Question 2 of 6: Effective stress under an added load — drained vs undrained

Question 2: Effective stress under an added load — drained vs undrained (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A saturated soil element (S = 100 %) carrying total vertical stress σ, with a static pore-water pressure u_s registered by the stand-pipe; a further total stress Δσ is then applied at the surface.

Find. The effective stress σ' immediately after loading (undrained) and after full drainage (drained), expressed through σ, Δσ, u_s and u_e, with commentary.

Approach. Apply Terzaghi's effective-stress principle σ' = σ − u, and split the pore pressure into the static part u_s plus the load-induced excess part u_e; for a saturated soil the pore-pressure parameter B = 1, so the instant of loading transfers all of Δσ to the water.

1. Initial (reference) state. Before Δσ, the pore pressure is purely static and $$\sigma' = \sigma - u_s.$$ This is the effective stress the skeleton already carries.
2. Undrained instant — the water takes the load. The container is saturated and closed, so no water can leave. For a saturated soil the excess pore pressure equals the applied total-stress increment (B = 1): $$u_e = \Delta\sigma .$$ The pore pressure becomes u = u_s + u_e = u_s + Δσ, and the total stress becomes σ + Δσ. Hence $$\sigma'_{\text{undrained}} = (\sigma + \Delta\sigma) - (u_s + \Delta\sigma) = \boxed{\,\sigma - u_s\,}.$$ The effective stress is unchanged from the initial value; the entire increment Δσ is momentarily carried by the pore water.
3. Drained end state — the excess dissipates. Given time, the excess pressure drains away, u_e → 0, and the pore pressure returns to the static value u_s. With the total stress still σ + Δσ, $$\sigma'_{\text{drained}} = (\sigma + \Delta\sigma) - u_s = \boxed{\,(\sigma - u_s) + \Delta\sigma\,}.$$ The effective stress has now risen by exactly Δσ.

Comparing the two: the added stress Δσ is first parked entirely in the pore water and only later transferred to the soil skeleton as the water drains. Because it is the effective stress that controls volume change and shear strength, the soil experiences no immediate compression or strength gain (undrained), whereas the long-term effective stress increases by the full Δσ, producing the consolidation settlement and strength gain. This is precisely Terzaghi's one-dimensional consolidation picture: settlement is time-dependent because σ' grows only as u_e decays.