8/14/2019 deflection of Simply supported beam.docx

1/3

Introduction

We want to be able to predict the deection of beams in bending, because many

applications have limitations on the amount of deection that can be tolerated.

Another common need for deection analysis arises from materials testing, in

which the transverse deection induced by a bending load is measured. If weknow the relation expected between the load and the deection, we can back

out" the material properties (specially the modulus) from the measurement. Wewill show, for instance, that t he deection at the midpoint of a beam subjected tothree -point bending" (beam loaded at its center and simply supported at itsedges) is

= WL3/(48EI)

where the length L and the moment of inertia I are geometrical parameters. If theratio of P to P is measured experimentally, the modulus E can be determined. Astiffness measured this way is called the exural modulus .

As an illustration of this process, consider the case of \three-point bending"shown in Fig. 1. This geometry is often used in materials testing, as it avoids theneed to clamp the specimen to the testing apparatus. If the load P is applied at

8/14/2019 deflection of Simply supported beam.docx

2/3

the midpoint, the reaction forces at A and B are equal to half the applied load.The loading function is then

Integrating according to the above scheme:

From symmetry, the beam has zero slope at the midpoint. Hence v ,x =0 @ x = L/2,so c3 can be found to be PL2/16. Integrating again:

The deection is zero at the left end, so c4 = 0. Rearranging, the beam deectionis given by

The ma ximum de ection occurs at x = L/2, which we can evaluate just before thesingularity term activates. Then

8/14/2019 deflection of Simply supported beam.docx

3/3