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 x_c by the following formula,

σ_cbc/ (σ_st/m) = x_c/(d-x_c)

Step two:

Find x using the following formula,

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

Step three:

Determine whether the section is under-reinforced or over-reinforced.

If x < xe; then the section is said to be under-reinforced

If x > xe; then the section is said to be over reinforced.

Step four:

And finally the last step would be to calculate the Moment of resistance depending on the type of section (under reinforced or over reinforced)

For under reinforced sections, σ_st is already and σ_cbc is calculated by the following mentioned formula;

σ_cbc/ (σ_st/m) = x/(d-x)

Also,

σ_{cbc =} σ_cbc [(x – d’)/x]

To find actual stress in concrete σ_sc

σ_sc = 1.5m σ_cbc

If actual σ_sc < permissible compressive stress in steel, then take the actual σ_sc; if it is more, then take permissible compressive stress in steel.

Taking moment about tensile steel

Mr = bx(σ_cbc/2)[d – (x/3)] + (1.5m -1) A_sc.σ_cbc (d – d’)

For over-reinforced sections, σ_cbc is known. Find σ_cbc by the following formula;

σ_cbc = σ_cbc [(x – d’)/x]

 Problem type 2

To find actual compressive stress in concrete

We would be discussing a detailed step by step procedure for finding the actual compressive stresses in steel and concrete.

Problem type 3

Design of the section

I think this should be enough for all of you to be able to grasp at once. I suggest that you practice regularly until you get perfect with the calculations. The simplest thing you could do is follow the step by step procedure given in the post. Breaking down complex procedure into small steps is the easiest way to learn complex stuff. We would further proceed with a numerical example in which we would follow the above mentioned procedure and calculate the moment of resistance, compressive stresses in steel and concrete.