Page 108 - Aircraft Stuctures for Engineering Student

92 Energy methods of structural analysis

Table 4.5 (Tension positive)
0 0 0 @
Member Length Area F

AB =I2 AB -R/2 -1 12 ~ 1 4 ~ ~
BC Ll2 AB -R/2 -112 ~ 1 4 ~ ~
CD LI2 A R 1 RIA
DE LIZ A R 1 RIA
BD LIZ A -R -1 RIA
EB LI2 A -R -1 RIA
A€ LI2 A R 1 RIA

The two terms in Eq. (iii) may be evaluated separately, bearing in mind that only the
beam ABC contributes to the first term while the complete structure contributes to the
second. Evaluating the summation term by a tabular process we have Table 4.5.
Summation of column @ in Table 4.5 gives

r=l
The bending moment at any section of the beam between A and F is
3 d 3 dM d3
M=-Pz--Rz, hence -= --
4 2 dR 2z
between F and B
P d3 dM d3
M = - (L - z) - -Rz, hence - = - -2
4 2 dR 2
and between B and C
P fi dM d3
M=-(L-z) --R(L-z), hence -= --(L-z)
4 2 dR 2
Thus
&=A{

giving
-ll&PL3 RL3
dz= +-
768EI 16EI
Substituting from Eqs (iv) and (v) into Eq. (iii)
m3
11d3~~~ RL A+IOAB
- +-
768EI 16EI+4E ( ABA ) =O