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.

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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 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

Doubly reinforced section diagram
Doubly reinforced section diagram

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)Asc(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

Guide to design of Doubly reinforced Sections | Civil Engineering

What are Doubly reinforced sections?

Sections that have tensile as well as compressive reinforcement are called doubly reinforced sections.

Necessity of design of doubly reinforced sections

When the dimensions of the beam are restricted for architectural or structural considerations, the section has insufficient area of concrete which results in inability of the beam to take sufficient compressive stresses. If not paid attention to, it could result in structural failure.

To solve this problem, steel is placed in the compressive area of the section to help the concrete section in resisting compressive stresses. (Steel is good at taking up both compression and tension.)

In this way, the moment of resistance of the section is increased without altering its dimensions.

Three important conditions where doubly reinforced sections are to be used:

1)       When the dimensions of the beam are restricted for architectural or structural purposes.

2)       Sections that are subjected to the reversal of bending moment (piles, braces in water towers etc.

3)       The portion of the beam over middle support in continuous T beams has to be designed as doubly reinforced section.

We are now going to begin with a series of articles on “Design of Doubly reinforced sections”. In our previous series of articles for “Singly reinforced sections“, we have covered every step in detail for the design and analysis of Singly reinforced sections.

We would be covering the following for “Doubly reinforced Sections”:

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

So let us begin with understanding the methods for determining the neutral axis for doubly reinforced sections.

Methods of determining Neutral axis for doubly reinforced sections

METHOD ONE:

Given that:

Dimensions of the beam:

b = width of the beam, d = depth of the beam

Permissible stresses in concrete = σcbc

Permissible stress in steel = σst

Modular ratio = m

From similar triangles in the equivalent concrete stress diagram,

σcbc/ (σst/m) = xc/(d – xc)

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