Deflection Coefficients
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.Example - Example 1
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
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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
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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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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
- 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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