Design of Doubly reinforced sections | Numerical example

7 step design procedure for Doubly reinforced sections

In our article series for “Design of Doubly reinforced sections”, we 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

6 step prodecure for determining stresses in steel and concrete 

Numerical example | Stresses in steel and concrete

7 step procedure for designing doubly reinforced sections

We shall now proceed with a numerical example “Design of Doubly reinforced sections” using the 7-step procedure we discussed in the previous article.

Numerical problem:

A doubly reinforced concrete beam 250mm wide and 600mm deep overall has to resist an external bending moment of 95kN-m. Find the amount of tensile and compressive steel required, if cover to the centre of steel on both sides is 50mm. σcbc = 5 N/mm2, σst = 140 N/mm2, m = 18.66

Given that:

Breadth of the beam = b = 250mm

Effective depth of the beam = d = 600 – 50 = 550mm

Distance of compressive steel from the top edge of the beam to the centre of the steel = d’ = 50mm

Permissible stress in concrete = σcbc = 5 N/mm2

Permissible stress in steel = σst = 140 N/mm2

Modular ratio = m = 18.66

Bending moment = M = 95 kN-m

Step one:

To find xc

σcbc/ (σst/m) = xc/(d – xc)

5/(140/18.66) = xc/(550 – xc)

xc = 219.95 =  220mm

Read moreDesign of Doubly reinforced sections | Numerical example

Types of Problems | Design of Doubly Reinforced Sections

Stepwise procedure for calculating Moment of resistance and compressive stresses in steel and concrete

While we proceed with the article series for “Doubly reinforced sections”, I would like to categorize the problems into different types. This will make your understanding of the concept better and concrete. I recommend that you practice enough to be able to understand and confidently solve the problems. This will also help you in real time when you would get into practice.

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.

So let’s begin with different types of problems for “Doubly reinforced sections”.

Problem type 1

To find Moment of resistance (Mr)

In a problem where we have to find Mr, specific data is given so that you could calculate the Moment of resistance. The following data will be given in the problem. I suggest that you make notes of the points below.

Breadth of the beam = b

Effective depth of the beam = d

Area of tensile steel = Ast

Area of compressive steel = Asc

Permissible stress in concrete = σcbc

Permissible stress in steel = σst

Modular ratio = m

Four – step procedure to solving the problem:

Step one:

Find xc by the following formula,

σcbc/ (σst/m) = xc/(d-xc)

Step two:

Find x using the following formula,

bxx/2 + (1.5m – 1)Asc (x – d’) = m Ast (d – x)

Read moreTypes of Problems | Design of Doubly Reinforced Sections

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.

Diagram for determining Moment of resistance
Diagram for determining Moment of resistance

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.

Read moreMoment of Resistance | Doubly reinforced Sections

Neutral Axis – Solved Example | Doubly reinforced Sections

Guide to design of 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 … Read moreNeutral Axis – Solved Example | Doubly reinforced Sections