WORLD FOR CIVIL

Topic 8.5: Special Topics I - Mohr's Circle

The equations for the axial and shear stress at any plane in a structural element, and the equations for the principal stress present in Topic 8.3 Plane Stresses are accurate and useful, however they are not easily remembered. A very useful way of expressing and visualizing the plane stresses in a loaded structural element is method known as Mohr's Circle, developed by a German engineer, Otto Mohr (1835 - 1918).

The method for drawing Mohr's Circle is as follows:

1. We draw a coordinate system with the x-axis representing the normal stresses, and the y-axis representing the shear stresses.

2. Using the values from a given structural element (Diagram 1), we graph two initial points. Point A with coordinates (), and Point B with coordinates () as shown in Diagram 2. [According to our sign convention, the normal x-stress is positive (tension) ,as is the shear stress (counterclockwise) on the associated face. Thus Point A is above the x-axis as shown. The normal y-stress in Diagram 1 is also positive, but the shear stress on that face is negative, so Point B is below the x-axis in the Mohr Circle Diagram.}]

3. We now connect points A and B. The line connecting points A and B intersects the x-axis. This is the center of Mohr's Circle.

In Mohr's Circle, the principal plane is represented by the line ED, which has an angle of zero and zero shear stress. The distance from the origin to Points D and E are the values of the maximum and minimum principal stresses, as shown in Diagram 2.

The radius of the circle is given by

R = ,
which is also equal to the maximum shear stress. In Diagram 2, the maximum shear stress is represented by line CF. We note that the plane represented by line FCG make an angle of 90^o with respect to the principal plane (ACB). This Mohr's Circle angle however is twice the angle in real space, so the angle the plane of maximum shear stress makes is actually 45^o different from the angle of the principal plane.
We also note that the location of the center of Mohr's Circle is from the origin.

Structural Element Sign Conventions
1. Tensile Stress will be considered positive, and Compressive Stresses will be considered negative.
2. The Shear Stress will be considered positive when a pair of shear stress acting on opposite sides of the element will produce a counterclockwise torque (couple). (Some text use the opposite direction for the positive shear stress. This changes a sign in several equations, so we must be somewhat careful of signs when working problems and examples.)
3. The incline plane angle will be measure from the vertical, counterclockwise to the plane. This will be the positive direction for the angle.

We will now work a Mohr's Circle example.

Example. In Diagram 3 a structural element is shown with axial and shear stress of normal x-stress = 4000 lb/in², normal y-stress = 3000 lb/in², shear stress = 1000 lb/in². We would like to find the principal planes, principal stresses, and the maximum shear stress.

We begin by drawing Mohr's Circle for this problem. Point A (+4000 lb/in², +1000 lb/in²), and Point B (3000 lb/in² , -1000 lb/in²) are drawn and connected. We also calculate the radius of the circle from
R = = {[(4000 lb/in² -3000 lb/in²)/2]² + (1000 lb/in²)²}^1/2
R = 1118 lb/in². This value is also the value of the maximum shear stress as we see from Mohr's Circle.
We next calculate the location of the center of Mohr's Circle = (4000 lb/in² +3000 lb/in²)/2 = 3500 lb/in². We then use the value of the radius and the location of the center of the circle to find the principal stresses. From Mohr's Circle we see that:

Maximum Stress = Location of Center + Radius = 3500 lb/in² + 1118 lb/in² = 4618 lb/in²
and
Minimum Stress = Location of Center - Radius = 3500 lb/in² - 1118 lb/in² = 2382 lb/in²
And from the geometry of the circle we can determine the angle the principal axis makes with respect to the element from Tan (2*theta) = (1000 lb/in²)/(4000 lb/in² - 3500 lb/in²), and solving for(2 theta), we find (2 theta) = 63.4^o, and 243.4^o, then theta = 31.7^o, and 121.7^o. The Mohr Circle angles, 63.4^o, and 243.4^o , are double the angles in real space, so the actual angles the principal planes make are 31.7^o, and 121.7^o.

The other important point with respect to the angles is that due to the way the initial points were chosen for Mohr's Circle, and due to the sign conventions used, the angles in Mohr's circle are clockwise from the structural element. In real space they will be in the opposite direction, counterclockwise from the vertical. (See Diagram 5.)