Page 125 - Aircraft Stuctures for Engineering Student

References 109

or
(4.34)

wher the bending moment at any section due to th unit load. Substituting for
de from Eq. (4.33) we have
ATe,B = IL at dz (4.35)

where t can vary arbitrarily along the span of the beam, but only linearly with depth.
For a beam supporting some form of external loading the total deflection is given by
the superposition of the temperature deflection from Eq. (4.35) and the bending
deflection from Eqs (4.27); thus

(4.36)

Example 4.17
Determine the deflection of the tip of the cantilever in Fig. 4.30 with the temperature
gradient shown.

Spanwise variation of t

Fig. 4.30 Beam of Example 4.1 1

Applying a unit load vertically downwards at B, MI = 1 x z. Also the temperature
t at a section z is to(I - z)/Z. Substituting in Eq. (4.35) gives

Integrating Eq. (i) gives

(i.e. downwards)

1 Charlton, T. M., Energy Principles in Applied Statics, Blackie, London, 1959.
2 Gregory, M. S., Introduction to Extremum Principles, Buttenvorths, London, 1969.
3 Megson, T. H. G., Structural and Stress Analysis, Arnold, London, 1996.