Page 24 - Aircraft Stuctures for Engineering Student

1.5 Boundary conditions 9

This condition is known as plane stress; the equilibrium equations then simplify to
do, k y
-+-+x=o
ax ay I
aoy dry,
-+-+Y=O
dy ax

1.5 Boi

The equations of equilibrium (1.5) (and also (1.6) for a two-dimensional system)
satisfy the requirements of equilibrium at all internal points of the body. Equilibrium
must also be satisfied at all positions on the boundary of the body where the compo-
nents of the surface force per unit area are 2, r and Z. Thus, the triangular element
of Fig. 1.7 at the boundary of a two-dimensional body of unit thickness is in equili-
brium under the action of surface forces on the element AB of the boundary and
internal forces on internal faces AC and CB.
Summation of forces in the x direction gives
26s - O.,6,V - Ty.,6X + xi 6X6y = 0
which, by taking the limit as Sx approaches zero, becomes

- dY dx
X=o,-+rVx-
ds . ds
The derivatives dylds and dxlds are the direction cosines 1 and m of the angles that
a normal to AB makes with the x and y axes respectively. Hence
-
X = oxl + ryxm
-
and in a similar manner Y = uym + rJ.
A relatively simple extension of this analysis produces the boundary conditions for
a three-dimensional body, namely

Yt

01 - x
Fig. 1.7 Stresses on the faces of an element at the boundaly of a two-dimensional body.