. By the levelling

of

load transfer due to injection,

the load-bearing capacitity of the entire masonry

is

obvi

ously increased, although the former estimate whereby



REPAIR OF MASONRY

STRU TURES

227

Fig. 1 Cracks and cavities are filled with suspension dark colour) to

enable the transfer of forces. The scale

is

graduated in cm.

Fig. 2 Cracks and cavities are filled with suspension dark colour) to

enable the transfer of forces.

Fig. 3

In this case of dense old

mortar

white gypsum mortar) the flow of

the suspension dark colour) was confined to the drill-hole. The scale

is

graduated in cm

injecting increases the quality of the

mortar

itself cannot be

confirmed.

Taking the results of the preliminary surveys

and

the

parameters described above into account, masonry

structures can be injected successfully.

f

cavities

and

faulty

areas are sufficiently well interconnected, if the cross-

Fig.

4

Surface area: 40 x 40cm

2

of the hardened suspension which

had

only plain contact to the detached

mortar

above plane view).

Fig. 5 The cement grout has been taken

off

The contact between the

cement

and

the old

mortar

was only slight.

Fig.

6

Old

mortar

which contained too little binder was washed out in

the course of drilling and gave space for an a ccumulation o f grout which

was not desired. The scale

is

graduated in

cm

sections of the flow paths allow for easy passage, and if the

old

mortar

does

not

absorb too much mixing water, then

there will be an even distribution of suspension.

n contrast to the apprehension on the

part

of monu

ment preservation, it has been discovered

that

hardly any

new injection material intrudes into the old mortar. The



228

FRITZ WENZEL AND

HELMUT

MAUS

grout fills the cracks, voids

and

cavities, basically remains

in the damaged

and

faulty areas of the masonry, and does

not penetrate the old mortar in the sense of a mixture.

Only in the vicinity of injection holes may cement particles

be washed into the old mortar fabric when the water

content

is

high

and

sufficiently high pressure

is

applied.

With this exception, areas of contact between both

materials are limited to the surfaces of cracks, cavities and

drill-holes resulting in more

or

less abrupt and plane

marginal zones. There

is

a distinct separation between

cement, stone

and

old mortar.

In

addition, follow-up

surveys have not revealed any old masonry which might

have been turned to concrete .

Since the injection grout does not penetrate the old

mortar the authors believe that the assumption has to

be

reconsidered whereby old mortar can be damaged by the

two minerals ettringite and thaumasite in a form which

they acquire in a secondary phase. t

is

not disputed that,

in special circumstances, the original volume of the

gypsum-containing mortar

is

increased by the creation of

ettringite and

that

cement stone

is

destroyed by the

creation of thaumasite. The circumstances needed for these

processes certainly do not occur in every building contain

ing gypsum mortar. The most important condition for the

transformation process

is

a continuous supply of water,

which

is

usually not provided in a fairly well maintained

building with sealed outer joints. In addition, this process

obviously takes place at a very slow pace within the

masonry and the injection cement

is

too dense for laminar

flow. In

the case of mineralizing,

as

can happen when

shotcrete

is

applied to the inner side and the moisture

disposition within the masonry

is

disturbed considerably,

the resulting layer at the areas of contact between old

mortar

and

new grout

is

only a

few

millimetres thick as the

surveys have shown. The expansion itself

is

considerably

smaller.

To

obtain deformations and cracks

that

doubt

lessly result from ettringite

or

thaumasite, the old mortar

would have to be soaked with cement paste. This may

happen locally due to areas of extremely lean old mortar,

but

this

is

especially not the rule with buildings containing

gypsum mortar. Caution must prevail when damage

is

being connected with the creation of expansive minerals.

Often other causes exist and are of greater significance.

THE REINFORCEMENT OF OLD M SONRY

The Purpose o Stitching

In general, stitching as subsequent reinforcement happens

where tension

or

thrust occurs which the masonry cannot

withstand. Examples can be found in

ahmann

[5J and

Pieper [2]. Stitching

is

always connected with grout

injection to form the bond between steel and masonry as

well as to provide corrosion protection.

In

multi-leaf

masonry the reinforcement bars connect the two outer

leaves through the inner filling which was strengthened by

injection.

As

the outer leaves are usually only one stone

thick, special attention must be paid to the anchorage

of

the bars.

Reinforcement Bars

As

a rule, the bars which are employed are made

of

ribbed

reinforcement steel with a d iameter of 8 to 20 mm, mainly

12

to

16

mm, with anchorage by bond. Also, steel with

through rolled thread ribs has proved itself. With long

anchor bars a sleeve joint may be used; and with short

anchor lengths, an additional end anchorage with washer

and nut,

or

with a special end-piece,

is

customary. When

the danger of corrosion

is

regarded

as

extreme, rustproof

steel

is

sometimes used, for example,

or

strongly moistu

rized structural elements. Steel with a smooth non-profiled

surface should not be used as the grip

is

weak.

The Purpose

o

Prestressing

Old masonry

is

grouted

and

prestressed if strongly torn

walls and pillars must be joined to regain their com

pression strength and thrust strength and, in addition, to

withstand tensile strength; if the masonry itself, without

auxiliary constructions of steel

or

reinforced concrete,

must span openings unsupported;

or

if masonry buildings,

because of an irregular subsoil, must act as stiff structures

to force even settlements. When the causes for the cracks

are removed, e.g. by improvement of the subsoil

or

by

reinforcement

of

the foundation, a loose armouring can

be

sufficient for further securing.

As

a rule, prestressing

is

only

applied in the case of severe damage to the masonry. With

the help of prestressing the force flow may be corrected in

old masonry constructions; in exceptional cases it may

even be changed in direction.

Prestressed nchor Ties

The most frequently used stressing tendons are steel rods

with through rolled thread ribs

on

both sides, of

15

to

36

mm diameter, with a steel quality about 850/1050 to

1100/1350. Such steel rods allow shortening at the con

struction site with a separator and joining with a thread

sleeve so

that

they can

be

added to long pretensioning

anchors.

f

the design stress

is

only used between two

thirds

and

three-quarters, this leaves reserves in case the

anchor force should increase over the force of prestressing,

e.g.

because of changes of load

or

movements in the

subsoil.

In

addition, there

is

no stress corrosion cracking at

this point because of the decreased utilization factor.

Performing long drillings up to

30

metres and more in

masonry with a drift of less than

0.3 is

not unusual for



REPAIR O MASONRY STRUCTURES

229

specialized companies. The anchor heads are manufac

tured of reinforced concrete or steel Wenzel [7]).

Rustproof steel with low design stress is not as suitable

for prestressed anchor ties. A larger diameter

is

needed,

which causes a larger interference in the old substance.

Recently, rustproof steel with higher strength has become

available as

well

Repair Concept

To strengthen old masonry new structural systems are

often inserted into parts of the existing fabric. Both the old

masonry and the new systems must be compatible with

each other. All proposed measures must add up to a sound

concept. Influence from the subsoil, from earlier rearrange

ments

or

damage, as well as from the repair steps, must be

taken into account and integrated into the concept.

Appropriate structural measures need the results of com

prehensive preliminary surveys and of careful planning.

One main reason is that, in engineer-style strengthened old

buildings, different structures and materials meet which

have their distinct weak spots and various limits in load

transfer capacity.

Reinforcement Concept

f

new structural elements, introduced to damaged or weak

old masonry, gather and transfer tensile forces

or

create

pressure in the cross-section, this may not only reinforce

stability and balance but may also create differences in

rigidity

and

in the

flow

of forces within the structure.

or

this reason the reinforcement concept should be structur

ally sound and correspond with the circumstances of the

particular building, especially with non-homogenous

multi-leaf masonry. Discontinuity in the

flow

of

forces,

alterations in the existing structural systems and local

differences in rigidity could lead to shifts in load transfer

that inevitably become visible as cracks.

In the case of local crack reinforcement with crosswise

inserted bars, only additional measures preventing re

newed movement in this area

of

the masonry can help

avoid any new considerable damage.

Near

the reinforce

ment the masonry fabric proves to

be

coherent. Outside

this area only the tensile strength

of

the injected masonry is

effective and fine new cracks may become visible. Preten

sioning the masonry can help to minimize

or

prevent this

predictable crack pattern,

but

in many cases it is more

appropriate to leave the old cracks

as

they are, to let them

work as natural movement joints, to seal them from time

to time

but

not to reinforce them.

Multi-leaf masonry often consists of a thicker outer leaf

and a thinner inner one with a core filling between them).

f the leaves are to be connected with bars and one is

requested to drill from outside and not to penetrate the

inner leaf entirely, the bonding between the connecting

bars and the inner leaf may prove to be insufficient. Should

the connection of the two leaves be structurally necessary

and unavoidable, the recommended method

is

to drill and

insert reinforcement bars crosswise from inside and out

side at the same height. Thus there

is

a greater possibility

of achieving a sufficient bond between the outer

and

inner

leaf and the injected core fillings as well

If their spread

is

disturbed, the forces

of

pretensioning

rods may cause local peaks in tension which may

overstrain the masonry. This should therefore be taken

into account when planning the reinforcement. If neces

sary, additional steps should be taken such as the infilling

of openings which disturb the flow of forces in the wall.

Determining the Necessary Reinforcement

Those tensile forces determined by engineering calculation

must be absorbed by the reinforcement and safely transfer

red and anchored. Information on the dimensioning of

tensioning rods and stitching bars, as well as on the

permitted masonry pressures - especially those parallel to

the course joints

and

on partial areas - are given in

research papers presented by Haller [6] and Dahmann [5]

and summarized by Wenzel

[7].

f

severely cracked masonry walls are to be strengthened

by prestressing, the lateral masonry pressure of

0 1 MN/m2 recommended for structural purposes in the

publications mentioned above has obviously proved effi-

cient in buildings which have been examined, and has been

widely accepted in practice.

Horizontal pretensioning rods in the upper third of

walls may cause lateral cracks in course joints. If the

masonry fabric had been

of

high quality,

or

if additional

vertical stitching bars had been inserted,

or

if pretension

ing measures had been taken as described by Haller, then

the areas were free of damage. Due to the increasing

vertical load in the middle and bottom parts

of

prestressed

walls, lateral cracks did not appear.

Insertion

of

Reinforcement see Figures 7-10)

Building surveys have shown that the technical procedures

which were used well into the 1970s have often been

responsible for corrosion of reinforced bars, for lack of

sufficient covering of hardened cement paste and for

poor

bonding between anchors and the masonry.

To

prevent this in future repair work, drill-holes for

reinforcement in masonry should allow for at least

2

cm

cement covering all round the steel bars. Nearly the same

is recommended in the case of sleeve connections. Struc

tural steel has proved to be suitable for these repair

purposes for decades. The bars and rods must be cen

tralized by using spacers to allow for sufficient protective



230

FRITZ WENZEL AND HELMUT MAUS

Fig. 7 This 60 years old prestressed bar diameter

32

mm) was lacking

protection against corrosion because of too much sand in the sur

rounding grout.

Fig.

8

The reinforcement bar covered by

1 5

em grout and centred in

the drill-hole provided good bond and sufficient corrosion protection as

we ll Today spacers are usual and indispensable to fit reinforcement in the

middle of the hole.

Th

e scale

is

graduated in cm .

Fig.

9

Efficient bond between steel diameter 12 mm) and cement as

well

as cement and masonry.

mortar

covering. The suspensions used to grout stitching

bars and pretensioning rods must contain cement binders

able to create a lasting alkaline environment of at least

pH 10

. Otherwise there is danger

of

corrosion such as

with suspensions having a high water/cement ratio.

f a different suspension is to be used for masonry that,

for instance, has less pressure strength than a cement

suspension, the pretensioning rod can be installed in a tube

and can be grouted with cement paste separately after

pretensioning.

Caving can reduce substantially the cross-section of the

drill-hole during insertion of reinforcement.

In

this case it

is

necessary to drill

tap

holes along the anchor channel and

grout the area before repeating the drilling for the rein

forcement rod.

To ensure

an

adequate bonding length of the stitching

bars, the drill-hole should be arranged in the stone rather

than in the joint. After removing the injection socket,

which should

not

be kept in place with gypsum, the drill

holes should be carefully sealed with a not-too-lean

mortar.

GENER L

PRINCIPLES

ND EXPERIENCE

Research results show that injecting and reinforcing old

masonry enables us to create technically reliable, economi

cally sound solutions which are acceptable to monument

preservation if the experience described above is followed.

Not

the perfectly and entirely grouted wall,

but

the

carefully considered repair of masonry may be the goal if

the valuable original substance

is

to be preserved.

The civil engineer has to approach an old structure from

his specific point of view. He is interested in conditions and

details which are of no concern and are not visible to

others. Surveys made available to the civil engineer, for

example by surveyors, can be helpful. They may provide

general information and measurements,

but

they cannot

relieve the engineer from his own thorough examination

and documentation.

The deformation-true survey

is an

important tool

both

for building research purposes and for documentation.

For

the experienced engineer, though, sometimes a

few

measurements taken by himself are more suitable to show

him what has happened to the building substance and how

aid can be applied. Too many measurements often ob

struct the view

of

the substantial information.

t may not be customary for the civil engineer to study

the history of the building he is supposed to help; but it is

very importan t for him to do so. Knowledge of the original

structure and of damage and alternations in the past will

enable him to assess the present condition more ac

curately.

By

studying the success and failure of past

attempts at rehabilitation, he can gain knowledge that

may contribute to the success of his own solution.

As

civil engineers

we

have to discover how the ageing,

ailing building helped itself and what hidden systems and

structures it has in reserve. We must also try to bring the

statical calculations into line with the damage record.

Otherwise it

will

not be possible for

us

to give reliable

information about the dangerous condition of the build-



REPAIR OF MASONRY STRUCTURES

231

300

200

4

100

157 1977 1518 1979 2980 1981 1982 1983

1981.;

1985 1985 2.987 1988 1989 1990 1991 ahr

1 2 Natursteinwand ganzflachig verpreBt

Langzeitmessungen

3 4 Natursteinwand bzw.

Ziegelwand

beide

nur

m

Bereich

der Spannglieder verpreBt

SDannankerver

laufe

telflPeraturberelnlgtJ

Fig.

10.

Anchor forces remain relatively constant.

ing. Usually it is advisable not

to

intervene in the existing

flow of forces, even if this pattern

is

not the original one,

but has been developed later on. The existing conditions

may be improved by sealing cracks, by decreasing the

eccentricity of forces, and by installing anchors or bracing

elements. But substantial shifts in load transfer should

remain exceptions

and

should be avoided whenever possi

ble. Why give

up

consolidation of the building fabric unde r

the present flow of forces and risk the revival of this

process elsewhere in the building with new deformations

and

cracks?

When dealing with a historically important building it

is

not sufficient

to

content oneself with the results of a finite

element overall analysis. We must try to translate the

results into realistic proposals for the repair. That may be

difficult

but

it is necessary as a presupposition for the

computational work to be considered

at

all. As we know,

the final aim of an engineer s work in preservation

is

the

building, not the calculation.

When repairing old buildings, the engineer has to tell

the architect what the old substance

is

still able to bear,

where which use is appropriate

and

where not. The more

the engineer understands design work, the better will

become the coopera tion with the architect. Influencing the

design concept helps

to

solve most of the structural

problems of old buildings more than statical contortions.

Design and structure of a building - structure being

regarded here in the sense of a conceptional order and

not

only as

an

assembly of bearing elements - are identical in

many

ways. To preserve the concept of design and

structure which the building or its alterations follow can

be of

no

slighter interest

than

the preservation of the

material used to realize it. Actually the civil engineer

is

the

person who should know best the answers

to

questions

about

structures

and

who should contribute this know

ledge

to

the discussion over the future of building

monuments.

A civil engineer called upon to help repair historically

important

buildings

is

not only expected

to

deliver techni

cally sound solutions. He

is

also expected to suggest

methods and procedures

that

are compatible with the

monumental value of the old structure. Intervention

and

destruction of the building substance as well as the

addition of technical aid must be kept to the necessary

minimum. Restraint must be exercised when using modern

technology. Too much aid can destroy what

is

meant to be

protected.

The wish for reversibility of engineering strengthening

measures

is

often misunderstood to mean that the only

acceptable remedies are those which can be removed and

replaced by better ones some day. This

cannot

be the

point. Should it be necessary to give technical assistance to

a monument then - above

all-

it is reasonable

to

look for

the most appropriate solution, for the minimum inter

vention and addition. This necessary minimum, however,

is

to

be inserted as a durable addition of

our

time.

Reversibility may be helpful, but it means rather the

possibility of repairing the inserted elements as well as

replacing them in case off ailure

or

deterioration, however,

it does

not

mean that they should be exchangeable every

time a new technical solution is available.

The question has often been raised whether the practical



232

FRITZ WENZEL

AND HELMUT

MAUS

experience and the results of scientific work

on

securing

old buildings could be embodied by standards. The answer

can only be: recommendations yes

but

standards no.

Every old building and each defect

is

a special case

of

its

own. The techniques applied have to be specially chosen to

meet the requirements of the particular building.

If

there

were standards the engineer would easily be tempted

primarily to meet these standards

but

to neglect the special

situation

of

the specific project.

During the repair of old buildings and masonry

structures the frequent presence of an experienced engineer

at the construction site is necessary. As the investigations

of our research group in Karlsruhe show, carelessness in

execution is responsible for most of the defects in repair

work - a fact we also know from new buildings erected in

the past decades.

To do only what is absolutely necessary in repair work

is not so

bad

since many buildings have experienced losses

in monumental value due to exaggerated application of

technical means. A thoughtless and presumptuous state

ment still heard today is

that

the following generations are

to be relieved of the burden of the historical substance

once and for all. This goal cannot be reached by technical

repair measures; fortunately science and technology are

not able to achieve it. I say, fortunately, because the care of

every generation over its historical heritage

is an import-

ant

link to its history and a basic stimulus to monument

conservation in the broadest sense.

REFEREN ES

1. Pieper, K. and Hempel, R. Schiiden und SicherungsmaBnahmen an

Bauten mit Gipsmortel , in

Erhalten historisch bedeutsamer Bauwerke.

Jahrbuch des Sonderforschungsbereiches

315, 1987. Berlin, 1988, pp.

73 88.

2.

Pieper, K.,

Sicherung historischer Bauten

Berlin, 1983.

3. Maus, H. and Wenzel, F., Zementhaltiges Injektionsgut und Beweh

rungsstiihle in altem Mauerwerk. Zustand, Wirkung, Dauerhaftigkeit ,

in

Erhalten historisch bedeutsamer Bauwerke. J ahrbuch

es

Sonderfors-

chungsbereichs 315, 1990.

Berlin,

1992.

4. Ullrich, M. and Wenzel, F., IngenieurmiiBige Bestandsuntersuchun

gen an sanierten Mauerwerksbauten , in

Erhalten historisch

bedeutsamer Bauwerke. J ahrbuch des Sonderforschungsbereichs 315

1990. Berlin, 1992.

5.

Dahmann, W. Untersuchungen zum Verbessern von mehrschaligem

Mauerwerk durch Vernadeln und Injizieren , Dissertation Universitiit

Karlsruhe, 1983;

us Forschung und Lehre

Institut fiir Trag

konstruktionen, Universitiit Karlsruhe, Heft 19, Karlsruhe, 1985.

6. Haller, J., Untersuchungen zum Vorspannen von Mauerwerk histor

ischer Bauten , Disserta tion Universitiit Karlsruhe, 1981;

Aus For-

schung und Lehre

Institut fiir Tragkonstruktionen, Universitiit

Karlsruhe, Heft 9, Karlsruhe, 1982.

7.

Wenzel, F., Verpressen, Vernadeln und Vorspannen von Mauerwerk

historischer Bauten. Stand der Forschung, Regeln fiir die Praxis , in

Erhalten historisch bedeutsamer Bauwerke. Jahrbuch des Sonderfors-

chungsbereiches

315, 1987. Berlin, 1988, pp. 53 72.



WEATHERING OF ROCK CORROSION OF STONE AND

RUSTING OF IRON

J.

E.

HARRIS

University

of

Manchester Institute of Science and Technology. Corrosion and Protection Centre.

PO Box

88.

Manchester M60 lQD. England


ABSTRACT. The central purpose of this paper is to present a survey of the extrinsic and intrinsic factors which

influence the durability of masonry.

In

approaching this subject other themes are developed; in particular a study is

made of the damage due to the volume changes which accompany all biological. physical and chemical changes. Light

can be thrown on the corrosion o f stone from a knowledge of the weathering of rocks thro ughout geological time, and

this aspect is explored in the opening section of the paper. The final

part

of the paper consists of a study of the stresses

and cracking of

stone

which can result from the expansive rusting of iron or steel reinforcements. Although mechanical

damage dominates the discussion some comments are made

on

the staining and dissolution

o

stone and examples are

illustrated.

SOMMARIO

Scopo del presente articolo e present are

una

rassegna dei fattori int rinsed ed estr insed che influenzano

la durabilita della muratura. All interno di tale rassegna si sviluppano altri temi, ed in particolare uno studio sui dann i

provocati dai cambiamenti di volume che accompagnano

Ie

modificazioni biologiche, fisiche, chimiche. Nella prima

parte dell articolo ci si sofferma sulle modificazioni degli ammassi rocciosi durant e Ie ere geologiche, e sui nesso tra i

dati desunti da tale analisi e il problem a della corroasione della pietra. La parte finale dell articolo e invece dedicata

ad uno studio degli stati tensionali e fessurativi risultanti dall espansione lega ta alla corrosione di elementi metallici di

rinforzo posti all interno della mu ratur a. Benche il danno strut turale sia l argomento centrale della discussione,

vengono anche presi in considerazione i probl emi conness alla dissoluzione e alla com pars a di macchie, e si illustrano

alcuni.

KEY WORDS: Stone, Metals, Weathering, Corrosion, Rusting, Mechanics of masonry.

INTRODUCTION

All reactions in the solid state whether they be biological,

physical, chemical or nuclear e.g. transmutations and

fissioning) involve a rearrangement of atoms in space, and

hence a change in specific volume.

Put

more simply, most

reactions are accompanied by either an expansion or a

contraction in volume.

Where these volume changes are constrained, stresses

will be developed and the question arises whether these

stresses are of sufficient magnitude to deform or fracture

either the object undergoing change itself or any sur

rounding medium. Expressed another way, if

iF is

the

energy released by the expansive reaction and if the

volume increase is

i V,

then deformation

or

fracture

is

possible providing

iF (J i V,

where

J is

the deformation

or fracture stress of the surrounding medium. Considering

chemical reactions first, and substituting typical values for

the above parameters, then it can be shown that for both

metals and stone the LHS of the inequality exceeds the

RHS by up to three orders of magnitude [1]. This

indicates that deformation or fracture is readily possible.

Although the energies released by physical reactions are

at least an order of magnitude lower, they are still of

sufficient magnitude to cause damage as householders

with burst pipes due to frost damage can verify). I t

is

not

convenient to deal with biological processes in this manner

M

eccanica 27:

233-250, 1992

©


but it has been demonstrated that very large stresses can

be generated by growing material, and everyone is familiar

with rocks being disturbed

or

even fractured by the

growing roots of trees. Nuclear reactions of course release

huge amounts of energy, sufficient in many cases to

vaporize all known materials; however, they are only of

very specialized interest in the present context so are only

discussed briefly.

We will begin by an account of the weathering of

outcrops

of

rock, that

is

to say the natural geological

processes which have been occurring over aeons. When the

rock is quarried and cut into suitable shapes and used for

building

or

sculpture,

we

will refer to it as stone and call

its deterioration as a result of reacting with the at

mosphere, corrosion .

Our

purpose here

is

to draw a

further (albeit contrived) distinction between geological

processes and the decay of

our buildings and statues. In

the section dealing with metals

we

will refer to their

deterio ration as rusting as in most cases the metal under

consideration is iron or steel.

The first section deals with the weathering of rock, the

second with the corrosion of stone, the third with how the

rusting of iron and steel inserts disrupt stonework.

Examples of damage to, mostly famous, buildings will be

presented throughout the text. Although surface erosion

and staining will be illustrated, attention will be focused on

mechanical damage due to volume-changing reactions.



234 1.

E.

HARRIS

WEATHERING OF

ROCK

Throughout time rock has been praised for its per

manence;

our

church

is

founded

upon

a rock

and

we sing

of the Rock of Ages . Emily Bronte wrote of

The

steadfast

rock of immortality . This is in spite of the fact that every

outcrop of rock in the countryside shows evidence of

weathering,

and

all stone buildings in due time start to

decay. (In this paper we will tend to assume that the terms

weathering , decay and corrosion are synonyms

and

hence interchangeable.)

Conventionally the corrosion of rock is divided into two

categories: disintegration (mechanical weathering) and

decomposition (chemical weathering). This is

not

in fact a

particularly useful dichotomy-chemical weathering, for

example, always leads to volume changes and this often

results in mechanical damage

and

disintegration. We will

not in fact use such a classification

but

discuss weathering

in relation to whether its primary cause is biological,

physical, chemical

or

nuclear (the last named is only

included for completeness: it has no relevance to the

durability of ordinary building stone).

Biological Weathering

Plants growing in fissures in rocks exert a pressure such

that the fissure sometimes expands into a crack which

propagates

and

this can lead to complete failure. Measure

ments of the pressures exerted by growing plants indicate

that these can exceed those generated by the freezing of

water. The effects can be dramatic;

Kernar

[2J has de

scribed how

an

alpine larch

had

split a block of schist

and

raised the upper part, weighing

about

1.4 tonnes, by

0.3

metres. Equally astonishing is the report [3J by the

horticulturalist, Anthony Huxley, of the lifting of a large

concrete paving slab by a horse mushroom.

In

this case the

expansion was due

to

hydraulic pressure the preformed

cells swelling rapidly as they took up water.

Huxley [3J has also drawn attention

to

the shoots from

daffodil bulbs forcing their way throu gh 8 cm o f tarmac

and suckering shoots of Rosa hispica penetrating 20 cm of

rubble, tarmac and gravel.

All

those with metalled drive

ways can quote similar examples. Huge forces can be

generated during the germination of

seeds pressures

up

to

2000 atmospheres have been recorded [4J.

Weathering by Physical Processes

Primitive farmers, when they wanted to clear large rocks

from their fields, used to light large fires above them and

when the rocks were very

hot

they were quenched with

water. Thermal gradient stresses

and

the variations in

degrees of contraction by differing constituents of the

rocks caused cracks to appear and such weakening as

sisted subsequent fragmentation

and

removal.

For

many

years it was thought that the large diurnal

temperature changes (up to 70 DC , which occur in desert

regions, could cause similar expansion cracking in exposed

rocks,

and

was in fact a

common

cause of weathering.

However, laboratory experiments in which granite

samples were subjected to such temperature changes

yielded

no

evidence for damage, even when cycles corre

sponding to 250 years exposure were imposed. This does

not, of course, rule out the possibility that thermal cycling

can be damaging in combination with some other weather

ing process.

Where diurnal temperature changes are important is

where they criss-cross the freezing point of water, for

example along the snow line on mountains with exposed

rocks. In these circumstances, water is absorbed into the

pores of the rocks during the warmth of the day

and then

frozen

at

night. The associated 9 volume increase is

thought

to

crack the rock - the cracks being filled with

water the following day and the process repeated until

complete fragmentation occurs. The intimae shattering

produced in this way is known as nivation

The actual processes involved may be more complicated

than

those

just

described - there are some puzzling

features.

t

has, for example, been found

that

liquids which

do not expand on freezing (e.g. nitrobenzene) can neverthe

less cause damage

to

rocks if they are injected into stones

and

frozen [5].

t

is now thoug ht that, as well as expansion

forces of ice itself, the hydraulic pressure in the rem ant

pore water forced ahead of the advanc ing ice/water front,

plays a role in the damaging processes. The conversion of

pore water to ordered water may also be important.

Finally, pore water supercooled well below the normal

freezing tempera ture may also exert damaging expansion

forces.

Another physical process of importan ce in weathering is

the stresses generated by the crystallization of salts nor

mally in solution in pore water. These salts can arise from

the stone itself or have been deposited as aerosols, as

happens with sea water close to the coast.

As was the case with the mechanism of generating frost

damage, the processes involved in producing crystalliza

tion forces are not straightforward. Damage can occur

even where there is

no

net increase in volume. An early

illustration of this phenomenon is due to the distinguished

metallurgist, C. H. Desch, who in 1914 discovered that

Plaster of Paris may break a test tube as it sets even

though there is

an

accompanying

reduction

in volume of

about

7 .

t

appears that in nature, where a new phase

forms in a confined space, irrespective of the sign of the

volume change, stresses will be generated, which can be

damaging.

Chemical Weathering

This is

the fragmentation and breakdown of rocks due to

chemical reaction between its constituents and ground



WEATHERING

CORROSION

AND RUSTING

235

water or the atmosphere [6]. The most

common

chemical

reactions are oxidation, hydration and carbonation. The

volume changes accompanying these reactions accelerate

disintegration.

As far as igneous rocks are concerned, if they are listed

according to their increasing resistance to chemical weath

ering then the sequence follows the order of crystallization

from

magma

[7].

With

olivine the most susceptible

to

corrosion, and quartz the least, the order is:

Olivine --> calcic plagioclase --> pyroxene --> intermediate

plagioclase

and

hornblende --> sodic

plagioclase --> biotite --> orthoclase --> quartz

The reason for this is that minerals which crystallize at

high temperatures are the furthest removed from equilib

rium conditions when they are exposed

to

the atmosphere

at ambient temperatures. As a general rule, as the igneous

rocks have never before been exposed

to

the atmosphere

they should react with it faster

than

the sedimentary rocks

(this breaks down in the case of the carbonates).

It is worth mentioning in passing the feldspars which are

very susceptible

to

weathering;

and

their decomposition

is

associated with a volume increase of sufficient magnitude

to cause granular disintegration.

Unlike the silicates, the common carbonate minerals

dissolve readily, depending on the carbon dioxide content

of the water-the formation of limestone caverns and

gorges

is

evidence of this. The

carbon

dioxide reacts with

the water to form carbonic acid which, in turn, converts

calcite

to

the more-soluble calcium bicarbonate. Specta

cular examples of weathering due

to both

physical

and

chemical processes are shown in Figures I and 2.

The weathering o f iron minerals

is

of even more funda

mental importance -life on earth could hardly have

evolved without it. The weathering products of iron

minerals dominate the colour of

our environment-earth

clay, bricks, even farmyard manure and

our

own blood

owe their redness (or brownness) to the presence of iron

atoms which started their lives as constituents of igneous

rock.

One

reason for the importance of iron

is that

it

is

ubiquitous; it constitutes no less than 5 of the earth's

crust. Rocks which contain more than about 50 of iron

minerals (and hence are heavy

and

dark) are known as

malfic. Such igneous iron minerals are: the pyroxenes,

amphiboles, olivine, biotite, magnetite

and

ilmenite.

The reaction of such minerals with air

and

moisture

cause the oxidation of ferrous ions

to

the ferric state

insoluble ferric hydroxide (Fe(OHh) is formed which

converts to a metacolloid, from which crystallizes goethite

(FeOOH), the chief ingredient oflimonite. (In former times

limonite was thoug ht to be a distinct mineral

but

today the

term is used to describe any weathering aggregate which

contains a high fraction of ferric oxide compounds; in

addition

to

goethite, limonite

may

contain either amor

phous ferric hydroxide or hematite.) Further dehydration

changes goethite

to

hematite. Similar processes take place

Fig. 1. Elephantine shape formed by weathering of limestone

(KCTSjSeattle,

he

Miracle Planet .

Fig. 2. Bryce Canyon: rock pinnacles of spectacular coloration produced

by the weathering of Wasatch beds (Eocene) in south-central Utah,

U S.A.



236

J E HARRIS

when metallic iron inserts in stone are corroded some

practical consequences of this are illustrated in a later

section of this paper.

Nuclear Weathering

Damage due

to

atomic irradiation is not relevant

to

the

subject of durability of normal building stone and a

discussion of it is only included here for completeness. The

process

is

of importance in the rather esoteric activity of

constructing stores for highly active nuclear waste. Where

these are located deep underground they are surrounded

by rocks which could lose their crystallinity due to

irradiation. The result is the formation of an amorphous

rocky material which will have a greater chemical activity

and

a greater solubility in ground water. Clearly

both

these properties are undesirable because they can ac

celerate the release of radioactivity to the environment. A

similar process occurs in natural minerals which contain

percent quantities of uran ium or thorium. The destruction

of crystallinity in these cases

is

known as metamictization.

The Extent of Natural Weathering

Those responsible for the preservation of old buildings are

concerned about weathering processes which may only

penetrate a few millimetres below the original stone

surface. Geology

is

on a grander scale.

and

in the huge

timescales involved, weathering has taken place

to

con

siderable depths: to 100 m or more in the case of shales in

Brazil and basalts in India, and to 60 m for the limestone of

Georgia,

U.S.A. Much of the earth s surface

is

covered by

the products of weathering unconsol idated rock debris

and the soil which has made the evolution of human life

possible. This covering

is

collectively know as the mantle.

Weathering of rocks has also resulted in the concen

tration of useful minerals into exploitable deposits. Essen

tially there are two processes: either the soluble constitu

ents are subsequently precipitated in concentrated form, or

the insoluble component left behind becomes sufficiently

pure to be designated an ore. The former process has given

rise, for example, to valuable deposits of silver, copper and

uranium. The latter process leads

to

the formation of

lateritic soils which can be a rich source of iron; deposits o f

nickel

and aluminium form simi lar residues from selective

dissolution processes.

Originally o f course all rocks were igneous, that is to say

they were formed from cooling magma. It

is

weathering

that

produced the material for the sedimentary rocks,

some of which transformed

to

metamorphic rock. The

weathering products of sedimentary, metamorphic and

igneous rock combine to form further sedimentary rock on

the floors of the oceans. When a

proportion

of this is

subducted it provides the constituents for new igenous

rock.

The whole comprises a majestic cycle of rock decay and

rebirth.

It

is apparently timeless,

but

it

is

not

timeless.

Behind it all lurks the Second Law of Thermodynamics;

the inexorable increase of entropy leading in due time to

Heat Death. Edington likened entropy to beauty and

melody because all three are connected with arrangements

and organization; Auden considered entropy to be

another

word for despair.

ORROSION OF STONE

The most

common

stones used for building are limestone,

marble, sandstone, basalt and granite. In the relatively

warm and damp British climate the first two are the most

vulnerable due

to

the solubility

and

chemical instability of

the mineral calcite (though with some sandstones their

silica particles are cemented together by calcite so this

stone too can be subject to similar decay processes).

Biological Corrosion

Biological processes can accelerate corrosion of stone

buildings

and

the role of lichens, algae

and

bacteria in such

decay is currently being studied at Britain s Building

Research Establishment and elsewhere. There have been

reports from Cologne Cathedral of bacteria in the stone

converting pollutants into nitric acid which accelerates

decay. Skoulikidis [8] has indicated the possibility of

sulphur-oxidizing bacteria accelerating the corrosion of

the Pentelic marble on the buildings on the Acropolis.

In a recent edition of Endeavour, Marco del

Monte

[9]

has reported an interesting example of biological attack

which is actually inhibited by sulphur dioxide in the

atmosphere. He has studied the pink-brown patina which

formerly covered the surface of the famous Trajan column

in Rome, but which

is

now disappearing under the action

of Rome s polluted atmosphere. The oxalate layer which

constitutes the patina (known in Italy as sCialbatura

is

the

result of a nat ural chemical process due to the colonization

of the surface of the column by epilithic and endolithic

crustose lichens. The lichen also caused holes to appear in

the surface of the stone.

Fortunately a number of plaster casts of the column

have been taken, including the excellent specimen in the

Victoria and Albert Museum in

London

[10]. The taking

of casts allows the degree of subsequent corrosion to be

assessed. In fact the lichen has now retreated on the

column, but a similar process can be identified on other

monuments

and

natural carbonate outcrops in clean rural

areas. Figure 3 is a photograph of lichen growing on a

churchyard statue.

Desiccated lichen if wetted will increase its weight by up



WEATHERING,

CORROSION AND

RUSTING

237

Fig. 3. Lichen growing on marble angel over a child s grave in a country

churchyard.

In

due time the lichen may damage the

stone-it

would not

have grown had the level of pollution been higher.

to

50

with

an

associated expansion in volume, but there

have been no reports of this swelling damaging stonework.

There are often-quoted, rather romantic, examples of

jungle plants invading and destroying ancient buildings in

the forests of Central America

and

Cambodia. Closer to

home, trees tend to damage stone buildings by robbing

their foundations of moisture and thereby causing sub

sidence. Complex chemical reactions can occur in some

cases between the minerals in stone and such climbers as

Boston Ivy and Virginia Creeper. Damage can also result

from the moisture retained almost permanently

at

stone/plant interfaces.

Bird droppings contain phosphoric and nitric acid

which react with carbonates to form calcium phosphates

and nitrates, and these processes can be destructive.

Corrosion

by

Physical Processes

Frost damage to stone is not always easy to distinguish

from

that

due to other causes, for example salt precipita

tion. t can be seen to occur extensively

on

masonry close

to the water line on rivers, etc. Large sections of frost

damaged stones sometimes split away in a characteristic

fashion.

The expansive force of freezing water has been used to

advantage in the past. In her beautifully written book

A

Land Jaquetta

Hawkes [11] describes how quarrymen

at

Stonesfield, near Oxford, used the forces of freezing water

to split the local limestone blocks during the manufacture

of Cotswold roofing tiles. The massive stone blocks were

quarried during the summer months and subsequently

exposed to the low temperatures of winter which froze the

quarry water in the stone which in turn split the blocks

into thin sheets suitable for their purpose. During par

ticularly mild winters, when no tiles could be produced, the

blocks had to be buried so that they did not lose by

evaporation during the subsequent summer months the

precious quarry water. The following winter the blocks

were dug

up

and re-exposed to the cold weather.

Quarry

water slowly diffuses to the surface of freshly

quarried building blocks; it then evaporates and deposits

its dissolved salts. This process can be damaging; crystalli

zation forces

can

cause surface cracking.

On

the other

hand, the process can harden the stone and improve its

durability. t is generally good practice to age stones after

quarrying,

i.e.

delay their use for building purposes for

several years. Christopher Wren was most careful to

ensure that the Portland stones used for the construction

of St Paul s Cathedral were adequately aged. t is par

ticularly important that stone intended for use for statues

is

well

aged, otherwise moisture migrating from the deeper

regions can deposit its solute and hence damage external

protruberances, such as noses.

Changes in ambient moisture levels can cause the

expansion and contraction

of

layers of clay incorporated

within the structure of some limestones, and this can be

very damaging. Such distress has occurred extensively in

the stonework of Leon Cathedral, one of Spain s most

important Gothic monuments [12].

Perhaps

not

surprisingly the most durable building

materials are those which have already suffered from a

high degree of natural weathering. Extreme examples are

the lateritic soils produced in warm humid climates where

the soluble minerals have been leached away and effective

ly all that is left is, for example, the insoluble oxide of iron,

Fe

z

3

• All that

is necessary is to fashion such clays into

suitable shapes and harden them in the sun (the word

laterite

is

derived from the Latin word for brick ).

Lateritic building material was used extensively in the

construction

of

the temples

at

Angkor Wat in Kampuchea,

and this explains the remarkable durability of parts of

these notable buildings.

Chemical Corrosion

In

our

cities, corrosion of limestone and marble has been

accelerated by increased levels of carbon dioxide, sulphur

dioxide and the oxides of nitrogen (though the influence of

the latter

is

far from being quantified

or

understood).



238

J E. HARRIS

Particularly damaging

is

sulphur dioxide, a produc t of the

burning of many fossilized fuels.

t

reacts with the stone

producing gypsum (calcium sulphate) which is thirty times

more soluble in water than calcite under natural con

ditions. The volume change on forming gypsum causes

cracking and accelerates the disintegration of the surface

of the limestone.

Skoulikidis [8J has reported gypsum layers as deep as

1.5 cm on the surfaces of marble blocks and statues on the

buildings of the Acropolis.

In

such circumstances removal

of the gypsum would completely destroy the remaining

decoration

on

stone

or

features

on

a statue and Skoulikidis

recommends that steps should be taken to preserve or

transform the gypsum layer

n

situ see Figure

4.

The difficulty in determining the influence of pollution

in urban environments on the corrosion rate of limestone

is

that

there is a large variation in the rates of weathering

of naturally occurring outcrops of limestone rocks.

ne

study, reported in the 1988 Watt Committee Report on air

pollution and acid rain [5J, indicates a variation in

natural rates between the very wide limits of 3 to

88

lm

per year.

In

1987 Jaynes and Cooke [13J compared the

corrosion rates of limestone in central London with those

of a variety of suburban, coastal and rural sites. A rather

surprising result from their work was that the corrosion

Fig. 4. The Caryatid from the Erechtheum in the British Museum.

Remarkable detail is preserved in spite of over two millennia exposure to

the atmosphere. Her nose, however, has disappeared.

rate in central London was only

about 25

greater

than

that of a rural area in spite of the fact that the London

atmosphere contained three or four times as much sulphur

dioxide.

In wet areas limestone decomposes quickly, whereas in

dry regions corrosion is greatly retarded. This difference

has been famously demonstrated by the transfer in

1881

from Egypt to New York of the obelisk

of

Thutmose

III

Having survived in Egypt for many centuries without

damage, it soon began to decay in its new surroundings

and special protection methods became necessary. Phys

ical as well as chemical processes contributed to its decay.

t

appears that the obelisk had lain on its side on the

ground for

about

500 years in Egypt before its transfer to

the New World, and during this time salts must have been

absorbed from the earth into the body of the stone.

In

the

damp New York atmosphere the solution and migration

of these salts, and their subsequent precipitation

at

the

surface of the stone, must have accelerated the corrosion

process.

The chemical weathering of iron minerals, particularly

when they are present as minor constituents in building

stone, can be an embarrassment. Quite frequently such

stones, after exposure to moisture

and

air, are discoloured

by rust marks

and

there exists the possibility

of

damage

due to expansion forces.

In 1932 Kieslinger reported the oxidation of small

grains of siderite (ferrous carbonate) in the surface of the

Pentelic marble blocks of the Parthenon. Such weathering

is not always displeasing-some dolomites when freshly

quarried are an uninteresting grey colour

but

weather to

an

attractive yellow

or

buff colour due to the trans

formatir

•

of ferrous carbonate to ferrous hydroxide. The

yellowing of some types

of

white marble by a similar

process can be regarded favourably - as a sign of mellow

ing with age. However, changes in colour

of the Taj Mahal

are taken as an indication of attack by acid rain.

Protection against orrosion

Methods used to arrest the decay of limestone and marble

statues can be controversial.

In

the 1960s it became

obvious

that

work was necessary to preserve the West

front of Wells Cathedral with its 300 medieval figures. A

small number of statues were treated with alkoxysilanes,

and this was much criticized. Others were given the lime

treatment consisting of up to 50 coatings of calcium

hydroxide. By a mechanism which is not understood, this

appears to consolidate the limestone,

but

there is little

information on the durability of the stone following such

treatments.

Another technique which has been employed for a

century

or

more is to treat the stone surface with barium

hydroxide solution. Bar ium replaces the calcium in calcite

or gypsum forming more durable compounds. The main



WEATHERING, CORROSION ND RUSTING

239

disadvantage of the process

is

the very slow rate of

chemical reaction so that in practice very little of the

barium compounds can be formed within a reasonable

timescale.

Yet another method of preservation

is

under active

development at, for example, the University of Louisville

[14]. This involves the injection of organic monomers or

prepolymers into the surface of a stone followed by a

polymerizing process accelerated by a curing agent. While

the synthetic polymers so formed can consolidate the

stone, there are worries over whether or not they are stable

under, for example, the action of sunlight. t is also

important to

make sure

that

the polymer itself does

not

absorb active gases. In all such treatments care must be

taken to ensure that the surfaces or the stone are not

hermetically sealed - if they are, moisture in the interior of

the stone

is

trapped

and

this can lead

to

problems.

Corrosion-staining

of

arble Statues

Returning

to

the subject of atmospheric attack on marble

or limestone, it can give rise to calcite itself going into

solution

and

being redeposited on a different

part

of the

surface of the stone (this is the self-same process which

gives rise to stalagmites

and

stalactites). In

urban

environ

ments the calcite reacts with sulphur dioxide in the

atmosphere to form the much more soluble calcium

sulphate (gypsum). The dissolution of gypsum on exposed

surfaces is a relatively rapid process

but

it does produce

shiny white surfaces which, in the early stages, are quite

acceptable until it is realized that all the detail

is

being

washed away. In more sheltered regions the combination

of calcite, gypsum

and

street dirt, produces a black

appearance. While this is often ugly, it can in certain

instances add drama to an otherwise uninteresting piece of

sculpture. Examples are shown in Figures 5 to 8.

a)

Fig. 6. The pod ium frieze on the Albert Memorial showing exposed areas

washed clean while protected regions are black due to incorporated dirt.

The figure in low relief on the left-hand side

is

the Monument s architect,

Gilbert Scott, who

is

behind his mentor, Pugin.

Fig.

7.

Face

of

an angel

in

Brompton Cemetery.

b)

Fig. 5. Two statues of female figures on Admiralty Arch in London, represent a) Navigation and b) Gunnery. They face down the Mall towards

Buckingham Palace. Navigation, not having a hat, has her face washed white by the rain and has suffered dissolution. In contr ast, Gunnery s face,

protected by the brim of her hat, is black - the undissolved calcium sulphate has incorporated into its structure the street soot.



240

J

E. HARRIS

Fig.

8.

Gravestone in Arlington Cemetery, Washington DC, U.S.A.

al

Some Examples

of

Chemical Corrosion

t Paul s Cathedral, London

The coping stone for St Paul s, and some of their statues,

had holes drilled in their upper surfaces into which were

attached lifting tackle. After being lifted into place and the

tackle removed, the holes were filled with lead. Subsequent

corrosion of the stone has caused the lead plugs

to

stand

proud thereby giving a reference permitting the measure

ment of the average rate of corrosion of the stone since the

Cathedral was built. After 262 years oflife, 233 plugs stood

proud

of

the stone by an average

of 20-38

mm.

As

a very

rough guide, limestone corrosion rates

of

the order of a

centimetre a century must be expected in such urban areas.

Bird s Statue

of t

Andrew

A

12

foot high Portland stone statue of St Andrew,

sculpted by Francis Bird in 1724, was so badly corroded it

was removed in

1923

from its exposed position on the

fayade of St Paul s. t was vacuum impregnated with silane

resin

and

is now on display in the forecourt of the

Cathedral (though behind the railings). t provides a vivid

demonstration of the ravages arising from exposure to the

London atmosphere for a 200-year period.

On

the

top

of

the statue s head a hole had been filled with lead and now

stands proud and provides a lead plug index measure

ment

of

the depth of corrosion, see Figures

9 a)

and b).

bl

Fig. 9. a) Francis Bird s Portland stone statue

ofSt

Andrew from thefa9ade

ofSt

Paul s Cathedral in London.

b)

The height

of

the lead plug on top of the

statue s head reveals the extent of the corrosion.



WEATHERING,

CORROSION

AND RUSTING

241

Although the author could not get close enough to the

statue to measure the height of the lead plug, the extent of

corrosion appears to be consistent with the rates quoted

above.

St Andrew's features are,

of

course, devastated. Using

half

or

one centimetre per century as a guide, it

is

safe to

assume

that

any limestone

or

marble statue in London

which has been exposed to the atmosphere for more than,

say, 80 years

or

so will by now have suffered severe distress.

Prominent features such as noses are especially vulnerable.

Queen Victoria Statues

The Victoria Memorial by Thomas Brock occupies a

dominant position in the Mall opposite the main fayade of

Buckingham Palace.

t was unveiled

on

May 16, 1911, by

King George V and cost £350,000.

As

with the Albert

Memorial, the design is a complex allegory.

It

is domi

nated

by

a marble statue of the Queen on an elaborate

pedestal decorated by numerous subsidiary statues

symbolizing Power, Peace, Progress, Manufacture, Agri

culture, British Sea Power, Painting, Architecture, Ship

building, War, Truth, Justice, Motherhood, Courage,

Constancy and Winged Victory

Figure

100a) is

a close-up view of the face

of

the Queen,

taken by a Times photographer. The ravages

of

corrosion

are very evident and it can

be

seen

that

it has been

necessary to make a new nose, the whiteness of which

contrasts uncomfortably with the remainder of the statue.

Another statue of the Queen, also by Brock, now stands

outside the annexe to the National Portrait Gallery

in

Carlton House Terrace. It, too, has a damaged nose, the

repair to which became badly

stained another

repair

is

currently being carried out, see Figure

100b).

Interestingly,

Brock assisted Foley with the statue of Albert for the

Albert Memorial, and took over the task when Foley died.

A more flattering sculpture, this time of the young

Queen in her coronation robes, stands in Kensington

Gardens close to the palace where she spent her childhood.

The sculptress was her own daughter, Princess Louise.

Since its unveling in

1893

it has

of

course suffered corro

sion

but

it is not discoloured and the attack is spread more

evenly, although again it has been necessary to replace the

nose,

see

Figure

lO c).

A rather beautiful statue of con

siderable historic

and

sentimental interest

is

gradually

being dissolved away; there might

be

a case for moving

it

indoors.

Altogether, some 150 outdoor statues of Queen Victoria

were erected, including

40

in India. A number of these were

sculpted by Brock, including statues in Hove, Worcester,

Birmingham, Liverpool, Carlisle, Belfast, Cape Town,

Agra, Bangalore

and

Calcutta. A comprehensive study

of

the corrosion of these would be of interest.

The Cologne ngel

n 1842

Cologne Cathedral was already one of the largest

b)

c)

Fig.

10.

Corrosion and destruction

of

Queen Victoria's nose on:

a)

Brock's Queen Victoria Memorial outside Buckingham Palace; b)

Brock's National Portrait Gallery'S statue; and

c)

The Young Victoria

statue outside Kensington Palace sculpted by Princess Louise.



242 J E.

HARRIS

a)

b)

Fig.

11.

a) Limestone statue of Angel carved by Peter Fuchs in 1880 and installed on a fa9ade of Cologne Cath edral and b) the same statue a hundred

years later. (Courtesy Dr Arnold Wolff.)

ecclesiastical buildings in Europe, yet it was only half

finished Building was restarted in tha t year and the shell

of the final structure was completed in 1880, when it

became the tallest

bu

ilding in the world.

Most

of the 800

statues

on

the outside of the Cathedral were made from a

soft limestone from France, including an angel carved by

the Cathedral sculptor, Peter Fuchs.

Fortunately, a photograph was taken

of

the angel

shortly after completion and when this is placed alongside

a modern photograph of the statue

see

Figure

11)

a vivid

illustration is provided of the ravages of a hundred years of

exposure to the polluted atmosphere of Cologne. These

photographs were incorporated into a poster which

caught the imagination of the German people and received

widespread publicity on radio, TV and in the press.

It

attracted money for the conservation programme, but

more importantly, it drew people's attention to the im

portance

of

taking care

of

the environment. (What

of

course was not available was evidence of how the statue

would have fared had it been exposed to 'clean' country air

for a similar period of time.)

Incidentally,

Dr

Arnold Wolff, the Cathedral Architect

at Cologne, has set his face firmly against cleaning up the

surface of the Cathedral, arguing that this will simply

expose fresh surfaces to attack from the city's pollutants.

This is not the policy followed in this country - note the

cleaning operation currently underway

at

Westminster

Abbey. An argument in favour of the latter policy is that in

central London since the start of the 1960s, in terms of

micrograms per cubic metre, the sulphur dioxide level has

fallen from 400 units to

60

units.

RUSTING O MET L

INSERTS

So far

we

have considered the corrosion of stone

p r s

and

discussed how various biological processes and physical

and chemical reactions can accelerate decay. We now turn

to another important damaging process, the rusting

and

associated expansion of iron

or

steel components in

corporated into the masonry or concrete to impart

strength

or

stability. The damaging process is known as

'oxide jacking' by corrosion scientists

and

as 'rust burst' by

architects.

The first systematic study of the volume changes which

accompany the rusting of metals was carried out in

1923

by two Bristish chemists, Pilling and Bedworth [16]. They

defined a parameter which has since become known as the

Pilling- Bedworth Ratio (PBR); it is the ratio between the

average volume occupied by a metal atom in the rust and



WEATHERING

CORROSION AND

RUSTING 243

Ml 'tal

o 0 0 0

o Oxygen atom

• Mt tal atom

- - ~ ~ ~ + - - < ~ ~ - 4 - - --

---

>

d

,

Pilling Bedworth

Ratio ¢ d mo

dl,

Fig.

12

Schematic representation of the volume expansion which occurs

on oxidizing (rusting) and idealized cubic metal to a cubic oxide. The

diagram illustrates how the Pilling- Bedworth Ratio

is

calculated.

the average volume occupied by a metal atom in the metal

lattice. An idealized example is shown in Figure 12 In

nearly all cases the rust occupies a greater volume than

that released by the metal ato ms consumed in the process,

i e

an expansion occurs because the PBR >

1

Although rust burst has been known by architects

and

conservators for a very long time, it was first studied in

detail by nuclear scientists when the phenomenon was

found to be very damaging in nuclear reactors. Figure 13 is

a vivid example of the effects observed when a bolted

assembly is heated to a high temperature under oxidizing

conditions. We will now give some examples of its occur

rence in various historic buildings

and

monuments.

Fig.

13

Two steel bolts initially identical,

but

the one on the right-hand

side has been exposed to high-pressure carbon dioxide gas at 500 cC for

4000 hours. Rusting of the washers has elongated the bolt by 40 (P.

Rowlands).

Fig.

14

Illustrating the technique employed by Sir Christopher Wren to

bind together with iron cramps the masonry blocks

of

St Paul's

Cathedral.

St Paul's Cathedral

For the construction of St Paul's, Christopher Wren

had

specified the use of wrought iron cramp s to bind together

the large blocks of

Portland

stone. These were set in

grooves so that the top of each cramp was flush with the

top of its stone block, see Figure 14 Although they

were sup posed

to

be set in lead, in

many

cases the lead did

not cover completely the surface of the cramps. Over the

years water has penetrated between the masonry blocks

and has caused the iron cramps to rust - see Figure 15 The

associated expansion has forced the blocks apart and

actually lifted the Cathedral. In the 1970s it was decided to

carry out the major task of removing the rusty iron cramps

and

replacing them with cramps manufactured from

an

austenitic stainless steel alloy.

Fig. 15 Three iron masonry cramps from St Paul's Cathedral. The centre

one has rusted very little, the

top

one quite extensively and the

bottom

one has completely transformed to rust. Notice the increase in thickness

which

is

responsible for the jacking.



244

J E HARRIS

Fig. 16 Illustrating how the ancient Greeks used iron cramps and dowels

to hold together the Pentelic marble blocks used in construction of the

Parthenon. The cramps were set in lead.

Buildings

on

the cropolis

Iron cramps set in lead and iron dowels were used

extensively by the Ancient Greeks to join together the

Pentelic marble blocks used to construct the buildings

on

the Acropolis at

Athens-see

Figure

16

Under normal

conditions these iron fastenings were not stressed and one

of their functions (especially the dowels) may have been to

resist lateral forces during earthquakes.

As

far as can be

ascertained they have performed well over a period of time

approaching two and a half millennia.

Nevertheless the buildings have suffered severe de

privations and during the period 1898 to 1933 extensive

repair and reconstruction was undertaken under the gen

eral direction of the Greek architect, Nicholas Belanos.

Much

of

the stonework was reinforced with

I

shaped steel

implants which were grouted into position; the technique

he employed is illustrated in Figure 17 Within a few

decades of these repairs having been carried out, the steel

implants have corroded and the associated expansion is

cracking the stonework all over the Acropolis, see Figure

18

t is puzzling that more care had not been taken.

Belanos himself

had

experienced the damage

that

could be

caused by using untreated iron inserts in stone - he had

been responsible for replacing rusting ironwork installed

in the Caryatid Porch by one

of

his predecessors, the

architect A Paccard. For these repairs Belanos used

replacements made from brass.

Although the corrosion of Belanos s steel takes second

place in importance to the attack

of

the surfaces of the

stonework by the polluted Athenian atmosphere, it nev

ertheless constitutes one

ofthe

best-known examples of the

destructive power of expanding rust.

Fig. 17 The insertion of an untreated steel girder into a marble block

from the Erechtheum during the repairs carried out by Belanos in the

period

1902

to

1908

Fig.

18

Double stone panel from the Propylaea which

is

badly cracked

due, in part, to the rusting of modern steel inserts.

Choice

o

Cramp Materials nd the Example o Castle

Coole

There is no general agreement on the best metal

or

alloy to

use for masonry cramps.

As

mentioned above,

an

austeni

tic stainless steel alloy was used for St Paul s. The Greek

restorers

at

Athens, on the other hand, have rejected the



WEATHERING,

CORROSION AND

RUSTING

245

use of stainless for repairs to the buildings

on

the Acropolis

because they fear that stress corrosion cracking may occur

in the polluted marine Athenian atmosphere. In fact this

type

of

failure should not occur providing the stainless

steel has been correctly heat treated during manufacture

and has not been accidentally 'sensitized' subsequently.

(Sensitizing occurs when chromium-rich precipitates form

at

the grain boundaries during faulty heat treatment, thus

forming a region close to the grain boundaries depleted of

chromium and hence susceptible to rapid localized

corrosion.)

Probably the best choice of stainless steel is molyb

denum stabilized 316 (Fe-17 Cr- 12 Ni-2.5 Mo)

which is known to offer good corrosion resistance even in

marine environments. Another good choice, if cost

is not

too important, is a Nimonic alloy such as Inconel. The

Greek workers have chosen a Ti- 4 Mn-4 Al alloy

because of its excellent resistance to all kinds of corrosive

atmospheres. This

is,

of course, also a very expensive

alternative, but no doubt justified in view

of

the unique

importance of the Athenian buildings.

What is quite certain, in the present state of our

knowledge, is

that

one should not adopt a 'belt and braces'

approach

and

set the stainless steel cramps in lead. This

was done some ten years ago in restoration work carried

out on Castle Coole, Enniskillen. This is an eighteenth

century House designed by Wyatt which incorporated a

Portland stone facing (100-150 mm thick) with a sub

stantial brick backing. The Portland stone was suffering

from the classic problem of oxide jacking from the original

wrought iron cramps which had been set in lead.

In the period 1982 to 1988 the whole of the Portland

stone face was taken down a nd rebuilt using both old and

new stone. All the old wrought iron cramps were cut

out

and the new wall assembled with 304 stainless steel turn

down cramps between stones and stainless steel ties back

to the main brick wall. The stainless steel cramps were set

into generous quantities of lead.

An inspection of the property early in 1990 revealed

evidence of damage to the stones surrounding the cramps.

More detailed examination showed that the stainless steel

cramps remained in good condition; it was the lead which

had corroded. There is in fact a 50 expansion on forming

red lead monoxide and as a great deal of lead had been

used during construction, it is

not

perhaps surprising

that

the masonry had been damaged.

A debate on the causes of the problem was launched in

the correspondence columns of

Bristish Corrosion Journal.

Some authorities argued that as lead is known to corrode

readily in strongly alkaline environments then the possible

use of lime mortar

in

the reconstruction could be the

source of the problem.

Of

course wrought iron cramps

have been set in lead for two a nd a half millennia and some

observers pointed

out

that it is strange that there haven't

been more examples

of

corrosion of the lead in alkaline

environments.

The solution to the problem appears to be

that

while

wrought iron is anodic with respect to lead, stainless steel

is cathoidic. In other words, in a corrosive environment

wrought iron set in lead will corrode preferentially thereby

protecting the lead (though this

is not

a problem in

alkaline environments because the corrosion rates will be

extremely slow - hence the success of ancient cramps). In

the case

of

stainless steel set in lead, the latter

is not

protected and extensive corrosion of the lead can occur.

Architects are advised not to set stainless steel in lead,

especially in strongly alkaline environments. When asked

what setting medium should be used one first enquires if

any medium is actually required. To go further, it is worth

considering if the cramps themselves are essential; the

Washington Memorial, the tallest masonry structure in

the world, was built without benefit of metal cramps.

The Taj Mahal

The Taj Mahal is not one of mankind's oldest monuments,

nor is it particularly important in the sweep of global

history,

but

it is, without question, one of the world's most

beautiful buildings. The mausoleum has suffered a mini

mum of decay in its 350 years oflife (compare, for example,

the degree of corrosion of Taj's marble with that of the

stonework of the, slightly younger, St Paul's Cathedral in

London).

Nevertheless, it has some problems and these could

increase as the subcontinent becomes progressively more

industrialized.

In

an

article by Christopher Thomas in The

Times it was revealed that numerous iron cramps holding

the marble blocks together

had

rusted and in some cases

cracked the stonework. These were being replaced by

cramps manufactured from stainless steel and titanium. An

example of damage to one

of

the associated buildings due

to the rusting

of an

insert is shown in Figure

19.

There have also been reports of the mausoleum's marble

changing colour, but this could be a consequence of

natural ageing processes. Nevertheless, the Indian Govern-

Fig.

19.

Taj Mahal: damage to decorated stonework due to the rusting

of

an iron support



246

J. E. HARRIS

ment has ban ned new industrial activity within a radius of

35 miles around the mausoleum, but it is doing nothing

about

the almost two hundred iron foundries already

operating nearby.

There are circumstances where the norma lly dry desert

like atmosphere of

northern

India can be kind to its

monuments note the extraordinary preservation of the

famous iron pillar of Delhi, which is still bright and shiny

after exposure

to

the atmosphere for

no

less

than

1600

years, see Figure

20.

Rusting

of

Reinforcement Bars

n

Concrete

In this paper we have concentra ted on the damage suffered

by historic monuments due

to

the expansive force of

rusting. In passing we should also mention the economi

cally

more

damaging process of disruption of concrete

structures due

to

the rusting of their reinforcement bars

(see Figure

21 .

This

is

a world-wide problem costing ·

billions of poun ds in repairs.

In fact, steel in concrete should be protected from

corrosive attack by the alkalinity of the cement paste.

Initially this is indeed the case, but over a period of time, if

the concrete is porous, it is attacked by the carbonic acid in

rainwater

and

the pH decreases towards acid conditions.

Fig.

20.

The famous Iron Pillar of Delhi which is completely free of rust

and

is

shiny in spite of 1600 years' exposure to the atmosphere.

Fig. 21. Spalling of reinforced concrete fence due to rusting of its steel

bars.

The end result

is

rusting of the steel

and

its associated

expansion causing spalling of the concrete. This problem

can be largely overcome by burying the steel beneath a

thick

and

impervious layer

of cement in

other words

demanding a high quality of workmanship.

'Acid Rain' is often blamed for the deterioration of

concrete buildings in our cities but the low level of sulphur

dioxide (say 20ppb) compared to

that

of

carbon

dioxide

(0.03 ) in even

urban

atmospheres, makes it unlikely that

the former impurity plays a significant role in lowering the

pH. Skoulikidis [8J has demonst rated that the thickness of

rust formed on steel reinforcements in concrete tends to be

independent of the level of pollution o f the atmosphere.

Carbonation by atmospheric carbon dioxide is in any

event a slow process. More rapid damage arises when salt

solutions are allowed to penetrate the concrete. This

occurs in marine environments and where salt is used for

de-icing reinforced concrete roads

and

bridges. The latter

problem is particularly acute in the northern states of the

U.S.A., where the amount of salt used on their highways

increased by an order of magnitude in the period 1955 to

1975. The damage this causes has been described as having

reached crisis proportions.

Albert Memorial

The Albert Memoria l is Britain's most famous monument,

and

it is the most complex

and

difficult

to

maintain. t was



WEATHERING, CORROSION AND RUSTING

247

designed in the Gothic style by Sir Giles Gilbert Scott and

constructed in the period 1864 to 1876

Its centrepiece is a colossal bronze statue by

J.

Foley of

the seated Prince holding in his right

hand

the catalogue of

the

Great

Exhibition of 1851 (on the site of which the

Memorial stands). The statue is partly protected from the

weather by an elaborate canopy, above which rises a richly

decorated spire consisting

of

a wrought and cast iron

frame clad with lead in which

is

embedded semi-precious

stones.

Beginning

at

the level of the Prince and moving upwards

there is a succession of 24 bronze statues starting with

representations of the Sciences. Then, on the outside of the

spire are bronze statues of the Christian Moral Virtues and

above these angels in various attitudes suggesting the

shedding of worldly honours in preparation for the greater

glory beyond the grave.

In the gables above the canopy, and in the spandrels, are

constructed elaborate and very beautiful mosaics of female

allegorical figures representing

Poetry, Architecture,

Sculpture

and

Painting.

The complete structure

is

raised

on

a shallow pyramid

of granite steps.

All the statuary below the Prince is of

marble. At the base of the structure four groups of marble

statues represent the Industrial Arts: Agriculture, Manu

facturing, Commerce and Engineering. Along the same

diagonals at the four corners of the granite steps are

groups of marble figures representing the continents:

Europe, Asia, America and Africa.

Most remarkable of all

is

the frieze

around

the podium

which is a High Victorian view

of

a Parnassus of the Arts:

169 marble figures carved in high relief by Armstead and

Philip represent leading architects, artists, painters, sculp

tors, poets and musicians throughout the ages.

From

the very commencement it was realized that the

extreme richness of the Memorial made it vulnerable to

corrosion in the polluted

London

atmosphere. So worried

was the Queen s secretary, Lord Grey, that he proposed

that the whole Memorial should be enclosed in glass. Such

is

the concern over its current condition that this idea was

resurrected a few years ago - the construction of a massive

glass pyramid to cover the monument was suggested.

Gilbert Scott s original idea was tha t the statue

of

the

Prince should be in Sicilian marble, but when he was

shown how badly this marble had corroded on another

London monument, Marble Arch, he decided that the

central statue would in fact be cast in bronze. His concern

over the durability of Sicilian marble also led him to

specify Campanella marble for the frieze

and

other statues.

The best preserved part is the Prince Consort statue

itself, which has been protected by the canopy and has

been regularly treated with lanolin, which has imparted a

pleasing

dark

patina to its surface (the statue was origin

ally gilded, as were other parts of the Memorial).

Throughout its history, corrosion of the Memorial has

given rise to comment

and

concern, and repairs and

Fig.

22

One of the bronze statues from the outside of the Albert

Memorial being removed for repair and restoration during the current

refurbishing campaign.

restoration work have from time to time been undertaken.

It

was the falling down of a large piece of lead cornice in

1983 which finally alerted the authorities to the parlous

state of the Memorial. In the period 1984/5 scaffolding was

erected and a comprehensive inspection carried out, and as

a result of this the decision was taken to undertake a major

overhaul and repair. This will involve removing and

refurbishing all the external bronze statues, cleaning and

preserving the marble statues and stripping the lead

cladding from the iron frame of the spire, removing the

rust and treating the iron before replacing the renovated

cladding. The work is underway, see Figure 22 . The

Monume nt has disappeared under a shroud of scaffolding,

and will

not

re-appear for a number of years.

Space does not permit a detailed account of the corro

sion of the stone and brickwork, the tesserae and the

metalwork, and the staining by salts of copper and iron of

the frieze, the damage by vandals and tourists, the attack

from algae and pigeon droppings and the war damage. A

major problem

is

the distortion of the mosaics due to

water ingress into the underlying plaster and its con

sequent swelling the expansion possibly being the result

of chemical changes to the hydration products, see Figure

23

Rusting of the ironwork gives much cause for concern.

The lead cladding was attached to the iron supporting

frame by means of bronze screws so there are rather

obvious possibilities for galvanic coupling and associated

accelerated corrosion.

In

any event water had penetrated

to the interface between the lead and the iron and the

rusting and associated expansion has burst the cladding

in many places, see Figures 24 and 25 There are numerous

cracks in the lead and these may have been a consequence

of a thermal ratchetting process as the original design did

not make provision for the take-up of strains arising from

differential expansion and contraction between the lead

and the iron.

It is reassuring to see that a start has been made on an

ambitious programme of repairs - the Memorial

is

well



248

J.

E. HARRIS

Fig. 23. Mosaic on Albert Memorial; note the missing tesserae and the

large patch which has been lost altogether. Moisture has penetrated

behind the mosaic and the resultant expansion of the support plaster may

have contributed to the damage.

worth preserving: no other building

or

monument en

capsulates so completely the spirit of the Victorians.

The Statue o iberty

As has been described, the spire of the Albert Memorial

consists of

an

iron frame with a lead covering; the Statue of

Liberty also has an iron frame, but with a copper skin

attached to it. The Statue s iron support, the armature or

crinoline, was designed by Gustave Eiffel (of Eiffel Tower

fame).

Not

surprisingly, after close to a hundred years of

life

in the polluted marine environment of New York

Harbour, the armature

had

rusted very badly.

n

contrast,

the copper skin has proved remarkably durable, losing

rather less than 10 of its thickness throughout its long

life (Figure

26).

The Americans instituted a massive repair programme

in the early 1980s in anticipation of the centenary an

niversary celebrations in 1986. n characteristic fashion

they decided on radical actions - the complete replacement

Fig. 24. The rusting and expansion of the iron support structure has burst

the lead covering of the Albert Memorial.

Fig. 25. Illustrating the reverse faces of two sections of lead cladding from

the Albert Memorial. The expansive force from the rusting iron support

structure has drawn the heads of the bronze bolts through the cladding

creating large holes.

of the iron armature with one manufactured from stainless

steel. (They could be so bold because they were backed by

a public collection which approached 100 million dollars,

and the stainless steel alloys were a gift from American

Indust ry - how one wishes similar largesse was available to

those responsible for the Albert Memorial repairs.)

A particular problem which occurred with the original

ironwork is germane to this paper. The copper skin was

Fig. 26. A Texas Instruments scientist measuring the thickness of the

patina on the nose of the Statue

of

Liberty.



WEATHERING,

CORROSION

ND RUSTING

249

Fig.

27

Illustrating how the rusting and expansion

of

the iron support

structure has damaged the copper skin of the Statue of Liberty. (After

Jacques Girardon.)

attached to the iron armature by means of saddles which

looped

around

the armature and was flush riveted to the

copper plates.

To

avoid electrolytic cells being set up the

copper was insulated from the iron by means of an

insulating material variously described as impregnated

asbestos

or

felt. Nevertheless, over the years the armature

has rusted and the associated expansion had torn out the

rivets, as illustrated in Figure

27

n choosing the stainless

steel for the new armature the repairers were careful to

choose alloys which were compatible with the copper - i e

was close to copper on the electrochemical scale.

There

is

every expectation that the refurbished Statue

will

last for several more centuries. We hope and expect

that the current repair programme with the Albert

Memorial will have

an

equally successful outcome.

Rusting

o

ron Supports in

Cemeteries

A fitting conclusion to a paper on decay

is

a short

discussion of the deterioration

of

gravestones and tombs

due to the rusting of their iron and steel inserts.

Graveyards and the older cemeteries are in fact excellent

places to gather information on relative rates of corrosion

and rusting. A variety

of

different types of stone are used

for the monuments, and each gravestone carries the date of

the death of the deceased so that the length of time the

stone has been exposed to the elements can be deduced. n

view of this it

is

not perhaps surprising that just over a

century ago Sir Archibald Geikie carried out the first

systematic study of the corrosion

of

stone when he in

vestigated the condition of Edinburgh s churchyards [17].

By studying graveyards in various parts of the world,

then, information can be obtained on the influence of

climate

on

the durability of different types of stone. t

is

quite clear that in our relatively warm and moist climate

granite shows good durability whereas some sandstones

are unstable.

From

the earlier discussion on the properties

of limestone it is evident that this stone

is

not suitable for

monuments expected to last a century

or

more. n view of

this it

is

perhaps surprising

that

Portland stone was chosen

for the monuments and gravestones in the very beautiful

American Military Cemetery near Cambridge, England. A

Fig.

28

The rusting and expansion of internal iron supports has damaged

this tomb in Brompton Cemetery, London.

Portland stone wall has the names carved into its surface

of all those for whom there are no known graves. From

corrosion

data

quoted earlier in this paper, it seems likely

that an extensive recarving

of

the names

will

be necessary

quite early in the next century. The granite wall of the

Vietnam Memorial in Washington

will

have a much

longer life

We end this paper with a quotat ion - the dying words of

the Buddha: All compound things are subject to decay .

Fig.

29

Disintegration of gravestones in Montmartre Cemetery, Paris.



250

1. E.

HARRIS

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Masonry Construction Structural Mechanics and Other Aspects -Jacques Heyman

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