Modulus of Elasticity of Concrete,explained detailly!!

MODULUS OF ELASTICITY OF CONCRETE

It’s difficult task to find out exactly the modulus of elasticity of concrete because when a load is applied to a concrete it shows two types of deformations (or strains).

1) Short term strain or deformation: It occurs immediately after the load is applied to the concrete.

This strain is also linear in nature.

2) Long term strain or the effective strain: It occurs due to the effect of creep and shrinkage of concrete. It’s time dependent deformation.This strain is non-linear in nature.

For this reason we introduce two types of modulus of concrete .i.e. the short term elastic modulus of concrete (Ec) and long term modulus of concrete or effective modulus of concrete (Ece).

Figure below shows the stress-strain graph of concrete obtained experimentally.

Assumptions:

As we can clearly observe from the stress vs strain diagram of concrete, while loading, the concrete shows an instantaneous deformation and the deformation is assumed to be linear, there is a point in the curve denoted by ‘A’ up to which the concrete shows it’s linear nature. After point ‘A’ due to dominant effect of creep and shrinkage the curve follow the non-linear pattern ( i.e. non-linear nature).. Also while unloading, it doesn’t return back following original path rather it return the path showed by the dotted line and sets permanent deformation to concrete as shown in figure.(i.e. non- elastic nature.)

Thus the graph so obtained is called non-linear and non-elastic graph.

Mathematics used to obtain short term and effective modulus of concrete:

The common mathematics used is Hooke’s law.

Mathematically,

Modulus of elasticity= stress/strain.

Methods used for different modulus of elasticity are: There are 3 general methods commonly used to find the modulus of elasticity of concrete. They are:

1) Initial tangent modulus: This method is used to find the short term elastic modulus of concrete (Ec). As the initial portion of curve is linear, so for this linear behavior we draw a straight line from origin tangent to initial points of curve. The slope of the line gives short term elastic modulus of concrete.
The figure below will help you for better understanding.

2 2) Tangent modulus method: This method is generally used to find out the long term modulus of concrete. For this we fix a point in non-linear portion of concrete say point E as shown in figure below. Then we draw a line tangent to point E. The slope of the line gives the long term or effective modulus of concrete (Ece). The figure below will help you for better understanding.

3 3) Secant method: This method is also suitable for effective modulus of elasticity of concrete.
Here a point say E is chosen and a secant line is drawn down to from origin to point E. The slope of the curve gives the modulus of concrete. Figure below helps you for more understanding.

Empirical relations of Ec and Ece:

Is Code 456:2000 provides a empirical formulas to find out the value of Ec short term and Ece effective modulus of concrete.

For short term modulus of concrete: Ec = 5000* (fck)^1/2 N/mm².

Where, fck = the characteristic cube strength of concrete in N/mm².

For example, if we have concrete of M20 mix, (1: 1.5: 3 proportion), then the value of fck = 20 and the value of Ec obtained is Ec = 5000* (fck)^0.5 N/mm². = 5000*(20)^0.5 N/mm² = 2.24 *10⁴ MPa.

Actual measured values may differ by -20% to +20% from values obtained from the above empirical relation.
(As per IS code 456:2000 clause 6.2.3.1).

For long term modulus of concrete: Ece = Ec/(1+a) .

Where,

a =creep co-efficient .
It’s values are 2.2, 1.6, 1.1 for 7^th, 28^th and one year period of loading respectively.

It can also be found on IS code clause 6.2.5.1.the values of creep co-efficient for different age of loading.

Thank you for reading……..

All kinds of feedbacks are heartly welcomed…….

Milan Tomic

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