Numerical examples for practice – Civil Engineering Projects

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)

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 of resistance

Types of problems in Doubly reinforced sections

Determining stresses in steel and concrete

Numerical example | Stresses in steel and concrete

Now we shall move on with a solved example. This will help you understand the methods in a better way. I suggest that you do them yourselves too. Practice will help you make your concepts more concrete and clear.

Example:

An reinforced concrete beam 200mm x 400mm overall is reinforced with 4 – 22mm⏀ bars with centres 30mm from the bottom edge and 3 – 20mm⏀ bars with centres 25mm from the top edge. Find the neutral axis of the beam, if m = 18.66

Given that,

Width of the beam = 200mm

Effective depth of the beam = 400 – 30 = 370mm

Distance of compressive steel from the top edge of the beam to the centre of the steel = d’ = 25mm

Modular ratio = m = 18.66

Area of concrete = Asc = 3 x π/4 x (20)² = 942 mm²

Area of tensile steel = Ast = 4 x π/4 x (22)² = 1520 mm²

To find x:

Equating moment of area on compression and tension sides about N.A.

bxx/2 + (1.5m – 1)A_sc(x – d’) = mAst (d – x)

200x²/2 + (1.5 x 18.66 – 1) 942 (x – 25)

= 18.66 x 1520 (370 – x)

Therefore, x2 + 537.87x – 111299 = 0

Solving the above equation, we get,

x = 159.579mm

Examples for practice

An reinforced concrete beam 300mm x 600mm overall is reinforced with 6 – 22mm⏀ bars with centres 30mm from the bottom edge and 5 – 20mm⏀ bars with centres 25mm from the top edge. Find the neutral axis of the beam, if m = 18.66

An reinforced concrete beam 300mm x 600mm overall is reinforced with 4 – 20mm⏀ bars with centres 25mm from the bottom edge and 6 – 20mm⏀ bars with centres 25mm from the top edge. Find the neutral axis of the beam, if m = 18.66

Types of Problems | Design of Doubly Reinforced Sections