# Why does the max principle stress at yielding for a ductile material equal to its yield strength in a uniaxial tensile test?

It is thought that the maximum principal stress for a ductile material is equal to its yield strength, in a uniaxial tensile test. This is also shown in the attached picture. I want to ask why is this the case? I mean why not is max shear stress at yielding equal to the yield strength of the material?

Actually it is.

When a material is loaded the stress state in the material is the same - it does not depend on the orientation that the stresses are analysed.

Mohr's circle is a way of visualising a stress state at different orientations/planes.

At yielding of the material, the stress state is that the maximum shear strength at yielding is equal to the shear stress in the 45 degree plane.

The same stress state at the plane which is parallel to the uniaxial tensile load, is equal to the tensile yield stress.

Your graph tells a different story.

From the property of the Mohr's Circle,

$$\tau_{max} = \dfrac {\sigma_1 - \sigma_2}{2}$$

For $$\sigma_2 = 0$$, (uniaxial loading)

$$\sigma_1 = 2\tau_{max}$$, or as indicated in the graph, $$\sigma_y = 2\tau_y$$

In which, $$\sigma_1$$ is the tensile stress. Does this answer your question?