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

# 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 90^{o} 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 39.8^{o} ccw from point B in the Mohr's circle diagram. Therefore, we draw a stress element oriented 19.9^{o} (half of 39.8^{o}) ccw from x as shown in the figure above.