In this article we will discuss about:- 1. Definition of Modulus of Elasticity 2. Determination of Modulus of Elasticity 3. Measurement of Strains in Concrete 4. Types of Young Elastic Modulus 5. Relation with Strength 6. Factors Affecting 7. Uses.
In the theory of reinforced concrete, it is assumed that concrete is elastic, isotropic and homogeneous and obeys Hooke’s law. Actually none of these assumptions is strictly true and concrete is not perfectly elastic material. By definition of elasticity, strain appears on the application of stress or force and disappears on removal of stress. If the stress-strain cure is straight as shown in Fig.15.1 then the material is elastic.
On the other hand if the curve is as shown in Fig.15.2 then the material is not perfectly elastic. In case of concrete, it deforms on the application of load, but this deformation does not follow any set rule. The deformation of concrete depends upon the magnitude of the load, the rate of applying the load and the time elapsed after which the observations are recorded. Thus the deformation behaviour of concrete is quite complex.
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In order to calculate the deflection of structures and design of concrete members with respect to their sections, quantity of steel etc., the knowledge of deformation properties is necessary. At the time of design of a reinforced concrete structure, it is assumed that the bond between concrete and steel is perfect and the stress in steel is ‘m’ times the stress in concrete, where m is the ratio of modulus of elasticity of steel and concrete. This ratio is known as modular ratio. The accuracy of design will depend upon the value of modulus of elasticity of concrete, as the modulus of steel more or less is a definite quantity.
Definition of Modulus of Elasticity:
It can be defined as the slope of the relation between the stress and strain. It can also be defined as the change of stress with respect to the elastic strain and may be computed by the following relation.
Modulus of elasticity = unit stress/unit strain
It is a measure of stiffness or resistance to deformation of a material. The terms elastic modulus or Young’s modulus of elasticity can be applied strictly to linear relationship i.e., straight part of stress strain curve. The magnitude of the observed strains and the curvature of the stress-strain relation depend on the rate of application of stress. When the load is applied extremely rapidly, the recorded strains are greatly reduced and the curvature of stress strain curve is reduced to a very small value.
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By slowing down the rate of loading i.e., by increasing the time of loading from 5 seconds to about 2 minutes, the increase in the strain is found to go up by 15%, but at normal rate of loading, normally 2 to 10 minutes time is required to test a specimen in an ordinary testing machine, the increase in strain is very small. Hence the degree of nonlinear behaviour is also very small.
Determination of Modulus of Elasticity:
The modulus of elasticity is determined by subjecting a cylinder of 15 cm diameter and 30 cm length or 15 cm cube to uniaxial compression usually in U.T.M. (Universal testing machine) and measuring the strains or deformations by strain gauges or dial gauges fixed at certain gauge length. The value of strain is calculated by dividing the gauge readings by gauge length. The stress will be obtained by dividing the load by the area of cross-section of the specimen. A stress-strain curve is drawn with the help of values of stress and strain obtained.
The modulus of elasticity so obtained from actual loading is called static modulus of elasticity. It has been observed that even under short term loading concrete does not behave as an elastic material. However upto about 10 to 15% of the ultimate strength of concrete, the stress-strain curve is not much curved and more accurate values of modulus of elasticity may be obtained.
For higher stresses, the stress-strain curve will be more curved and will give inaccurate results. Stress-strain curve for various mix proportioned concrete are shown in Fig. 15.3. The modulus of elasticity of concrete may be measured in compression, tension or shear. The modulus of elasticity in tension is equal to the modulus of elasticity in compression.
Stress-Strain Relationship of Aggregate and Cement Paste:
The curve drawn between stress and strain of aggregate alone is found a fairly good straight line. Similarly the curve of stress-strain of cement paste alone shows a fairly good straight line. But the stress-strain curve of concrete, which is a combination of aggregate and cement paste gives a curved curve.
Perhaps this is due to the development of fine or micro cracks at the interface of the aggregate and cement paste. This failure of bond at the interface increases at a faster rate than that due to applied stress. Thus the stress-strain curve continues to bend faster than the increase of stress. Stress-strain relationship of aggregate, cement paste and concrete are shown in Fig. 15.4.
Measurement of Strains in Concrete:
The measurements of strains in concrete are not easy, but within limits it can be determined by Lamb’s roller extensometer. In this method the extensometer is fixed on a 15 cms x 30 cms cylinder and placed in compression testing machine and loaded at a rate of 140 kg/cm2 per minute. The load on cylinder increased till it reaches to 1/3rd of cube strength plus 7 kg/cm2. Now this load is sustained for 1 minute. After sustained loading for one minute the load is released gradually at a rate of 1.5 kg/cm2.
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In the second operation, the reading of the extensometer is noted and it is again loaded till the load reaches to 1/3rd of the cube strength plus 1.5 kg/cm2. The reading of extensometer is noted and the load released slowly till it reaches to a value of 1.5 kg/cm2 on the cylinder specimen.
In the third cycle, the load from zero position to 1/3rd of the cube strength plus 1.5 kg/cm2 is divided into 10 intervals. The cylinder is loaded with the standard rate and at the end of each interval the reading of extensometer noted. The difference between the strains of second and third observations should not be more than 5%. These strains are plotted against stress as shown in Fig. 15.5.
Types of Young Elastic Modulus:
Elastic modulus of concrete can be classified into two main groups as:
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1. Static modulus.
2. Dynamic modulus.
1. Static Elastic Modulus:
The strains obtained as above are plotted against stress and a curve is obtained as shown in Fig. 15.5. As concrete is an imperfect elastic material, stress strain diagram is a curved line. Hence three methods can be used to determine the modulus of elasticity.
(a) Initial tangent modulus.
(b) Tangent modulus.
(c) Secant modulus.
(a) Initial Tangent Modulus:
It is represented by the slope of a tangent to the stress-strain curve drawn passing through the origin. This modulus has significance only for low stresses and thus is of limited value and not easy to determine. It is represented by line OA in the Fig. 15.5.
(b) Tangent Modulus:
It is represented by the slope of the line drawn tangent to the stress-strain curve at any point on the curve, but this modulus applies only to very small changes in load above or below the load at which the tangent modulus is considered. Secondly it is difficult to determine tangent modulus with accuracy as the tangent to the curve is drawn by the eye judgment.
(c) Secant Modulus:
It is represented by the slope of a line drawn from the origin to any point C on the curve. This method is most practical and is in most general use as it represents the actual deformation at the selected point and no uncertainties are involved in its determination. Secant modulus is found to decrease with the increase in stress hence stress at which it has been determined should be stated.
2. Dynamic Modulus:
The value of modulus of elasticity Ec determined by actual loading of concrete is known as static modulus of elasticity. This method of testing is known as destructive method as the specimen is stressed or loaded till its failure. The static modulus of elasticity does not represent the true elastic behaviour of concrete due to the phenomenon of creep. At higher stresses the modulus of elasticity is affected more seriously.
Thus a non-destructive method of testing known as dynamic method is adopted for determining the modulus of elasticity. In this case no stress is applied on the specimen. The modulus of elasticity is determined by subjecting the specimen to longitudinal vibration at their natural frequency that is why this is known as dynamic modulus.
In this method either the resonant frequency through a specimen of concrete or pulse velocity travelling through the concrete is measured. From the known values of length of specimen, density of concrete and resonant frequency the value of dynamic modulus in S.I. units is determined from the relation-
Ed = K.n2L2 ρ
where,
Ed = dynamic modulus of elasticity
K = a constant
n = resonant frequency
L = length of specimen
ρ = density of concrete
If length of specimen is measured in mm and density ρ in kg/m3 then-
Ed = 4n2L2 ρ x 10–15 GPa
The value of dynamic modulus of elasticity can also be determined from the relation-
Ed = ρv2 [(1 + µ)(1 – 2µ)/(1 – µ)]
where,
v = pulse velocity in mm/s
ρ = density of concrete kg/m3
µ = poisson’s ratio.
The value of dynamic modulus of elasticity computed from ultrasonic pulse velocity method is somewhat higher than static modulus of elasticity as the creep remains unaffected in dynamic modulus. Creep also does not significantly affect the initial tangent modulus. Thus the value of initial tangent modulus and dynamic modulus is approximately the same, but the value of dynamic modulus is appreciably higher than secant modulus. The relation between static and dynamic modulic is given by the following relation in G.N/m2.
Ec = 1.25 Ed – 19 …(i)
This relation is not applicable to very rich concrete with cement content more than 500 kg/m3 and light weight concrete. For light weight concrete relation is-
Ec = 1.04 Ed – 4.1 …(ii)
Relation between Modulus of Elasticity and Strength:
It has been observed that for the same stress-strength ratio, the stronger the concrete, higher the strain, On the contrary stronger the concrete higher the modulus of elasticity. This may be due to the fact that for stronger concretes its gel is also stronger, hence there is less strain for a given load. This lower strain gives the higher values of modulus of elasticity. In international system of units, (SI units) the unit of modulus of elasticity is GPa. (Gega Pascal)
ISI-456-2000 has suggested the relation between static modulus of elasticity Ec and the characteristic strength of concrete as follows-
Ee = 5000 √fck
where Ec is in N/mm2 units called (GPa), and fck concrete cylinder strength of 28 days.
Some values of modulus of elasticity are shown in Table 15.1:
In SI units Ee = 9.1 (fck)1/3 for concrete density 2320 kg/m3.
Note:
Actual measured values may differ ± 20% from the values obtained from the above relation.
Factors Affecting the Modulus of Elasticity:
Following factors affect the value of modulus of elasticity:
1. Strength of Concrete:
It is one of the most important factors which affect the modulus of elasticity. Higher strengths give higher value of modulus of elasticity.
2. State of Wetness of Concrete:
The value of modulus of elasticity of wet specimen is found higher from 3 to 4 GPa (3.2 to 4.3 x 104 kg/cm2) than dry specimen i.e., the modulus of elasticity of wet concrete is higher by 16.3% to 7.5 according to its compressive strength. Higher value 16.3% is observed for lower strength i.e., 21 MPa and 7.5% increase for 70 MPa strength, while the strength of wet concrete is found less than dry concrete. The strain of wet concrete is found less than for dry concrete, hence the modulus of elasticity is higher for wet concrete than for dry concrete.
The influence of moisture condition at the time of test for secant modulus at different ages is shown in Fig. 15.6.
3. Properties of Aggregate:
The modulus of aggregate and its volumetric proportion affect the modulus of elasticity of concrete as follows:
(a) Higher the modulus of the aggregate, the higher the modulus of elasticity of concrete. The modulus of aggregate is higher than the modulus of cement paste.
(b) The greater the volume of aggregate, the higher the modulus of elasticity of concrete. However strength of concrete is not affected much by these properties.
4. The Effect of Age:
The modulus of elasticity of concrete increases more rapidly with age than the strength of concrete. Thus the relation between the modulus of elasticity of concrete and its strength depends on age.
5. Mix Proportion:
It has been observed that richer mixes have higher modulus of elasticity of concrete i.e., higher the amount of cement; higher the modulus of elasticity. The value of modulus of elasticity of concrete of mix proportion 1:1.67:2 are found 31.9 GPa while for a mix of 1:2.5:3 is 25 GPa for the same age and wet conditions.
The modulus of elasticity of light weight aggregate concrete usually varies from 40 to 80% of the modulus of normal weight concrete of the same strength in fact it is similar to that of cement paste.
Shape of Stress-Strain Curve:
The shape of stress-strain curve affects the static modulus of elasticity of concrete Ec, but not the dynamic modulus Ed hence the ratio of Ec and Ed is not fixed. The relationship between the modulus of elasticity and strength is not much affected by the temperature upto 230°C as both these properties vary with temperature in the same way. Steam cured concrete has slightly lower modulus than water cured concrete of the same strength.
Uses of Modulus of Elasticity:
Though concrete is not an elastic material in true sense, yet within limits it is considered an elastic material. The modulus of elasticity of concrete is used in the calculations of structural deformations. In case of reinforced concrete structures it is used to determine the stresses developed in simple elements and also to determine the moments, deflections and stresses in more complicated structures.
The dynamic modulus is used to determine the relative durability of concrete when exposed to severe climatic conditions as the dynamic modulus of concrete changes with the quality of concrete. This method is very useful to determine the quality of concrete, when it is subjected to alternate freezing and thawing.