Page 123 - Aircraft Stuctures for Engineering Student

4.1 1 Temperature effects 107
since the end A is restrained from rotation. Similarly the rotation at B is given by

MB MA
-oc+-62+-eB=o
A4 M M
Solving Eqs (iv) and (v) for MA gives

The fact that the arbitrary moment M does not appear in the expression for the
restraining moment at A (similarly it does not appear in MB), produced by the
load W, indicates an extremely useful application of the reciprocal theorem,
namely the model analysis of statically indeterminate structures. For example, the
fixed beam of Fig. 4.26(c) could possibly be a full-scale bridge girder. It is then
only necessary to construct a model, say of Perspex, having the same flexural rigidity
EZ as the full-scale beam and measure rotations and displacements produced by an
arbitrary moment M to obtain fixing moments in the full-scale beam supporting a
full-scale load.

F

A uniform temperature applied across a beam section produces an expansion of the
beam, as shown in Fig. 4.27, provided there are no constraints. However, a linear
temperature gradient across the beam section causes the upper fibres of the beam
to expand more than the lower ones, producing a bending strain as shown in
Fig. 4.28 without the associated bending stresses, again provided no constraints are
present.
Consider an element of the beam of depth h and length 6z subjected to a linear
temperature gradient over its depth, as shown in Fig. 4.29(a). The upper surface of
P

Expansion

Fig. 4.27 Expansion of beam due to uniform temperature.

I%

Fig. 4.28 Bending of beam due to linear temperature gradient.