The objective is to relate the new stress in x' and y' coordinate
system to the original stresses in the x and y coordinate. To do this, the original
stress element is sliced at an angle θ, as shown
in the diagram at the left. The stresses on the cut plane must be in equilibrium
with the stresses on the outside surfaces of the stress element. Remember, nothing
is moving, so all stresses and their associated forces must obey static equilibrium
equations, ΣF
= 0 and ΣM = 0.
Before the stresses are actually summed, the area on each surface needs to be
defined. The plane section
at the angle θ is assumed to have a basic area of dA.
The stress element is really just a point, so the area is infinitesimal, or just
dA. The other two surfaces are based on dA. The bottom surface will be 'sinθ dA'
and the left surface will be 'cosθ dA', which are shown
in the diagram at the left.
Summing the forces in each direction gives
ΣF_{x} = 0 =
(σ_{x´} dA) cosθ -
(τ_{x´y´} dA)
sinθ
- σ_{x} (cosθ dA) - τ_{xy} (sinθ dA)
ΣF_{y} = 0 = (σ_{x´} dA) sinθ - (τ_{x´y´} dA)
cosθ
- σ_{y} (sinθ dA) - τ_{xy} (cosθ dA)
There are two unknowns, σ_{x´} and τ_{x´y´} and two equations, so they can be determined, giving
σ_{x´} =
σ_{x} cos^{2}θ + σ_{y}
sin^{2}θ + 2 τ_{xy} sinθ cosθ
τ_{x´y´} =
- (σ_{x} - σ_{y} ) sinθ cosθ +τ_{xy} (cos^{2}θ - sin^{2}θ)
The y' direction can be developed in the same way, but the section plane is
90^{o} offset. The final equation is
σ_{y´} = σ_{x} sin^{2}θ + σ_{y} cos^{2}θ -
2 τ_{xy} sinθ cosθ |