Mohr's Circle

The Purpose

We know the stress state (${\sigma }_x,{\sigma }_y,{\tau }_{xy}$) on the horizontal/vertical faces. Mohr’s Circle allows us to find the stress on any inclined plane ($\theta$) graphically, without solving complex rotation equations.

Most importantly, it reveals the Principal Stresses (Max/Min normal stress) and the Maximum Shear Stress.

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The Sign Convention

• Normal Stress ($\sigma$):
  • Tension (+): Pulling away from the element (Right on the graph).
  • Compression (-): Pushing into the element (Left on the graph).
• Shear Stress ($\tau$):
  • Standard Convention:
    • Face X: If $\tau$ rotates the element Counter-Clockwise $\to$ Plot Down (-).
    • Face X: If $\tau$ rotates the element Clockwise $\to$ Plot Up (+).
  • Note: The vertical axis is usually $\tau$ (positive down) or $\tau$ (positive up).

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Construction Steps

1. Identify the Points:

• Point X (Face X): Plot $({\sigma }_x,-{\tau }_{xy})$
• Point Y (Face Y): Plot $({\sigma }_y,{\tau }_{xy})$
• Note: One point will be above the axis, one below.

2. Find the Centre ($C$): The circle lies on the $\sigma$ axis (horizontal).

$$C=\frac{{\sigma }_x+{\sigma }_y}{2}$$

• This is the Average Normal Stress (${\sigma }_{avg}$).

3. Calculate the Radius ($R$):

Distance from Centre $C$ to Point X.

$$R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

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Critical Values

Once drawn, the circle reveals the failure points immediately:

1. Principal Stresses (${\sigma }_1,{\sigma }_2$): The points where the circle crosses the horizontal $\sigma$-axis. Shear is Zero here.

• Max Tension/Compression:

$${\sigma }_{1,2}=C\pm R$$

2. Maximum Shear Stress (${\tau }_{max}$): The highest/lowest point of the circle (top/bottom).

$${\tau }_{max}=R$$

3. Principal Orientation (${\theta }_p$): The angle on the circle ($2\theta$) is double the physical angle ($\theta$).

• If you rotate $90^{\circ }$ on the circle (from Point X to ${\sigma }_1$), you rotate $45^{\circ }$ on the real element.

$$\tan (2{\theta }_p)=\frac{2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}$$

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Special Cases

• Uniaxial Tension: Circle touches the origin. ${\sigma }_2=0$.
• Pure Shear: Centre is at origin ($C=0$). ${\sigma }_1=-{\sigma }_2$.
• Hydrostatic Pressure: ${\sigma }_x={\sigma }_y$. Radius = 0. The circle is a Dot.

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Mohr’s Circle (Inverse Construction)

The Goal

You start with the Principal Stresses (${\sigma }_1,{\sigma }_2$) and want to find the normal ($\sigma$) and shear ($\tau$) stress on a plane inclined at a specific angle $\theta$.

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The Setup

1. Plot Principal Stresses

• Mark ${\sigma }_1$ (Max) and ${\sigma }_2$ (Min) on the horizontal $\sigma$-axis.
• Note: Shear is zero at these points.

2. Draw the Circle

• Centre ($C$): Midpoint between ${\sigma }_1$ and ${\sigma }_2$.
• Radius ($R$): Distance from $C$ to ${\sigma }_1$.
• Draw the full circle.

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The Rotation Rule ($2\theta$)

The Golden Rule of Mohr

Real World vs. Circle World

• Real Element: Angle is $\theta$.
• Mohr’s Circle: Angle is $2\theta$.
• Direction: The direction is the SAME.
  • If you rotate the cut Counter-Clockwise (CCW) by $\theta$ in reality, you rotate the radius Counter-Clockwise (CCW) by $2\theta$ on the circle.

The Procedure

1. Start at the Reference Point: Usually Point 1 (${\sigma }_1,0$), which represents the Principal Plane (where max stress acts).
2. Rotate: Measure an angle of $2\theta$ from the horizontal axis in the correct direction.
3. Find Point P: The end of this new radius is your state $({\sigma }_{\theta },{\tau }_{\theta })$.
4. Read Values: The coordinates of P are the Normal and Shear stress on that inclined plane.

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The “Pole” Method (Origin of Planes)

This is a graphical trick to find the stress on any plane without calculating angles first. It is often faster for complex problems.

1. Find the Pole ($P$)

• Start at a known point on the circle (e.g., the Principal Stress point ${\sigma }_1,0$).
• Draw a line through that point PARALLEL to the physical face it represents (e.g., since ${\sigma }_1$ acts on a vertical plane, draw a vertical line through the point).
• Where this line crosses the circle again is the Pole ($P$).

2. Use the Pole

• Draw a line from the Pole ($P$) parallel to any physical cut you want to analyse (angle $\theta$).
• The point where this line hits the circle gives you the exact $(\sigma ,\tau )$ for that cut.