Bending of Beams - Structural and Stress Analysis - page 226

a lateral contraction of the lower. The section does not therefore remain rectangular
but distorts as shown in Fig. 9.14(c); the effect is known as anticlastic bending .

Anticlastic bending is of interest in the analysis of thin-walled box beams in which the
cross sections aremaintained by stiffening ribs. The prevention of anticlastic distortion
induces local variations in stress distributions in the webs and covers of the box beam
and also in the stiffening ribs.

9.4 S TRAIN E NERGY IN B ENDING

A positive bending moment applied to a length of beam causes the upper longitudinal
fibres to be compressed and the lower ones to stretch as shown in Fig. 9.5(a). The
bending moment therefore does work on the length of beam and this work is absorbed
by the beam as strain energy.

Suppose that the bending moment, M , in Fig. 9.5(a) is gradually applied so that when
it reaches its final value the angle subtended at the centre of curvature by the element
δ x is δ

θ. From Fig. 9.5(a) we see that

R δ θ = δ x

Substituting in Eq. (9.7) for R we obtain

EI z
δ

M =

x δ

θ (9.19)

so that θ is a linear function of M . It follows that the work done by the gradually

applied moment M is M

δ

δ

θ/ 2 subject to the condition that the limit of proportionality
is not exceeded. The strain energy,

δ U , of the elemental length of beam is therefore

given by

1
2 M

δ U = δ θ (9.20)

or, substituting for δ θ from Eq. (9.19) in Eq. (9.20)

M 2
EI z δ

1
2

δ U = x

The total strain energy, U , due to bending in a beam of length L is therefore

M 2
2 EI z

U = d x (9.21)

L

9.5 U NSYMMETRICAL B ENDING

Frequently in civil engineering construction beam sections do not possess any axes of
symmetry. Typical examples are shown in Fig. 9.15 where the angle section has legs of
unequal length and the Z-section possesses anti- or skew symmetry about a horizontal