Page 68 - Aircraft Stuctures for Engineering Student

3.1 Prandtl stress function solution 53

Fig. 3.2 Formation of the direction cosines I and rn of the normal to the surface of the bar.

where Vz is the two-dimensional Laplacian operator

The parameter V2q5 is therefore constant at any section of the bar so that the function
q5 must satisfy the equation
$4 $4
+
- - constant = F (say) (3-4)
=
ax2 ay2
at all points within the bar.
Finally we must ensure that 4 fals the boundary conditions specified by Eqs (1.7).
On the cylindrical surface of the bar there are no externally applied forces so that
X = Y = = 0. The direction cosine n is also zero and therefore the first two
equations of Eqs (1.7) are identically satisfied, leaving the third equation as the
boundary condition, viz.
ryzm + I-J 0 (3.5)
=
The direction cosines I and m of the normal N to any point on the surface of the bar
are, by reference to Fig. 3.2
dx
I=- dY m=--
ds ' dr
Substituting Eqs (3.2) and (3.6) into Eq. (3.5) we have
-- +--=o
84 dx d+dy
dx ds dy ds
or

Thus 4 is constant on the surface of the bar and since the actual value of this constant
does not affect the stresses of Eq. (3.2) we may conveniently take the constant to be
zero. Hence on the cylindrical surface of the bar we have the boundary condition
$=O (3-7)