1. Wedge Equilibrium Set Up

a) Draw the free-body diagram of a wedge for determining the stresses acting on the inclined face.  Let the inclined face of the wedge have a unit length.  Define axis x' to be perpendicular to the inclined face, and pointing outward.  Label the unknown forces.  Write the equilibrium equations, and determine the stresses.

b) In the preceding part, the free body you have chosen lies on one side of the inclined plane. Repeat that part by choosing the other side.  Check that the same stresses are obtained.

a) Show on the figure a set of  (x', y') axes you have chosen earlier, and indicate the angle q of the stress transformation formulas.  Fill in below the values of sinq and cosq, then the terms of the transformation formula for t_x'y'.  Check the value obtained for t_x'y'.

sinq =

cosq =

t_x'y' =

b)  Re-label axis y as axis x, and let the new y-axis be oriented leftward, so counterclockwise rotation remains positive. Redo part a), and check the answer for t_x'y'. 

3.1. Draw to approximate scale the Mohr circle for the stresses of part 2a.  Indicate points X and Y, the center C, and the origin O of the stress axes. 

a) Show on the figure s_av, and visually estimate the radius R. Deduce an estimate of  s₁, s₂, and t_max.

b) Show on the Mohr circle the angle q_p defining the orientation of the principal axes (x_p, y_p) relative to the (x, y) axes.

c) Using the geometry of the Mohr circle, compute, without reference to established formulas,  R then s₁, s₂, and t_max.

d) Using the origin-of-axes property of the leftmost point of the Mohr circle, compute  tanq_p using the geometry of the Mohr circle, and observing the sign convention for q_p.  Determine q_p. 

e) Show 2q_p on the Mohr circle, and compute tan2q_p using the geometry of the Mohr circle, and observing the sign convention for q_p.  Determine q_p so  that it looks the same as in the Mohr circle..