# Sample question for GATE Mechanical Engineering on Strength of Materials-Complex Stresses

Question:

For a stress element having  and , find the principal stresses and plot the principal directions on a stress element correctly aligned wrt x-y coordinate system. In addition to this, plot the maximum and minimum shear stresses  and , respectively, on another stress element and find the corresponding normal stresses.

Solution:

Step 1: Draw the Mohr's circle as per the given data.

Step 2: Use the Mohr's circle to find out the stress components.

The firrst step to construct Mohr's diagram is to draw the  and  axes and locate points A of  and C of  on the  axis. Then, we represent  in the cw direction and  in the ccw direction. Hence, point B has the coordinates ,  and point D the coordinates , .

The line BD is the diameter and point E the center of the Mohr's circle. The intersection of the circle with the  axis gives the principal stresses  and  at points F and G, respectively.

The x axis of the stress elements is line EB and the y axis line ED. The segments BA and AE have the length of 60 and 50 MPa, respectively. The length of segment BE is

Since the intersection E is 50 MPa from the origin, the principal stresses are

The angle  wrt the x axis cw to  is:

To draw the principal stress element, we start with the x and y axes parallel to the original axes as shown in figure below. The angle  is in the same direction as the angle  in the Mohr's circle diagram. Thus, measuring  (half of ) clockwise from x axis, we can locate the  axis. The  axis will be at 90o with respect to the  axis, as shown in figures below.

To draw the second stress element, we note that the two extreme shear stresses occur at the points H and I in the Mohr's circle above. The two normal stresses corresponding to these shear stresses are each equal to 50 MPa. Point H is 39.8o ccw from point B in the Mohr's circle diagram. Therefore, we draw a stress element oriented 19.9o (half of 39.8o) ccw from x as shown in the figure above.