Page 180 - Aircraft Stuctures for Engineering Student
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164 Structural instability
Y+
Fig. 6.10 Beam-column supporting a point load.
Eq. (6.34) becomes
d2v
Wa
-+A v= --
dz2 EII
the general solution of which is
Wa
w = A cos Xz + Bsin XZ - -z (6.36)
PI
Similarly, the general solution of Eq. (6.35) is
W
(I
v = C cos Xz + D sin Az - - - a)(, - z) (6.37)
PI
where A, By C and D are constants which are found from the boundary conditions as
follows.
When z = 0, v = 0, therefore from Eq. (6.36) A = 0. At z = I, w = 0 giving, from
Eq. (6.37), C = -DtanXI. At the point of application of the load the deflection
and slope of the beam given by Eqs (6.36) and (6.37) must be the same. Hence,
equating deflections
Wa Wa
B sin X(1 - a) - - (I - a) = D[sin X(1- a) - tan XI cos X(1 - a)] - - (I - a)
PI PI
and equating slopes
Wa W
BXcosX(1-a) --= DX[cosX(I-a)+tanXIsinX(I-a)] +-(I-a)
PI PI
Solving the above equations for B and D and substituting for A, By C and D in Eqs
(6.36) and (6.37) we have
W sin Xa Wa
V= sink--z for z < I-a (6.38)
PA sin XI PI
W sin X(1 - a) W
V= sinX(I-2)--(I-a)(/-z) for z2l-a (6.39)
PA sin XI PI
These equations for the beam-column deflection enable the bending moment and
resulting bending stresses to be found at all sections.
A particular case arises when the load is applied at the centre of the span.
The deflection curve is then symmetrical with a maximum deflection under the