Question 8.13: Figure P.8.13 illustrates an idealized representation of par......
Figure P.8.13 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. 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.8.13, occurs when P has such a value that
tan λa = λ(a + b)
where
λ² = P/EI
The forces acting on the members AB and BC are shown in Fig. S.8.13.
Considering first the moment equilibrium of BC about C,
from which
v_{\mathrm{B}}={\frac{V b}{P}} (i)
For the member AB and from Eq. (8.1),
E I{\cfrac{\mathrm{d}^{2}ν}{\mathrm{d}z^{2}}}=-P_{\mathrm{CR}}ν (8.1)
E I{\frac{\mathrm{d}^{2}v}{\mathrm{d}z^{2}}}=-P v-Vz
or
{\frac{\mathrm{d}^{2}v}{\mathrm{d}z^{2}}}+{\frac{P}{E I}}v=-{\frac{V z}{E I}} (ii)
The solution of Eq. (ii) is
\upsilon=A\cos\lambda z+B\sin\lambda z-{\frac{V z}{P}} (iii)
When z = 0, υ = 0 so that A = 0. Also when z=a, dυ/dz = 0, hence
0=\lambda B\cos\lambda a-{\frac{V}{P}}from which
B={\frac{V}{\lambda P\cos\lambda a}}and Eq. (iii) becomes
\ v={\frac{V}{P}}\left({\frac{\sin\lambda z}{\lambda\cos\lambda a}}-z\right)When z=a, υ = υ_{B} = Vb/P from Eq. (i). Thus,
{\frac{V b}{P}}={\frac{V}{P}}{\bigg(}{\frac{\sin\lambda a}{\lambda\cos\lambda a}}-a{\bigg)}from which
\lambda(a+b)=\tan\lambda a