Page 198 - Aircraft Stuctures for Engineering Student

182 Structural instability

total movement of Cy uc, in the x direction is given by
1”l I1
uc = u+ C’D = u+ C’C”sina (S C C N 90”)
But
c‘c” = clsle = cse

Hence
uC =u+BCSsina=u+ysf3 (6.70)

Also the total movement of C in the y direction is
vc =v-DC”=v-C’C1’co~~=v-BCSco~a

so that
vc = v - xse (6.71)
Since at this particular cross-section of the column the centroidal axis has been
displaced, the axial load P produces bending moments about the displaced x and y
axes given, respectively, by
M, = pVc = P(V - xse) (6.72)

and
iwY = pUc = P(U + yse) (6.73)
From simple beam theory (Section 9.1)

(6.74)
and

d2u
EI - = -M - -p( u+Yse) (6.75)
yy dz2
Y -
where I,, and Iyy are the second moments of area of the cross-section of the column
about the principal centroidal axes, E is Young’s modulus for the material of the
column and z is measured along the centroidal longitudinal axis.
The axial load P on the column will, at any cross-section, be distributed as a
uniform direct stress CT. Thus, the direct load on any element of length 6s at a point
B(xB,~B) atds acting in a direction parallel to the longitudinal axis of the
is
column. In a similar manner to the movement of C to C” the point B will be displaced
to B”. The horizontal movement of B in the x direction is then
UB =u+~‘~=~+~l~’l~~~p

But
BIB” = S’B’B = SB8
Hence
UB = u+OSBcosP