Plane Stress Transformation

σ_{x^{'}} = \frac{σ _{x} + σ _{y}}{2} + \frac{σ _{x} - σ _{y}}{2} cos 2 θ + τ_{x y} sin 2 θ

Stresses on a face at angle θ (Handbook sign convention) follow the transformation equations. τx'y' uses a similar formula with a minus cosine term. Equivalent to reading Mohr's circle at 2θ.

$θ$: Element rotation
$σ_{x^{'}}$: Normal stress on the rotated face

Step-by-step solved example

σx=100, σy=0, τxy=0. Face at θ=30°. σx'?

A rotation 2θ on Mohr is θ on the element.

1. Plug in
σx'=50 + 50 cos60° + 0 = 50+25=75.
$σ_{x^{'}} = 75$

Answer: 75 (same units as σx)

10 practice questions

0/10 correct

1.θ=0 must give σx'=

2.θ=90° gives σx'=

3.The transformation uses

4.Uniaxial σx=σ, others 0, θ=45°: σx'=

5.These equations assume

6.τx'y' = 0 on principal planes by definition, so σx' becomes

7.Sign of τ in the Handbook must be

8.If σx=σy and τ=0, σx' is

9.The invariant σx+σy equals

10.These are NOT valid as written for