Problem $1$

Show that the yield strength in shear $(S_{sy})$ according to MSS and DE theories is obtained as

$S_{sy}^{MSS}=\frac{S_y}{2}$

$S_{sy}^{DE}=\frac{S_y}{\sqrt[\phantom{2}]{3}}$

Note that $S_{sy}$ is defined as the shear stress $$ at which material yields under pure shear state of

stress.

Answer all the following questions:

a$.$ Pure shear is when you have plane stress state of stress with only shear stresses, all

other normal stresses are zero. Draw a stress element under pure shear stress state

and denote the shear stress by $.$

b$.$ Determine the principal stress components ${}_i$ and ${}_j$ assuming plane$-$stress with

${}_k=0.$

c$.$ Draw the MSS and DE yield surfaces on the ${}_A-{}_B$ plane.

d$.$ Draw the load$-$line corresponding to the pure$-$shear stress state.

e$.$ Mark the points at which this line intersects MSS and DE yield surfaces.

f$.$ Determine the value of $$ at which the MSS criteria is satisfied