Then, we can draw a picture of the wedge with appropriate stresses on each face and use equilibrium equations to relate the stresses.

 

 

 

 

 

 

 

To write equilibrium equations, we must multiply the stresses by the area on which they act to get forces.  Assuming the area of the cut face is A, the horizontal face of the wedge will have an area of A sinq and the vertical face will have an area of A cosq.

 

Now, considering equilibrium in the x’ direction

 

SF_x’ = 0 :             s_x’A - (s_x Acosq) cosq -

(t_xy Acosq) sinq - (t_xy Asinq) cosq -

(s_y Asinq) sinq = 0

 

Rearranging and dividing out the common A, we get

 

s_x’ = s_x cos²q + 2 t_xy sinq cosq + s_y sin²q

 

Using the following trig identities, we can rewrite this equation

 

cos²q = (1 + cos2q)/2                sin²q = (1 - cos2q)/2                 2sinq cosq = sin2q

 

s_x’ = (s_x + s_y)/2 + ((s_x - s_y)/2) cos2q + t_xy sin2q

 

If we consider the y’ direction,

 

SF_y’ = 0

 

t_x’y’ A + (s_x Acosq) sinq - (t_xy Acosq) cosq + (t_xy Asinq) sinq - (s_y Asinq) cosq = 0

 

Rearranging and dividing out A, we get

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