Concept of Failure and Equivalent Stresses:
While we are testing materials for strength, usually, we apply one uniaxial component of stress generally in tension or compression and attain a value for the limiting state of yield of failure and let a solid to have failed if that state is attained.
What is meant by reaching the restricting state? Let us go into the question, a little deeper. While we conduct a tension test on a mild steel rod and seek out that it yields when the stress attain a value of 250 MPa, this must be the limiting state as it is achieved by experimental. But one might put a question whether the material has yielded because the normal stress attained a limit of 260 MPa or on the contrary because the normal strain has attained a value of 0.0013. The question need not stop here. If you determine the maximum shear stress, you might also ask whether reaching a shear stress of 130 MPa is the limit for yielding of the material. Further question might be "whether there are any other criteria for yield?"
All independent answer to this question had led to a failure theory and now we shall learn a few of these theories.