Flexure & Shear in RC Members

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7 December 2004 – The Structural Engineer|25

technical note: flexure & shear

IntroductionDuring the past 100 years, methods of

design of reinforced concrete structureshave been reviewed and improved

through research,using analytical as

well as experimental techniques. For

most common design situations, horizon-

tal elements of the structure, beams or

slabs, transfer loads to vertical members

or columns through two commonly

understood mechanisms – shear and

flexure.Although they must occur simul-

taneously in theory, it is often considered

convenient to treat them separately in

zones where they appear to be predomi-

nant. Researchers have developed

analytical methods for design against

flexure, both for general design and forcomplex and rigorous design situations.

A significant proportion of research on

shear design, however, has opted for

development of empirical methods for

estimating the shear resistance. These

methods are based on using a number of

parameters and they are supported by a

large number of tests on beams and

slabs. However, these empirical methods

could not be verified satisfactorily using

a theoretical approach, in any way

similar to the verification of rigorous

analysis methods used in flexural design.

This paper examines the development of

shear design methods and discusses thefeasibility of a viewing shear and flexure

together.

Common failure mechanismsIn a simply supported beam, a load at

mid-span and the resulting flexure is

resisted by a couple, tension at the

bottom and compression at the top,

giving smooth trajectories of stresses in

the mid-span region.The beam could fail

if the magnitude of the load exceeds a

certain critical limit and the resulting

failure mechanism has been sufficiently

investigated. Most limit state design

methods include idealisation of compres-

sion stress block, e.g.a rectangular block

with its depth limited to half the effec-

tive depth.The general aim is to induce

yielding of tension steel, which would

lead to a ductile failure and provide suffi-

cient warning, in preference to failure of

concrete in compression, which could be

brittle and sudden.

For the part of beam near the support,

the stresses could reach a disturbed

state, contrary to the smooth trajectories

of stresses parallel to the top and bottom

faces of the beam.The mode of failure in

this region is generally known as the

diagonal shear failure and it has not

been analytically explained to any

degree of satisfaction, despite several

decades of study. The general limit state

design rules are aimed at avoiding the

shear failure,which is brittle and

sudden, using different parameters asso-

ciated with the so-called shear capacity

of a reinforced concrete member, for

example, the strength of concrete, beam

dimensions,amount of tension steel,

amount of web reinforcement or links,

distance of the applied load from the

support, etc. The common design

methods use the parameter ‘shear stress’as a means of ensuring adequate margin

of safety against the undesirable failure,

whereby an applied shear stress has to

be less than the limiting design shear

stress.Such shear stress does not really

exist across the cross-section of a rein-

forced concrete beam.The concept of

‘shear stress’ perhaps belongs to the very

early stages of reinforced concrete

design, which were influenced by proce-

dures for designing timber joists.

Initial concepts of design againstshear

Importance of concrete strength and function of links as the tension member of

a truss

Mörsch truss analogy was introduced in

about 19032 and it was aimed at estimat-

ing the shear resistance of a concrete

section. If the applied shear exceeded the

shear resistance of concrete, the early

classical Mörsch truss analogy method

required provision of shear reinforce-

ment in the form of links for the entire

applied shear. The beam was treated as a

cracked beam,acting like a truss with

the compression block and the tension

steel as the two chords. The diagonal

compression struts (inclined at 45°) were

provided by concrete strips in between

the cracks and the vertical links

provided the tension members (Fig 1).

The entire applied shear was carried by

tension in the links subjected to a

permissible tensile stress, a fraction of

the yield stress of steel.

Mörsch3 commented in 1922 that it

was not possible to carry out a mathe-

matical evaluation of the slope of a shear

crack, which determined the inclination

of concrete struts. He accepted the value

of 45° for this slope and arrived at the

usual calculation for links, 45° being an

assumption as unfavourable as possible

for all practical purposes. Even today,

evaluation of the slope of a shear crack

continues to be the main difficulty in

attempting a rigorous analysis and

design against shear.

It was recognised in 1907 by Talbot4

that the shear strength depended on the

strength of concrete, the tension rein-

forcement and the length of beam. He

concluded that the stirrups did not actu-

ally develop stresses as high as predicted

by the 45° truss analogy. He deduced,

therefore, that part of the shear force

must be carried by concrete. Similar

observations were made by Richard5 in

1927.

Significance of shear span

Kani developed a model called ‘Tooth

Model’ during the 1960s5,6. Kani’s

concept was based on the idealisation of flexural shear failure mechanism as

breaking off a concrete tooth between

two flexural cracks. Kani looked upon a

concrete beam with cracks as a ‘comb’,

the ‘teeth’ being the segments of concrete

between the cracks and the ‘spine’ being

the uncracked compression zone.

The tension steel was at the lower

edge of the teeth and the bond forces at

this level applied the load to the teeth.

This load varied linearly from zero at the

support to the maximum where the

bending moment applied to the beam

was the maximum, generally at the point

of application of the load (Fig 2). Shear-cracking was assumed to be the result of

a flexural failure of the teeth subjected to

this loading and a long beam would fail

immediately if the teeth broke.A short

beam, however, would continue to

perform as an arch and support the

applied external load.

Kani produced two relations,which

were supported by tests and showed that

the shear strength interacted with ‘shear

span’, distance between the applied load

and the support.The line showing capac-

ity of teeth assumes linear variation of

the applied ‘bond force load’.The line

representing the arch action strengthwas derived from a geometrical consider-

ation that the beam strength was a func-

tion of the compression block at the load

point.The ratio of arch strength to the

beam flexural strength was related to

Flexure andshear in

reinforcedconcretemembersDr Satish Desai sets out the history

and development of the design of reinforced concrete membersinfluenced in the main by their

flexure and shear crackingproperties. The methods of calculating these properties over the last 100 years are highlighted

Fig 1.Mörsch truss



26|The Structural Engineer – 7 December 2004


the reduction of the depth of compression

block, which, in its turn, was related to

the shear-span ratio or ratio of the ‘shear

span’ to the effective depth of beam. Kaniused this approach to obtain the ratio of

the ultimate bending resistance (Mu) and

the theoretical flexural capacity (Mfl),

(Fig 3). Kani did not consider the effects

of dowel action of tension steel and

aggregate interlock across the crack,

which have been identified as contribut-

ing factors to shear resistance of a beam

(Fig 4).

Aggregate interlock

The aggregate interlock is the resistance

to slippage,attributed to friction along

the shear crack.This friction is generated

after the crack is initiated by an appliedshear exceeding the shear-cracking load.

Gravel aggregate performs better than

limestone and lightweight aggregate,

since the strength of aggregate and

matrix within the concrete is an influen-

tial parameter. The beneficial effect of

aggregate interlock increases with the

increase in size of aggregate.The effect of

the size of aggregate on aggregate inter-

lock and,hence,on the shear resistance of

a member is represented by a multiplier,

commonly known as the ‘depth factor’. If

the size of aggregate is the same, the

aggregate interlock will have a greater

benefit for shallower sections comparedwith the benefit for deeper sections.The

size of coarse aggregate (say 20mm) is

normally the same for different strengths

of concrete,used in beams with different

depths. An allowance is made,therefore,

to the shear strength based purely on the

compressive strength of concrete and

without any regard to the size of aggre-

gate.

(The depth factor is also meant to

account for higher shear carrying capac-

ity of shallower members, according to

fracture energy principle.)

Dowel action of tension steel

The contribution of dowel action to the

shear resistance of a beam is mobilised

when the shear crack crosses the tension

steel.As the shear force increases, the

diagonal crack opens up.This action of

the increasing shear force produces

tensile stresses in concrete surrounding

the tension steel and an increase in the

dowel force.This combination produces

splitting cracks in concrete along the line

of the tension steel and a reduction in

the bond between concrete and the steel.

This triggers redistribution of stresses,

as the stiffness of the dowel bar is

rapidly lost.This loss of dowel stiffness

reduces the resistance afforded by the

dowel to the rotation of beam segments

on either side of the crack.The dowel

splitting is accelerated as the initial

crack opens up with further increase in

shear, leading to the final failure.

Development of shear design rulesin the UKThe Institution of Structural Engineers7

produced a report in 1969, giving a

consolidated resume of research in shear

design.This report has led the way to

development of shear design rules in theUK.The IStructE report has illustrated

a number of theoretical approaches to

shear design,which have led to some

important principles as given below:

• Incorporation of shear-span ratio in

shear design rules would present prob-

lems related to beams continuous over

supports or fixed at their ends. For

calculating the shear-span ratio for

such beams, it would be necessary to

assume a distribution of moments

using elastic analysis.However, this

distribution would be unlikely to

correspond to the actual distribution of

moments at failure and, hence, the value of shear-span ratio used in the

design would be incorrect and this

could undermine the basis of design.

• The shear failure is influenced by the

inclination of compressive stresses and

the resulting principal tensile stresses.

With the increase in applied shear, the

depth of neutral axis or the compres-

sion block is reduced and, if the tensile

stresses exceed the tensile strength of

concrete in the neutral axis region,

shear failure will occur.

• It is impracticable to estimate individ-

ual contributions to the shear capacity,

provided by strength of concrete,aggregate interlock and dowel action

of tension steel.Calculation of crack-

width and the control of inclined crack

widths are not suitable for generalisa-

tion and for developing practical guid-

ance and design rules.Additionally,

links would have influence on aggre-

gate interlock and, even more substan-

tially, on the dowel action, which

makes an explicit evaluation of aggre-

gate interlock and dowel action very

complex, as structural beams would

invariably have links.The same is true

for slabs,as the cross reinforcement

influences the dowel action, similarly

to the influence of links in beams. The

major factors governing aggregate

interlock and dowel action of tension

steel are the strength of concrete and

the amount of tension steel. If a rule

for estimating shear capacity includes

concrete strength and amount of steel,

the benefit afforded by these mecha-

nisms can be accounted for by adjust-

ing a constant multiplier in the rule.

Placas and Regan8 studied different

modes of shear failure and derived a

semi-empirical equation for shear crack-

ing resistance (V cr, in psi units) as

follows:

V cr=8(f ckρ )1/3 bd ≤ 12(f ck)

1/3 bd

This rule excludes any parameters

representing aggregate interlock and

dowel action of tension steel.It has an

empirically adjusted constant, evaluated

as ‘8’ from results of tests on beams with

breadth ‘b’, effective depth ‘d’, cylinder

strength of concrete ‘f ck’ and the percent-

age of tension reinforcement ‘ρ ’.With

these parameters, it is considered that

the aggregate interlock and the dowelaction effects are accounted for, without

any separate quantification of these

mechanisms.

Zsutty9 proposed the following rule for

V cr in psi units,which introduced the

shear-span ratio ‘a’:

V a

f bd 60

100

/

cr

ck1 3

= tf p

Zsutty’s rule agrees with Regan’s rule

when the value of shear-span ratio (a) is

‘4.22’ and, for a value of ‘a’ as 2.5, it gives

an estimate of V cr 20% higher than that

given by Regan’s rule. Regan’s rule,

therefore, seems to cover the critical caseof shear-span ratio with an extra reserve,

without having to evaluate the ratio.

Regan’s equation resembles the rule

for design concrete shear stress given in

the current British Standard,BS 811010,

which does not include the parameter ‘a’

or the shear-span ratio but uses it as a

limit for allowing enhanced shear stress

for sections close to the support. The BS

8110 rule is used to arrive at the contri-

bution of concrete to the shear resistance

(V C) or shear capacity of a beam without

any links:

.V f d bd kN a

0 27 400

1000 2/

/

cm

cu1 3

1 4

$= c t_ d _i n i

If ‘a’ is less than 2,V c is increased by

multiplying the value given by the above

equation by a factor of (2/a).The other

parameters in this rule are given as

Fig 2. (Left)Kani’s ToothModel

Fig 3. (Below)Comparisonbetween capacityof arch andcapacity of teeth

Fig 4. (Above)Shear resistancemechanisms





follows:

.bd

A100 3 0

st#= t

A st = Amount of tension steel (mm2)

f cu < 40 N/mm2

400/d > 1.0 (depth factor d being the

effective depth)

γ m = partial factor for materials = 1.25

BS 8110 does not allow the applied

shear stress, v, to exceed 0.8(f cu)0.5 or

5N/mm2, whichever is the lesser. Beyond

this limit, the shear carrying capacity of

the member cannot be enhanced with

provision of shear reinforcement. This is

to ensure a safe limit on the compressive

stresses in the web concrete.

Rules in ACI code11 are similar to the

BS 8110 rules,except for the use of cylin-

der strength (f’c) with a power of 0.5 in

place of ‘1/3’,which reflects the difference

in the implied tensile-compressive rela-

tionship.Equation 11.5 includes shear-

span ratio concept and it applies to

beams subjected to shear and flexure.

For beams with ratio of span to effective

depth (d) greater than or equal to 5.0, ‘V c’

is given in kN as shown below:

. .V f bd A

M V

d bd

0 158 17 241000

c cs

u

u= +l< F

kN

The ACI code also gives a simplified

Equation 11-3,which can be written as

follows:

V c = 0.166 √f’c bd/1000 kN

Both BS 8110 and the ACI code follow

the addition principle for estimating

shear capacity of a beam with links.The

total ultimate shear capacity (V u) is

given as addition of the shear capacity

contribution of concrete to that of the

links (V L).

V L = f yv × A sw × d/s

Where A sw is the area of cross-section

of links and ‘s’ is the spacing of links.

It must be noted that the ‘addition

principle’ is a compromise, since the

links perform a multiple function in a

sense of qualitative improvement in the

concrete section.The links make the

concrete section ‘ductile’ and provide

resistance to tensile stresses across the

cracks.They bridge over any local weak-

ness over the depth and avoid triggering

of a premature failure. The links also

enhance the bond of concrete around the

tension steel and increase its effective-

ness in providing dowel action and

clamping action to arrest the progress of

a predominant shear crack. On the other

hand, influence of aggregate interlock is

reduced and the size effect becomes less

significant as confinement resulting from

high strain gradients in shallower beams

is mitigated with provision of links. In

conclusion, the character of concrete

section is fundamentally changed with

introduction of links and it is not possi-

ble to quantify all benefits with any

satisfactory degree of precision.

Furthermore, it is questionable whether

the links would always reach the limit-

ing tensile stress in steel that is

normally used in evaluation of ‘V L’.

However, the sum ‘V c + V L’ is assumed to

account for the overall shear strength of

a member provided with links, so that all

qualitative benefits of links are included

in the estimate of shear strength.This

has been supported by tests on beams

representing common structural

members subjected to shear and flexure.

Truss analogies and combinedconsideration of flexure and shearCollins developed the diagonal compres-

sion field theory12, which was based on

compatibility conditions for strains in the

compression struts and the transverse

and longitudinal steel.The theory, in its

simplified form,assumes that the longi-

tudinal steel is symmetrically placed and

the web steel is vertical.The effect of

bending moment on truss members is

not considered.This approach considers

average values of stresses and strains

and ignores any local effects.

Additionally, it is based on the following

underlying principles, which could be

valid only under certain idealised condi-

tions:

• Disregarding any tensile strength of

concrete after cracking,

• Attributing resistance to shear to the

diagonal compression field,and

• setting an upper limit for shear capac-

ity at the ultimate load, by assuming

yielding of the longitudinal steel.

Vecchio and Collins13 modified the

compression field theory with general

improvements in the following aspects:

• Consideration of the presence of

tensile stresses between cracks as well

as studying the state of stresses influ-

encing the compressive strength of

concrete;

• Assumption that the strain in concrete

is equal to that in the steel;

• Assumption that the principal strain

axis is coincidental with the principal

stress axis;

• Evaluation of the relationship of both

tensile and compressive stresses with

the corresponding strains;

• Inclination of the compressive fields as

a function of the longitudinal, diagonal

and transverse strains in the concrete;

and

• Treating the principal compressive

stress as a function of compressive

strain and the corresponding tensilestrain.

A simultaneous consideration of axial

forces, bending,shear and torsion could

be vital for designing the walls and

shells of structures, such as those of

submerged containers, offshore plat-

forms and nuclear container vessels.A

combined application of these actions on

a two-dimensional element produces an

important state of stress known as the

membrane stress. Hsu14 has described

this two-dimensional element as the

membrane element, which forms the

basic building block of a large variety of structures. Computer analysis and the

design of structural frames, comprising

an assembly of members with such

membrane elements, could be carried out

to meet the fundamental compliance

criteria: stress equilibrium,strain

compatibility and the constitutive laws of

mechanics of materials (steel and

concrete). Hsu has described application

of various unified theory models to the

design of reinforced concrete members,

mostly similar to those given above and

with an addition of strut-and-tie model

and Softened Truss Model.

Strut-and-tie model is particularlyuseful for designing knee-joints of a

portal frame, corbels, openings in beams,

articulated or halved joints, etc.The

strut-and-tie model is based on arrang-

ing struts and ties within the member in

Fig 5.Strut-and-tiemodel



28|The Structural Engineer – 7 December 2004


such a way that the internal forces are in

equilibrium with the boundary forces

(Fig 5).This technique is very well illus-

trated by Schlaich et al15 with many

examples of application of this model.

This model is suitable for estimating

shear resistance as well as flexural

resistance as shown in Fig 6. The model

combines the contribution of diagonal

concrete struts as well as the vertical

tension in links for resistance to shear.

The inclination of concrete compression

struts is α , the same as the angle

assumed to be made by the inclined

cracks with respect to the longitudinal

axis of the beam. If ‘d v’ is the lever arm of

the truss, each cell of the truss will have

a horizontal dimension of d vcotα , except

for the end cell, which will have half this

dimension.This model has been refined

with considerable research, resulting in

an improved understanding of shear

flow, the behaviour of the nodes where

the struts and ties intersect and sizing

the dimensions of the struts and ties.

The strut-and-tie method has its limi-

tations as it complies only with equilib-

rium condition and, where necessary,

supplementary calculations are needed

for considerations of the compatibility

conditions.It is recommended that it

should be used only by skilled and expe-

rienced designers, who have an under-

standing of stress flows, bond between

steel and concrete,and anchorage of steel

in local regions.If such aspects are inad-

equately considered,serviceability condi-

tions may not be complied with and

premature failures could occur, as theapplication of this model by itself does

not cover these points.

Softened Truss Model employs the

actual stress–strain relationship for the

materials, instead of the linear one corre-

sponding to Hooke’s law for concrete and

steel.For concrete, the stress-strain curve

has two characteristics: first, it is non-

linear and second,as a result of cracking,

the compression in concrete is ‘softened’

due to tensile stresses, which are gener-

ated in the perpendicular direction (Fig

6).This model uses the softened biaxial

constitutive law of concrete and can

predict shear and torsional strengths,aswell as the corresponding load-deforma-

tion behaviour of a structure throughout

its post-cracking loading history.

Hsu has proposed a number of simpli-

fications to the theoretical use of the

models, subject to certain limitations.

However, he has recommended that

these simplifications should be used only

by designers who know the subject, in

order to avoid unsafe solutions through

incorrect applications.

Future for general design rules forreinforced concrete

The current practice tends to usecomputer software for rigorous analyses

against flexure but design against shear

continues to use empirical rules,which

serve the purpose of reducing the risk of a

sudden and brittle failure.Although the

shear failure is known to be a function of

tensile strength of concrete, the empirical

rules use compressive strength as an

input parameter, implying a direct rela-

tionship between the tensile strength and

the measured compressive strength.The

writer has drawn attention to potential

benefits of using measured tensile

strength,particularly for specially

designed mixes16. Such mixes would have

improved durability, owing to their

improved microstructure, i.e. reduction in

voids and permeability.These attributesare shown to result in improved tensile-

compressive relationship as well,giving

higher shear resistance contribution of

concrete.

The current BS 8110 and the ACI Code

have continued to use the addition princi-

ple,which is a compromise solution for a

complex problem.Hsu has acknowledged

the validity of ‘contribution of concrete’

(V C) and explained difficulties in its math-

ematical derivation14. He has observed

from test results that the shear strength of

membrane elements is made up of two

terms,one attributable to steel and the

other attributable to concrete,V C. He hasremarked that the existence of the term

V C is apparently caused by the fact that

the actual direction of cracks is different

from the assumed direction of post-crack-

ing principal stresses and strains.A theo-

retical approach to account for this actual

direction of cracks would require incorpo-

ration of the constitutive law relating

shear stress to the shear strain in the

direction of the cracks.This approach

would also require very complex equilib-

rium and compatibility equations.Hsu has

conceded that efficient algorithms to solve

the complex equations are needed before

the ‘contribution of concrete’ can bederived mathematically.This has led to

some approximations and compromises

implied in the ‘addition principle model’,

which assumes a fixed angle of inclination

of cracks and accounts for the contribution

of concrete to the shear resistance together

with the contribution of links suitably

spaced across the crack.

Eurocode EC217 may opt for the ‘vari-

able strut inclination method’.This

method allows variation in inclination of

concrete struts subject to certain condi-

tions.This could enable reduced provision

of links resulting from inclination of

compressive struts assumed to be shal-

lower than 45°,which is a compromise not

satisfactorily supportable by any scientific

analysis.

Variable strut inclination method is

associated with plastic theory, which could

apparently eliminate any gaps in under-

standing structural behavior of concrete,

particularly against the action of shear,

and it does not recognise the tensile

strength of concrete.The theory can lead

to an apparently coherent method for

dealing with shear and flexure, for

example,using a plasticity truss model.

This model is based on the principle

whereby both the longitudinal and the

transverse steel must yield before failure

and the role of concrete is limited to provi-

sion of diagonal struts.

According to Hsu, truss models cannot

account for the ‘contribution of concrete’,

which is a real part of shear resistance of

concrete elements.Additionally, the vari-

able strut inclination method is imperfect,

as it cannot be applied consistently to all

concrete elements, for example, flat slabs

and beams without shear reinforcement.

Perhaps future research could lead to

increase in tensile strength of concrete and

to development of the field of ‘concretetensile ties’, similar to that related to

compressive struts.In the meantime,it

remains questionable whether a truss

model could suitably lead to formulation of

general design rules, if it cannot be applied

to design of all structural elements in a

consistent manner.

Most structural beams are required to

have nominal links,to avoid any marginal

and unforeseen increase in the applied

shear and for the convenience of forming

reinforcement cages that would stay in

shape during concreting.Slabs,particu-

larly flat slabs,are often constructed

without shear reinforcement and provisionof depth depends on the limit on nominal

shear strength of concrete or V C. Depth of

a flat slab has considerable influence on

planning of a building and on economy of

construction.Any marginal reduction in

depth can provide better use of floor-to-

ceiling space and reduce the overall height

of the building, volume of concrete and

loads on columns and the substructure.

This could be achieved by using higher

grades of concrete and by using tensile

strength of special mixes as a parameter

in the shear design rule.16

ConclusionsSimultaneous consideration of axial forces,

bending,shear and torsion can be

achieved by using certain truss models,

which could serve specific and specialist

design situations.However, such analyses

Fig 6.Stress–strainrelationship inmembraneelements





must be done carefully by designers with

skills and experience in the fields of stress

flows,bond between steel and concrete,

and anchorage of steel in local regions,

Otherwise,serviceability conditions may

not be complied with and premature fail-

ures could occur, as the application of this

model by itself does not cover these points.

In the writer’s opinion, truss models

could not lead to a rational approach for

general use or for arriving at a solution for

the shear resistance of concrete suitable

for common structural design.Concrete is

a structural material with its own charac-

ter, which should not be ignored simply

because a failure mechanism is not fully

understood or it cannot be modelled to

mathematical perfection. It would be

unacceptable if one would adopt a model

for general design rules, similar to the

plasticity truss model, simply because it

appears to apply coherently for combined

consideration of shear and flexure.

Furthermore,abandoning shear strength

of concrete and reliance on imaginary

concrete struts is not viable for flat slabs

where links are often omitted.This seems

to be an anomaly, which may give rise to

an unreasonable degree of inconsistency

between shear design rules for beams and

flat slabs.The right way forward must be

to improve the tensile strength of concrete

and develop better techniques for its

measurement,so that it will continue to

have a role to play in assessment of

strength of concrete members. se

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7. Shear Study Group Report : The Institution of Structural Engineers, London, Jan. 1969

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