Module 9 · Lesson 9.2

Stress paths

Tracking the stress point through a loading; total and effective paths.

Why this matters

Two soil elements can end at the same failure stress yet behave completely differently on the way, because they travelled different paths. The stress path records that journey, and — crucially — the effective stress path is what the soil actually feels. Reading it tells you how close a loading is driving the soil towards failure.

From first principles

Pore pressure shifts the path sideways, not up

We want to show: Show that changing pore pressure moves the stress point horizontally (mean stress) but not vertically (deviatoric stress).

Pore pressure is a hydrostatic (all-round) stress. It pushes equally on every face, so it cannot create any shear — it can only change the mean stress. On a plot of shear against mean stress, that means a purely sideways move.

Now the two loading cases fall out:

Drained loading — the soil drains as it is loaded, so no excess pore pressure develops ( $u$ stays at its static value). The ESP runs parallel to and a fixed distance left of the TSP; both climb as the load adds shear.
Undrained loading — the water cannot escape, so shearing generates excess pore pressure that changes with the load. The ESP peels away from the TSP: it bends left if the soil wants to contract (loose or normally consolidated, positive excess $u$ ), or right if it wants to dilate (dense or heavily overconsolidated, negative excess $u$ ).

Failure occurs when the effective stress path reaches the failure line — never the total path.

Practice

During undrained loading a clay element reaches total stresses σ1 = 250 kPa, σ3 = 130 kPa with an excess pore pressure u = 70 kPa. What is the effective mean-stress invariant s′, in kPa?

Summary

A stress path is the track of the stress point through a loading.
ESP = TSP shifted left by the pore pressure u; the two are the same height (t, q).
Drained: no excess u, ESP parallel to TSP. Undrained: excess u develops, ESP peels left (contractive) or right (dilative).
Failure is when the effective stress path reaches the failure line — never the total path.

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint