# Maximum Shear Stress for Plane Stress

When a body is under plane stress,  determining maximum shear stress can depends on the stress state.  In this article we'll look at the conditions that determine whether in-plane shear stress or out of plane shear stress is the maximum.

## Plane Stress

Consider an element where applied stresses are confined to the xy plane.

In this condition σz = τxz = τzx = τyz = τzy = 0.  Since σz = 0 the out of plane principal stress is also zero.

Before we go any further, let's cover the numbering of principal stresses.  Generally the in-plane principal stresses are calculated as σ1 and σ2.  Once the principal stresses are calculated, they are renumbered according to σ1 > σ2 > σ3.

## In-Plane Principal Stresses Have the Same Sign

First we'll look at the case where the in-plane principal stresses have the same sign.  In the example, both principal stresses are positive (tensile).  The same is true when both are negative (compressive).

Consider the stress element below.  The max in-plane shear stress is 6.3 ksi.  The max out of plane shear stress is 10.2 ksi.

We can see why out of plane shear stress is the maximum by looking at Mohr's circle.

The inner blue circle radius is the max in-plane shear stress (6.3 ksi).  The inner green circle radius is the max shear stress for plane 2-3 and is less than the in-plane shear stress, so we disregard it.  The outer circle radius is the max overall shear stress (10.2 ksi) for plane 1-3 and is out of plane.

Therefore, the out of plane shear stress governs.

## In-Plane Principal Stresses Have Opposite Signs

For this case, Mohr's circle is drawn through σ1 and σ3.  σ2 = 0.  The circle radius is the max in-plane shear stress.  We could draw circles for planes 1-2 and 2-3.  The circle for 1-2 would pass through 0 and σ1, and the circle for 2-3 would pass through σ3 and 0.  Both of these circles have smaller radii than the in-plane circle so they are disregarded.

Therefore, in-plane shear stress governs.

## Summary of Maximum Shear Stress

When in-plane principal stresses are both positive or both negative, then the max shear stress is out of plane.

When in-plane principal stresses have opposite signs, then the max shear stress is in-plane.