Stress Transformation and Mohr's Circle

Hello! Welcome to the next lesson in your study of stress analysis.

In the previous lesson, we learned how to use the stress transformation equations to find the normal (${\sigma }_{x^{\prime }}$) and shear (${\tau }_{x^{\prime }{y}^{\prime }}$) stresses on any plane rotated by an angle $\theta$. This was a crucial step in understanding that stress is a directional quantity.

Today, we address the learning outcome: Calculate principal stresses and maximum in-plane shear stress using transformation equations or Mohr's circle. We'll focus on finding the specific orientations where the normal and shear stresses reach their maximum and minimum values. These values, particularly the principal stresses (max/min normal stresses) and the maximum in-plane shear stress, are fundamental to predicting when a material will yield or fracture. This is a critical concept in engineering design, especially in aerospace, where ensuring structural integrity under complex loading is paramount.

We will cover two powerful methods to find these critical stress values:

1. Analytical Method: Using direct formulas derived from the transformation equations.
2. Graphical Method: Using a visual tool called Mohr's Circle.

1. Defining Principal Stresses and Maximum Shear Stress

As we rotate a stress element, the normal and shear stresses on its faces change. At certain angles, the normal stresses will reach their maximum and minimum values. At these specific angles, the shear stress becomes zero.

• Principal Stresses (${\sigma }_1,{\sigma }_2$): The maximum and minimum normal stresses at a point. They occur on planes where the shear stress is zero. These are often denoted as ${\sigma }_1$ (maximum) and ${\sigma }_2$ (minimum).
• Maximum In-plane Shear Stress (${\tau }_{max}$): The absolute maximum shear stress at a point in the 2D plane. This occurs on planes that are 45° away from the principal planes.

The following video provides an excellent visual introduction to these concepts.

2. The Analytical Method: Formulas for Principal Stresses

Since you prefer a formula-based approach, let's start there. The stress transformation equations from our last lesson can be used to derive direct formulas for the principal stresses and the maximum in-plane shear stress.

Here is a summary of those essential formulas:

Principal Stresses:
The two principal stresses, ${\sigma }_1$ and ${\sigma }_2$, are calculated using:
${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$
The + sign gives the maximum principal stress (${\sigma }_1$), and the - sign gives the minimum principal stress (${\sigma }_2$).

Maximum In-plane Shear Stress:
The maximum in-plane shear stress is equal to the square root term from the principal stress formula:
${\tau }_{max}=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$
On the planes of maximum shear stress, the normal stress is not zero. It is equal to the average normal stress:
${\sigma }_{avg}=\frac{{\sigma }_x+{\sigma }_y}{2}$

Orientation of Principal Planes:
The angle ${\theta }_p$ to the principal planes is found with:
$\tan (2{\theta }_p)=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$
This equation gives two angles for $2{\theta }_p$, 180° apart, which correspond to two principal planes 90° apart.

3. The Graphical Method: Mohr's Circle

While the equations are direct, the graphical method known as Mohr's Circle provides a more intuitive way to visualize stress transformation. It is a plot of the transformation equations, where every point on the circle represents the ($\sigma$, $\tau$) state on a specific plane.

The next video gives a concise overview of how to construct the circle.

4. A Complete Worked Example using Mohr's Circle

The best way to learn Mohr's Circle is to work through a problem from start to finish. The following video is a comprehensive tutorial that does just that.

Key Steps for Constructing Mohr's Circle:

1. Set up Axes: $\sigma$ on the horizontal axis (positive to the right) and $\tau$ on the vertical axis (positive downwards).
2. Identify Stresses: From the stress element, find ${\sigma }_x$, ${\sigma }_y$, and ${\tau }_{xy}$. Use the convention that a shear stress causing counter-clockwise rotation of the element is positive.
3. Plot Points:
   • Plot the "X-face" point at $({\sigma }_x,{\tau }_{xy})$.
   • Plot the "Y-face" point at $({\sigma }_y,-{\tau }_{xy})$.
4. Draw the Circle: The line connecting these two points is the diameter. The point where this line crosses the $\sigma$-axis is the center of the circle. Draw the circle through points X and Y.
5. Calculate Center and Radius:
   • Center $C={\sigma }_{avg}=({\sigma }_x+{\sigma }_y)/2$.
   • Radius $R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$.
6. Find Stresses:
   • ${\sigma }_1=C+R$ and ${\sigma }_2=C-R$.
   • ${\tau }_{max}=R$.

Conclusion

Today you've learned two methods for finding the most critical stress values in a 2D stress state: the principal stresses and the maximum in-plane shear stress. Mastering this is a significant step towards analyzing and predicting how components will behave under real-world loads.

Key Takeaways:

• Principal stresses (${\sigma }_1,{\sigma }_2$) are the maximum and minimum normal stresses, occurring on planes with zero shear stress.
• The maximum in-plane shear stress (${\tau }_{max}$) occurs on planes oriented 45° from the principal planes.
• You can find these values using either direct formulas or the graphical Mohr's Circle method.
• Mohr's circle provides a powerful visual tool for understanding the relationship between $\sigma$ and $\tau$ as a function of orientation.

Next Lesson Preview:
So far, we have started with a given stress state (${\sigma }_x,{\sigma }_y,{\tau }_{xy}$). But where do these stresses come from in a real object? In the next lesson, we will analyze states of combined loading. You will learn how to determine the stress state at a point in a component that is simultaneously subjected to axial forces, bending moments, and torsion. Once you determine that initial stress state, you can then apply the methods learned today to find the principal stresses and ultimate safety of the part.