Page 29 - Aircraft Stuctures for Engineering Student

14 Basic elasticity

which, on rearranging, becomes

on = ox cos2 e + uy sin2 e + Txy sin 2e
as in Eq. (1.8). Similarly it may be shown that

Q'N = rxu cos 28 - ( g~ ") sin20 = --7
~
as in Eq. (1.9). Note that the construction of Fig. 1.9(b) corresponds to the stress
system of Fig. 1.9(a) so that any sign reversal must be allowed for. Also, the On
and OT axes must be constructed to the same scale or the equation of the circle is
not represented.
The maximum and minimum values of the direct stress, viz. the major and minor
principal stresses nI and q1, occur when N (and Q') coincide with B and A respec-
tively. Thus
q = OC + radius of circle

or

and in the same fashion

The principal planes are then given by 28 = P(q) and 28 = P + r(uI1).
Also the maximum and minimum values of shear stress occur when Q' coincides
with D and E at the upper and lower extremities of the circle.
At these points Q'N is equal to the radius of the circle which is given by

f
as
Hence ~,,,,h = f z/(a, - cy)' + 4~2~ before. The planes of maximum and mini-
mum shear stress are given by 28 = P + r/2 and 28 = ,8 + 3~12, these being inclined
at 45" to the principal planes.
Example 1.1
Direct stresses of 160 N/mm2, tension, and 120 N/mm2, compression, are applied at a
particular point in an elastic material on two mutually perpendicular planes. The
principal stress in the material is limited to 200 N/mm2, tension. Calculate the allow-
able value of shear stress at the point on the given planes. Determine also the value of
the other principal stress and the maximum value of shear stress at the point. Verify
your answer using Mohr's circle.