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2.3.2 Constructing Mohr's Circle of Stress
Let's draw a Mohr's circle!
Things are about to get exciting – in the truest sense of the word! We're diving deeper into the world of Mohr's circle of stress, a brilliant tool for understanding stresses in a material.
Don't worry, it's not as complicated as it looks!
First, let's take a quick look at the uniaxial stress state. This is easy: imagine pulling on a bar or compressing it. In this case, there is only one stress component, the normal stress \(\sigma_x\).
Now here's the clever part: With Mohr's circle of stress, we can visualize this stress. And at the same time, we get the stress states for all arbitrary cutting angles included! We just have to read them off. Brilliant, right?
Here's how it works:
- Draw a coordinate system. The \(\sigma\)-axis points horizontally to the right, the \(\tau\)-axis normally vertically upwards (positive). But hey, you can also turn it downwards if you like. Both variants are mathematically equivalent. But note: The choice of the \(\tau\)-axis direction influences the direction of counting the angles in the stress circle!
- Mark \(\sigma_x\) on the \(\sigma\)-axis. Pay attention to the sign: tensile stress is positive, compressive stress is negative. Choose a suitable scale, which must be the same for the \(\sigma\)- and \(\tau\)-axes, e.g. 1 cm per 10 MPa or so. No matter, as long as it's the same and the circle fits on the paper!
- The center of the circle is exactly in the middle between \(\sigma_x\) and the \(\tau\)-axis.
- Draw a circle with a compass around the center. The radius is equal to half the magnitude of \(\sigma_x\).
Boom! Your first Mohr's circle of stress is ready!
Fancy an example? Here are four variants of a stress circle:
Here is the description of what is drawn above:
- \(\sigma_x\) is positive (tensile stress, a, b, c),
- \(\sigma_x\) is negative (compressive stress, d, e, f),
- both variants optionally with positive \(\tau\)-axis vertically upwards or downwards positive.
What you need to know now:
- The circle shows you all the stresses in the material. Each stress has a specific point on the circle.
- The angle between two points on the circle is directly related to the cutting angle between the corresponding stresses.
It's that simple! With Mohr's circle of stress, you've got stresses under control.
So what are you waiting for? Get your compass out and let's go!