Question 7 of 9: Friction Angle from SPT and Design of a 2.5 m Square Footing

Question 7: Friction Angle from SPT and Design of a 2.5 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.

Find. A representative design value of $\phi'$ for the bearing stratum from the corrected SPT profile, and then the allowable bearing pressure and allowable column load for a 2.5 m square footing, with the governing limit state identified.

Approach. Normalise each $N_{60}$ to a vertical effective stress of 100 kPa, convert the normalised counts to $\phi'$ with a published correlation and adopt a representative value over the footing's influence zone; then compute the ultimate bearing capacity from the general bearing-capacity equation, and separately the pressure that limits settlement to 25 mm, and design to whichever is smaller.

1. Part 1 — compute the vertical effective stress at each test depth. The water table stands at 18 m, below every test, so total and effective stresses are identical and no buoyancy correction arises. Taking $\gamma = 18.5$ kN/m$^3$ over the upper 6 m and 20.4 kN/m$^3$ below it, $\sigma'_v = 18.5z$ for $z \le 6$ m and $\sigma'_v = 111.0 + 20.4(z-6)$ below. At 3 m this gives 55.5 kPa and at 15 m, 294.6 kPa.
2. Apply the overburden correction. Using the Liao and Whitman (1986) form, which is the one recommended for routine work by CFEM, $$C_N=\sqrt{\frac{p_a}{\sigma'_v}}\le 1.7,\qquad (N_1)_{60}=C_N N_{60}$$ Applying this to each row gives the following table. The raw counts rise from 6 to 26 over the profile, a factor of more than four, while the corrected counts rise only from 8 to 16 — the deposit is far more uniform than the field record suggests.

   Depth (m)	$N_{60}$	$\sigma'_v$ (kPa)	$C_N$	$(N_1)_{60}$	$\phi'$, Wolff (deg)	$\phi'$, Hatanaka (deg)
   3.0	6	55.5	1.342	8.1	29.5	32.7
   5.0	10	92.5	1.040	10.4	30.2	34.4
   6.0	12	111.0	0.949	11.4	30.5	35.1
   8.0	15	151.8	0.812	12.2	30.7	35.6
   10.0	20	192.6	0.721	14.4	31.3	37.0
   12.0	24	233.4	0.655	15.7	31.7	37.7
   13.5	25	264.0	0.616	15.4	31.6	37.5
   15.0	26	294.6	0.583	15.2	31.5	37.4

3. Convert to a friction angle and select a design value. The technique adopted is the Peck, Hanson and Thornburn (1974) relation as fitted by Wolff (1989) and tabulated in Das, Principles of Foundation Engineering, Eq. (2.28): $$\phi'=27.1+0.30\,(N_1)_{60}-0.00054\,\left[(N_1)_{60}\right]^{2}$$ Over the footing's influence zone — from the base at 2.0 m to $D_f + 2B = 7.0$ m, which is where practically all of the bearing and settlement response is generated — the three tests at 3, 5 and 6 m give $\phi'$ of 29.5, 30.2 and 30.5 degrees, a mean of 30.0 degrees. Over the whole profile the mean is 30.9 degrees. The Hatanaka and Uchida (1996) expression $\phi'=\sqrt{20(N_1)_{60}}+20$ returns 33 to 37 degrees over the same data, so it is the more generous correlation; the conservative choice is therefore $$\boxed{\phi'_{design}=30^{\circ}}$$ which corresponds in the CFEM density table to a medium-dense sand with a relative density of roughly 40 to 50 percent, consistent with $(N_1)_{60}$ of 8 to 16.
4. Part 2 — bearing capacity factors for the design friction angle. From the general bearing-capacity equation (Das Table 3.3, Prandtl–Reissner $N_q$ with Vesic's $N_{\gamma}$): $$N_q=e^{\pi\tan\phi'}\tan^{2}\!\left(45^{\circ}+\frac{\phi'}{2}\right)=18.40,\qquad N_c=(N_q-1)\cot\phi'=30.14,\qquad N_{\gamma}=2(N_q+1)\tan\phi'=22.40$$ Since $c' = 0$ in a clean sand the $N_c$ term drops out of the calculation, but it is quoted because the shape factor for the surcharge term is written in terms of it.
5. Shape and depth factors. For a square footing (De Beer, 1970) and $D_f/B = 2.0/2.5 = 0.8 \le 1$ (Hansen, 1970): $$F_{qs}=1+\frac{B}{L}\tan\phi'=1.577,\qquad F_{\gamma s}=1-0.4\frac{B}{L}=0.60,\qquad F_{qd}=1+2\tan\phi'(1-\sin\phi')^{2}\frac{D_f}{B}=1.231,\qquad F_{\gamma d}=1$$
6. Ultimate and allowable bearing capacity. The surcharge at founding level is $q=\gamma D_f=18.5\times2.0=37.0$ kPa, and with $c'=0$ $$q_u=qN_qF_{qs}F_{qd}+\tfrac12\gamma BN_{\gamma}F_{\gamma s}F_{\gamma d} =1322.0+310.8=\boxed{1633\ \text{kPa}}$$ The net ultimate value is $q_{u(net)}=1633-37=1596$ kPa, and at a factor of safety of 3 $$q_{all(net)}=\frac{1596}{3}=532\ \text{kPa}\quad\Rightarrow\quad Q_{all}=532\times(2.5)^{2}=3325\ \text{kN}$$
7. Part 3 — the settlement limit, which will govern. A medium-dense sand fails at a very high pressure but compresses long before it gets there, so the serviceability limit must be checked independently. Meyerhof's (1965) SPT expression for $B > 1.22$ m (Das Eq. 5.42) gives the pressure that produces a chosen settlement: $$q_{all}=11.98\,N_{60}\left(\frac{3.28B+1}{3.28B}\right)^{2}\left(\frac{S_e}{25}\right)$$ Taking the mean $N_{60}$ of 6, 10 and 12 over the influence zone, that is $N_{60}\approx9$, with $B = 2.5$ m and $S_e = 25$ mm, $$q_{all}=11.98(9)\left(\frac{9.2}{8.2}\right)^{2}(1)=107.8\times1.259=\boxed{136\ \text{kPa}}$$ Das's form of the expression also carries a depth factor $F_d=1+0.33D_f/B=1.26$ (limited to 1.33); it is conservatively taken as 1.0 here, so the value above is a lower bound. This is a quarter of the bearing-capacity value, so settlement governs the design by a wide margin — the expected result for a square footing of this size on a medium-dense sand.
8. Cross-check the settlement by an independent method. Meyerhof's expression is known to be conservative, so the design pressure is confirmed with Schmertmann's strain-influence method using $E_s=500(N_{60}+15)$ kPa. At $\Delta q=136$ kPa the peak influence factor and the depth-embedment correction are $$I_{zp}=0.5+0.1\sqrt{\frac{\Delta q}{\sigma'_{zp}}}=0.5+0.1\sqrt{\frac{136}{60.1}}=0.650,\qquad C_1=1-0.5\frac{q}{\Delta q}=1-0.5\frac{37.0}{136}=0.864$$ Integrating $I_z/E_s$ over the three sublayers 2.0–3.25 m, 3.25–5.0 m and 5.0–7.0 m with $N_{60}$ of 6, 10 and 13 gives $S_e = C_1C_2\Delta q\sum(I_z/E_s)\Delta z = 16.3$ mm immediately, rising to 22.8 mm at ten years with the creep factor $C_2=1+0.2\log_{10}(t/0.1)=1.4$ (19.6 mm at one year, $C_2 = 1.2$), where $\sigma'_{zp}=18.5\times3.25=60.1$ kPa is the overburden at $B/2$ below the base. Both are below the 25 mm target, which confirms that 136 kPa is a safe rather than an optimistic design pressure.
9. State the design. Adopt a 2.5 m square pad founded at $D_f = 2.0$ m — below the frost depth typical of most of southern Canada and deep enough to develop the surcharge term — carrying an allowable net bearing pressure of 135 kPa and hence a service column load of $$Q_{all}=135\times(2.5)^{2}=\boxed{844\ \text{kN}}$$ with a factor of safety against bearing failure of $1596/135 = 11.8$ and a predicted settlement of 16 mm immediately and about 23 mm after ten years.