# Deflection Coefficients

**Contents**

## The Method Of Deflection Coefficients.

It can be seen that any beam of length and flexural rigidity which carries a load (no mattter how it is distributed), will have a maximum deflection of ; where is a constant which depends upon the type of loading and supports. The value of has been found for the standard cases of a cantilever and a simply supported beam (See Deflection of Beams Part 1 Example 4 and Part 3 Example 1), and the deflection in other cases may frequently be built up by superposition.**Deflection**is a term that is used to describe the degree to which a structural element is displaced under a load.

**The Principle of Superposition**: This states that where a number of loads act together on an elastic material, the resulting strain is the sum of the individual strains caused by each load acting separately.

##### Example - Example 1

**show**that the the load carried by the prop If and Find the

**position**and

**value of the maximum Bending Moment**. If is the load on the prop, then its deflection is carries a uniformly distributed load of 5 tons over its length of 10 ft. The beam is supported by three vertical steel tie rods each 6 ft. long, one at each end and one in the middle, the end rods having diameters of 1 in. and the centre rod 1.25 in. Calculate the

**deflection**at the centre of the beam below the end points and the stress in each tie rod.

Substituting the numerical values given: The reaction at the end supportsis given by: And for , For a maximum And

- The
**deflection**is - The
**value**of the maximum Bending Moment is

## Deflection Due To Shear

**cantilever**is a beam anchored at only one end. The beam carries the load to the support where it is resisted by moment and shear stress. Cantilever construction allows for overhanging structures without external bracing.

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### For A Cantilever With A Load Of W At The Free End W = F

- Thus from equation (3), If is the deflection due to shear, then

### For A Cantilever With A Uniformly Distributed Load.

- A load acting on a length (situated at a distance from the fixed end) will produce a deflection due to shear at this point of . For this load alone the distortion produced is indicated in the diagram, and is uniform for the shear force over the length and zero over the rest of the beam . Hence the total deflection due to shear for all the distributed load is given by:
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### For A Simply Supported Beam With Central Load W

**beam**is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a

**bending moment**.

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### A Simply Supported Beam With A Uniformly Distributed Load

- Considering a load only at a distance from one end the deflection at the load will be: Note this has already been proved in equation (5)
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**23287/Deflection-Coefficients-006.png**cannot be found in /users/23287/Deflection-Coefficients-006.png. Please contact the submission author.By proportion the deflection at the centre of the beam : Then the total central deflection due to shear is:

### I-section

- The shear force is treated as being uniformly distributed over the web area.
Thus and and using equation (2) By similar methods to those used from a rectangular section the deflections due to shear may be obtained as follows:
- Cantilever with end load
- Cantilever with distributed load
- Simply supported beam with central load
- Simply supported beam with distributed load

The Strain Energy method known as**{Castigliano's**Theorem} (See Bending of Curved Bars) may be used where a number of loads exist concurrently, or to find the distributed load by imposing a concentrated load at a deflection point; the latter giving it a value of zero. i.e.

##### Example - Example 1

**show**that if and are the deflections due to shear and Bending due to a concentrated load at the free end, and find the

**value for**for steel. . Hence find the

**least value of**if the deflection due to shear is not to exceed 1% of the total.

- is
- The
**least value of**is

## Deflection By Graphical Method

It was shown in the pages on**"Shearing force and Bending Moment"**that a

**Funicular Polygon**could be used to perform a double integration of the load curve and this would produce the Bending Moment diagram.

produce the

**Deflection Curve**.

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