Stresses on oblique plane
Stresses on oblique plane: Till now we have dealt with either pure normal direct stress or pure shear stress. In many instances, however both direct and shear stresses acts and the resultant stress across any section will be neither normal nor tangential to the plane.
A plane stse of stress is a 2 dimensional stae of stress in a sense that the stress components in one direction are all zero i.e
sz = tyz = tzx = 0
examples of plane state of stress includes plates and shells.
Consider the general case of a bar under direct load F giving rise to a stress sy vertically
The stress acting at a point is represented by the stresses acting on the faces of the element enclosing the point.
The stresses change with the inclination of the planes passing through that point i.e. the stress on the faces of the element vary as the angular position of the element changes.
Let the block be of unit depth now considering the equilibrium of forces on the triangle portion ABC
Resolving forces perpendicular to BC, gives
sq.BC.1 = sysinq . AB . 1
but AB/BC = sinq or AB = BCsinq
Substituting this value in the above equation, we get
sq.BC.1 = sysinq . BCsinq . 1 or (1)
Now resolving the forces parallel to BC
tq.BC.1 = sy cosq . ABsinq . 1
again AB = BCcosq
tq.BC.1 = sycosq . BCsinq . 1 or tq = sysinqcosq
If q = 900 the BC will be parallel to AB and tq = 0, i.e. there will be only direct stress or normal stress.
By examining the equations (1) and (2), the following conclusions may be drawn
(i) The value of direct stress sq is maximum and is equal to sy when q = 900.
(ii) The shear stress tq has a maximum value of 0.5 sy when q = 450
(iii) The stresses sq and sq are not simply the resolution of sy
Material subjected to pure shear: Consider the element shown to which shear stresses have been applied to the sides AB and DC
Complementary shear stresses of equal value but of opposite effect are then set up on the sides AD and BC in order to prevent the rotation of the element. Since the applied and complementary shear stresses are of equal value on the x and y planes. Therefore, they are both represented by the symbol txy.