16-Civ-B3 Geotechnical Design · May 2015

Question 7 of 9: Load Carrying Capacity of a Sixteen-Pile Group in Soft Clay

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

Notes on this paper

Paper format. Professional Engineers Ontario / Engineers Canada National Examinations, May 2015 — 98-Civ-B3 Geotechnical Design. Three hours, OPEN BOOK, any non-communicating calculator. Section A carries five discussion questions of 7 marks each (answer any four); Section B carries four design questions of 24 marks each (answer any three); 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 timed attempt.

Reference texts (98-Civ-B3 / 16-Civ-B3 Geotechnical Design).

B. M. Das, Principles of Foundation Engineering, 9th ed. — bearing capacity (Ch. 3), stress increase in a soil mass (Ch. 6), retaining walls (Ch. 8), pile foundations (Ch. 11), subsurface exploration (Ch. 2).
B. M. Das, Principles of Geotechnical Engineering, 9th ed. — consolidation (Ch. 11), shear strength (Ch. 12), lateral earth pressure (Ch. 13).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for site investigation, bearing resistance, deep foundations and earth-retaining structures.
R. F. Craig, Craig's Soil Mechanics, 9th ed. — earth pressure theory, effective stress and slope stability.
D. P. Coduto, Foundation Design: Principles and Practices, 3rd ed. — in-situ testing, settlement of shallow and deep foundations.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice, 6th ed. — shaft adhesion in clay, pile-group behaviour, load testing.

Sources of design charts and assumed values (page-1 Note 6). Note 6 of this paper requires the candidate to identify the source of every design chart used and of every value assumed in the absence of data. They are named where used and collected here:

Q6 — bearing capacity factors $N_c$, $N_q$, $N_{\gamma}$ from Das, Principles of Foundation Engineering, Table 3.3 (Prandtl–Reissner $N_q$, Vesic $N_{\gamma}=2(N_q+1)\tan\phi'$); shape factors after De Beer (1970) and depth factors after Hansen (1970), Das Table 3.4. Assumed: unit weight of water $\gamma_w = 9.81$ kN/m$^3$; general shear failure; the sand extends at least $2B$ below the base.
Q7 — adhesion factor $\alpha = 1.0$ for soft clay ($c_u \le 50$ kPa) from NAVFAC DM-7.2 Fig. 1 and Tomlinson & Woodward Table 4.6; $\lambda = 0.24$ at an embedded length of 12 m from Vijayvergiya & Focht (1972) as tabulated by Das, Table 11.7; bearing factor $N_c^{*}=9$ for $L/D \ge 4$ (Skempton). Assumed: pile spacing $s = 3d = 1.5$ m centre to centre, driven closed-end concrete piles, clay $\gamma_{sat} = 17$ kN/m$^3$ with the water table at ground level.
Q8 — compression index from the Skempton correlation $C_c = 0.009\,(LL-10)$, Das Principles of Geotechnical Engineering Eq. (11.42); $2{:}1$ stress distribution, Das Eq. (6.31); Boussinesq rectangular influence factor (Das Table 6.6) used as the cross-check.
Q9 — Coulomb active pressure coefficient, Das Principles of Foundation Engineering Eq. (8.13) with the wall friction angle prescribed by the question, $\delta = 0.6\phi' = 18^{\circ}$. Assumed: unit weight of reinforced concrete $\gamma_c = 24$ kN/m$^3$ (CSA A23.3 nominal); passive resistance in front of the toe neglected; the backfill surface is horizontal and carries no surcharge.

Question 7: Load Carrying Capacity of a Sixteen-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.

Quantity	Symbol	Value
Number of piles, arrangement	$m \times n$	$4 \times 4 = 16$
Pile length (embedded)	$L$	12 m
Pile diameter	$d$	0.5 m
Average undrained strength of clay	$c_u$	50 kPa
Pile spacing (assumed)	$s$	$3d = 1.5$ m
Adhesion factor (assumed, soft clay)	$\alpha$	1.0
Toe bearing factor	$N_c^{*}$	9
Clay saturated unit weight (assumed)	$\gamma_{sat}$	17 kN/m$^3$

Find. The ultimate load carrying capacity of the 16-pile group, taking the smaller of the sum of the individual pile capacities and the capacity of the group acting as a single block, and the corresponding allowable load at a factor of safety of 3.

Sixteen-pile group: plan of the cap at $s = 3d$ centres (left) and section through one row (right). Only four of the sixteen piles appear in the section.

Approach. Compute the ultimate capacity of a single pile by the $\alpha$ (total-stress) method, multiply by 16 for the sum of the individual capacities, compute the capacity of the group acting as a single block of plan $B_g \times L_g$ and depth $L$, and take the smaller of the two as the group capacity; then check the shaft resistance against the independent $\lambda$-method.

Geometry of a single pile and of the group. The cross-sectional area at the toe is $A_p = \pi d^2/4 = \pi(0.5)^2/4 = 0.19635$ m$^2$ and the perimeter is $p = \pi d = \pi(0.5) = 1.5708$ m. Adopting the standard minimum spacing $s = 3d = 1.5$ m (closer spacing causes driving interference and heave in soft clay; wider spacing wastes cap concrete), a $4\times4$ group with the cap projecting $d/2$ beyond the outer piles has plan dimensions $B_g = L_g = 3s + d = 3(1.5)+0.5 = 5.0$ m.
End bearing of a single pile. In a saturated clay loaded undrained the toe resistance is $Q_p = A_p N_c^{*} c_u$ with $N_c^{*}=9$ for an embedment ratio $L/d = 24 \gg 4$ (Skempton): $$Q_p = 0.19635 \times 9 \times 50 = 88.4\ \text{kN}$$ The toe contributes less than one tenth of the capacity, as expected for a long friction pile in soft clay.
Shaft resistance of a single pile by the alpha method. For a soft clay with $c_u \le 50$ kPa the adhesion factor is taken as $\alpha = 1.0$ (NAVFAC DM-7.2 adhesion chart for driven concrete piles; Tomlinson & Woodward Table 4.6). Then $$Q_s = \alpha\,c_u\,p\,L = 1.0 \times 50 \times 1.5708 \times 12 = 942.5\ \text{kN}$$ and the ultimate capacity of the single pile is $$\boxed{Q_{u(\text{single})} = 88.4 + 942.5 = 1030.8\ \text{kN}}$$
Sum of the individual pile capacities. Taken as sixteen independent piles, $$\sum Q_u = 16 \times 1030.8 = 16{,}493\ \text{kN}$$ This is the upper bound: it presumes that each pile can mobilise its full shaft and toe resistance without interfering with its neighbours.
Capacity of the group acting as a block. The competing failure mode is that the whole $5.0 \times 5.0 \times 12$ m block of clay and piles punches down as one body, shearing on its perimeter at the full undrained strength of the clay (soil against soil) and bearing at its base: $$Q_{g(\text{block})} = 2(B_g+L_g)\,c_u L + 9\,c_u\,B_g L_g$$ Substituting, the perimeter term is $2(5.0+5.0)(50)(12) = 12{,}000$ kN and the base term is $9(50)(5.0)(5.0) = 11{,}250$ kN, so $Q_{g(\text{block})} = 23{,}250$ kN.
Governing group capacity and group efficiency. The design value is the smaller of the two, $$Q_{g(u)} = \min\{16{,}493;\ 23{,}250\} = 16{,}493\ \text{kN}$$ so the individual-pile mode governs and the group efficiency is $\eta = 16{,}493/16{,}493 = 1.0$. At the customary factor of safety of 3 on ultimate capacity, $$\boxed{Q_{g(\text{all})} = \frac{16{,}493}{3} = 5{,}500\ \text{kN}\ \ (\approx 5.5\ \text{MN})}$$ which is about 344 kN of working load per pile.
Independent check on the shaft resistance by the lambda method. Because $\alpha = 1.0$ is the most generous defensible value, the answer should be bracketed. With the water table at ground level and $\gamma_{sat} = 17$ kN/m$^3$, the mean effective vertical stress over the shaft is $\bar\sigma'_v = (17-9.81)(12)/2 = 43.1$ kPa. Vijayvergiya and Focht give $\lambda = 0.24$ at an embedded length of 12 m, so $$f_{av} = \lambda(\bar\sigma'_v + 2\bar c_u) = 0.24\,(43.1 + 100) = 34.4\ \text{kPa}$$ giving $Q_s = 34.4(1.5708)(12) = 647.6$ kN, a single-pile capacity of 735.9 kN, a group ultimate of 11,775 kN and an allowable load of 3,925 kN. The two methods bracket the answer between roughly 3.9 MN and 5.5 MN, which is the honest precision of a capacity predicted from $c_u$ alone.
Empirical efficiency formulae, and why they are not used here. The Converse–Labarre expression $\eta = 1 - \theta[(n-1)m+(m-1)n]/(90mn)$ with $\theta = \tan^{-1}(d/s) = 18.44^{\circ}$ returns $\eta = 0.693$ and hence 11,426 kN. Formulae of this kind were calibrated on driven piles in sand and have no theoretical basis in clay; current practice (Das §11.19, CFEM Ch. 18) is the block-versus-sum comparison used in step 6, and the Converse–Labarre value is quoted here only to show that it happens to fall inside the bracket established by the two strength methods.

Quantity	Value
Toe area $A_p$ / perimeter $p$	0.19635 m$^2$ / 1.5708 m
End bearing per pile, $Q_p$	88.4 kN
Shaft resistance per pile ($\alpha = 1.0$), $Q_s$	942.5 kN
Ultimate capacity, single pile	1030.8 kN
Sum of 16 individual piles	16,493 kN
Group as a block ($5.0 \times 5.0 \times 12$ m)	23,250 kN
Governing ultimate group capacity	16,493 kN
Group efficiency, $\eta$	1.00 (individual mode governs)
Allowable group load at FS = 3	5,500 kN
Lambda-method bracket (ultimate / allowable)	11,775 kN / 3,925 kN

Other criteria to be considered in the design of this pile group. Capacity is only the first of several limit states, and for a friction group in soft clay it is rarely the one that governs:

Settlement. A group settles far more than a single pile at the same load per pile, because the stress bulbs of the individual piles overlap into one large bulb that reaches roughly $1.5B_g = 7.5$ m below the toe. Settlement must be computed by the equivalent-raft method — a raft at $2L/3 = 8$ m depth of plan $B_g \times L_g$ spreading at 2:1 — adding elastic settlement and the consolidation settlement of every compressible layer within the bulb. In soft clay this normally governs the design.
Negative skin friction (downdrag). If fill is placed over the site, if the water table is lowered, or if the soft clay is still consolidating under its own weight, the clay moves down relative to the piles and the shaft friction reverses over the affected length, adding load instead of carrying it. This is the classic hazard of a friction group in soft clay and can add several hundred kilonewtons per pile.
What lies below the toe. A 12 m pile in soft clay must be proved to have competent ground beneath it; if the clay continues, the group is a floating foundation and the consolidation settlement of the underlying clay controls.
Installation effects. Driving sixteen displacement piles at 1.5 m centres in soft clay displaces about 38 m$^3$ of soil, causing ground heave, lateral displacement and large excess pore pressures that can lift previously driven piles and damage adjacent structures or services. Redriving checks, a sensible driving sequence (inside outwards), and allowance for set-up (capacity may double over weeks) are all needed.
Lateral load, moment and eccentricity from wind, earth pressure or crane surcharge, checked by $p$–$y$ analysis or Broms' charts, and the resulting tension in the trailing piles.
Structural design of the pile section and of the cap for punching, bending and shear, plus the connection detail; and durability of the pile material in a soft clay that may be sulphate-bearing or contain organic acids.
Verification. Because $c_u$-based predictions carry a wide scatter, a preliminary static load test to failure plus dynamic testing (PDA with CAPWAP) on about one production pile in ten is normal Canadian practice and is the only way to justify a factor of safety at the lower end of the range.

Topic. Ultimate capacity of a friction pile group in saturated clay: individual-pile mode against block mode.

Key relations. $Q_u = Q_p + Q_s = A_pN_c^{*}c_u + \sum\alpha c_u p\Delta L$; block capacity $Q_g = 2(B_g+L_g)c_uL + N_c^{*}c_uB_gL_g$; group efficiency $\eta = Q_{g}/\sum Q_u$; $\lambda$-method $f_{av} = \lambda(\bar\sigma'_v+2\bar c_u)$; equivalent raft at $2L/3$ for group settlement.

Why this works. Two kinematically admissible failure mechanisms exist for a group: each pile can slip through the soil independently, or the piles and the clay between them can move as one block that shears on its outer surface. Whichever requires less load is the one that will occur, so the capacity is the smaller of the two. The block mode matters only when the piles are close together: at $s = 3d$ the block perimeter $4 \times 5.0 = 20$ m is much less than the sum of the sixteen pile perimeters $16 \times 1.571 = 25.1$ m, but the block also gains a very large base area, and here that is enough to make the block mode the stronger one.

Common pitfall. Reporting the block capacity as the answer without comparing it to the sum, or applying an empirical efficiency formula such as Converse–Labarre to a clay for which it was never calibrated. A second trap is to declare the design complete once capacity is checked: settlement and downdrag, not capacity, usually size a friction group in soft clay.

Source. Das, Principles of Foundation Engineering 9th ed. §11.12 ($\alpha$ method), §11.14 ($\lambda$ method), §11.19–11.21 (group capacity and settlement); CFEM 4th ed. Ch. 18.

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