## P

Example 8.3

A three-dimensional complex stress system has principal stress values of 280 MN/m2, 50 MN/m2 and - 120 MN/m2. Determine (a) analytically and (b) graphically:

(i) the limiting value of the maximum shear stress;

(ii) the values of the octahedral normal and shear stresses.

Solution (a): Analytical

(i) The limiting value of the maximum shear stress is the greatest value obtained in any plane of the three-dimensional system. In terms of the principal stresses this is given by

(ii) The octahedral normal stress is given by o-0ct = 5 [cri + <j2 + 0-3]

(iii) The octahedral shear stress is

Toct = 5 [(cr, - <72 )2 + (<72 - <r3 )2 + (<^3 - <7, )2]

Solution (b): Graphical

(i) The graphical solution is obtained by constructing the three-dimensional Mohr's representation of Fig. 8.43. The limiting value of the maximum shear stress is then equal to the radius of the principal circle.

(ii) The direction cosines of the octahedral planes are

The values of the normal and shear stresses on these planes are then obtained using the procedures of §8.7. Example 8.4

A rectangular strain gauge rosette bonded at a point on the surface of an engineering component gave the following readings at peak load during test trials:

£0 = 1240 x 10~6, £45 = 400 x 10"6, £90 = 200 x 10~6

Determine the magnitude and direction of the principal stresses present at the point, and hence construct the full three-dimensional Mohr representations of the stress and strain systems present. E = 210 GN/m2, v = 0.3.

### Solution

The two-dimensional Mohr's strain circle representing strain conditions in the plane of the surface at the point in question is drawn using the procedure of §14.14^ (Fig. 8.44). Fig. 8.44.

This establishes the values of the principal strains in the surface plane as 1330 fi£ and

The relevant two-dimensional stress circle can then be superimposed as described in §14.13 using the relationships:

radius of stress circle = --- x radius of strain circle

0 0