Page 23 - Aircraft Stuctures for Engineering Student

8 Basic elasticity

Taking moments about an axis through the centre of the element parallel to the z
axis
SX

- (ry,.~6y)6x6z-=0 6Y
2

which simplifies to

Dividing through by SxSySz and taking the limit as 6x and Sy approach zero

rxy = Tyx I
Similarly 7x2 = 7,x (1.4)
Tyz = 7.y
Now considering the equilibrium of the element in the x direction

- ryxSxSz+ (., +-SZ 2 ) SxSy
r
- rzXSxSy + XSxSySz = 0
which gives

Or, writing T~~ = T~~ and T,, = r,, from Eqs (1.4)

Similarly
80, a7,,
%y
-+-+-+z=o J
dz ax ay
The equations of equilibrium must be satisfied at all interior points in a deformable
body under a three-dimensional force system.

Most aircraft structural components are fabricated from thin metal sheet so that
stresses across the thickness of the sheet are usually negligible. Assuming, say, that
the z axis is in the direction of the thickness then the three-dimensional case of
Section 1.3 reduces to a two-dimensional case in which c,, r,, and ryz are all zero.