Design Stress-Strain Curve of Concrete and Steel

Welcome to civil engineering fanatics. Today, we will discuss the design stress-strain curve of concrete and steel which is the basis for the design theory behind reinforced concrete structures. We know that most of the structures are designed based on the Limit state method (LSM). The design assumptions and related equations are based on this idealized stress-strain curve of concrete and steel which is as suggested by the Indian standard code IS:456-2000.

Before we move to the stress-strain analysis, let's have an overview of concrete and steel reinforcement.

We know that concrete is weak in tension and strong in compression. While steel is strong in tension, we incorporate it in concrete to impart tensile strength to concrete. Hence, we have reinforced concrete structure.

So in stress-strain analysis, concrete is tested in compression (Compression Test) and the steel is tested for tension (Tension Test). So the stress-strain curve of concrete is as per the compression test and that of steel is as per the tension test.

Now, let's study the stress-strain curve of concrete and steel in detail.

Design Stress-Strain Curve of Concrete – Idealized Stress-Strain Curve of Concrete

The idealized stress-strain curve for concrete is as prescribed by IS:456-2000. The curve represents the elastic and post-elastic behavior of concrete and is used for the design of concrete structures.

This curve shows the stress-strain curve for a concrete with compressive strength of f_ck, called the characteristic compressive strength. The test sample used here is a 150 mm cube. This curve is an idealized stress-strain curve with two shapes, parabolic shape and rectangular shape.

In the curve, you can see, the stress increases with the strain until a point of strain value equal to 0.002. Beyond this point, with the stress remaining constant, the strain increases to a value of 0.0035.  Till OA it is a parabolic curve, and beyond A it is straight. So the curve obtained is the parabolic-rectangular curve. 

But, when we design a full structure (buildings), the compressive strength is reduced due to the influence of size. Hence, IS:456 recommends a value of compressive strength as 0.67fck. And the obtained curve is as shown.

But in actual design, we do not use this value, because in a limit state design theory, the design strength of the material is obtained by dividing it by a partial safety factor. As per the Indian Standard code, this partial safety factor for concrete is 1.5 (γ_c). That gives the compressive strength of 0.67f_ck/γ_c = 0.67f_ck/1.5 = 0.45f_ck. And the corresponding curve is shown. 

So in the design problem, for concrete, we take the design strength as 0.45f_ck and the ultimate strain as 0.0035. The ultimate strength of concrete is reached at a strain value of 0.0035 for all the concrete samples. 

Design Stress-Strain Curve for Reinforcing Steel

As you know, two types of steel are used for reinforced structures: Mild steel and HYSD steel. In the case of steel, the yield strength of steel is the characteristic strength of steel, represented as fy. 

Design Stress-Strain Curve for Mild Steel

Let's check the stress-strain curve of mild steel. Here, until a point stress is directly proportional to strain. Here, like concrete, there is no influence of shape or size of steel on the stress-strain curve (elastic limit or yield point). Steel is elastic and linear. This curve is a characteristic curve. 

Fig.2. Design Stress-Strain Curve for Mild Steel

But this shows the design curve, which is obtained by dividing the yield strength with the partial safety factor of steel γs. Which is fy/γs = fy/1.15 = 0.87fy. 

This curve forms the design curve. The yield point of the steel can be obtained from the graph. 

At point A', 

Modulus of elasticity E = Yield Stress/Yield Strain; 

Then Yield Strain ε_y = Yield Stress/ E_s = 0.87fy/E_s;

E_s for steel = 2 x 10⁵ MPa; 

Design Stress-Strain Curve for HYSD Steel Bars

The stress-strain curve for HYSD bars is shown. There is no specific yield point for HYSD bars. The stress corresponding to a strain of 0.002 is known as the yield stress fy of the HYSD bars.

Fig.3. Design Stress-Strain Curve for HYSD Steel Bars

The design curve is similar to mild steel with the same design stress as 0.87fy. Hence, comparing this with that of mild steel, the strain at that portion will be 0.87fy/Es. Hence total strain will be, 

ε = 0.002 + 0.87fy/Es

So finally in the design problem, we use ε = 0.002 + 0.87fy/Es, and design stress as 0.87fy, even in the case of mild steel. Even if mild steel has a specific yield point, but IS code specifies uniform criteria for all grades of steel. Because we demand the steel to yield at the ultimate limit of strain. During ultimate strain, the Failure will be ductile in nature. Ductile failure shows ample warning of impending collapse-show signs of failure before complete failure.

Key Point

Finally, if we compare the stress-strain curve for concrete and steel, in the first case we apply the partial safety factor for concrete at all the stress levels. But in the case of steel, the partial safety factor is only for the inelastic region. 

This is because, the modulus of elasticity of concrete is dependent on the compressive strength of concrete, given by the formula Ec = 5000 sqrt (fck) MPa. But, Es for steel is 2 x 10⁵ MPa irrespective of the stress level. Hence we apply the partial safety factor only for the non-linear portion of stress in steel. 

Also Read: Working Stress Method (WSM) vs Limit State Method (LSM) in Structural Engineering