9 P

P= 137.4 kN Substituting for ax in either of Eqs (i) or (ii) gives txy = 29.7 N/mm2 From the theory of the torsion of circular section bars rv, = 29.7 N/mm2 = - = T * ^ X} ' J 7T x 504/32

from which

Note that P could have been found directly in this particular case from the axial strain. Thus, from the first of Eqs (1.47)

as before.

References

1 Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd edition, McGraw-Hill Book Company, New York, 1951.

2 Wang, C. T., Applied Elasticity. McGraw-Hill Book Company, New York, 1953.

Problems

P.l.l A structural member supports loads which produce, at a particular point, a direct tensile stress of 80 N/mm2 and a shear stress of 45 N/mm2 on the same plane. Calculate the values and directions of the principal stresses at the point and also the maximum shear stress, stating on which planes this will act.

Problems 33

Ans.

<rj = 100.2N/mm2, 6 = 24° 11' o„ = -20.2 N/mm2, 6 = 114° 11' tmax = 60.2 N/mm2, at 45° to principal planes

P.1.2 At a point in an elastic material there are two mutually perpendicular planes, one of which carries a direct tensile stress at 50 N/mm2 and a shear stress of 40 N/mm2, while the other plane is subjected to a direct compressive stress of 35 N/mm2 and a complementary shear stress of 40 N/mm2. Determine the principal stresses at the point, the position of the planes on which they act and the position of the planes on which there is no normal stress.

Ans. £7j = 65.9N/mm2, 9 = 21° 38' on = -50.9N/mm2, 0= 111038' No normal stress on planes at 70° 21' and —27° 5' to vertical.

P.1.3 Listed below are varying combinations of stresses acting at a point and referred to axes x and y in an elastic material. Using Mohr's circle of stress determine the principal stresses at the point and their directions for each combination.

 (i) +54 +30 +5
0 0