Soil Mechanics / module 9
Stress paths, drainage and loading
Following the stress state as a load is applied: the Mohr circle and its invariants, total and effective stress paths, and why drained and undrained loading trace different routes to failure.
What you will understand
- Represent a stress state with the Mohr circle and its centre/radius invariants
- Compute the s–t and p–q invariants and relate them to total and effective stress
- Show that pore pressure shifts the mean stress but not the deviatoric stress
- Draw total and effective stress paths for a loading
- Explain how drained and undrained paths differ and reach the failure line differently
Lessons
Representing a stress state
The Mohr circle, the s–t and p–q invariants, and the total–effective split.
Stress paths
Tracking the stress point through a loading; total and effective paths.
Why the path matters
The same soil, two loadings, two routes to failure.
Module summary
- The Mohr circle and its s–t / p–q invariants represent a stress state as a point that moves along a stress path.
- Pore pressure shifts the mean stress (s′ = s − u) but not the deviatoric stress (t′ = t): ESP = TSP shifted left by u.
- Drained loading keeps ESP parallel to TSP; undrained loading makes the ESP peel towards or away from failure.
- Failure is governed by the effective stress path; loading on clay is short-term critical, unloading long-term critical.
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
A stress state has total principal stresses σ1 = 300 kPa and σ3 = 120 kPa with pore pressure u = 50 kPa. What is the effective mean-stress invariant s′, in kPa?
Check yourself
On a plot of shear (t) against mean stress (s), increasing the pore pressure at constant total stress moves the effective stress point:
Practice
For total stresses σ1 = 180 kPa and σ3 = 60 kPa, what is the deviatoric invariant t (the Mohr-circle radius), in kPa? (It is the same in total and effective stress.)