Structural Analysis Fundamentals / module 14
Complex stress and strain
Stress on any plane through a point, principal stresses, Mohr's circle, strain gauges and when materials yield.
What you will understand
- Describe the state of stress at a point using components on two perpendicular planes
- Derive the transformation equations from equilibrium of a wedge
- Find principal stresses, their directions and the maximum shear stress
- Use Mohr's circle and connect every construction to a physical plane
- Convert strain-gauge rosette readings into stresses
- Apply the Tresca and von Mises yield criteria
Lessons
Stress on an inclined plane
Why the answer depends on the plane, and deriving the transformation equations.
Principal stresses and Mohr's circle
The planes with no shear, the largest shear, and the picture that ties it all together.
Strain gauges and yield criteria
Measuring strain on a real structure, and deciding whether the material will yield.
Stress trajectories
Following the principal stresses through a beam, and why cracks go where they go.
Strain transformation and Mohr's circle of strain
The same circle again, with one factor of two that catches everybody.
Module summary
- Stress at a point is not a single number — it depends on the plane you look at
- σ and τ on an inclined plane follow from equilibrium of a wedge alone
- Principal planes carry no shear stress; the principal stresses are the largest and smallest normal stresses
- Maximum shear stress is (σ1 − σ2)/2, on planes 45° from the principal ones
- Mohr's circle is a picture of the transformation equations: a physical rotation of θ is 2θ on the circle
- Rosettes measure three strains so the full in-plane strain state can be recovered
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
At a point σx = 100 N/mm², σy = 40 N/mm² and τxy = 40 N/mm². What is the major principal stress?
Practice
For that same state, what is the maximum in-plane shear stress?
Check yourself
Rotating a physical plane by 20° moves you how far around Mohr's circle?