07-Str-B5 · December 2015

Question 5 of 6: Deep Foundations: driven pipe piles

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

Notes on this paper

Paper format: National Exams, December 2015 — 07-Str-B5 Foundation Engineering. Three hours, open book, any non-communicating calculator permitted. Six questions of equal value (30 marks each); five constitute a complete paper and only the first five in the answer book are marked. All six are solved here. This subject is pure geotechnical engineering.

Reference texts (the books an open-book candidate should have on the desk for this subject):

Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the Canadian limit-states framework, geotechnical resistance factors, pile design.
B. M. Das, Principles of Foundation Engineering, 9th ed. — bearing capacity, shape/depth/inclination factors, retaining walls, pile groups.
R. F. Craig / J. Knappett, Craig's Soil Mechanics, 9th ed. — Skempton's bearing-capacity factors, Skempton–Bjerrum settlement correction, slope-stability charts.
D. W. Taylor, Fundamentals of Soil Mechanics — the stability-number charts used in Question 3.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice, 6th ed. — adhesion factors, equivalent-raft settlement, block failure of pile groups.

Check — assumptions adopted across this paper. The examination omits several parameters that a foundation designer must have; each is adopted explicitly here and flagged again where it is used. (1) Question 2 and Question 5 give bulk unit weights but no groundwater table — effective stresses are computed with the quoted unit weights acting as effective weights (i.e. no free water in the profile); if instead the water table stood at ground level the effective stresses roughly halve and the computed consolidation settlements roughly double, and that sensitivity is reported. (2) Question 2 gives no allowable settlement — 50 mm total is adopted, the upper end of the CFEM 4th ed. range for framed structures. (3) Question 4 gives only the submerged unit weight of the foundation sand; the moist unit weight above the water table is taken as 20.0 kN/m³, consistent with the quoted $\gamma^{\prime} = 10.2$ kN/m³. (4) Concrete unit weight is taken as 24 kN/m³ throughout. (5) Where a question states only a factor of safety, the gross definition is used for shallow foundations sized on total load and the net definition where the question separates net pressure, and the convention used is stated in each answer.

Question 5 — Deep Foundations: driven pipe piles (30 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. Closed-ended 406 mm steel pipe piles, 20 m long, driven through 8 m of normally consolidated clay into a slightly overconsolidated clay that extends to 26 m, above very dense sand.

Given data — Question 5
Quantity	Symbol	Value
Pile diameter, wall thickness, length	d, t, L	406 mm, 12.7 mm, 20 m (closed-ended)
Upper clay: thickness, unit weight, strength	—	8 m, 16 kN/m³, c_u = 60 kPa (NC)
Lower clay: thickness, unit weight, strength	—	18 m, 18 kN/m³, c_u = 110 kPa (lightly OC)
Upper clay compressibility	C_c, e₀	0.35, 0.95
Lower clay compressibility	C_c, e₀	0.15, 0.60
Specified dead and live load	DL, LL	1.4 MN, 0.6 MN
Factor of safety on the single pile	F	2.5

Find. The allowable single-pile capacity, the number and layout of piles needed for the specified loads, the consolidation settlement of that group by the equivalent-raft method, and whether the group satisfies the ULS under factored loads.

Figure 5.1 — Driven pipe pile group and the equivalent raft used for the settlement check. The raft falls inside the overconsolidated clay, so only that layer contributes to consolidation.

Approach. Use the alpha method for shaft adhesion with Das's $\alpha$ against $c_u/p_a$ table, add $9c_u$ end bearing on the closed end, divide by 2.5 for the allowable load, size the group on the specified loads, place an equivalent raft at two-thirds of the pile length for settlement, and finally re-check the same group against factored loads with a CFEM geotechnical resistance factor.

Part (a) — pile geometry. Driven closed-ended, so the full circular section carries end bearing and the outside surface carries adhesion: $$p = \pi d = \pi(0.406) = 1.2755\ \text{m},\qquad A_p = \frac{\pi}{4}(0.406)^{2} = 0.12946\ \text{m}^{2}$$
Part (a) — adhesion factors. Entering Das's table with $c_u/p_a$ (with $p_a = 100$ kPa): $$\frac{60}{100} = 0.60 \Rightarrow \alpha_1 = 0.62,\qquad \frac{110}{100} = 1.10 \Rightarrow \alpha_2 = 0.45$$ The stiffer clay attracts the lower adhesion factor, which is the usual pattern: a stiff clay is disturbed and remoulded more, relative to its intact strength, by the passage of a driven pile.
Part (a) — shaft resistance. The pile passes through 8 m of the upper clay and 12 m of the lower clay: $$Q_{s1} = \alpha_1c_{u1}p\,L_1 = 0.62 \times 60 \times 1.2755 \times 8 = 379.6\ \text{kN}$$ $$Q_{s2} = \alpha_2c_{u2}p\,L_2 = 0.45 \times 110 \times 1.2755 \times 12 = 757.6\ \text{kN}$$
Part (a) — end bearing and design capacity. The toe is in the lower clay, well below five diameters: $$Q_p = 9c_{u2}A_p = 9 \times 110 \times 0.12946 = 128.2\ \text{kN}$$ $$Q_u = 379.6 + 757.6 + 128.2 = 1265.4\ \text{kN}$$ $$\boxed{Q_{all} = \frac{Q_u}{2.5} = \frac{1265.4}{2.5} = 506\ \text{kN per pile}}$$ End bearing is only 10 per cent of the total, so this is a friction pile in all but name.
Part (b) — number of piles for the specified loads. The specified (unfactored) load is $1400 + 600 = 2000$ kN, so $$n = \frac{2000}{506.2} = 3.95 \quad\Rightarrow\quad \boxed{n = 4\ \text{piles in a } 2\times 2\ \text{group}}$$ Adopting a spacing of 1.25 m (3.1 pile diameters, above the usual 2.5 to 3 diameter minimum) gives a group plan of $B_g = L_g = 1.25 + 0.406 = 1.656$ m and a cap of about 2.1 m square. The achieved factor of safety on the group is $4 \times 1265.4/2000 = 2.53$.
Part (b) — confirm that individual failure, not block failure, governs. For the same group, $$Q_{blk} = 2(B_g+L_g)\sum c_{ui}L_i + B_gL_g(9c_{u2}) = 6.624 \times 1800 + 2.744 \times 990 = 14\,638\ \text{kN}$$ against $4 \times 1265.4 = 5062$ kN for the four piles acting individually. The block is nearly three times stronger, so the group efficiency is 1.0 and no reduction is required.
Part (c) — locate the equivalent raft. These are friction piles, so the load is taken to act on a fictitious raft at two-thirds of the embedded length, with the plan area of the pile group: $$z_{raft} = \tfrac{2}{3}L = \tfrac{2}{3}(20) = 13.33\ \text{m},\qquad 1.656 \times 1.656\ \text{m},\qquad q_r = \frac{2000}{1.656^{2}} = 729\ \text{kPa}$$ The raft lies at 13.33 m, i.e. within the overconsolidated clay, so the normally consolidated clay above it is carried by the piles and does not consolidate under this load; only the 12.67 m from 13.33 m to the dense sand at 26 m contributes, with $C_c/(1+e_0) = 0.15/1.60 = 0.09375$.
Part (c) — consolidation settlement. Spreading the load at 2:1 below the raft and summing over 0.26 m sublayers, with $\sigma^{\prime}_{v0} = 16(8) + 18(z-8)$: $$\Delta\sigma(z) = \frac{Q}{(B_g+z^{*})(L_g+z^{*})},\qquad s_c = \sum \frac{C_c}{1+e_0}\Delta z\,\log_{10}\frac{\sigma^{\prime}_{v0}+\Delta\sigma}{\sigma^{\prime}_{v0}}$$ where $z^{*}$ is measured below the raft. The result is $$\boxed{s_c = 116\ \text{mm}}$$ of which 84 mm occurs in the first 2 m below the raft, where the spread has barely begun and the stress increase is still several hundred kilopascals.
Part (c) — interpret the settlement. A 116 mm consolidation settlement is far beyond the 25 to 50 mm normally tolerated for a residential structure, and the reason is visible in the arithmetic: four piles are a very small footprint for 2 MN, so the equivalent raft is only 1.66 m square and carries 729 kPa. The equivalent-raft idealisation is deliberately conservative at this scale, and a rigorous pile-group interaction analysis would give less, but the message is unambiguous: the group must be spread out. Enlarging to the six-pile group demanded by part (d), at the same 1.25 m spacing, reduces the equivalent raft pressure to 416 kPa and the settlement to 91 mm; going to a nine-pile group at 1.5 m spacing, or lengthening the piles to bear in the dense sand at 26 m, is what would actually bring the settlement inside tolerance.
Part (d) — the ultimate limit state check. Under the National Building Code of Canada load combination for dead plus live load, $$\sum \alpha_iQ_i = 1.25(1400) + 1.5(600) = 1750 + 900 = 2650\ \text{kN}$$ The factored geotechnical resistance uses the CFEM 4th edition resistance factor for axial compression from a static analysis, $\varphi_{gu} = 0.4$: $$R_f = \varphi_{gu}\,n\,Q_u = 0.4 \times 4 \times 1265.4 = 2025\ \text{kN}\ < \ 2650\ \text{kN}$$ $$\boxed{\text{the 4-pile group does NOT satisfy the ULS}}$$
Part (d) — the group the ULS actually requires. Solving for the number of piles, $$n \ge \frac{2650}{0.4 \times 1265.4} = 5.24 \quad\Rightarrow\quad n = 6\ \text{piles (2 by 3)},\qquad R_f = 3037\ \text{kN} \ge 2650\ \checkmark$$ The discrepancy between parts (b) and (d) is instructive rather than contradictory. A global factor of safety of 2.5 corresponds to an implied resistance factor of $1/2.5 = 0.40$ on the working load, but the limit-states check applies 0.4 to the resistance while simultaneously raising the load by an average factor of 1.325. The limit-states route is therefore about 30 per cent more demanding here, which is why $\varphi_{gu} = 0.4$ is paired in CFEM with the encouragement to raise it to 0.5 or 0.6 when the capacity is confirmed by a static load test or by dynamic monitoring. At $\varphi_{gu} = 0.5$ the requirement would fall to 4.2 piles, so a load test on the first production pile would very likely justify a five-pile group.

Check — assumptions in Question 5. (1) No groundwater table is given; effective stresses are computed with the quoted bulk unit weights acting as effective weights. With the water table at the ground surface, $\sigma^{\prime}_{v0}$ at the raft would drop from 224 to 118 kPa and the consolidation settlement would rise to roughly 190 mm, so the conclusion is unchanged. (2) $\alpha$ is read from Das's table (0.62 and 0.45); the API expression $\alpha = 0.5(c_u/\sigma^{\prime}_v)^{-0.5}$ would give 0.52 and 0.73, a 32 per cent larger $Q_u$ and a three-pile group, so the choice of adhesion correlation matters and Das's values are the more conservative. (3) The lower clay is "slightly overconsolidated" but no preconsolidation pressure is given, so the full $C_c$ is used with no recompression branch. (4) Negative skin friction is not considered; if the site were being filled or the upper NC clay were still consolidating, downdrag would have to be added to the load in part (d).

Final results — Question 5
Quantity	Result
Adhesion factors, upper and lower clay	0.62 and 0.45
(a) Shaft resistance, upper and lower clay	379.6 kN and 757.6 kN
(a) End bearing	128.2 kN
(a) Ultimate and design axial capacity (F = 2.5)	1265 kN, 506 kN per pile
(b) Specified load, piles required	2000 kN, n = 3.95, use 4 piles (2 by 2 at 1.25 m)
(b) Block capacity check	14 638 kN, individual failure governs (η = 1.0)
(c) Equivalent raft	1.656 m square at 13.33 m, q = 729 kPa
(c) Consolidation settlement, 4-pile group	116 mm (91 mm for the 6-pile group)
(d) Factored load and factored resistance (n = 4)	2650 kN against 2025 kN, ULS not satisfied
(d) Piles required for the ULS	n = 5.24, use 6 piles (R_f = 3037 kN)

Topic. Alpha-method capacity of driven closed-ended pipe piles in layered clay, group sizing under working-stress and limit-states criteria, and equivalent-raft consolidation settlement.

Key relations. $Q_u = \sum \alpha_ic_{ui}p\,L_i + 9c_{u,toe}A_p$ with $\alpha$ from $c_u/p_a$. Working stress: $Q_{all} = Q_u/F$, $n \ge \sum Q_{spec}/Q_{all}$. Block failure $Q_{blk} = 2(B_g+L_g)\sum c_{ui}L_i + B_gL_g\,9c_u$. Limit states: $\varphi_{gu}nQ_u \ge 1.25DL + 1.5LL$ with $\varphi_{gu} = 0.4$ for static analysis. Equivalent raft at $2L/3$ with 2:1 spread and $s_c = \sum[C_c/(1+e_0)]H\log_{10}[(\sigma^{\prime}_{v0}+\Delta\sigma)/\sigma^{\prime}_{v0}]$.

Why this works. The alpha method sidesteps the fact that nobody can predict the effective stress acting on a pile shaft after driving. Installing a closed-ended pile displaces a full pile volume of clay, generating excess pore pressures of the order of the undrained strength itself, which then dissipate over weeks and leave the clay locally remoulded and reconsolidated. Rather than model that, the method correlates the mobilised shaft stress directly with the intact $c_u$ through a single empirical factor, and the factor falls with increasing strength because stiff clays lose proportionally more of their strength on remoulding. The equivalent raft then answers a completely different question. Capacity is about the soil immediately around the pile; settlement is about the soil beneath the group as a whole, which never sees an individual pile at all. Placing the fictitious raft at two-thirds depth is a way of saying that the load has been shed into the soil progressively down the shaft, and by two-thirds of the way down essentially all of it has arrived.

Common pitfall. The commonest slip is to compute the group settlement from the individual pile settlement, which misses the point entirely: a 2 by 2 group settles far more than a single pile at the same load per pile, because the stress bulbs overlap and the influence depth grows with the group width. The second is to mix the two design philosophies, for example checking a factored load against an allowable capacity, which double-counts the safety margin. The third, specific to this problem, is to include the normally consolidated clay in the settlement summation: it lies above the equivalent raft, is carried by shaft adhesion, and does not consolidate under this load, though it would if the site were surcharged.

Source. B. M. Das, Principles of Foundation Engineering, 9th ed., Ch. 12 (pile capacity by the alpha method, group efficiency, consolidation settlement of pile groups); Canadian Geotechnical Society, Canadian Foundation Engineering Manual, 4th ed., Ch. 18 (deep foundations) and Table 8.1 (geotechnical resistance factors); National Research Council, National Building Code of Canada 2020, Part 4 (load combinations); M. J. Tomlinson and J. Woodward, Pile Design and Construction Practice, 6th ed., Ch. 4 and 5.

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