Page 61 - Aircraft Stuctures for Engineering Student

46 Two-dimensional problems in elasticity

Separating the terms containing x and y in this equation and writing

we have

The term on the r.h.s. of this equation is a constant which means that F1 (x) and F2( y)
must be constants, otherwise a variation of either x or y would destroy the equality.
Denoting Fl (x) by C and F2( y) by D gives
Pb2
C+D=-- (viii)
8IG
and

so that

and
Py3 uPy3
h(Y) =a-=+DY+H

Therefore from Eqs (vii)
Px2y uPy3 Py3
+
u=---- + - Dy + H
2EI 6EI 6IG
uPxy2 Px3
v= - +-+Cx+F
2EI 6EI
The constants C, D, F and H are now determined from Eq. (viii) and the displacement
boundary conditions imposed by the support. system. Assuming that the support
prevents movement of the point K in the beam cross-section at the built-in end
then u = v = 0 at x = I, y = 0 and from Eqs (ix) and (x)

pi3
H=0, F=---Cl
6EI
If we now assume that the slope of the neutral plane is zero at the built-in end then
&/ax = 0 at x = I, y = 0 and from Eq. (x)

PI2
C=--
2EI
It follows immediately that