Question 7 of 10: Friction angle from SPT and design of a 2.0 m by 3.0 m footing

Question 7: Friction angle from SPT and design of a 2.0 m by 3.0 m 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. An SPT profile through a sandy deposit with the water table at 18 m depth, and a rectangular footing 2.0 m by 3.0 m founded at 1.5 m.

Find. A design value of the effective friction angle $\phi'$ from the blow counts, and then the allowable bearing pressure and column load for the 2.0 m by 3.0 m footing at 1.5 m depth, checking both bearing capacity and settlement.

Approach. Normalise each blow count for overburden to obtain $(N_1)_{60}$, average it over the footing's influence zone, convert to $\phi'$ with two published correlations, then run the Meyerhof general bearing-capacity equation for the design, as the question requires, and Meyerhof's settlement-controlled pressure only as a serviceability check, since that relation is itself a direct correlation to blow count.

1. Build the effective vertical stress profile. The water table at 18 m is below every test depth, so total and effective stresses coincide throughout the table. Accumulating the layer weights, $$\sigma'_v(z) = \sum \gamma_i \Delta z_i$$ gives, for example at 8 m, $\sigma'_v = 19.0(6) + 21.5(2) = 114 + 43 = 157\ \text{kPa}$. The full profile is tabulated in the Given data above.
2. Correct the blow counts for overburden. Blow count rises with confining stress even in a sand of constant density, so the raw values must be normalised to a reference stress of one atmosphere before any strength correlation is applied. Table I gives field blow counts $N_f$; no hammer type or energy measurement is reported, so the reference energy ratio of 60 per cent (a safety hammer) is assumed and $N_{60} = N_f\,\eta_H\eta_B\eta_S\eta_R/60 = N_f$. Using the Liao and Whitman (1986) correction, which is the form recommended in Das Ch. 2 and in CFEM, $$C_N = \sqrt{\frac{p_a}{\sigma'_v}}, \qquad p_a = 100\ \text{kPa}, \qquad (N_1)_{60} = C_N N_{60}$$ At 2 m, $C_N = \sqrt{100/38} = 1.622$ and $(N_1)_{60} = 6(1.622) = 9.7$; at 16 m, $C_N = \sqrt{100/328.2} = 0.552$ and $(N_1)_{60} = 26(0.552) = 14.4$. The corrected profile is nearly uniform below about 6 m, which tells us the deposit is a single medium-dense sand rather than a sequence of increasingly dense layers.
3. Choose the representative value for the footing. A footing of width $B = 2.0\ \text{m}$ founded at $D_f = 1.5\ \text{m}$ stresses the soil from the base down to roughly $2B$ below it, that is from 1.5 m to 5.5 m. The tests at 2, 4 and 6 m bracket that zone, so $$(N_1)_{60,\,av} = \frac{9.7 + 11.5 + 13.1}{3} = 11.4$$
4. Convert to a friction angle. Two independent correlations are used, both reproduced in Das, Principles of Foundation Engineering, Ch. 2. Wolff (1989) gives $$\phi' = 27.1 + 0.3 (N_1)_{60} - 0.00054 \left[(N_1)_{60}\right]^2 = 27.1 + 0.3(11.4) - 0.00054(11.4)^2 = 30.5^\circ$$ and Hatanaka and Uchida (1996) give $$\phi' = \sqrt{20 (N_1)_{60}} + 20 = \sqrt{20(11.4)} + 20 = 35.1^\circ$$ The Peck, Hanson and Thornburn curve reads about 31° at this blow count. Wolff is the lower bound and Hatanaka–Uchida the upper; adopting a value near the lower end of the band, $$\boxed{\phi'_{design} = 31^\circ}$$ which corresponds to a medium-dense sand and is consistent with a corrected blow count of about 11.
5. Bearing-capacity factors. Using the Meyerhof / Reissner–Vesic expressions with $\phi' = 31^\circ$, $$N_q = e^{\pi \tan\phi'}\tan^2\!\left(45^\circ + \frac{\phi'}{2}\right) = 20.63, \qquad N_c = (N_q - 1)\cot\phi' = 32.67$$ $$N_\gamma = 2\left(N_q + 1\right)\tan\phi' = 2(21.63)(0.6009) = 25.99$$
6. Shape and depth factors. With $B = 2.0\ \text{m}$, $L = 3.0\ \text{m}$ and $D_f/B = 0.75$, the De Beer shape factors and Hansen depth factors are $$F_{qs} = 1 + \frac{B}{L}\tan\phi' = 1 + 0.667(0.6009) = 1.401, \qquad F_{\gamma s} = 1 - 0.4\frac{B}{L} = 0.733$$ $$F_{qd} = 1 + 2\tan\phi'\left(1 - \sin\phi'\right)^2 \frac{D_f}{B} = 1 + 2(0.6009)(0.4850)^2(0.75) = 1.212, \qquad F_{\gamma d} = 1$$ The cohesion terms vanish because a clean sand has $c' = 0$.
7. Ultimate bearing capacity. The surcharge at founding level is $q = \gamma D_f = 19.0(1.5) = 28.5\ \text{kPa}$, and the water table is far below the failure zone so no buoyancy correction applies. Substituting into the general equation, $$q_u = q N_q F_{qs} F_{qd} + \tfrac{1}{2}\gamma B N_\gamma F_{\gamma s} F_{\gamma d}$$ $$q_u = 28.5(20.63)(1.401)(1.212) + \tfrac{1}{2}(19.0)(2.0)(25.99)(0.733) = 998.0 + 362.2 = 1360\ \text{kPa}$$ so that the net ultimate value is $$\boxed{q_{u(net)} = 1360 - 28.5 = 1332\ \text{kPa}}$$ and with the customary factor of safety of 3 on net bearing capacity, $q_{all(net)} = 1332/3 = 444\ \text{kPa}$, equivalent to a column load of $444(2.0)(3.0) = 2664\ \text{kN}$.
8. Serviceability check (not the design basis). The question's Note requires the design to rest on $\phi'$, and Meyerhof's settlement relation is itself a direct correlation of bearing pressure to blow count, so it is used here only to flag whether the strength-based pressure is serviceable. Meyerhof's settlement-controlled expression for $B > 1.22\ \text{m}$ (Das, Ch. 5) gives the net pressure producing 25 mm of settlement: $$q_{net(25)} = 7.99\, N_{60}\left(\frac{3.28B + 1}{3.28B}\right)^2 F_d\ \ [\text{kPa}], \qquad F_d = 1 + 0.33\frac{D_f}{B} \le 1.33$$ With the mean uncorrected $N_{60} = (6 + 10 + 14)/3 = 10$ over the influence zone, $F_d = 1 + 0.33(1.5/2.0) = 1.248$ and $\left[(3.28 \times 2 + 1)/(3.28 \times 2)\right]^2 = 1.328$, $$q_{net(25)} = 7.99(10)(1.328)(1.248) = 132\ \text{kPa}$$ Because the relation is linear in settlement ($q_{net} \propto S_e/25$), the strength-based 444 kPa would imply roughly $25(444/132.4) \approx 84\ \text{mm}$.
9. Adopt the design values. On the basis the question prescribes — $\phi' = 31^\circ$ in the general bearing-capacity equation with FS = 3 on the net ultimate value — the design is $$\boxed{q_{all(net)} = 444\ \text{kPa}, \qquad Q_{all} = 444(2.0)(3.0) = 2664\ \text{kN}}$$ The serviceability check shows that this pressure would settle of the order of 80 mm, so if the structure tolerates only the customary 25 mm the working pressure should be limited to about 132 kPa (column load about 794 kN, FS against bearing failure $1332/132 = 10.1$), or the settlement confirmed by a modulus-based analysis. That limit is a recommendation recorded alongside the prescribed design, not a replacement for it.

The result is the classic outcome for a footing on a medium-dense sand: the strength check is comfortably satisfied and serviceability, not shear failure, is what limits the working load in practice. If a larger column load is required, the efficient move is to widen the footing (which raises the settlement-limited pressure only slowly, since $q_{net(25)}$ tends to a constant for large $B$, but raises the total load in proportion to the area) or to deepen it, rather than to seek a higher factor of safety on bearing.