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Neenu
September 07, 2024

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.

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 (Î³_{c}).
That gives the compressive strength of 0.67f_{ck}/Î³_{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.

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.

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 Î³s. Which is fy/Î³s = 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 Îµ_{y} = Yield Stress/ E_{s}
= 0.87fy/E_{s};

E_{s}
for steel = 2 x 10^{5} MPa;

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,

Îµ = 0.002 + 0.87fy/Es

So finally in the
design problem, we use Îµ = 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.

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 10^{5 }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

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