# Concept of Moment Redistribution

### Introduction

Moment redistribution must not be confused with moment distribution, as the two terms are distinctly dissimilar in meaning. Moment distribution is an approximate method of elastic analysis while moment redistribution refers to behaviour indeterminate concrete structures that are not purely elastic.

This post presents the principle of moment redistribution, when and how to carry out redistribution and the possible benefits.

### Principles

In the design of concrete structures, we use some elastic method of elastic analysis to obtain the forces, despite the fact that reinforced concrete doesn’t behave like an elastic material near its ultimate load. The assumption of elastic behaviour is valid at low-stress levels but as a section approaches its ultimate moment of resistance, plastic deformation will occur. We can further visualise this by considering an inderterminate beam operating at the ultimate limit state.

When the beam is loaded, it moment capacity increases until it forms a sufficient number of plastic hinges to make it determinate. If we further increase the load on the beam, the moment at the locations of the plastic hinge do not increase. Instead, the increased load increases the moment in the less stressed portion of the beam. This concept of shifting the point of application of moment in a beam is termed Moment Redistribution.

### Application

In the design of reinforced concrete beams, it’s common to have larger moments at the support region of the beam. Thus, the support region will be congested with reinforcement bars if designed for such high moments. This could, in turn, lead to problems of internal cracks and problems during pouring and compaction of the concrete.

So, in order to solve this problem of congested reinforcement. We redistribute a proportion of the moment at the support, to location where the moment is reasonably low.

### Rules for Redistributing Moments

The British Standard as well as the Eurocode allows the use of moment redistribution up to a maximum of 30%. There are some specific rules, to be followed in-order to carry out moment redistribution properly. These are:

• The resulting distribution must remain in equilibrium with the applied loads
• The redistributed moment at any section must not be less than 70% of the elastic moment at any section
• There will be limitations on the depth of the neutral axis. Please verify with your chosen code of practice.
• Do not redistribute the column moments.
• When considering alternate spans loaded, do not move the unloaded span moment diagram. Only move the fully loaded diagram up or down. This is illustrated in figure 1.

### Worked Example

We are going to illustrate, this concept of moment redistribution using the beam shown in figure 2. This beam was the worked example in the post,Analysis using Moment Distribution. Please consider reviewing the post because the application of the sinking support was modelled wrongly, the correct analysis have been updated.

If we take a look at the bending moment diagram obtained from moment distribution analysis, we could see that the moment at support C is very high. If we attempt to design the beam using this moment diagram, there could be congested reinforcement at this support. For us to reduce this moment we are going to redistribute 20% of the moments from the support into the spans.

##### Redistribution

The process of redistributing moment is very simple. The first thing to do is to reduce the moment at the section by the percentage of redistribution. Secondly, we recalculate the corresponding moment at the other sections to that effect using simple rules of statics.

Figure 4 Shows the sections of the beam that will be affected by a reduction in the moment at point C. We analyze the sections of the beam discretely for the redistributed moment and shears. Span AB is not affected by this redistribution.

##### Redistributed Moments

Reduced Moment at point C is obtained as

{ M }_{ c }=(1-0.20)\times 341.45=273.16kN.m
##### Span B-C

We need to recalculate first the shear at supports and then the new span moments due to the moment reduction at point C. From equilibrum of section BC in figure 4, we have :

{ V }_{ BC }=\frac { { wl } }{ 2 } +\left( \frac { { M }_{ B }-{ M }_{ C } }{ l } \right) =\frac { 20\times 12 }{ 2 } +\left( \frac { 40.6-273.16 }{ 12 } \right) =100.62kN
{ V }_{ CB }=wl-{ V }_{ BC }=(20\times 12)-100.62=139.4kN

To determine the Moment in the span, we need to find out, at what point is it maximum

{ M }={ V }_{ BC }x-\frac { { wx }^{ 2 } }{ 2 } -{ M }_{ B }
\frac { dM }{ dx } ={ V }_{ BC }-wx=0
x=\frac { { V }_{ BC } }{ w } =\frac { 100.62 }{ 20 } =5.03m
{ M }_{ max }=(100.62\times 5.03)-\frac { 20\times 5.03^{ 2 } }{ 2 } -40.63=212.5kN.m
##### Span C-D

We follow the same procedure as with span B-C

{ V }_{ CD }=\frac { P }{ 2 } +\left( \frac { { M }_{ c }-{ M }_{ d } }{ l } \right) =\frac { 250 }{ 2 } +\left( \frac { 273.16-204.4 }{ 8 } \right) =133.6kN
{ V }_{ DC }=P-{ V }_{ DC }=250-133.6=116.4kN

For span C-D, the maximum moment will occur under the point load, therefore x=4m

{ M }_{ max }={ V }_{ CD }x-{ M }_{ CD }=(133.6\times 4)-264.62=269.8kN.m

If we compare the redistributed bending moment and shear diagrams with that obtained directly from moment distribution analysis. We will see that the redistributed diagrams would be easier to use for design as the moment at every section is not considerably high compared to others.

THANK YOU FOR READING. GOD BLESS!

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