Moment of Resistance | Doubly reinforced Sections

Methods for calculating Moment of resistance for Doubly reinforced beam

In our article series for “Doubly reinforced sections”, we have covered the following:

What are doubly reinforced sections?

Methods for determining Neutral Axis?

Solved numerical examples for determining Neutral Axis

Numerical examples for practice (Find Neutral axis)

Methods for calculating Moment of Resistance

Numerical example for calculating Moment of resistance

Types of problems in Doubly reinforced sections

Determining stresses in steel and concrete 

Numerical example | Stresses in steel and concrete

Also check out: “Singly reinforced Sections” article series.

Now, our next step would be to study different methods for calculating moment of resistance (MR).

Two methods for calculating Moment of Resistance

There are two methods for calculating the moment of resistance of doubly reinforced sections. They are as follows:

1. Method 1 – Elastic theory
2. Method 2 – Steel beam theory

Method 1

Method using Elastic theory

The moment of resistance is calculated by taking the moments of forces about the centre of gravity of the tensile steel.

From the figure above, we get,

Mr = Moment of compressive force of concrete about tensile steel + Moment of equivalent concrete force of compression steel about tensile steel

= bx(σcbc/2)(d – x/3) + (1.5m – 1)Asc. σcbc (d – d’)

Where, (1.5m – 1)Asc is the equivalent concrete area of compression steel.

σcbc is the compressive stress in the concrete at a depth d’ mm from the top of the beam, where Asc is provided.

To find σcbc

From the figure above, we get,

σcbc/x = σcbc (x – d’)

σcbc = σcbc [(x – d’)/x)]

Method 2

Method using the Steel beam theory

The moment of resistance can be calculated using this method. The calculations are approximate and not very accurate. The whole concrete is neglected. Asc and Ast are taken into consideration for the calculations purpose.

Three important assumptions:

• All the concrete area is neglected
• Permissible stresses in tensile and compressive steel are the same.
• Permissible stresses reach their maximum stress at the same time (Asc = Ast)

Taking the moment about compression steel,

Mr = Ast.σst (d – d’)