Gradation, Shear Strength and Permeability: 16-Civ-A4 Geotechnical Materials and Analysis, December 2018, Q3

Question 3: Gradation, Shear Strength and Permeability (15 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. Two sand gradation curves (Figure 2): Sand A plots to the right of Sand B and reaches the 10% ordinate at a larger particle size, i.e. $D_{10}(A)\gt D_{10}(B)$; Sand A is the coarser, more broadly (gap-)graded soil and Sand B the finer, more uniform one. Part (a) states that both sands have liquefied.

Find. (a) Which statement (i)–(iv) on the shear strength of the liquefied sands is true; (b) the sand with the higher saturated permeability, with reasons.

Figure Q3 — grain-size distribution. Sand A (red) is the coarser, more broadly graded soil: its D₁₀ is larger, so it has the higher permeability.

(a) Shear strength after liquefaction — answer (iv): both sand A and sand B have (essentially) no shear strength. A saturated cohesionless sand derives all of its strength from effective stress, $\tau_f = \sigma'\tan\phi'$ with $c'=0$. Liquefaction is, by definition, the state in which cyclic earthquake loading has driven the excess pore-water pressure up to the total stress, so $u = \sigma$ and $\sigma' = \sigma - u = 0$. Then $\tau_f = 0\times\tan\phi' = 0$ for both sands, whatever their friction angles — the grains are no longer pressed together and the soil behaves as a heavy fluid. Gradation still matters before liquefaction: the finer, uniform Sand B is the more liquefaction-susceptible and, drained, the coarser Sand A would mobilise the higher $\phi'$ — which is why (i) is the trap; once both have liquefied, (iv) is the true statement.

(b) Permeability — Sand A. By Hazen’s relation the saturated permeability scales with the effective size, $k \approx C\,D_{10}^{2}$, because the smallest 10% of the grains controls the size of the pore throats through which water must pass. Since $D_{10}(A)\gt D_{10}(B)$, Sand A has the larger pore channels and therefore the higher saturated coefficient of permeability. The finer Sand B, with smaller voids, impedes flow more strongly.

Topic: Liquefaction (zero effective stress) and reading a particle-size distribution for permeability.

Key relations: $\tau_f=\sigma'\tan\phi'$ with $\sigma'=\sigma-u$ (liquefied: $u=\sigma$, so $\tau_f=0$); uniformity $C_u=D_{60}/D_{10}$; Hazen $k\approx C\,D_{10}^{2}$.

Why this works: Sand strength is purely frictional, so it exists only while effective stress does; liquefaction removes the effective stress and with it all the strength, regardless of gradation. Permeability, by contrast, is a property of the pore geometry, and the curve’s $D_{10}$ measures the controlling pore-throat size, so the sand with the larger $D_{10}$ drains more freely.

Common pitfall: Answering (a) with the drained friction-angle argument (“the coarser sand is stronger”) — the question says both sands are liquefied, so $\sigma'=0$ and neither has shear strength. In (b), do not read “broadly graded” as “low permeability”; the larger $D_{10}$ decides it.

Source: Holtz, Kovacs & Sheahan, An Introduction to Geotechnical Engineering, chs. 2 & 7; Das, ch. 7 (Hazen).

Question 3 of 6: Gradation, Shear Strength and Permeability

Question 3: Gradation, Shear Strength and Permeability (15 marks)