07-Str-B1 · May 2015

Question 7 of 9: Load Carrying Capacity of a 4 × 4 Pile Group in Soft Clay

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

Notes on this paper

National Examinations — May 2015 — 07-Str-B1 Geotechnical Design. Three-hour, OPEN-BOOK exam; any non-communicating calculator permitted (the candidate must record its make and model). Format: Section A carries five discussion questions of 7 marks each, of which any FOUR are to be answered; Section B carries four design problems of 24 marks each, of which any THREE are to be answered — a marked total of 100. The paper instructs candidates to state any interpretive assumptions, to identify the source of every design chart and assumed value, and to exercise sound engineering judgment where data are absent. All nine printed questions are worked below, because the set is intended as a study resource.

Reference texts: Das, B.M., Principles of Foundation Engineering (9th ed., Cengage) — general bearing-capacity equation, pile and pile-group capacity, consolidation settlement of footings, retaining walls; Das, B.M., Principles of Geotechnical Engineering (9th ed., Cengage) — lateral earth pressure, effective stress, consolidation theory; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM, 4th ed., 2006) — Canadian practice for site investigation, SPT/CPT interpretation, pile design and tolerable settlement; Craig, R.F. / Knappett, J.A., Craig's Soil Mechanics (8th ed., CRC Press) — shear strength and earth-pressure theory; Duncan, J.M., Wright, S.G. & Brandon, T.L., Soil Strength and Slope Stability (2nd ed., Wiley) — fully softened and residual strengths for fissured and expansive clays; Fredlund, D.G., Rahardjo, H. & Fredlund, M.D., Unsaturated Soil Mechanics in Engineering Practice (Wiley) — swelling soils and matric suction.

Note — Figure 2 is printed over a coarse halftone. The soil-property annotations inside the photograph-style Figure 2 (Question 8) are printed over a coarse dot screen. The values used below are read from the printed figure and are: upper sand $\gamma = 15\ \text{kN/m}^3$ over 1.5 m, lower sand $\gamma_{sat} = 18\ \text{kN/m}^3$ over 1.5 m, normally consolidated clay 2.5 m thick with $w = 35\%$ and $LL = 48$, over sand; groundwater table at the underside of the footing.

Assumptions declared once, applied throughout. $\gamma_w = 9.81\ \text{kN/m}^3$; reinforced concrete $\gamma_c = 24\ \text{kN/m}^3$; specific gravity of soil solids $G_s = 2.70$ where a void ratio must be back-figured from water content; loads are vertical and concentric unless stated. Every assumption that changes a numerical answer is repeated in the question where it is used.

Question 7: Load Carrying Capacity of a 4 × 4 Pile Group in Soft Clay (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. Sixteen cylindrical piles in a square 4 × 4 arrangement, each 12 m long and 500 mm in diameter, embedded in a uniform soft clay of average undrained shear strength 50 kPa.

Given data and assumed values — Question 7
Quantity	Symbol	Value
Number of piles (4 × 4)	$n$	16
Pile diameter	$d$	0.500 m
Embedded length	$L$	12 m
Undrained strength of clay	$c_u$	50 kPa
Centre-to-centre spacing (assumed)	$s = 3d$	1.50 m
Adhesion factor (assumed, Das table)	$\alpha$	0.68
Toe bearing factor	$N^*_c$	9

Find. The ultimate and allowable load carrying capacity of the group, and the additional criteria that govern its design.

Assumptions, with justification. (i) Spacing $s = 3d = 1.5$ m. The question omits it. Three diameters is the normal minimum in codes and in CFEM: closer spacing makes block failure and installation heave critical, wider spacing gives an uneconomically large cap. (ii) $\alpha = 0.68$. With $c_u/p_a = 50/100 = 0.5$, Das's tabulation after Terzaghi, Peck and Mesri gives $\alpha = 0.68$ by interpolation between 0.74 at $c_u/p_a = 0.4$ and 0.62 at 0.6. (A designer treating the clay as genuinely soft and normally consolidated might take $\alpha \to 1.0$, which raises the single-pile capacity to 1031 kN; the value adopted here is the more conservative and is the tabulated one.) (iii) $N^*_c = 9$ for a pile whose embedment exceeds four diameters — the standard Skempton value. (iv) The clay is uniform over the full length and no soft or fissured layer underlies the toes. (v) Piles are driven, so no relaxation of shaft resistance from boring is applied, and capacity is assessed at long term after full set-up. (vi) The pile cap is not in contact with the ground, so it contributes no bearing. (vii) The group is loaded vertically and concentrically. (viii) A factor of safety of 3 on ultimate capacity, appropriate where no static load test is performed.

Figure 7.1 — Assumed group geometry: sixteen 500 mm piles at 3d = 1.5 m centres give a block 5.0 m square in plan and 12 m deep.

Approach. Compute the ultimate capacity of a single pile by the $\alpha$ (total-stress) method, multiply by 16 for the individual-pile failure mode, compute the capacity of the group failing as a single block, and take the smaller of the two as the group capacity, as Das and CFEM require for piles in clay.

Pile geometry. The cross-sectional area and perimeter of a 500 mm cylindrical pile are $$A_p = \frac{\pi d^2}{4} = \frac{\pi(0.5)^2}{4} = 0.1963\ \text{m}^2 \qquad p = \pi d = \pi(0.5) = 1.5708\ \text{m}$$
Toe (point) resistance of a single pile. In saturated clay loaded undrained, $Q_p = A_p N^*_c c_u$ with $N^*_c = 9$: $$Q_p = 0.1963(9)(50) = 88.4\ \text{kN}$$ The toe contributes little, which is characteristic of a friction pile in clay.
Shaft resistance of a single pile. By the $\alpha$ method, $Q_s = \alpha c_u p L$: $$Q_s = 0.68(50)(1.5708)(12) = 640.9\ \text{kN}$$
Ultimate capacity of a single pile. Summing the two contributions, $$Q_{u} = Q_p + Q_s = 88.4 + 640.9 = \boxed{729.2\ \text{kN per pile}}$$
Mode 1 — individual pile failure. If each pile punches through the clay independently, $$\sum Q_u = n\,Q_u = 16(729.2) = 11\,668\ \text{kN}$$
Group plan dimensions for the block mode. With four piles per side at 1.5 m centres, $$B_g = L_g = (n_1 - 1)s + d = 3(1.5) + 0.5 = 5.0\ \text{m}$$
Mode 2 — block failure. The block is sheared on its four faces at the full undrained strength (clay against clay) and bears on its base with $N^*_c = 9$: $$Q_{g(block)} = 9c_uB_gL_g + 2(B_g+L_g)L\,c_u$$ $$Q_{g(block)} = 9(50)(5.0)(5.0) + 2(10.0)(12)(50) = 11\,250 + 12\,000 = 23\,250\ \text{kN}$$
Governing mode and group efficiency. The smaller of the two controls: $$Q_{g(u)} = \min(11\,668,\ 23\,250) = \boxed{11\,668\ \text{kN}}$$ so individual-pile failure governs and the group efficiency is $\eta = 11\,668/11\,668 = 1.00$. At three-diameter spacing in clay the block is more than twice as strong as the sum of the piles, so no group reduction applies.
Allowable capacity. With a factor of safety of 3 on the ultimate group capacity, $$Q_{g(all)} = \frac{11\,668}{3} = \boxed{3889\ \text{kN}}$$
Cross-check with an empirical efficiency formula. The Converse–Labarre expression, with $\theta = \tan^{-1}(d/s) = 18.43^\circ$, gives $$\eta = 1 - \frac{\theta\left[(n_1-1)n_2 + (n_2-1)n_1\right]}{90\,n_1n_2} = 1 - \frac{18.43(24)}{90(16)} = 0.693$$ that is, 8084 kN. Converse–Labarre was calibrated on piles in sand and is not the recommended basis for clay, but it is a useful reminder that a designer who is uncertain about spacing effects has a conservative fallback.

Other criteria that govern the design of this group. Ultimate capacity is only the first of several checks.

Settlement. For a group in clay, settlement is estimated by placing an equivalent raft at two-thirds of the embedded depth (8 m here) and computing consolidation settlement of the clay beneath it under the group stress. Group settlement is typically several times the settlement of a single pile at the same average load, so serviceability frequently governs the pile count rather than capacity.
Negative skin friction (downdrag). If the site is filled, or if the clay is still consolidating under recent surcharge or dewatering, the upper shaft drags the pile down and must be treated as a load, not a resistance. Bitumen coating of the upper shaft is the usual mitigation.
Lateral load, moment and buckling. Wind, seismic and eccentric loads produce shear and moment at the pile heads; check by $p$–$y$ analysis and check the structural capacity of the pile section, which for a slender pile in very soft clay may also require a buckling check.
Uplift. Shaft resistance in tension is lower than in compression (typically 70 to 80 %) and the toe contributes nothing; check any corner pile that goes into tension under overturning.
Structural and constructional issues. Cap thickness and punching shear; pile installation tolerance and the additional moments from out-of-position piles; heave and lateral displacement of previously driven piles when driving in soft clay, which may require re-driving; installation-induced pore pressures and their effect on nearby structures; noise and vibration limits.
Verification. Static load testing of at least one pile, or dynamic testing with signal matching on a sample of production piles, is the only reliable confirmation of the assumed $\alpha$; the factor of safety may be reduced toward 2 where load testing is carried out.

Final results — Question 7
Quantity	Symbol	Value
Pile area / perimeter	$A_p$ / $p$	0.1963 m² / 1.5708 m
Toe resistance, single pile	$Q_p$	88.4 kN
Shaft resistance, single pile ($\alpha = 0.68$)	$Q_s$	640.9 kN
Ultimate capacity, single pile	$Q_u$	729.2 kN
Sum of individual capacities (16 piles)	$\sum Q_u$	11 668 kN
Group plan dimensions	$B_g \times L_g$	5.0 m × 5.0 m
Block failure capacity	$Q_{g(block)}$	23 250 kN
Ultimate group capacity (governing)	$Q_{g(u)}$	11 668 kN
Group efficiency	$\eta$	1.00
Allowable group capacity (FS = 3)	$Q_{g(all)}$	3889 kN

Topic. Ultimate capacity of a pile group in saturated clay and the two competing failure modes.

Key relations. $Q_u = A_pN^*_cc_u + \sum \alpha c_u p \Delta L$ with $N^*_c = 9$; group as individual piles $\sum Q_u = nQ_u$; group as a block $Q_{g} = 9c_uB_gL_g + 2(B_g+L_g)Lc_u$; $B_g = (n_1-1)s + d$; $\eta = Q_{g(u)}/nQ_u$; Converse–Labarre $\eta = 1 - \theta[(n_1-1)n_2+(n_2-1)n_1]/(90n_1n_2)$.

Why this works. A pile group can fail in only two kinematically admissible ways: each pile can punch through the soil on its own cylindrical surface, or the whole block of soil enclosed by the outer piles can move down as one, shearing on its perimeter and bearing on its base. Whichever requires less load is the real capacity, so the design value is the minimum of the two. At wide spacing the perimeter of $n$ separate cylinders greatly exceeds the perimeter of the block, so the block mode is stronger and individual failure governs; at close spacing the block perimeter is comparatively small and the block mode takes over, which is why efficiency falls below one only for tight groups.

Common pitfall. Applying a Converse–Labarre-type efficiency reduction to a clay group as well as checking block failure, which double-counts the group effect and under-designs nothing but wastes piles; or omitting the block check entirely for a tight group, which is unconservative. A separate and frequent error is quoting the group ultimate capacity as the design load without ever checking group settlement, which usually governs.

Source. Das, Principles of Foundation Engineering, 9th ed., §12.12–§12.16 ($\alpha$ method) and Ch. 13 (Pile-group efficiency, block failure, group settlement); CFEM 4th ed., §18.5 (Pile Groups).

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