Problem #$4$: $(30$ points$)$ At a point in a structural member, the normal stresses and the shear stress

are shown in Figure $4.$ In an x$-$y coordinate system $($not shown in the figure$),$ the positive x $-$

direction is horizontal to the right, and the positive y$-$direction is vertical upward.

a$)$ Draw Mohr's circle for this state of stress. Identify the points $($x$)$ and $($y$)$ on the circle that

correspond to the faces with normals in the x$-$and y$-$directions, respectively.

$[$Note: You should use the circle attached on the next page or you should draw one like it$.$

If you draw your own, then you must use a compass $($or some other circle template$)$ and a

straight edge to draw the circle and the axes. For readability and clarity, the circle should

be at least about $5$ inches in diameter.$]$

Use Mohr's circle to answer parts b$),$ c$),$ and d$)$:

b$)$ Find the principal stresses sigma$\_(1)$ and sigma$\_(2)($where sigma$\_(1)>$ sigma$\_(2)),$ draw the rotated element on which they

act, and compute the angle of rotation required from the original element shown in Figure $4.$

When drawing the rotated element, show the angle of rotation clearly and draw all the

stresses that act on the element.

c$)$ Find the magnitude of the maximum shear stress tau$\_("$max $"),$ draw the rotated element on which it

acts, and compute the angle of rotation required from the original element shown in Figure

When drawing the rotated element, show the angle of rotation clearly and draw all the

stresses that act on the element.

d$)$ Find the normal stress sigma$\_($n$)$ and shear stress tau$\_($nt$)$ on the dashed inclined plane a $-$ b shown in Figure

Draw the stresses on the plane.

Please show all details!