MODULE – I

STRESS – STRAIN CHARACTERISTIC OF CONCRETE UNDER SINGLE

AND MULTI – AXIAL STRESSES

BEHAVIOUR OF CONCRETE UNDER UNIAXIAL COMPRESSION

The strength of concrete under uniaxial compression is determined by loading ‘standard test cubes’ (150 mm size) to failure in a compression testing machine, as per IS 516 : 1959. The test specimens are generally tested 28 days after casting (and continuous curing). The loading is strain-controlled and generally applied at a uniform strain rate of 0.001 mm/mm per minute in a standard test. The maximum stress attained during the loading process is referred to as the cube strength of concrete. the cube strength is subject to variability; its characteristic (5-percentile) and mean values are denoted by fck and fcm respectively.

‘‘Characteristic strength is defined as the strength of material below which not more than 5 percent of the test results are expected to fall’’

Accordingly, the mean strength of the concrete fcm (as obtained from 28-day compression tests) has to be significantly greater than the 5 percentile characteristic strength fck that is specified by the designer

In some countries, ‘standard test cylinders’ (150 mm diameter and 300 mm high) are used instead of cubes. The cylinder strength is found to be invariably lower than the ‘cube strength’ for the same quality of concrete; its nominal value, termed as ‘specified cylinder strength’ by the ACI code , is denoted by . ′fc

It should be noted that among the various properties of concrete, the one that is actually measured in practice most often is the compressive strength. The measured value of compressive strength can be correlated to many other important properties such as tensile strength, shear strength, modulus of elasticity, etc.

INFLUENCE OF SIZE OF TEST SPECIMEN

It has been observed that the height/width ratio and the cross-sectional dimensions of the test

specimen have a pronounced effect on the compressive strength (maximum stress level)

obtained from the uniaxial compression test.

The standard test cylinder has a diameter of 150 mm and a height-diameter ratio equal to 2.0.

With reference to this ‘standard’, it is seen that, maintaining the same diameter of 150 mm,

the strength increases by about 80 percent as the height/diameter ratio is reduced from 2.0 to

0.5; also, maintaining the same height/diameter ratio of 2.0, the strength drops by about 17

percent as the diameter is increased from 150 mm to 900 mm. Although the real reasons for

this behaviour are not known with certainty, some plausible explanations that have been

proposed are discussed below.

Proper measure of uniaxial compressive stress can be obtained (in terms of load divided by

cross-sectional area) only if the stress is uniformly distributed across the cross-section of the

longitudinally loaded test specimen. Such a state of stress can be expected only at some

distance away from the top and bottom surfaces where the loading is applied (St. Venant’s

principle) — which is possible only if the height/width ratio of the specimen is sufficiently

large.

Uniaxial compression implies that the specimen is not subject to lateral loading or lateral

restraint. However, in practice, lateral restraint, known as platen restraint, is bound to

manifest owing to the friction between the end surfaces of the concrete specimen and the

adjacent steel platens of the testing machine. This introduces radial (inward) shear forces at

the top and bottom surfaces, resulting in restraint against free lateral displacements.

The effect of this lateral restraint is to enhance the compressive strength (maximum stress

prior to failure) in the longitudinal direction; this effect dies down with increasing distance

from the platen restraint. Thus, the value of the compressive strength depends on the

height/width ratio of the specimen; the greater this ratio, the less the strength, because the less

is the beneficial influence of the lateral restraint at the (weakest) section, located near the

mid-height of the specimen.

The reduction in compressive strength with increasing size, while maintaining the same

height/width ratio, is attributed to size effect — a phenomenon which requires a fracture

mechanics background for understanding.

Fracture mechanics is the field of mechanics concerned with the study of the propagation of

cracks in materials. It uses methods of analytical solid mechanics to calculate the driving

force on a crack and those of experimental solid mechanics to characterize the material's

resistance to fracture.

From the above, it also follows that the ‘standard test cube’ (which has a height/width ratio of

1.0) would register a compressive strength that is higher than that of the ‘standard test

cylinder’ (with a height/diameter ratio of 2.0), made of the same concrete, and that the

cylinder strength is closer to the true uniaxial compressive strength of concrete. The cube

strength is found to be approximately 1.25 times the cylinder strength , whereby f′c ≈ 0.8 fck.

STRESS STRAIN RELATIONSHIP OF CONCRETE

Concrete is assumed elastic, isotropic, and homogeneous & obeys Hooke's law (F∝x).

Actually none of these are strictly true & concrete is not an elastic material. The response of

concrete to applied load is quite complex. Stress - Strain relationship of aggregate alone/

cement paste alone are good straight line. But the stress-strain relationship of concrete -

aggregate +cement paste somewhat a curved relationship. It has been found that this

nonlinear pre-peak behaviour is caused by a progressive growth of three different types of

cracks:

• Cracks running through the mortar matrix;

• Cracks running through the aggregates;

• Cracks at the bond between mortar and aggregates.

To some extent the micro- cracks (mostly bond cracks) at the inter face of the aggregate and

cement paste - already presented in the unloaded concrete (due to shrinkage {effect due to

decrease in volume due to loss/consumption of water}, temperature effects {effect due to

difference coefficient of thermal expansion} in and other causes). Cracks running through the

aggregates are found when the difference in aggregate and matrix properties decreases, for

example, in high-strength concrete.

It is observed that the interface between aggregate and mortar has a very different structure

and represents the weakest link in normal-strength concrete (note: in the case of other types

of concrete this interface strength may be higher than for example - the aggregate strength. In

general it can be said that the ratio between aggregate, bond and cement paste strength

determinates the properties of a certain type of concrete.

Fig. Progressive failure of concrete under uniaxial compression

(from a to d, for increasing axial deformation)

The stress-strain curve is obtained from uniaxial compression tests. The total area under the

stress-strain curve can represent the amount of energy absorbed by the specimen under

loading. The compressive stress-strain behavior of concrete is a significant issue in the

flexural analysis of reinforced concrete beams and columns

From the graph it is clear that maximum stress is reached at a strain approximately equal to

0.002; beyond this point, an increase in strain is accompanied by a decrease in stress. Usually

the strain at failure is in the range of 0.003 to 0.005.

We can see that the stress-strain curve consists of four parts:

In initial stages the stress – strain curve is almost a straight line, hence behaving like

an elastic material, on which the working – stress method of design was based on.

The non-linearity comes when stress level exceeds about one-third to one-half of the

maximum. This is due to which the micro-cracks already presented in the unloaded

concrete start to propagate. At stresses between 0.5fc' and 0.7fc' adjacent bond cracks

at the interface of mortar and aggregates, caused by the different stiffness of the two

materials, start to bridge in the form of mortar cracks, due to stress concentrations at

the tips of bond cracks.

At 75 to 90% of the ultimate stress, the stress reaches a critical stress level for

spontaneous crack growth roughly parallel to the direction of the applied loading

under a sustained stress. Cracks propagate rapidly in both the matrix and the

transition zone. Failure occurs when the cracks join together and become continuous.

The descending part, along which strain increases while stress decreases (descending

branch). This phenomenon, called strain softening is attributed to the unstable

propagation of the internal cracks, i.e., the concrete tends to expand laterally, and

longitudinal cracks become visible when the lateral strain (due to the Poisson effect)

exceeds the limiting tensile strain of concrete (0.0001—0.0002). As a result of the

associated larger lateral extensions, the apparent Poisson’s ratio increases sharply ie,

laterally the dimension increases more.

It is not easy to trace the descending part of the stress –strain curve, since the

specimen fails explosively because it can’t absorb the release in strain energy from

the machine when the load decreases after max stress. And it is achieved only through

controlled application of the load through a rigid testing machine…

Although the shape of the descending part is generally not accounted in most routine

designs, for an accurate and rational design of the structures subjected to unusual

loading such as earthquakes, it is desirable to know the complete stress-strain curve.

The entire stress-strain curve of concrete is also useful for investigating the ductility

of concrete.

Regarding the ductility – we can note that, for higher strength concretes, the

compressive stresses drop faster than those of lower strength concretes after passing

the peak strengths, while considering the descending curve it is almost flat in low

grade concrete shows that it can be have long deformations for a small range of stress.

Accordingly, it can be concluded that lower strength concretes have more ductility

than higher strength concretes.

MODULUS OF ELASTICITY.

Modulus of elasticity of concrete, Ec, is a measure of the resistance to deformation

of concrete, which is subjected to compressive load. Its ratio b/w axial stress and

axial strain (in linear elastic range). Hence it is applicable only in the very initial

portion which is linear. Ie, when the load is low intensity, and of very short duration.

If the loading is sustained for a relatively long duration, inelastic creep effects come

into play, even at relatively low stress levels,Besides, non-linearities are also likely to

be introduced on account of creep and shrinkage.

Modulus of elasticity of concrete is effected by many factors, such as compressive

strength, density as well as degree and duration of loading. When concrete is loaded

in the service range under short-term loading, it can be assumed that concrete is an

elastic material and has a linear stress-strain relationship. On the other hand, when

concrete is subjected to sustained loading or repeated loading (e.g. Earthquake), the

stress-strain relationship is not linear anymore and engineers have to consider the

plastic strain or creep strain because of the gradual decreasing of modulus of elasticity

when concrete has more deformations.

There are three different kinds of modulus of elasticity: initial tangent modulus

(dynamic), secant modulus, and tangent modulus. Secant modulus is mostly used in

civil engineering. It is obtained by calculating the slope of the line linking the initial

point (the origin) and the point considered (usually at about 1/3 rd of the maximum

compressive stress).

Static Modulus of Elasticity - The short-term static modulus of elasticity (Ec) is used

in computing the ‘instantaneous’ elastic deflection

Secant modulus is mostly used in civil engineering It is obtained by calculating the

slope of the line linking the initial point (the origin) and the point considered (usually

at about 1/3rd of the maximum compressive stress).

The Code (Cl. 6.2.3.1) gives the following empirical expression for the static modulus

Ec (in MPa units) in terms of the characteristic cube strength fck (in MPa units):

Tangent Modulus - The slope of a line drawn tangent to the stress-strain curve at any

point on the curve.

Dynamic Modulus - The modulus of elasticity corresponding to a small instantaneous

strain. It can be approximated by the tangent modulus drawn at the origin. It finds

application in some cases of cyclic loading (wind- or earthquake-induced), where

long-term effects are negligible. However, even in such cases, the non-elastic

behaviour of concrete manifests, particularly if high intensity cyclic loads are

involved; in such cases, a pronounced hysterisis effect is observed, with each cycle of

loading producing incremental permanent deformation , which we will discuss in

cyclic loading.

Factors Affecting Modulus of Elasticity of Concrete

Effects of moisture condition

o Specimens tested in dry condition show about 15% decreases in elastic modulus as

compared to the wet specimens. This is explained by the fact that drying produces

more micro cracks in the transition zone, which affects the stress-strain behaviour of

the concrete.

This is opposite to its effects on compressive strength. The compressive

strength is increased by about 15% when tested dry as compared with the wet

specimens.

Effects of cement matrix ( Cement mortar paste)

o The lower the porosity of the cement paste, the higher the elastic modulus of the

cement paste.

o The higher the elastic modulus of the cement paste, the higher the elastic modulus of

the concrete.

Effects of transition zone

o The void spaces and the micro-cracks in the transition play a major role in affecting

the stress-strain behaviour of concrete.

o The transition zone characteristics affect the elastic modulus more than it affects the

compressive strength of concrete.

POISSON’S RATIO

This is another elastic constant, defined as the ratio of the lateral strain to the longitudinal

strain, under uniform axial stress. When a concrete prism is subjected to a uniaxial

compression test, the longitudinal compressive strains are accompanied by lateral tensile

strains. The prism as a whole also undergoes a volume change, which can be measured in

terms of volumetric strain.

Typical observed variations of longitudinal, lateral and volumetric strains are depicted in

Fig. It is seen that at a stress equal to about 80 percent of the compressive strength, there

is a point of inflection on the volumetric strain curve. As the stress is increased beyond

this point, the rate of volume reduction decreases; soon thereafter, the volume stops

decreasing, and in fact, starts increasing. It is believed that this inflection point coincides

with the initiation of major micro-cracking in the concrete, leading to large lateral

extensions. Poisson’s ratio appears to be essentially constant for stresses below the

inflection point. At higher stresses, the apparent Poisson’s ratio begins to increase

sharply.

Widely varying values of Poisson’s ratio have been obtained — in the range of 0.10 to

0.30. A value of about 0.2 is usually considered for design.

INFLUENCE OF DURATION OF LOADING ON STRESS-STRAIN CURVE

The standard compression test is usually completed in less than 10 minutes, the loading

being gradually applied at a uniform strain rate of 0.001 mm/mm per minute. When the

load is applied at a faster strain rate (which occurs, for instance, when an impact load is

suddenly applied), it is found that both the modulus of elasticity and the strength of

concrete increase, although the failure strain decreases

On the other hand, when the load is applied at a slow strain rate, such that the duration of

loading is increased from 10 minutes to as much as one year or more, there is a slight

reduction in compressive strength, accompanied by a decrease in the modulus of elasticity

and a significant increase in the failure strain, as depicted in Fig; the stress-strain curve

also becomes relatively flat after the maximum stress is reached.

MAXIMUM COMPRESSIVE STRESS OF CONCRETE IN DESIGN PRACTICE

The compressive strength of concrete in an actual concrete structure cannot be expected

to be exactly the same as that obtained from a standard uniaxial compression test for the

same quality of concrete. There are many factors responsible for this difference in

strength, mainly, the effects of duration of loading, size of the member (size effect) and

the strain gradient.

The value of the maximum compressive stress (strength) of concrete is generally taken as

0.85 times the ‘specified cylinder strength’ (f’c), for the design of reinforced concrete

structural members (compression members as well as flexural members). This works out

approximately to 0.67 times the ‘characteristic cube strength’ (fck) — as adopted by the

Code. The Code also limits the failure strain of concrete to 0.002 under direct

compression and 0.0035 under flexure.

BEHAVIOUR OF CONCRETE UNDER TENSION

Concrete is not normally designed to resist direct tension. However, tensile stresses do

develop in concrete members as a result of flexure, shrinkage and temperature changes.

Principal tensile stresses may also result from multi-axial states of stress.

Often cracking in concrete is a result of the tensile strength (or limiting tensile strain)

being exceeded. As pure shear causes tension on diagonal planes, knowledge of the direct

tensile strength of concrete is useful for estimating the shear strength of beams with

unreinforced webs, etc. Also, knowledge of the flexural tensile strength of concrete is

necessary for estimation of the ‘moment at first crack’, required for the computation of

deflections and crack widths in flexural members.

Concrete is very weak in tension, the direct tensile strength being only about 7 to 15

percent of the compressive strength. It is difficult to perform a direct tension test on a

concrete specimen, as it requires a purely axial tensile force to be applied, free of any

misalignment and secondary stress in the specimen at the grips of the testing machine.

Hence, indirect tension tests are resorted to, usually the flexure test or the cylinder

splitting test.

MODULUS OF RUPTURE

In the flexure test most commonly employed, a ‘standard’ plain concrete beam of a

square or rectangular cross-section is simply supported and subjected to third-points

loading until failure. Assuming a linear stress distribution across the cross-section, the

theoretical maximum tensile stress reached in the extreme fibre is termed the modulus of

rupture (fcr). It is obtained by applying the flexure formula:

where M is the bending moment causing failure, and Z is the section modulus

However, the actual stress distribution is not really linear, and the modulus of rupture so

computed is found to be greater than the direct tensile strength by as much as 60–100

percent. Nevertheless, fcr is the appropriate tensile strength to be considered in the

evaluation of the cracking moment (Mcr) of a beam by the flexure formula, as the same

assumptions are involved in its calculation.

The Code suggests the following empirical formula for estimating:

Where fcr and fck are in MPa units.

SPLITTING TENSILE STRENGTH

The cylinder splitting test is the easiest to perform and gives more uniform results

compared to other tension tests. In this test, a ‘standard’ plain concrete cylinder (of the

same type as used for the compression test) is loaded in compression on its side along a

diametral plane. Failure occurs by the splitting of the cylinder along the loaded plane. In

an elastic homogeneous cylinder, this loading produces a nearly uniform tensile stress

across the loaded plane as shown in Fig.

From theory of elasticity concepts, the following formula for the evaluation of the

splitting tensile strength fct is obtained:

where P is the maximum applied load, d is the diameter and L the length of the cylinder.

STRESS-STRAIN CURVE OF CONCRETE IN TENSION

Concrete has a low failure strain in uniaxial tension. It is found to be in the range of

0.0001 to 0.0002. The stress-strain curve in tension is generally approximated as a

straight line from the origin to the failure point. The modulus of elasticity in tension is

taken to be the same as that in compression. As the tensile strength of concrete is very

low, and often ignored in design, the tensile stress-strain relation is of little practical

value.

SHEAR STRENGTH AND TENSILE STRENGTH

Concrete is rarely subjected to conditions of pure shear; hence, the strength of concrete in

pure shear is of little practical relevance in design. Moreover, a state of pure shear is

accompanied by principal tensile stresses of equal magnitude on a diagonal plane, and

since the tensile strength of concrete is less than its shear strength, failure invariably

occurs in tension. This, incidentally, makes it difficult to experimentally determine the

resistance of concrete to pure shearing stresses. A reliable assessment of the shear

strength can be obtained only from tests under combined stresses. On the basis of such

studies, the strength of concrete in pure shear has been reported to be in the range of 10–

20 percent of its compressive strength. In normal design practice, the shear strength of

concrete is governed by its tensile strength, because of the associated principal tensile

(diagonal tension) stresses and the need to control cracking of concrete.

BEHAVIOUR OF CONCRETE UNDER COMBINED STRESSES

Structural members are usually subjected to various combinations of axial forces, bending

moments, transverse shear forces and twisting moments. The resulting three-dimensional

state of stress acting at any point on an element may be transformed into an equivalent set

of three normal stresses (principal stresses) acting in three orthogonal directions. When

one of these three principal stresses is zero, the state of stress is termed biaxial. The

failure strength of materials under combined stresses is normally defined by appropriate

failure criteria. However, as yet, there is no universally accepted criterion for describing

the failure of concrete.

BIAXIAL STATE OF STRESS

Concrete subjected to a biaxial state of stress has been studied extensively due to its

relative simplicity in comparison with the triaxial case, and because of its common

occurrence in flexural members, plates and thin shells. Figure shows the general shape of

the biaxial strength envelopes for concrete, obtained experimentally, along with proposed

approximations.

It is found that the strength of concrete in biaxial compression is greater than in uniaxial

compression by up to 27 percent. The biaxial tensile strength is nearly equal to its

uniaxial tensile strength. However, in the region of combined compression and tension,

the compressive strength decreases nearly linearly with an increase in the accompanying

tensile stress. Observed failure modes suggest that tensile strains are of vital importance

in the failure criteria and failure mechanism of concrete for both uniaxial and biaxial

states of stress.

INFLUENCE OF SHEAR STRESS

Normal stresses are accompanied by shear stresses on planes other than the principal

planes. For a prediction of the strength of concrete in a general biaxial state of stress,

Mohr’s theory of failure is sometimes used. A more accurate (experiment based) failure

envelope for the case of direct stress (compression or tension) in one direction, combined

with shear stress, is shown in Fig.

It is seen that the compressive strength (as well as the tensile strength) of concrete is

reduced by the presence of shear stress. Also, the shear strength of concrete is enhanced

by the application of direct compression (except in the extreme case of very high

compression), whereas it is (expectedly) reduced by the application of direct tension.

BEHAVIOUR UNDER TRIAXIAL COMPRESSION

When concrete is subject to compression in three orthogonal directions, its strength and

ductility are greatly enhanced. This effect is attributed to the all-round confinement of

concrete, which reduces significantly the tendency for internal cracking and volume

increase just prior to failure.`