ANALYSIS OF PLATES AI© SLABS ON

ELASTIC FOUNDATIONS

by

RALPH LYMAN MCGUIRE, JR., B.S.

A THESIS

IN

CIVIL. ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirement for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

Accepted

August, 1972

Co p. 3,

ACKNOWLEDGMENTS

I am deeply indebted to Professor Jimmy Hiram Smith, whose

ability as an educator, is directly responsible for the success

of this study. Dr. Smith's many hours of guidance has fostered

in me the true spirit of learning. Equal thanks are extended to

Professors James R. McDonald and C. V, G. Vallabhan whcse encour­

agement and advice aided immeasurably in the preparation of this

report. Also, thanks are extended to Dr. Ernst Kiesling; whose

policies in the Civil Engineering Department, make learning an

enjoyable process.

Finally, I will always owe a vote of thanks to the United

States Air Force for affording me the opportunity to pursue this

field of study.

11

/ TABLE OF CONTENTS

ACKNOWLEDGMENTS i i

LIST OF FIGURES iv

I . INTRODUCTION 1

I I . DEVELOPMENT OF THE MODEL 3 m

Plate Bendingi 5

Framework Bending , 7

Determination of Element Properties 15

Matrix Method of Solution l6

III. MODULUS OF SUBGRADE REACTION 20

Introduction 20

Accepted Methods 22

Incorporation of MSR into the Model. . . . . . . 25

IV. RESULTS 28

Plate and Slab Analysis 28

Comparison of Analytical and Experi­mental Results . . . . . . . 33

Conclusions 37

LIST OF REFERENCES 39

111

LIST OF FIGURES

FIGURE PAGE

1. Elemental Framework Concepts ^

2. Bending of Plate Element Showing Deflected Position 6

3t Deflection of Plate Under the Influence of Torques Applied Along All Edges . , 8

^. Manner of Beam Element Displacement Showing (a) Positive Internal Displacements and (b) Loading and Displacement Sequence . . . 9

5. Numbering System for Nodal Points and Members

of Framework 10

6. Master Stiffness Matrix For One Grid Element 12

7. Bending of Framework Under Moments Applied at the Nodes 13

8. Bending of Framework with Torque Applied at the Nodes 11-

9. Typical i ramework Showing Cross-Sectional Properties • 17

10. Load-Deformation Data for Two Feet of Clay (After Siddiqi and Hudson) 21

11. Typical Load-Deformation Curve Showing Tan­gent and Secant Modulus Relationships 22

12. Comparison of Equation 2^ with Laboratory Load-Deflection Data 26

13. Graphical Representation of Plate or Slab Resting on Elastic Foundation 26

1^. Plan View of Typical Framework Mesh Showing Shaded Area as Area of Inter­action of Node 5 a-id Foundation 27

IV

FIGURE PAGE

15» Typical Method of Sect ioning Body Along Lines of Symmetry for Analysis Under Center Load 29

16. Lr'ad-Deformation Curve For Concrete Runway Slab Foundation 31

17. Average Load-Deformation Curve for Clay (2 f t ; Showing Range of MSR Inves t iga ted 32

18. Def lec t ion of Concrete Runway Sect ion Under 30,000 Pound Center Load'. 3^

19. Def lec t ion of Aluminum P la t e Under 100 Pound Center Load 35

20. Def lect ion of Aluminum P la t e Under 200 Pound Center Load 36

2 1 . Change in Deflect ion Pred ic t ion with Varying MSR - 200 Pound Center Load on Aluminum P la t e • 38

CHAPTER I

INTRODUCTION

In 19^1* when Hrennikoff (l) presented his original method

for solution of elasticity problems by replacing the structure

with a representative framework, little attention was paid to the

model's potential. At that time analyses utilizing the framework

models required use of either the relaxation or moment distribu­

tion methods for solution and were therefore very cumbersome.

Furthermore, the less complicated framework models could not accomo­

date values of Poisson's Ratio C^) of other than l/3»

In 1965* Yettram and Husain (2) showed that, by considering

the torsional stiffness of the framework members, materials with

any value o f ^ could be represented with even the simplest of

Hrennikoff's models. Smith (3) extended the analysis to include

non-linear behavior in I968. Then in 1970 (4) he demonstrated an

additional application of the model by using it to analyze shell

structures. Smith's analysis of a ^8 ft by 7^ ft "quonset-type"

structure, fabricated of doubly corrugated steel panels, was in

excellent agreement with experimental results.

Realizing the potential of the framework method of analysis,

this study was established to investigate the possible extension

of the method to plates and slabs on elastic foundations. The

basic premise of this investigation is that the effect of the

foundation can be adequately treated by considering linear (or

non-linear) spring action at each node of the framework model.

Only linear spring action was investigated. It was felt that in­

corporating non-linear spring action into the model would not in­

crease the validity of the results obtained when comparing the

analysis with experimental data.

Comparison between analysis and experimental results required

that well documented field data be available. Early in the investi­

gation it became apparent that very few sources of simple load-

deformation curves for various foundation materials existed. It

would have been a simple matter to accept the published estimates

for the broad range of Moduli of Subgrade Reaction (MSR) or to

extract these estimates from the many available laboratory stress-

strain curves. However, rather than penalize the analysis as a

result of possible incorrect estimates, this type of data vras dis­

carded.

A thorough search of the literature produced two very complete

and well documented reports of plate load tests. Rao (9) reported

the results for deflection tests of a 30 ft x 30 ft x 1 ft concrete

runway slab subjected to concentrated loads of up to 30,000 pounds.

An equally well controlled series of tests was reported by Agarwal

and Hudson (5) who were investigating the deflection of 9 in x 9 in

x 0.125 in aluminum plates on clay foundations. These two series

of experimental results provided a very good test for comparison of

the analysis with field data.

CHAPTER II

DEVELOPMENT OF THE MODEL

Hrennikoff based his original work on the thesis that a solid

elastic body can be represented by a series of frameworks consisting

of beam elements. These elements are arranged in a definite pattern

and are chosen so that they have elastic properties suitable for

the particular problem under investigation. All load vectors must

be applied at the framework joints. The type of loading (i.e.

concentrated or uniform) will have an effect on the number of

frameworks required to represent the elastic body.

Hrennikoff discussed a number of different framework configura­

tions suitable for two dimensional analysis. Of all the configura­

tions presented by Hrennikoff, only one was capable of representing

materials having^ of other than l/3 (See Figure l(a)). Figure l(b)

illustrates the Hrennikoff model that Yettram and Husain modified

to accomodate different values o f ^ by including torsional stiff­

ness in the non-diagonal beam elements.

The Yettram-Husain modified model was selected for this present

study for the following reason. The simplified model has a smaller

number of nodes per framework element. Hence the amount of compu­

tational effort to obtain a solution is significantly less. For

example, if four framework elements are used to represent a continu­

ous plate, twenty-five nodes are required for the configuration

shown in Figure l(a) while only 9 nodes are required for the other

element configuration.

(a)

(t)

FIGURE 1. ELEMENTAL FRAMEWCRK CONCEPTS

5

In order for the model shown in Figure l(b) to react to any

loading condition in a manner identical to that of a plate of equal

size, proper cross-sectional properties must be selected for the

side and diagonal beam elements. Selection of these properties is

done by applying statically equivalent loading and support conditions

to the same size continuous plate and framework model. The result­

ing equivalent rotations are equated in order to solve for the

stiffnesses required to enforce identical deflection behavior of

the plate and model.

PLATE BENDING

Consider the plate element shown in Figure 2. The moment at

a section of the plate perpendicular to the x-axis is given by the

relationship (8)

' ' )

"^(JL) , n ^ C U

.3

,2

Mx = D V^>^ '^ -sy' y (1) where: cO = deflection

IVIx " moment in plate

D = plate flexural rigidity = •" \2{\-fJ>^)

Q = modulus of elastacity

f = plate thickness, and

f^ = Poisson's ratio

Under bending action of this type the plate forms an anticlastic

surface and

B^gj _ Mx a oj - H- Mx 1 -^,,2 r / 1 ..2 ^x^ D(l-^2) ' ^r D(l-A^")

o

M

PL,

M

g

H

EH

pq

CM

I \

1 1

7

Integrating both of these once over the length (/) of the plate

yields

^ ^ i 2 j y^ / ^ Q (33 dx E t ^^

dgj // 12 Mx / D f3) dy - ^-i7r-/=&pv 3

Likewise, applying equal twisting moments along the edge of the

plate yields (See Figure j)

^2

Mxy = D(I-/X) -^^^ M

This i s not an independent equation but follovfs from equation ( l ) .

Integrat ing t h i s re la t ionship once

duL.] J[ Mxy dy dx / 1 D( l - / ^ )

yields

FRAMEWORK BENDING

The framework chosen for this investigation is statically

indeterminate and matrix methods are utilized for the analysis.

The internal deformations as shown in Figure (a) must be deter­

mined for each beam element in the framework. The positive loading

and displacement sequence is as shown in Figure ^(b).

In matrix notation the member oriented stiffness of each beam

8

o

M

B

b O

o

Pi

C ^

w

o M

element i s expressed as

k=

4 E I Ji

- 2 EI

^ 2 E I i 4EI

0 L 0 shear modulus - p s i

0

0 GJ /J

where: G

E

I = moment of i n e r t i a - iri

= Young's modulus - p s i

'.J\

J = torsional constant - in

y, = beam length - in

(6)

VI

^z^^z M , 0, •y ' ^ y

-TT^ MX > • >

FIGURE 4- MANNER OF BEAM ELEMENT DISPLACEMENT SHOWING (a) POSITIVE

INTERNAL DISPLACEMENTS AND (b) LOADING AND DISPLACEMENT

SEQUENCE

Transformation of member displacements i n to system d i s p l a c e ­

ments i s accomplished through a geometry or compa t ib i l i t y r e l a t i o n ­

s h i p . Applying u n i t displacements according t o the sequence

10

established in Figure ^(b) and examining the internal displace­

ments, the compatability matrix is

B =

J/jl - SIN a cos a W

\

COS a SINQ^

-ly SIN a -cos a

0 -cosa -siNa

- (7)

Each element makes a contribution to the system stiffness in

accordance with the relationship B kB (l2)- For the member and

nodal numbering system shown in Figure 5> fJ e stiffness matrix of

the fraFiOwork can be assembled according to the relationship

{f}-[B]'[k][B](d> (8)

where: f is a vector of applied external forces

d is a vector of external displacements

k is the element stiffness matrix (Equation 6)

B is the compatibility matrix (Equation 7)

FIGURE 5. NUMBERING SYSTEM FOR NODAL POINTS AND MEMBERS OF FRAMEWORK

11

Because three degrees of freedom have been established for each

node, the resultant matrix is of order 12 x 12. The matrix (Figure 6)

is representative of a square framework only. Non-square frameworks

require a more general relationship where all member lengths are

related by a constant multiplier (See Reference 2).

VThen the framework is subjected to moments about the x-axis

as shown in Figure 7> the rotations about the x- and y-axes are

Qjy and Qjy^ respectively. If a load vector of moments — ^ is

applied at each node of the framework and the proper deformations

assignedf —^^ , — -—jthe matrix yields only two independent equa­

tions. These equations can then be solved f or fy and fx and

0fv . / ' f z J F i ^ Id ] Mx 2E V 2>IT I' -f 2 I Id y

y (9)

/ ( Id ) V 2 J ^ 1^+ 2ITd /

0fx . Mx " 2E \21Z 1^+ 2ITd

where: I = moment of i n e r t i a of- non-diagonal members

I J = moment of i n e r t i a of diagonal members

If torques a re appl ied t o the nodes of the model as in Figure 8

the r e s u l t i n g r o t a t i o n , ^fI , i s obtained as

Q.. = Mxy/<f2" (10) 2>f2" G J + 2EId

12

pp

pq

^CVJ

PI I

^

o

-51 P

.CM

I

i

O

JC^ PQ

\ o o o H=

o ^ ^

+ <

m 1

<

o 1

¥ Q +

P + IS

CM I

pq i

pq X I

X I

< : I o

X! C\2

W

> rs I p p P ¥

S

X

I

pq I

< I

o I

.CV2

P I

o I

o

?

p I

CM I

p I

1

p I

I

X CM

P

P

P

¥

P

P

?

CM

?

H

-;t

M

^ CM rH

»n Tzi

'^ S en

X n

pci

CO

ft EH

@ E H CO <

pc;

B M

X CM

CO

: s

VO

H ^

cn

K J

N CM "-3

O ^ CM

n n n < p :i c 3 C

I . 1

:

13

[3 g

EH

EH

P

g

p

g ^

CD

p

M

^

14

8

p p p

e

p

plH

5 CD

I—I

p

00

K P O

15

Also because the edges of the plate and framework remain

straight

^ft = - " / i (11)

DETERMINATION OF ELEMENT PROPERTIES

Equating the rotations of the plate to those of the model for

identical loading conditions yields the required cross sectional

properties. Equating these rotations

^PY ~ 9fy

epx = ^f X *> (12)

5pt = 0ft

From these relationships we determine that

I = /f ^4{\^•H•)

I = H-I2( l -yu.2)

(13)

(1^)

J = {\-ZH-)li 12(1-A^)

(15)

These relationships define the values of the cross sectional proper­

ties for the beam elements in the individual frameworks. Each member

of the framework contributes to the stiffness of the mesh and

16

t h e r e f o r e the values for t he non-diagonal members a re doubled for

those members i n t e r n a l t o the framework. The f u l l s ign i f i cance of

t h i s i s shown in Figure 9»

MATRIX METHOD _0F SOLUTION

Based on equi l ibr ium, s t r e s s - s t r a i n , and compat ib i l i ty r e l a ­

t i o n s h i p s of deformable bodies , a r e l a t i o n s h i p between a vec tor of

appl ied ex t e rna l forces f, and the r e s u l t i n g vector of i n t e r n a l

forces p can be developed. In matrix no ta t ion t h i s r e l a t i o n s h i p

i s

{')•[ ']« (16)

where: f i s a vector of applied external forces

p i s a vector of in ternal forces

A is a matrix enforcing equilibrium

From s t r e s s - s t r a i n re la t ionships , .the internal deformations

[v\ in terms of in ternal forces / p \ is

where! k , the element s t i f fness matrix, represents the s t ress-

s t r a i n r e l a t ionsh ip . Continuity demands r e su l t in a re la t ionship

between external deformations /d \ , and internal deformations {v\,

of the form

= B d

where B is the compatibility matrix.

17

FIGURE 9 . TYPICAL FRAMEWORK SHOWING CROSS-SECTIONAL PROPERTIES

18

If we now apply a set of virtual displacements to a structure

subjected to a system of real loads and introduce the theorem of

virtual work

/ 1 ( 1 ^ 1^\ / T ^ •®* ^ 'ternal work = internal work).

we find that the compatibility matrix B is the transpose of the

equilibrium matrix A , It follows then that

and

Proceeding

['1=['H']*H or

where B P k B is the stiffness matrix for the overall system

and is defined as K , Then

H ' HH ^^ [dj =[K]-l[fj (I7)b

and the relationship between internal forces and external deforma­

tions is

19

{p)=t][BJf) -- («) Equation (l7)b is the relationship that will be employed to

determine the displacements (d), for a structure under a system of

external loads <f>. Proper application of Equation 18 will yield

the internal forces developed as a result of. the original loading.

CHAPIER III

MODULUS OF SUBGRADE REACTION

INIRODUCTION

A large portion of this investigation was devoted to determin­

ing the manner in which a foundation subgrade resists defonnation

to various loads. All of the existing methods (8, 10, 11, 13, 1^,

15) are based on the well known Winkler hypothesis. In I867, Winkler

postulated that the deformation behavior of a subgrade material is

essentially that of a dense liquid. He also stated that this be­

havior can be represented by a bed of closely spaced springs. From

these assumptions he established the ratio

MSR = q/y

where: MSR = Modulus of Subgrade Reaction - pci

q = load intensity - psi

y = average settlement due to q - in

The values of MSR are determined from the slope of the q versus y

curve obtained from plate-load tests. Typically these plots are

nonlinear (See Figure lO) over the broad range of loads. Figure 10

is reproduced from the work by Siddiqi and Hudson (6) and was used

extensively in this investigation. They illustrate the non-linear

behavior of the soil and the effect of plate size on the tests.

An interesting feature of the load-deformation curves presented by

Siddiqi and Hudson is that even though the curves differ according

20

21

CM o

o CVJ

•

M

ci" M P 9 CO

p o

b

o CXD

O o • VO

o • LO

o • «>

o • CO

o •

CVJ

P

O 1 3

p

&

o M

<

g P I

O P

o H

o M Cm

isd - 3anss3yci

22

to plate size, each appears to be linear up to some unique value

of deflection. If the foundation deflections are within the linear

range the MSR is determined by analogy with the "tangent modulus"

concept. If deflections of the foundation are expected to exceed

the linear range of behavior then the "secant mcxiulus" concept is

usually employed. The meaning of these terms is shown in Figure 11.

CO

MSR (TANGENT) =

MSR (SECANT)

Aq. AYi

Ay

y - INCHES

FIGURE 11. TYPICAL LOAD-DEFORMATION CURVE SHOWING TANGENT

AND SECANT MODULUS RELATIONSHIPS

ACCEPTED METHODS

The value of MSR is a function of plate size used for the load-

deformation tests. Any value derived from load tests must then be

corrected to account for the actual size of the plate or slab under

investigation. Terzaghi (13) has suggested two relationships based

on experimental investigation that include this correction. For

unsaturated clays subjected to less than one-half the ultimate

bearing load the relationship is

23 t

..r^r-> M S R , V

'^^^B "" ~B— " ^

where: MSRg = MSR under a foot ing of s i z e B x B

MSR = MSR under a foo t ing 12 in x 12 in

Terzaghi r e a l i z e d t h a t fo r sands the se t t lement decreases as

the r e s i s t a n c e t o deformation ( E ^ ) i nc r ea se s . Following from t h i s

he proposed the fol lowing r a t i o , a l s o based on experimental i n ­

v e s t i g a t i o n , 2

MSR, = f B-f I ] M S R (20) B V 2B /

Equation (20) is valid for square footings only. To obtain the

MSR for a rectangular plate (i.e. b x nb) from data for a square

plate, Terzaghi suggested an empirical relationship

MSRn= M S R ( - V ^ ) (21) " ^ l.5n /

The limiting value of equation (21) when n becomes large is

MSRn = 0 . 6 6 7 MSR (22)

Terzaghi , however, recognized t h i s phenomenon and placed the l i m i t i n g

values for v a l i d i t y of equation (23) a t no more than one-half t he

bear ing capac i ty of t he subgrade.

Another more v e r s a t i l e expression for MSR was suggested by

Vesic .

^/WiT^)-MSR =0.65 V f ^ \T~jP~ j- (23)

24

where: ' B = foo t ing width

Eg = modulus of e l a s t i c i t y for s o i l

Ejj = modulus of e l a s t i c i t y for foot ing

I = moment of i n e r t i a of foot ing

f^ = Po i s son ' s r a t i o

The advantage of t h i s equation i s t h a t only Eg andyU are requi red

from l abora to ry t e s t s on the s o i l t o def ine the MSR. The v a l i d i t y

of equat ion (23) i s not diminished for values exceeding one-half

t he bear ing capac i ty of t he subgrade.

Using the theory of e l a s t i c i t y , Skempton ( l 6 ) showed t h a t for

t he same r a t i o of appl ied s t r e s s t o u l t ima te s t r e s s , the deforma­

t i o n of t he s o i l in a ciz 'cular p l a t e bear ing t e s t i s r e l a t e d t o the

s o i l ' s a x i a l s t r a i n in a t r i a x i a l compression t e s t by

V g = 0 . 2 9 4 5 Nc€ (2it)

where: y = foo t ing se t t l ement - in

B = foo t ing width - in

N = Pfy = bear ing capaci ty fac to r

€ = a x i a l s t r a i n of clay s o i l - i n / i n

Pf = u l t i m a t e bear ing capac i ty of c lay - p s i

C = apparent cohesion of the clay - p s i

Agarwal and Hudson (5) used t h i s r e l a t i o n s h i p in t h e i r i n ­

v e s t i g a t i o n with very good r e s u l t s . Using data 'from an unconfined

compression t e s t they generated a l oad -de f l ec t i on curve for a 9 in

diameter p l a t e and comjDared i t with t h e i r load-deformation curve

25

for a similar plate. These results are shown in Figure 12.

With the exception of equation (23) all of the previous rela­

tionships in this chapter are representative of clay soils. Little,

if any, similar information is available for sands or granular

soils. Granular soils therefore were not considered in the in­

vestigation.

INCORPORATION OF MSR INTO THE MODEL

Just as each member in the framework makes its contribution

to the stiffness of the framework, the springs in the model shown

in Figure 13 contribute to the ncxie stiffness according to the

character of the foundation represented.

The vertical resistance of the foundation to deformation can

be accounted for only at the ncdal points of the frameworks The

foundation stiffness at each node is proportional in magnitude to

the contributory area of the solid body it represents. An examina­

tion of the plan view of the framework in Figure l4 shows that

interior nodes (i.e. node 5) accept a stiffness in the vertical

direction in proportion to the shaded area. Corner nodes on the

boundary will react with the foundation one-quarter as much as

interior nodes and all other edge nodes one-half as much. The

springs representing the foundation at each node offer resistance

to only vertical displacement of that node and therefore must have

units of lb/in. The values of MSR are always given in terms of unit

area. By simple geometry then, the spring stiffness coefficient

for interior springs is

8.0

"^ 6.0 ex.

I

UJ

CO

[2 4-0 o.

2.0

0 0.04

— SKEMPTON (REF. 16)

— EXPERIMENTAL (REF. 5)

0.08 0.12 0.16 DEFLECTION - INCHES

26

0.20 0.24

FIGURE 12. COMPARISON OF EQUATION ( 2 ^ ) WITH LABORATORY LOAD-

DEFLECTION DATA

FIGURE 1 3 . GRAPHICAL REPRESENTATION OF PLATE OR SLAB RESTING ON

ELASTIC FOUNDATION

27

SPRING CONSTANT = ( M S R ) ( / ) 2 (25)

These values of the spring constants are then algebraically

added to the appropriate diagonal element of the stiffness matrix

for the framework under consideration.

k ^

FIGURE \h . PLAN VIEW OF TYPICAL FRAMEWORK MESH SHOWING SHADED

AREA AS AREA OF INTERACTION OF NODE 5 AND FOUNDATION

CHAPTER IV

RESULTS

PLATE AND SLAB ANALYSIS

In a previous s ec t i on of t h i s r e p o r t a framework model vras

s e l e c t ed which i s appropr ia te for t he ana lys i s of continuous p l a t e s

on e l a s t i c foundat ions . Appropriate c r o s s - s e c t i o n a l p rope r t i e s of

t he framework elements were determined in such a way t h a t de f l ec t i ons

of t he framework mcxiel a r e t he same as those of the continuous p l a t e

when t he two a re subjected t o s t a t i c a l l y equivalent l oads . In the

a n a l y s i s conducted here only one-quar ter of the p l a t e i s needed

s ince a l l loads and supports a r e symirietrical.

Figure 15 i s a plan view of the p l a t e and s l a b analyzed in t h i s

s tudy . The framework mesh shown has ^9 nodes. Remembering t h a t

each node has t h r e e degrees of freedom, i t i s evident t h a t t h e r e

a r e 1^7 ( i . e . 3 x 49) simultaneous equations t o be solved.

To f a c i l i t a t e the ana ly s i s a computer program was w r i t t e n t h a t

assembles the s t i f f n e s s matr ix for the s t r u c t u r e and solves the

r e s u l t i r g equat ions for de f l ec t i ons a t t he nodes. The program was

compiled in such a manner t h a t only the p l a t e or s l ab dimensions.

Young's modulus, shear modulus, Po isson ' s r a t i o , and the value of

MSR a r e r equ i r ed as input da ta so long as a ^9 node system i s

analyzed. Systems of s i z e s other than 49 nodes r e q u i r e t h a t a d d i ­

t i o n a l da ta be compiled t o account for any change in the number of

nodes.

The method of a n a l y s i s included here i s not l imi ted t o p l a t e s

28

29

FIGURE 1 5 . TYPICAL METHOD OF SECTIONING BODY ALONG LINES OF

SYMMETRY FOR ANALYSIS UNDER CENTER LOAD

30

or slabs under the action of a center load. Any type of loading

condition can be treated using the framework method. Uniform loads

are treated by distributing the total force to the nodal points of

the framework. Therefore it may be necessary to use a finer mesh

of frameworks for accurate simulation of uniform loads than is

necessary for concentrated loads.

As mentioned previously two series of tests were available

which were sufficiently documented to warrant comparison of results

from analysis and experiment. One test was conducted on a 30 x

30 x 1 ft concrete slab while the other was applied to a 9 x 9 x

0.125 in aluminum plate. Application of the framework method of

analysis to the plates on elastic foundations produced results

which were in excellent agreement with experimentally obtained

deformations. The same number of framework elements were used for

the concrete slab and the aluminum plate.

The load-deformation curve used to establish the MSR for the

concrete slab is shown in Figu.re I6. The data of Figure I6 was

taken from tests using a 12 in diameter plate placed on the surface

of a 6 in gravel base after the gravel had been compacted in place

on top of a 10 ft clay layer. The load-deformation curves for the

clay soil used in the evaluation of the aluminum plates is shown

in Figure 17. The various MSR shown in Figure 17 indicates an

attempt to match MSR with expected plate deflections. The results

of the use of different MSR for differeni. load conditions is dis­

cussed later.

31

o

S3

8 P < P CO

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COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS

Figures 18 through 20 are plots which compare the deflections

of the plate and slab as computed by the framework method of analysis

with the appropriate experimentally obtained deflections.

Figures 18 through 20 also show the results of analyses conducted

by the source organization (5, 9). Good agreement was obtained

in every case for the framework method.

Concrete Slab

In Figure 18 the deflections of the- 30 x 30 x 1 ft concrete

slab tested by Rao (9) are compared with those obtained from analysis

using the framework mcxiel. Several series of experimental tests

were run on the same concrete slab. It was evident that some per­

manent deformation had occurred. The properties of the subgrade

were subsequently altered as a result of repetitive loading. The

extent of this alteration could not be resolved for use in the frame­

work analysis. For this reason, only the data corresponding to

the first series of tests was used.

The fact that the measured deflection directly under the load

on the concrete did not agree with the analysis does not discredit

the study. An inconsistency of this magnitude can be the result

of cracked concrete, gage failure or other experimental error.

The continuity of the experimental results at all other locations

on the slab substantiate this reasoning.

Aluminum Plate

Figures 19 and 20 show a comparison between the analysis and

34

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experimental evaluation of the 9 in x 9 in x 0.125 in aluminum

plate. Much of the data from this series (Reference 5) was taken

for foundation reactions well into the non-linear range. As stated

earlier, the MSR varies according to the range of deformations

occurring in the foundation and therefore careful selection of

the MSR 3s dictated. Figure 21 illustrates the sensitivity of

the calculated deflection to the value of MSR selected. The curves

indicate that proper selection of the MSR is imperative if compa­

rable results are to be obtained.

CONCLUSIONS

Based on the results of the analysis, the following conclusions

are drawn:

(1) A framework of the Hrennikoff type as modified in Reference

2 and further modified in this thesis may be used to predict the

bending behavior of plates or slabs on elastic foundations.

(2) The full potential of the framework method of analysis

for plates and slabs on elastic foundation can not be realized with­

out accurate data reflecting the load-deformation behavior of the

subgrade material. Any type of material can be used for a subgrade

in an analysis using this method provided that some appropriate

information for determining MSR is available. In the absence of

load-deformation curves for a foundation material, any one of the

methods outlined in Chapter III can be used. However the accuracy

of the analysis results will be reflected in the accuracy of the

chosen value for MSR. ^

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LIST OF REFERENCES

1* Hrennikoff, A. "Solution of problems of elasticity by the framework method". Journal of the Applied Mechanics, ASME, Vol. 63., 1941, pp. A-I69-A-I73.

2. Yettram, A. and Husain, H. "Grid-framework method for plates in flexure". Journal of the Engineering Mechanics Division, ASCE, Vol. 91., No. EM3, I965, pp 53-64.

3. Smith, J. "A method for analyzing nonlinear plate structures". Doctoral Dissertation, University of Arizona, I968.

4. Smith, J. "Analysis and Full-Scale Testing of an Aircraft Shelter", AFWL-TR-70-170, Air Force Weapons Laboratory, 1971.

5. Agarwal, S. and Hudson, W. "Experimental Verification of Discrete-Element Solution for Plates and Pavement Slabs", Center for Highway Research, The University of Texas, Research Report 56-15, April 1970.

6. Siddiqi, Q, and Hudson, W. "Experimental Evaluation of Subgrade Modulus and Its Application in Model Slab Studies", Center for Highway Research, The University of Texas, Research Report 56-16, Jan. 1970.

7. Girijavallabhan, C. and Reese, L. "Finite-element method for problems in soil mechanics". Journal of the Soil Mechanics and Foundations Div. ASCE, Vol. 94., SMZ, 1968, pp. 473-496.

8. Timoshenko, S. and Woinovjsky-Krieger, S, "Theory of Plates and Shells", Engineering Societies Monograph 2nd Edition, McGraw-Hill Book Company, New York, 1959.

9. Rao, H. "Nondestructive Evaluation of Airfield Pavements (Phase 1)", University of New Mexico, CERF, Air Force Weapons Laboratory AFWL-TR-71-75, Dec. 1971.

10. Bowles, J. "Foundation Analysis and Design", McGraw-Hill Book Company, New York, I968.

11. Lambe, T. and Whitman, R. "Soil Mechanics", John Wiley and Sons, Inc., New York, Series in Soil Mechanics, I969.

12. Pestel, E. and Leckie, F., "Matrix Methods in Elastomechanics", McGraw-Hill Book Company, New York, 1963.

39

40

13. Terzaghi, K, "Evaluation of coefficient of subgrade reaction", Geotechnique, London, Vol. 5, No. 4, pp. 297-326, 1955.

14. Vesic, A. "Bending of beams resting on isotropic elastic solid. Journal of the Engineering Mechanics Div., ASCE, Vol. EI'IZ-87, pp. 35-53.

15. Hetenyi, M. "Beams on Elastic Foundation", The University of Michigan Press, Ann Arbor, Mich., 1946.

16. Skempton, A. W. "The Bearing Capacity of Clays", Building Research Congress, 1951.