Page 30 - Aircraft Stuctures for Engineering Student

1.8 Mohr's circle of stress 15

B

t
cy (-120N/mrn2)
Fig. 1.10 Stress system for Example 1 .I.

The stress system at the point in the material may be represented as shown in
Fig. 1.10 by considering the stresses to act uniformly over the sides of a triangular
element ABC of unit thickness. Suppose that the direct stress on the principal
plane AB is U. For horizontal equilibrium of the element

~
~
U~~ cos e = U x + Tx.,,~~
which simplifies to
rXy tan 8 = u - CJ~
Considering vertical equilibrium gives
uAB sin e = uyAC + T,,BC

or
Txy cot e = - (ii)

Hence from the product of Eqs (i) and (ii)
2 - Sx)(U - (4
Txy = (.
Now substituting the values nX = 160N/mm2, gy = -120N/mm 2 and u = c1 =
200 N/mm2 we have
T~~ = f 113 N/mm2

Replacing cot B in Eq. (ii) by 1/ tan 8 from Eq. (i) yields a quadratic equation in u
2 - U(Ux - Uy) + UxUy - Try = 0 (iii)
2
The numerical solutions of Eq. (iii) corresponding to the given values of ox, and 74,;
are the principal stresses at the point, namely
aI = 200 N/mm2 (given), aJr = - 160 N/mm 2