16-Civ-B3 Geotechnical Design · May 2016

Question 8 of 9: Ultimate capacity of a driven pile through clay into dense sand (24 marks)

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 2016 — 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. — subsurface exploration (Ch. 2), bearing capacity of shallow foundations (Ch. 3), settlement of shallow foundations including Schmertmann's method (Ch. 5), retaining walls (Ch. 8), pile foundations (Ch. 11).
B. M. Das, Principles of Geotechnical Engineering, 9th ed. — shear strength (Ch. 12), lateral earth pressure (Ch. 13), slope stability including the planar-surface analysis (Ch. 15).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for site investigation, in-situ testing, bearing resistance, deep foundations and earth-retaining structures.
R. F. Craig, Craig's Soil Mechanics, 9th ed. — effective stress, undrained versus drained behaviour, earth pressure and slope stability.
D. P. Coduto, Foundation Design: Principles and Practices, 3rd ed. — CPT correlations and settlement of shallow foundations on sand.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice, 6th ed. — shaft adhesion in clay and driven displacement piles in sand.

Sources of design charts and assumed values (page-1 Note 6). Note 6 requires the candidate to identify the source of every design chart used and of every value assumed where the paper gives none. They are named at the point of use and collected here:

Strain-influence diagram and the C₁, C₂ correction factors (Q6) — Schmertmann, Hartman and Brown (1978), as tabulated in Das, Principles of Foundation Engineering, 9th ed., Section 5.6.
Rankine active coefficient for an inclined backfill (Q7) — Das, Principles of Geotechnical Engineering, 9th ed., Eq. (13.35).
Bearing-capacity factors N₢, Nᵤ, Nγ and the depth and load- inclination factors (Q7) — Vesic / Meyerhof as tabulated in Das, Principles of Foundation Engineering, 9th ed., Tables 3.3 and 3.4, applied to a retaining-wall base in Section 8.6.
Meyerhof bearing-capacity factor Nᵤ* for a driven pile point and the limiting point resistance (Q8) — Das, 9th ed., Section 11.9 and its interpolated Nᵤ* table; Nᵤ* = 143 at φ′ = 35°.
Adhesion factor α against cu/p₀ (Q8) — Das, 9th ed., Table 11.6 (after Terzaghi, Peck and Mesri); α = 0.68 at cu/p₀ = 0.5.
Earth-pressure coefficient K and interface friction angle δ′ for a driven high-displacement pile (Q8) — Das, 9th ed., Section 11.11: K ≈ 1.4K₀ and δ′ ≈ 0.8φ′ are assumed, and the critical-depth rule L′ = 15D is Das Eq. (11.42).
Unit weight of water γᵣ = 9.81 kN/m³ and g = 9.81 m/s² throughout; atmospheric pressure p₀ = 100 kPa.

Section A — discussion questions (7 marks each)

Question 8 — Ultimate capacity of a driven pile through clay into dense sand (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. Read from Figure 3.

Quantity	Symbol	Value
Pile type	—	prefabricated (precast) concrete, driven — a high-displacement pile
Pile diameter	D	0.50 m
Upper layer: clay	c_u, γ	50 kPa, 17 kN/m³, thickness 5.0 m
Lower layer: dense sand	c′, φ′, γ	0, 35°, 20 kN/m³, thickness 8.0 m (pile fully embedded)
Total embedded length	L	13.0 m
Groundwater table	—	at the ground surface (Figure 3)
Unit weight of water	γ_w	9.81 kN/m³

Find. The ultimate bearing capacity $Q_u = Q_p + Q_s$ of the single pile, and the allowable load at a factor of safety of 3.

Figure Q8. Soil profile of Figure 3. The pile is driven through 5.0 m of clay and 8.0 m of dense sand; the water table stands at the ground surface, so both unit weights shown are saturated values and the vertical effective stress is computed with buoyant weights throughout.

Approach. Build the vertical effective-stress profile with buoyant unit weights, take the point resistance from Meyerhof's expression for a pile in sand together with its limiting value, take the shaft resistance in the clay by the $\alpha$ method and in the sand by the $\beta$-type expression $f = K\sigma'_v\tan\delta'$ with the critical-depth cut-off, and sum.

Pile geometry. $$A_p = \frac{\pi D^{2}}{4} = \frac{\pi (0.50)^{2}}{4} = 0.19635\ \text{m}^2, \qquad p = \pi D = 1.5708\ \text{m}.$$
Effective-stress profile. The water table is at the ground surface, so the given unit weights are saturated values and buoyant weights apply from the top down. At the base of the clay (5.0 m), $$\sigma'_v = \left(17 - 9.81\right)\times 5.0 = 35.95\ \text{kPa},$$ and at the pile tip (13.0 m), $$\sigma'_v = 35.95 + \left(20 - 9.81\right)\times 8.0 = 117.47\ \text{kPa}.$$
Point resistance: Meyerhof's expression. The tip is in dense sand, so the point resistance is governed entirely by the frictional term $$Q_p = A_p q' N_q^{*},$$ with $N_q^{*} = 143$ at $\phi' = 35^\circ$ from Meyerhof's chart as tabulated in the reference text. Substituting, $$Q_p = 0.19635 \times 117.47 \times 143 = 3298\ \text{kN}.$$
Apply Meyerhof's limiting point resistance. Meyerhof observed that the unit point resistance does not grow without limit with depth; it is capped at $$q_p \le 0.5\,p_a N_q^{*}\tan\phi' = 0.5 \times 100 \times 143 \times \tan 35^\circ = 5006\ \text{kPa},$$ against the unlimited value $q'N_q^{*} = 117.47 \times 143 = 16\,798$ kPa. The cap governs by a wide margin, so $$Q_p = A_p \times 5006 = 0.19635 \times 5006 = \boxed{Q_p = 983\ \text{kN}}.$$ This is the usual outcome for a slender pile: the embedment ratio into the sand is 8.0/0.5 = 16 and the overall ratio is 26, both beyond the point where the limit bites.
Shaft resistance in the clay: the $\alpha$ method. $$Q_{s,\text{clay}} = \alpha c_u\,p\,L_{\text{clay}} .$$ The adhesion factor is read against the normalised strength $c_u/p_a = 50/100 = 0.5$, which gives $\alpha = 0.68$ by linear interpolation in the reference table (0.74 at 0.4 and 0.62 at 0.6). Hence $$Q_{s,\text{clay}} = 0.68 \times 50 \times 1.5708 \times 5.0 = 267.0\ \text{kN}.$$
Set up the shaft resistance in the sand. The unit friction is $f = K\sigma'_v\tan\delta'$. For a driven high-displacement pile the reference text recommends an earth-pressure coefficient between $K_0$ and $1.8K_0$; the mid-range value is adopted, $$K = 1.4K_0 = 1.4\left(1 - \sin 35^\circ\right) = 1.4(0.42642) = 0.5970,$$ with an interface friction angle $\delta' = 0.8\phi' = 28^\circ$ for a precast concrete surface, so $\tan\delta' = 0.53171$.
Apply the critical-depth rule. Unit shaft friction in sand increases with depth only to a critical depth $$L' = 15D = 15(0.50) = 7.50\ \text{m}$$ below the ground surface, and is constant below it. The effective stress at that depth is $$\sigma'_{v}\left(7.5\ \text{m}\right) = 35.95 + 10.19\times2.5 = 61.43\ \text{kPa}.$$ The sand shaft therefore splits at 7.5 m.
Shaft resistance from 5.0 m to 7.5 m. The effective stress rises linearly from 35.95 to 61.43 kPa, so the average unit friction over this 2.5 m is $$f_{\text{av}} = K\left(\frac{35.95 + 61.43}{2}\right)\tan\delta' = 0.5970 \times 48.69 \times 0.53171 = 15.45\ \text{kPa},$$ $$Q_{s,1} = 15.45 \times 1.5708 \times 2.5 = 60.7\ \text{kN}.$$
Shaft resistance from 7.5 m to 13.0 m. Below the critical depth the unit friction holds at its 7.5 m value, $$f = 0.5970 \times 61.43 \times 0.53171 = 19.50\ \text{kPa},$$ $$Q_{s,2} = 19.50 \times 1.5708 \times 5.5 = 168.5\ \text{kN}.$$
Total shaft and ultimate capacity. $$Q_s = 267.0 + 60.7 + 168.5 = 496.2\ \text{kN},$$ $$Q_u = Q_p + Q_s = 983.0 + 496.2 = \boxed{Q_u \approx 1479\ \text{kN}} .$$
Allowable load. Applying the customary overall factor of safety of 3 to a single driven pile designed without a static load test, $$Q_{\text{all}} = \frac{1479}{3} \approx 493\ \text{kN}.$$ Two thirds of the ultimate capacity comes from the point and one third from the shaft, so this is an end-bearing pile with useful shaft assistance rather than a friction pile; the capacity is therefore sensitive to the assumed $N_q^{*}$ and should be confirmed by a static load test or by dynamic monitoring on the first piles driven.

Quantity	Value
Cross-sectional area, A_p	0.19635 m²
Perimeter, p	1.5708 m
Effective stress at the clay/sand contact (5.0 m)	35.95 kPa
Effective stress at the tip (13.0 m)	117.47 kPa
Meyerhof factor N_q* at φ′ = 35°	143
Unlimited point resistance	16 798 kPa (3298 kN) — not attainable
Limiting point resistance, 0.5p_aN_q*tanφ′	5006 kPa — governs
Point resistance, Q_p	983 kN
Adhesion factor in the clay, α	0.68
Shaft resistance in the clay (0–5 m)	267.0 kN
Shaft resistance in the sand (5–7.5 m)	60.7 kN
Shaft resistance in the sand (7.5–13 m)	168.5 kN
Total shaft resistance, Q_s	496.2 kN
Ultimate bearing capacity, Q_u	1479 kN
Allowable load at FS = 3	493 kN

Check: assumptions recorded under page-1 Notes 1, 6 and 7. The paper gives no pile length below the sand contact other than the 8.0 m dimensioned in Figure 3, no interface friction angle and no earth-pressure coefficient, so the following are assumed and their sources named.

Water table. Figure 3 shows the water-table symbol at the ground surface, so the two unit weights are treated as saturated and buoyant weights are used throughout. Had the profile been dry, σ′_v at the tip would be 245 kPa instead of 117 kPa — but the point resistance would not change, because the limiting value still governs; only the sand shaft would grow, from 229 kN to 507 kN, raising Q_u to about 1757 kN. The water table therefore costs about 16 per cent of the dry-profile capacity.
K and δ′. K = 1.4K₀ and δ′ = 0.8φ′ are mid-range values from Das Section 11.11 for a driven high-displacement pile. The defensible range K = K₀ to 1.8K₀ spans a sand shaft of 164 to 295 kN, that is Q_u = 1414 to 1545 kN — a spread of only 9 per cent, because the point dominates.
Critical depth. The L′ = 15D rule is measured from the ground surface, following the reference text. Measuring it from the top of the sand instead would place the cut-off at 12.5 m and raise Q_s by about 76 kN (about 5 per cent of Q_u).
α against a driven displacement pile. Driving remoulds a sensitive clay and the adhesion recovers over weeks as the excess pore pressures set up by driving dissipate. The value quoted is the long-term (set-up) value; capacity measured at the end of driving would be lower, which is the usual reason a re-strike is specified.
Not included. No group effects, no negative skin friction (no fill or consolidating layer is indicated), and no allowance for the weight of the pile itself, which is conventionally offset against the soil displaced.

Topic. Static capacity of a driven displacement pile crossing a cohesive layer into a granular bearing stratum.

Key relations. $Q_u = Q_p + Q_s$; $Q_p = A_p q' N_q^{*} \le A_p\left(0.5p_aN_q^{*}\tan\phi'\right)$ (Das Eqs. 11.19 and 11.20); $Q_{s,\text{clay}} = \sum \alpha c_u p \Delta L$ with $\alpha$ from $c_u/p_a$; $Q_{s,\text{sand}} = \sum K\sigma'_v\tan\delta'\,p\,\Delta L$ with $\sigma'_v$ capped at the critical depth $L' = 15D$.

Why this works. The two soils fail in different ways and must be treated separately. In the clay the pile is loaded fast enough to be undrained, so the shaft resistance is an adhesion proportional to $c_u$, and $\alpha$ falls as the clay gets stronger because a stiff clay is more disturbed by driving and generates a larger gap and remoulded annulus relative to its intact strength. In the sand the response is drained and frictional, so resistance is proportional to effective stress — but only up to a point: arching around the shaft and grain crushing at the tip mean that measured resistances stop increasing below roughly 15 to 20 diameters, which is what the limiting $q_p$ and the critical depth $L'$ encode. Both are empirical caps on an otherwise unbounded linear theory.

Common pitfall. Reporting the unlimited $A_pq'N_q^{*}$ — here 3298 kN, more than three times the true value — because the limiting check was skipped. The second is using total rather than buoyant unit weights when the water table is at the surface, which inflates the sand shaft by a factor of two. The third is applying the sand expression to the clay layer, or the $\alpha$ method to the sand; each layer gets the formulation matched to its drainage condition.

Source. Das, Principles of Foundation Engineering, 9th ed., Ch. 11 — Section 11.9 (Meyerhof's method for the point resistance in sand), Section 11.11 (frictional resistance in sand, $K$, $\delta'$ and the critical depth) and Section 11.12 (the $\alpha$ method in clay, Table 11.6); Tomlinson and Woodward, Pile Design and Construction Practice, 6th ed., Ch. 4, for shaft adhesion and set-up after driving.

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