Page 181 - Aircraft Stuctures for Engineering Student

6.5 Energy method 165

Y+
Fig. 6.1 1 Beam-column supporting end moments.
load of
w XI Wl
v,,, =-tan---
2PX 2 4P
Finally, we consider a beam-column subjected to end moments MA and MB in
addition to an axial load P (Fig. 6.11). The deflected form of the beam-column
may be found by using the principle of superposition and the results of the previous
case. First, we imagine that MB acts alone with the axial load P. If we assume that the
point load W moves towards B and simultaneously increases so that the product
Wu = constant = MB then, in the limit as a tends to zero, we have the moment MB
applied at B. The deflection curve is then obtained from Eq. (6.38) by substituting
Xu for sin Xa (since Xu is now very small) and MB for Wa. Thus
Xz
sin
V =% (7&)
(6.40)
In a similar way, we find the deflection curve corresponding to MA acting alone. Sup-
pose that W moves towards A such that the product W(I - a) = constant = MA.
Then as (I - a) tends to zero we have sin X(1- a) = X(I - a) and Eq. (6.39) becomes
sin XI --I
MA sinX(1- z) (I - z)
v=-[ P I (6.41)
The effect of the two moments acting simultaneously is obtained by superposition of
the results of Eqs (6.40) and (6.41). Hence for the beam-column of Fig. 6.11
sin XI --I
v=- MB (sinXz z) I 7 [ sinX(1 -z) (I-z) (6.42)
P sinX1 I 1
Equation (6.42) is also the deflected form of a beam-column supporting eccentrically
applied end loads at A and B. For example, if eA and eB are the eccentricities of P at
the ends A and B respectively, then MA = PeA, MB = PeB, giving a deflected form of
sin XI --I
sin Xz sin X(1 - z) (I - z) (6.43)
v=eB(,Xr-5) 1
Other beam-column configurations featuring a variety of end conditions and
loading regimes may be analysed by a similar procedure.

The fact that the total potential energy of an elastic body possesses a stationary value
in an equilibrium state may be used to investigate the neutral equilibrium of a buckled