Material subjected to combined direct and shear stresses
Material subjected to combined direct and shear stresses: Now consider a complex stress system shown below, acting on an element of material.
The stresses sx and sy may be compressive or tensile and may be the result of direct forces or as a result of bending.The shear stresses may be as shown or completely reversed and occur as a result of either shear force or torsion as shown in the figure below:
As per the double subscript notation the shear stress on the face BC should be notified as tyx , however, we have already seen that for a pair of shear stresses there is a set of complementary shear stresses generated such that tyx = txy
By looking at this state of stress, it may be observed that this state of stress is combination of two different cases:
(i) Material subjected to pure stae of stress shear. In this case the various formulas deserved are as follows
sq = tyx sin2 q
tq = - tyx cos 2 q
(ii) Material subjected to two mutually perpendicular direct stresses. In this case the various formula's derived are as follows.
To get the required equations for the case under consideration,let us add the respective equations for the above two cases such that
These are the equilibrium equations for stresses at a point. They do not depend on material proportions and are equally valid for elastic and inelastic behaviour
This eqn gives two values of 2q that differ by 1800 .Hence the planes on which maximum and minimum normal stresses occurate 900 apart.
From the triangle it may be determined