Deflection of Beams - Structural and Stress Analysis - page 323

Chapter 13 / Deflection of Beams

In Chapters 9, 10 and 11 we investigated the strength of beams in terms of the stresses
produced by the action of bending, shear and torsion, respectively. An associated
problem is the determination of the deflections of beams caused by different loads
for, in addition to strength, a beam must possess sufficient stiffness so that excessive
deflections do not have an adverse effect on adjacent structural members. In many
cases, maximum allowable deflections are specified by Codes of Practice in terms of
the dimensions of the beam, particularly the span; typical values are quoted in Section
8.7. We also saw in Section 8.7 that beams may be designed using either elastic or
plastic analysis. However, since beam deflections must always occur within the elastic
limit of the material of a beam they are determined using elastic theory.

There are several different methods of obtaining deflections in beams, the choice
depending upon the type of problembeing solved. For example, the double integration
method gives the complete shape of a beamwhereas themoment-areamethod can only
be used to determine the deflection at a particular beam section. The latter method,
however, is also useful in the analysis of statically indeterminate beams.

Generally beam deflections are caused primarily by the bending action of applied
loads. In some instances, however, where a beam's cross-sectional dimensions are not
small compared with its length, deflections due to shear become significant and must
be calculated. We shall consider beam deflections due to shear in addition to those
produced by bending. We shall also include deflections due to unsymmetrical bending.

13.1 D IFFERENTIAL E QUATION OF S YMMETRICAL B ENDING

In Chapter 9 we developed an expression relating the curvature, 1 / R , of a beam to the
applied bending moment, M , and flexural rigidity, EI , i.e.

1
R =

M
EI

(Eq. (9.11))

For a beam of a given material and cross section, EI is constant so that the curvature
is directly proportional to the bending moment. We have also shown that bending
moments produced by shear loads vary along the length of a beam, which implies that

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