Mohr's Circle 2D Stress Transformation Engineering Reference Guide
Peer-reviewed mathematical formulation, verified step-by-step worked engineering calculations, design codes, and practical guidelines.
Governing Theory & Derivation
Analytical physics and classical mechanics foundations
Mohr's Circle, developed by German civil engineer Christian Otto Mohr in 1882, is a graphical and analytical method for transforming plane stress tensors. Given normal stresses (σ_x, σ_y) and shear stress (τ_xy) on an orthogonal element, it determines the maximum normal (principal) stresses, maximum in-plane shear stress, and principal plane orientations.
1. Coordinate Transformation Equations
Equilibrium of an inclined wedged element rotated by an angle θ yields the normal and shear stress on the rotated plane:
2. Circle Radius and Center
Squaring and adding these equations yields the equation of a circle (σ - σ_avg)² + τ² = R² centered on the normal stress axis:
3. Principal Stresses and Maximum Shear
The principal stresses occur where shear stress is identically zero. Maximum in-plane shear stress equals the circle radius R:
Step-by-Step Worked Example
Biaxial Stress Transformation in a Pressurized Thin-Walled Vessel
Problem Statement
A strain gauge rosette on a pressure vessel skin reveals a plane stress state of σ_x = 80 MPa (longitudinal tension), σ_y = -40 MPa (transverse compression), and τ_xy = 35 MPa. Determine the principal stresses, maximum shear stress, principal angle, and Von Mises equivalent stress.
Given Parameters
Detailed Calculation Steps
The circle center is at (20 MPa, 0) with a radius of 69.46 MPa.
σ_1 represents the peak tensile stress on the material, and σ_2 represents peak compressive stress.
Rotating the element 15.13° counter-clockwise aligns the element with the principal normal axes where shear stress vanishes.
Von Mises stress compares multi-axial stresses directly against a uniaxial tensile yield strength test.
Applicable Standards & Design Codes
Recognized engineering guidelines and criteria
Rules for Construction of Pressure Vessels
Uses stress intensity (Tresca criterion) and distortion energy (Von Mises) for allowable stress limits.
Standard Test Methods for Tension Testing of Metallic Materials
Standard benchmark for uniaxial tensile yields used in equivalent stress comparisons.
Assumptions & Practical Limitations
Critical constraints for professional applications
- Strictly 2D plane stress formulation (σ_z = τ_xz = τ_yz = 0). For 3D stress states, the out-of-plane principal stress (σ_3 = 0) must be checked for absolute maximum shear.
- Applies only to linear, elastic, homogeneous, and isotropic materials.
- Does not predict crack propagation or fracture mechanics in brittle materials with sharp stress raisers.
Frequently Asked Questions
Common technical engineering inquiries
Q:Why is Mohr's circle represented as 2θ instead of θ?
In coordinate transformations, the trigonometrical identity for transformed shear and normal stress involves double angles (cos 2θ and sin 2θ). Therefore, an actual physical rotation of angle θ on the material corresponds to a 2θ rotation around the center on Mohr's circle.
Q:How does Von Mises stress relate to Mohr's circle?
Mohr's circle identifies the principal stresses (σ_1 and σ_2). Von Mises stress takes these principal stresses and calculates the octahedral shear stress (distortion energy) representing the yield threshold for ductile materials.