Page 55 - Aircraft Stuctures for Engineering Student

40 Two-dimensional problems in elasticity

A tfttt'.
4
4 A _I -7-
ux=22c -- -- I
+-

I -c i
+-

I- X
*
, J + S t t + l rxy=-B

Example 2.2
A more complex polynomial for the stress function is
Ax3 Bx2y Cxy2 Dy3
4=---+--- +-+-
6 2 2 6
As before
&4-
a~ ax~ay2 ay4= O

so that the compatibility equation (2.9) is identically satisfied. The stresses are given by

We may choose any number of values of the coefficients A, B, C and D to produce a
variety of loading conditions on a rectangular plate. For example, if we assume
A = B = C = 0 then uX = Dy, uy = 0 and rxy = 0, so that for axes referred to an
origin at the mid-point of a vertical side of the plate we obtain the state of pure
bending shown in Fig. 2.2(a). Alternatively, Fig. 2.2(b) shows the loading conditions
corresponding to A = C = D = 0 in which a, = 0, ay = By and rxy = -Bx.
By assuming polynomials of the second or third degree for the stress function we
ensure that the compatibility equation is identically satisfied whatever the values of
the coefficients. For polynomials of higher degrees, compatibility is satisfied only if