What Is Mohr's Circle and How Does It Work?

TL;DR

Mohr's circle is a graphical tool for calculating principal normal stresses and maximal shear stress from a given stress state. It simplifies stress analysis by visually representing these stresses, with a 2D circle for two-dimensional cases and three circles for three-dimensional cases, highlighting the relationship between shear stress and the principal stresses.

Transcript

NICHOLAS BURNAND: Hello and welcome to this short video on Mohr's circle. The goal of this video is to show you how Mohr's circle works, how they could be useful for you, and to make some interesting observations as well. So first of all, Mohr's circle is a graphical way to find the principle normal stresses and the maximal shear stress of a given ... Read More

Key Insights

• 🔨 Mohr's circle is a graphical tool that simplifies stress analysis by representing the principle normal stresses and maximal shear stress visually.
• ⭕ The 2D Mohr's circle involves a square stress element with two normal forces and shear stress.
• 🔵 In 3D, three Mohr's circles are used to represent the principal stresses, and the maximal shear stress depends on the difference between the smallest and largest principal stresses.
• 💼 Shear stresses can exist even without applied shear forces, and the angle of maximal shear stress is different in 2D and 3D cases.
• 👷 Mohr's circle can also be constructed by projecting a 3D stress tensor onto different axes, providing an alternative method to find the principal stresses.

Questions & Answers

Q: What is the purpose of Mohr's circle in stress analysis?

Mohr's circle is a graphical tool used to determine the principle normal stresses and maximal shear stress, providing valuable information for stress analysis. It helps engineers analyze the behavior of materials under different stress conditions.

Q: How do you construct a 2D Mohr's circle?

To construct a 2D Mohr's circle, you need to plot the normal stresses (sigma x and sigma y) on the x-axis and the shear stress (tau x,y) on the y-axis. The diameter of the circle represents the difference between the normal stresses, and the angle of the circle provides the angle of rotation required to align the principal normal stresses.

Q: How does the maximal shear stress change in a 3D stress state?

The maximal shear stress in a 3D stress state depends on the difference between the smallest and largest principal stresses. It is independent of the middle principal stress (sigma 2). Changing the value of sigma 2 does not impact the maximal shear stress.

Q: How can projections of a 3D stress tensor be used to determine the principal normal stresses?

By projecting the 3D stress tensor onto the x, y, and z-axis, you can create three 2D Mohr's circles. The endpoints of these circles represent the principal normal stresses, and the maximal shear stress can also be determined. By rotating the stress element using the angles obtained from the projections, the principal normal stresses can be obtained.

Summary & Key Takeaways

• Mohr's circle is a graphical method to determine the principle normal stresses and maximal shear stress in a stress state.

• The 2D Mohr's circle involves a square stress element with normal forces and shear stress.

• In 3D, three Mohr's circles are used to represent the principal stresses, and the maximal shear stress depends on the difference between the smallest and largest principal stresses.