Page 165 - Aircraft Stuctures for Engineering Student

Problems 149

giving
16qoa4b4 TX . 7ry
W= sin - sin -
T6D(a2 +b2)2 a b
At the centre of the plate w is a maximum and

16qoa4b4
Wmax =
7r6D(a2 + b2)2
For a square plate and assuming v = 0.3

a4
wmaX = 0.0455qo -
Et3
which compares favourably with the result of Example 5.1.
In this chapter we have dealt exclusively with small deflections of thin plates. For a
plate subjected to large deflections the middle plane will be stretched due to bending so
that Eq. (5.33) requires modification. The relevant theory is outside the scope of this
book but may be found in a variety of references.

Jaeger, J. C., Elementary Theory of Elastic Plates, Pergamon Press, New York, 1964.
Timoshenko, S. P. and Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd edition,
McGraw-Hill Book Company, New York, 1959.
Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd edition, McGraw-Hill Book
Company, New York, 1961.
Wang, Chi-Teh, Applied Elasticity, McGraw-Hill Book Company, New York, 1953.

P.5.1 A plate 10 mm thick is subjected to bending moments M, equal to 10 N m/mm
and My equal to 5 N m/mm. Calculate the maximum direct stresses in the plate.
Ans. a,,,,, = f600N/mm2, = f300N/mm2.
P.5.2 For the plate and loading of problem P.5.1 find the maximum twisting
moment per unit length in the plate and the direction of the planes on which this
occurs.
Ans. 2.5 Nm/mm at 45" to the x and y axes.

P.5.3 The plate of the previous two problems is subjected to a twisting moment of
5 N m/mm along each edge, in addition to the bending moments of M, = 10 N mjmm
and My = 5 N m/mm. Determine the principal moments in the plate, the planes on
which they act and the corresponding principal stresses.
Ans. 13.1 Nm/mm, 1.9Nm/mm, a = -31.7", a = +58.3", *786N/mm2,
fl 14N/mm2.