Mohr's Circle & Strain Gauge Rosette Stress Calculator

Comprehensive Solid Mechanics: Mohr's Circle & Stress Transformation Guide

In continuum mechanics and structural engineering, determining the state of stress at a critical point of a loaded component (such as a pressure vessel, drive shaft, turbine rotor, or bridge girder) requires analyzing how normal and shear stresses transform as the observation plane rotates. First published by German civil engineer Christian Otto Mohr in 1882, **Mohr's Circle** provides a geometric representation of coordinate transformation equations for second-order symmetric tensors.

1. Plane Stress State and 2D Transformation Equations

Consider a differential square material element subjected to plane stress ($sigma_z = 0, tau_{xz} = 0, tau_{yz} = 0$). The planar stress state is described by normal stresses $sigma_x, sigma_y$ and shear stress $tau_{xy} = tau_{yx}$.

When the reference coordinate frame is rotated counterclockwise by an angle $theta$ to form new axes $(x', y')$, static equilibrium of a cut triangular wedge yields the fundamental **stress transformation equations**:

Notice that the sum of the orthogonal normal stresses is an invariant under any rotation:

2. Derivation of the Equation of the Circle

Squaring both sides of the transformation equations and adding them eliminates the angle $2theta$ using the Pythagorean identity $cos^2(2theta) + sin^2(2theta) = 1$:

This matches the standard Cartesian equation of a circle $(x - x_c)^2 + y^2 = R^2$ in the $sigma - tau$ plane, where:

• Center Coordinates: $C = (sigma_{avg}, 0) = left(frac{sigma_x + sigma_y}{2}, 0right)$
• Radius ($R$): $R = sqrt{left(frac{sigma_x - sigma_y}{2}right)^2 + tau_{xy}^2}$

3. Principal Stresses & Principal Orientation Angles

The **principal stresses** ($sigma_1, sigma_2$) are the extreme values of normal stress on the circle. They occur at the intersections of Mohr's circle with the horizontal $sigma$-axis, where shear stress is identically zero ($tau = 0$):

The orientation of the principal planes $theta_p$ is found by differentiating $sigma_{x'}$ with respect to $theta$ and setting the derivative to zero:

4. Maximum Shear Stresses & 3D Absolute Maximum

The maximum in-plane shear stress equals the radius of Mohr's circle and acts on planes oriented at $45^circ$ to the principal planes:

However, in structural design under plane stress where $sigma_3 = 0$, three 3D Mohr's circles exist (connecting pairs $(sigma_1, sigma_2)$, $(sigma_2, sigma_3)$, and $(sigma_1, sigma_3)$). The **absolute maximum shear stress ($tau_{abs,max}$)** governing ductile slip dislocation depends on the signs of $sigma_1$ and $sigma_2$:

• If $sigma_1 > 0$ and $sigma_2 < 0$: $tau_{abs,max} = frac{sigma_1 - sigma_2}{2} = tau_{max,in-plane}$
• If both $sigma_1, sigma_2 > 0$: $tau_{abs,max} = frac{sigma_1 - sigma_3}{2} = frac{sigma_1}{2}$
• If both $sigma_1, sigma_2 < 0$: $tau_{abs,max} = frac{sigma_3 - sigma_2}{2} = frac{|sigma_2|}{2}$

5. Yield Criteria: Von Mises vs. Tresca

For ductile metallic alloys (steel, aluminum, titanium), failure under multiaxial stress is governed by yield criteria:

• Tresca (Maximum Shear Stress Criterion): Yielding occurs when $tau_{abs,max} ge frac{sigma_y}{2} implies sigma_1 - sigma_3 ge sigma_y$.
• Von Mises (Distortional Energy Criterion): Yielding occurs when distortional strain energy reaches the uniaxial threshold:
  $$sigma_{vm} = sqrt{sigma_1^2 - sigma_1 sigma_2 + sigma_2^2} = sqrt{sigma_x^2 - sigma_x sigma_y + sigma_y^2 + 3tau_{xy}^2} le sigma_y$$

6. Strain Gauge Rosette Reductions

Because stress cannot be measured directly with physical sensors, electrical resistance strain gauges are bonded to surfaces to record normal strains. A single gauge measures strain in only one direction. To fully reconstruct the 2D stress tensor, a three-element **strain gauge rosette** is used:

Rectangular Rosette ($0^circ / 45^circ / 90^circ$)

With three gauges aligned at angles $theta_a = 0^circ$, $theta_b = 45^circ$, and $theta_c = 90^circ$:

Applying Hooke's Law for isotropic plane stress with Young's modulus $E$ and Poisson's ratio $nu$:

Delta Rosette ($0^circ / 60^circ / 120^circ$ Equiangular Configuration)

In applications where physical packaging demands symmetric gauge distributions, an equiangular rosette spaces three filaments at $0^circ, 60^circ$, and $120^circ$. The reduction equations are:

7. Hydrostatic Stress Tensor Decomposition & Octahedral Shear Stress

Any 3D Cauchy stress tensor $boldsymbol{sigma}$ can be uniquely decomposed into a volumetric (hydrostatic) spherical tensor $boldsymbol{sigma}_{hyd}$ that causes purely elastic volume changes, and a deviatoric stress tensor $boldsymbol{s}$ that drives plastic shape distortion:

The **octahedral shear stress** ($tau_{oct}$) acts on planes whose unit normals make equal angles with all three principal axes ($cosalpha = 1/sqrt{3}$). It is directly proportional to the von Mises equivalent stress:

This identity provides the thermodynamic foundation for the distortional energy failure hypothesis, confirming that ductile metallic crystals undergo dislocation slip solely due to octahedral shear stresses, remaining impervious to hydrostatic pressures up to hundreds of gigapascals.

8. Step-by-Step Worked Mohr's Circle Example

Analyze a beam subjected to combined bending and torsion with stress state:

• $sigma_x = 80.0text{ MPa}$ (Tension)
• $sigma_y = -40.0text{ MPa}$ (Compression)
• $tau_{xy} = 35.0text{ MPa}$ (Clockwise on right face)
• Find stresses on a plane inclined at $theta = 30.0^circ$

Step 1: Compute Center and Radius

Step 2: Calculate Principal Stresses

Step 3: Calculate Principal Orientation Angle

Step 4: Transformed Stresses at $theta = 30.0^circ$ ($2theta = 60.0^circ$)

Step 5: Von Mises Equivalent Stress