16-Civ-B3 Geotechnical Design · December 2017

Question 9 of 9: Friction angle from the SPT and design of a 3 m square footing

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, December 2017 — 16-Civ-B3 Geotechnical Design; three hours, open book, any non-communicating calculator. Section A holds five discussion questions worth 7 marks each of which four are marked; Section B holds four design questions worth 24 marks each of which three are marked, so the examinable total is 4 × 7 + 3 × 24 = 100 marks. All nine questions are worked below, because the set is a study resource rather than a marked script.

Reference texts. B. M. Das, Principles of Foundation Engineering, 8th–9th ed. (Cengage); B. M. Das, Principles of Geotechnical Engineering, 9th ed.; R. F. Craig / J. Knappett, Craig's Soil Mechanics, 9th ed.; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed.; D. P. Coduto, Foundation Design: Principles and Practices; J. E. Bowles, Foundation Analysis and Design; ASTM D1586 (SPT), D5778 (CPTu), D2573 (field vane).

Source of design charts and assumed values (page 1, Note 6). Rankine active and passive coefficients from Das, Principles of Foundation Engineering, Ch. 7 (Eqs. 7.11 and 7.30); cantilever-wall stability procedure from Das Ch. 8 (Eqs. 8.11–8.14). Drilled-shaft adhesion factor alpha* = 0.55 and the 1.5 m surface exclusion from Reese & O'Neill (1989), tabulated in Das Ch. 12; bearing factor Nc* = 9 from Skempton (1951); block-failure check from Das Ch. 11 (Eq. 11.55). Compression index from Skempton's Cc = 0.009(LL − 10). Strain-influence factors and the C1, C2 corrections from Schmertmann, Hartman & Brown (1978) as presented in Das Ch. 5; Es = 500(N60 + 15) kPa from Das Table 5.7 (Bowles). Friction angle from the SPT via Wolff (1989), phi' = 27.1 + 0.3(N1)60 − 0.00054[(N1)60]², with the Liao & Whitman (1986) overburden correction; Vesic bearing-capacity, shape and depth factors from Das Tables 4.2 and 4.3. Every assumed value (unit weights of concrete, specific gravity for the void ratio, pile spacing, factors of safety) is stated in a callout beside the step that uses it.

Question 9: Friction angle from the SPT and design of a 3 m square footing (24 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 sandy deposit with the SPT record below, the water table at 18 m, and a square footing 3.0 m × 3.0 m founded at D_f = 2.5 m.

Depth (m)	2	4	6	8	10	12	14	16
gamma (kN/m³)	19.0	19.0	19.0	21.5	21.4	21.4	21.4	21.4
N₆₀	6	10	14	18	20	24	25	26

Find. A design friction angle from the blow counts, with its source named, and hence the allowable bearing pressure and column load for the 3 m square footing, checking both bearing capacity and settlement.

The 3 m square footing at 2.5 m, its influence zone of 2B = 6 m, and the blow-count record. The dashed curve is N₆₀ corrected for overburden by the Liao & Whitman relation; note how nearly constant (N₁)₆₀ is below 4 m, which is what identifies a uniform deposit of medium density.

Approach. Correct the field blow counts for effective overburden, average the corrected values over the depth of influence beneath the footing, convert to phi' by Wolff's correlation, then use that phi' in the general (Vesic) bearing-capacity equation for the strength check and in Schmertmann's method with an N-derived modulus for the settlement check, and adopt the lower of the two allowable pressures.

Complying with the exam's note. The note forbids methods that correlate bearing capacity directly to a penetration index — Meyerhof's q_all = 11.98N₆₀[(3.28B + 1)/3.28B]² rule and Terzaghi–Peck's charts are the ones it has in mind. It does not forbid using N to estimate a soil property. The route taken here is N₆₀ → (N₁)₆₀ → phi' → bearing-capacity factors for the strength check, and N₆₀ → E_s → Schmertmann for the settlement check; the modulus correlation is a deformation property, not a capacity, so the settlement calculation is admissible. Meyerhof's direct rule is quoted at the end only as an independent order-of-magnitude check, never as the design basis.

Effective overburden at each test depth. The water table is at 18 m, below every test, so total and effective stresses are equal and each layer's tabulated unit weight is applied over the 2 m above its test depth: $$\sigma'_0(2)=2(19.0)=38.0\ \text{kPa}, \quad \sigma'_0(4)=76.0, \quad \sigma'_0(6)=114.0, \quad \sigma'_0(8)=114.0+2(21.5)=157.0\ \text{kPa}$$ and so on to 328.2 kPa at 16 m.
Overburden correction. Liao and Whitman's form, $C_N=\sqrt{p_a/\sigma'_0}$ with $p_a = 100$ kPa, capped at 1.7, gives the corrected blow counts $(N_1)_{60}=C_NN_{60}$.

Depth (m)	2	4	6	8	10	12	14	16
sigma'₀ (kPa)	38.0	76.0	114.0	157.0	199.8	242.6	285.4	328.2
C_N	1.622	1.147	0.937	0.798	0.707	0.642	0.592	0.552
N₆₀	6	10	14	18	20	24	25	26
(N₁)₆₀	9.7	11.5	13.1	14.4	14.2	15.4	14.8	14.4

Average over the zone that matters. The bearing-capacity mechanism and most of the settlement develop between the base at 2.5 m and a depth of $2B = 6$ m below it, that is from 2.5 m to 8.5 m. The three tests inside that window are at 4, 6 and 8 m, giving $$(N_1)_{60,avg}=\frac{11.47+13.11+14.37}{3}=13.0, \qquad N_{60,avg}=\frac{10+14+18}{3}=14$$ Corrected values below 4 m sit between 13 and 15 throughout, so the deposit is uniform medium-dense sand and the average is representative rather than a smoothing of scattered data.
Friction angle. Wolff's (1989) correlation, as tabulated by Das, converts the corrected blow count to a friction angle: $$\phi' = 27.1+0.3(N_1)_{60}-0.00054\left[(N_1)_{60}\right]^{2}=27.1+0.3(13.0)-0.00054(13.0)^{2}=\boxed{30.9^{\circ}\approx 31^{\circ}}$$ Peck, Hanson and Thornburn's chart gives the same value to within a degree for this blow count. Kulhawy and Mayne's expression, $\phi'=\tan^{-1}\left[N_{60}/\left(12.2+20.3\,\sigma'_0/p_a\right)\right]^{0.34}$, returns 35 to 36° over the same three depths; the lower Wolff value is adopted because it is the conservative end of the defensible range and because a 5° difference in phi' changes N_gamma by more than a factor of two.
Bearing-capacity factors at phi' = 31°. Using the Reissner, Prandtl and Vesic expressions, $$N_q = e^{\pi\tan 31^{\circ}}\tan^{2}\!\left(45^{\circ}+\tfrac{31^{\circ}}{2}\right)=20.63, \qquad N_\gamma = 2(N_q+1)\tan 31^{\circ}=25.99$$ with $N_c = (N_q-1)\cot\phi' = 32.67$, which plays no part because $c' = 0$.
Shape and depth factors. For a square footing, $B/L = 1$: $$F_{qs}=1+\frac{B}{L}\tan\phi'=1.601, \qquad F_{\gamma s}=1-0.4\frac{B}{L}=0.600$$ and with $D_f/B = 2.5/3.0 = 0.833 \le 1$, $$F_{qd}=1+2\tan\phi'\left(1-\sin\phi'\right)^{2}\frac{D_f}{B}=1+2(0.6009)(0.4850)^{2}(0.8333)=1.236, \qquad F_{\gamma d}=1$$
Ultimate and allowable bearing capacity. The surcharge at founding level is $q = 19.0(2.5)=47.5$ kPa, so $$q_u=qN_qF_{qs}F_{qd}+\tfrac{1}{2}\gamma BN_\gamma F_{\gamma s}F_{\gamma d}=47.5(20.63)(1.601)(1.236)+\tfrac{1}{2}(19.0)(3.0)(25.99)(0.600)=1938+445=2383\ \text{kPa}$$ Taking a factor of safety of 3 on the net capacity, $$q_{all(net)}=\frac{q_u-q}{3}=\frac{2383-47.5}{3}=778\ \text{kPa}$$
Soil modulus for the settlement check. Bowles' correlation for sand, tabulated by Das, gives $E_s = 500(N_{60}+15)$ kPa, so the 2 m band centred on each test yields 10 500 kPa for the 2 m test, 12 500 kPa at 4 m, 14 500 kPa at 6 m and 16 500 kPa at 8 m.
Schmertmann settlement for a square footing. Here $L/B = 1$, so the axisymmetric diagram applies: $I_z = 0.1$ at the base, peak at $z = B/2 = 1.5$ m, zero at $z = 2B = 6.0$ m. The effective stress at the depth of the peak, 4.0 m, is $\sigma'_{zp}=19.0(4.0)=76.0$ kPa, and the peak ordinate is $$I_{zp}=0.5+0.1\sqrt{\frac{\Delta q}{\sigma'_{zp}}}$$ which contains the unknown, so the calculation is iterative. Integrating $I_z/E_s$ over the five sublayers gives $\sum (I_z/E_s)\Delta z = 1.538\times10^{-4}$ m³/kN at convergence, with $I_{zp}=0.657$.
Solve for the pressure that gives 25 mm. With no time specified, take the immediate settlement, $C_2 = 1.0$, and $C_1 = 1-0.5(47.5)/\Delta q$: $$S_e = C_1(1.0)\Delta q\left(1.538\times10^{-4}\right)=0.025\ \text{m} \Rightarrow \Delta q = 186\ \text{kPa}, \quad C_1 = 0.873$$ so the settlement-controlled net pressure is $$\boxed{q_{all(net)}=186\ \text{kPa}, \quad \bar{q}=186+47.5=234\ \text{kPa gross}}$$
Adopt the design values. Settlement governs by a factor of more than four over the strength criterion, so the footing is designed for a net allowable pressure of about 185 kPa, giving a permissible column load of $$Q_{all}=q_{all(net)}\,B L = 186(3.0)(3.0)=1676\ \text{kN}, \ \text{say } 1650\ \text{kN}$$ At that pressure the factor of safety against a bearing failure is $2335/186 = 12.5$, which is the signature of a settlement-governed design on medium-dense sand.
Independent check on the settlement answer. Meyerhof's direct 25 mm rule, $q_{all(net)}=11.98N_{60}\left[(3.28B+1)/(3.28B)\right]^{2}$, gives $11.98(14)(1.214)=204$ kPa for this footing, about 10 per cent above the Schmertmann value of 186 kPa. It is quoted only as corroboration, since the exam's note excludes it as a design basis; agreement at this level is useful evidence that the modulus correlation and the influence diagram have been applied correctly, and the lower Schmertmann figure is the one adopted.

Quantity	Value
Design zone (base to 2B below)	2.5 m to 8.5 m
Average N₆₀ / (N₁)₆₀ in the zone	14 / 13.0
Design friction angle (Wolff, 1989)	31°
N_q / N_gamma (Reissner, Vesic)	20.63 / 25.99
Shape / depth factors F_qs, F_{gamma s}, F_qd	1.601, 0.600, 1.236
Ultimate bearing capacity q_u	2383 kPa
Net allowable from strength (FS = 3)	778 kPa — not critical
Σ(I_z/E_s)Δz / I_zp	1.538 × 10⁻⁴ m³/kN / 0.657
Net allowable from the 25 mm limit	186 kPa (governs)
Gross pressure at founding level	234 kPa
Allowable column load on the 3.0 m square footing	1676 kN, say 1650 kN
Meyerhof 25 mm rule (check only)	204 kPa

Check — judgements behind the design. (1) The unit weight tabulated at a given depth is taken to apply to the 2 m of soil above that depth, which is the only self-consistent reading of a table that starts at 2 m; the jump from 19.0 to 21.5 kN/m³ between 6 and 8 m marks a denser sub-stratum and is consistent with the rise in N₆₀. (2) The correlation source is named as the exam requires: Wolff (1989) via Das, Table 2.5, with the Liao & Whitman (1986) overburden correction; a chart source would have been Peck, Hanson & Thornburn (1974). (3) The 25 mm settlement limit is the conventional criterion for an isolated footing on sand (CFEM Ch. 8); if the structure tolerates 40 mm the permissible net pressure rises to about 270 kPa and the strength criterion is still not reached. (4) The ten-year creep factor C₂ = 1.4 was not applied because the question sets no time; including it would reduce the net allowable pressure to 143 kPa and a column load of 1290 kN, and for a permanent structure that is the value worth carrying into design.

Topic. Deriving a design friction angle from corrected SPT blow counts, and using it in a bearing-capacity and settlement design.

Key relations. C_N = (p_a/sigma'₀)^0.5 ≤ 1.7 (Liao & Whitman); (N₁)₆₀ = C_NN₆₀; phi' = 27.1 + 0.3(N₁)₆₀ − 0.00054[(N₁)₆₀]² (Wolff); general bearing-capacity equation with Vesic factors; E_s = 500(N₆₀ + 15) kPa; Schmertmann S_e = C₁C₂ΔqΣ(I_z/E_s)Δz with I_zp = 0.5 + 0.1(Δq/sigma'_zp)^0.5.

Why this works. The blow count measures the resistance of the sand to dynamic penetration, which depends on both its relative density and the confining stress acting at the time of the test. Dividing out the confining-stress effect with C_N leaves a number that tracks relative density alone, and relative density is what determines the peak friction angle. That is why the raw record here rises steadily with depth while the corrected record is nearly flat: the sand is of uniform density and it is the increasing overburden, not increasing density, that produced most of the trend. Averaging over the zone that the footing actually stresses, rather than over the whole borehole, is the second half of getting a representative value.

Common pitfall. Averaging blow counts over the full depth of the borehole, which here would give phi' about 2° higher; applying C_N twice by using a correlation that already expects (N₁)₆₀ on data that has been corrected; taking sigma'_zp at the base rather than at the depth of the peak influence factor; and stopping at the bearing-capacity check on a sand, where settlement almost always governs.

Source. Das, Principles of Foundation Engineering, Ch. 2 (Section 2.7, Eqs. 2.11–2.15 and Table 2.5) for the SPT corrections and phi' correlations, Ch. 4 (Table 4.2, Eqs. 4.26–4.28) for the general bearing-capacity equation, and Ch. 5 (Section 5.7, Table 5.7) for E_s and the strain-influence method; Canadian Foundation Engineering Manual, 4th ed., Ch. 4 and 10.

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