Modulus of Elasticity of Concrete - Initial Tangential Modulus, Tangent Modulus and Secant Modulus

The modulus of elasticity of concrete is given by three types of namely Initial Tangential Modulus, Tangent Modulus and Secant Modulus. Unlike steel, we cannot determine the modulus of elasticity by the formula E = Stress/Strain. This is because concrete is an inelastic material and therefore the stress-strain curve is very uneven in nature. So at different points of the curve, different slope (elastic modulus) is determined. So we have three elastic modulus of concrete. 

Let's learn in detail the three types of modulus of elasticity of concrete using a general stress-strain curve representation of a concrete. 

Types of Modulus of Elasticity of Concrete

The three types of elastic modulus of concrete are:

1. Initial Tangential Modulus (Eit) or Dynamic Modulus
2. Tangent Modulus (Et) 
3. Secant Modulus (Es)

Let's consider the stress-strain curve of a hardened concrete as shown in figure-1.   

Stress-Strain Curve of Normal Concrete Representing Determination of Modulus of Elasticity

1. Initial Tangent Modulus (Eit) or Dynamic Modulus

The initial tangent modulus of concrete can be determined by taking the slope of the tangent drawn at the origin of the stress-stress curve of the concrete. In the figure, the tangent drawn at the origin O is OA. The slope is Eit. 

Initial Tangent modulus is used in:

🔴Theoretical Analyis

🔴Material Modeling

🔴Early-stage elastic response studies

📊 Useful for understanding, not for routine design.

2. Tangent Modulus (Es)

Tangent modulus is the slope of tangent drawn at any point on the stress-strain curve of the concrete. Consider a point B on the curve and draw a tangent. The slope of this tangent is the tangent modulus at B.

Where it is used
🔴Advanced structural analysis
🔴Nonlinear finite element analysis (FEA)
🔴Research and academic studies

🔬 This modulus lives more in software and labs than on site.

3. Secant Modulus (Es)

Secant modulus (Es) is the slope of line joining the origin and any point on the stress-strain curve of the concrete. 

Where it is used
🔴Design calculations in practice
🔴Serviceability limit state analysis
🔴IS 456 explicitly uses this concept

📐 It gives a realistic stiffness for working loads.

Short-Term (Ec) and Long-Term Elastic Modulus (Ecl)

Short term and long term elastic modulus is specified in IS:456-2000, 

Short-Term Elastic Modulus (Ec)

Short term modulus of elasticity (Eci) is the value of the secant modulus at the point located at 1/3rd of the maximum compressive strength. 

As per IS:456-2000,

  Ec = 5000 √fck  (fck in N/mm2)

📌 This is the modulus usually mentioned in design codes.

Secant Modulus of Concrete

Long Term Elastic Modulus (Ecl)

This elastic modulus considers creep and shrinkage of the concrete in long term. An adjusted modulus that considers creep and sustained loading over time

Ecl = Ec / (1+θ)

Ec =  Short Term Elastic Modulus

θ = Creep Coefficient

⏳ Concrete slowly relaxes under long-term loads, and this modulus captures that reality.

Both short and long term elastic modulus are modulus values taken at the initial stage of concrete and long term age of concrete respectively. 

In everyday design, static (secant) modulus is king. But when you move into testing, diagnostics, earthquakes, or advanced analysis, other moduli quietly step onto the stage.

Read More On: Stress-Strain Curve of Concrete and Steel