Page 217 - Aircraft Stuctures for Engineering Student

Problems 201

Fig. P.6.9
where
(Y = P/P~, P, = ~EI/P

Assume r is small compared with d so that the following relationships are applicable:
Cross-sectional area of tube = rdt.
Second moment of area of tube = rd3t/8.

P.6.10 Figure P.6.10 illustrates an idealized representation of part of an aircraft
control circuit. A uniform, straight bar of length a and flexural stiffness EI is built-
in at the end A and hinged at B to a link BC, of length b, whose other end C is
pinned so that it is free to slide along the line ABC between smooth, rigid guides.
A, B and C are initially in a straight line and the system carries a compression
force P, as shown.

Fig. P.6.10
Assuming that the link BC has a sufficiently high flexural stiffness to prevent its
buckling as a pin-ended strut, show, by setting up and solving the differential
equation for flexure of AB, that buckling of the system, of the type illustrated in
Fig. P.6.10, occurs when P has such a value that
tan Xa = X(a + b)
where
X2 = P/EI

P.6.11 A pin-ended column of length I has its central portion reinforced, the
second moment of its area being I2 while that of the end portions, each of length a,
is II. Use the energy method to determine the critical load of the column, assuming
that its centre-line deflects into the parabola w = kz(1- z) and taking the more
accurate of the two expressions for the bending moment.