SlipSurface IQ Pile — theory

What the program computes, how, and where each expression comes from. Symbols: D the pile diameter (or side), L its length below the pile head, Ab the base area, p the perimeter, σ′v the vertical effective stress, cu the undrained shear strength, φ′ the effective friction angle, pa = 101.325 kPa the atmospheric pressure. Stresses in kPa, forces in kN.

1. Stresses

The profile is a stack of layers from the ground surface down; the last one continues below the profile where a stress is asked for deeper. σv0 = Σ γ·h above the water table and Σ γsat·h below it; u0 = γw·(z − zw) below the water table; σ′v0 = σv0 − u0. The shaft is cut into slices no thicker than 0.25 m that never straddle a layer boundary, and every integral below is the midpoint sum over them.

2. Shaft friction

Qs = Σ fs·p·Δz

Granular layers

fs = K·σ′v·tan δ,  K = (K/K0)·K0,  K0 = 1 − sin φ′,  δ = (δ/φ′)·φ′

K/K0 is 1.0 for a bored pile, 1.2 for a driven pile with a small displacement and 1.4 for one with a large displacement, unless entered (the middle of Das's ranges K0, K0–1.4·K0 and K0–1.8·K0). With the critical depth switched on, σ′v in sand is held at its value at zc = (zc/D)·D below the ground surface (Meyerhof 1976; Vesić 1967), 15·D by default.

Cohesive layers

method	fs
API RP 2A (1987)	α·cu, α = 0.5·ψ^-0.5 (ψ ≤ 1), 0.5·ψ^-0.25 (ψ > 1), ψ = cu/σ′v, α ≤ 1
Kulhawy & Phoon (1993)	α·cu, α = 0.21 + 0.26·pa/cu ≤ 1 (drilled shafts)
Sladen (1992)	α·cu, α = C·(σ′v/cu)^0.45 ≤ 1, C = 0.4 bored, 0.5 driven
β — Burland (1973), Meyerhof (1976)	(1 − sin φ′)·tan φ′·√OCR·σ′v
λ — Vijayvergiya & Focht (1972)	λ·(σ′v + 2cu), λ from the pile's penetration

The λ table runs from 0.5 at the surface to 0.110 below 70 m (Das). Applied slice by slice it gives λ·(σ̄′v + 2c̄u) over the clay, the original form.

SPT (Meyerhof 1976)

fs = 0.02·pa·N60 for a large displacement pile, 0.01·pa·N60 otherwise; clay slices keep the chosen clay method. It is derived for driven piles, and shown beside the others.

3. Base resistance

Qb = qb·Ab

The tip is in the layer at its depth (the one below, if it sits on a boundary). A weaker layer within 3·D below it is reported.

Sand

• Meyerhof (1976): qb = σ′v·Nq* ≤ ql = 0.5·pa·Nq*·tan φ′, Nq* from Das's table of Meyerhof's values (12.4 at 20°, 56.7 at 30°, 346 at 40°), interpolated logarithmically.
• Vesić (1977): qb = σ′o·Nσ* = σ′v·Nq*, Nq* = 3/(3 − sin φ′)·e^((π/2 − φ′)tan φ′)·tan²(45 + φ′/2)·Irr^(4 sin φ′/(3(1 + sin φ′))), Ir = Es/(2(1 + ν)·σ′v·tan φ′), Irr = Ir/(1 + Ir·Δ), Δ = 0.005·(1 − (φ′ − 25)/20)·σ′v/pa.
• Janbu (1976): qb = σ′v·Nq*, Nq* = (tan φ′ + √(1 + tan² φ′))²·e^(2η′·tan φ′), η′ from 60° (soft) to 105° (dense), 90° by default.

σ′v at the tip is the capped one when the critical depth is on.

Clay (undrained, net of the overburden)

• Skempton / Meyerhof: qb = 9·cu
• Vesić: qb = Nc*·cu, Nc* = 4/3·(ln Ir + 1) + π/2 + 1, Ir = Es/(3·cu)
• Janbu at φ = 0: Nc* = lim (Nq* − 1)·cot φ = 2 + 2η′ (5.14 at η′ = 90°)

The overburden term q·Nq = q is left out, as is customary, because it is balanced by the pile's own weight; the weight is still subtracted if asked (conservative).

SPT (Meyerhof 1976)

qb = 0.4·pa·N60·Lb/D ≤ 4·pa·N60, Lb the embedment in the bearing layer.

4. Weight, capacity and the check

W = Ab·[γp·(length above the water table) + (γp − γw)·(length below it)]
Qult = Qs + Qb,  Qult,net = Qult − W,  Qall = Qult,net / FS

The buoyant weight is used on request (the default); the weight is taken off on request (the default). The load per pile is Q/n; the pile passes when Q/n ≤ Qall. Every combination of a shaft method with a base method is reported, and the chosen pair makes the checks.

5. Groups

A rectangular group of n1 × n2 piles at spacings sx and sy (s is their mean where both matter), outline Bg × Lg = [(n1 − 1)sx + D] × [(n2 − 1)sy + D].

method	η
Converse–Labarre	1 − θ·[(n1 − 1)n2 + (n2 − 1)n1]/(90·n1·n2), θ = arctan(D/s) [°]
Los Angeles Group	1 − D/(π·s·n1·n2)·[n1(n2 − 1) + n2(n1 − 1) + √2(n1 − 1)(n2 − 1)]
Seiler–Keeney	1 − [36s/(75s² − 7)]·(n1 + n2 − 2)/(n1 + n2 − 1) + 0.3/(n1 + n2), s in m
Feld	1 − (number of neighbours, straight and diagonal)/16, averaged over the group

Each η is capped at 1. Block failure takes the group as one block: shaft 2(Bg + Lg)·Σ fs·Δz with fs = cu in clay and K0·σ′v·tan φ′ in sand (soil on soil), base Bg·Lg·qb with qb = Nc·cu, Nc = 5(1 + 0.2Bg/Lg)(1 + 0.2·z/Bg), z/Bg ≤ 2.5 (Skempton) in clay and the chosen method's unit base resistance in sand.

Qg,ult = min(η·n·Qult, Qblock),  Qg,all = (Qg,ult − n·W)/FS,  Q ≤ Qg,all

6. Seismic load case: compression and uplift

The seismic combination at the underside of the cap — the vertical load V and the overturning moments M_B and M_L, from the structural analysis — is shared among the piles by a rigid cap:

P_i = V/n + M_B·x_i/Σx² + M_L·y_i/Σy²

x_i along B and y_i along L from the centre of the group; P > 0 is compression. A single pile has no lever arm, so its moments are left out (with a warning).

• Compression: the most loaded pile, Pmax ≤ Qult,net / FS_E,c.

• Uplift of a pile: the largest tension T = −Pmin is resisted by the shaft friction in tension and by the pile's own weight,

  Qs,t = Σ λt·ζ·fs·p·Δz,   Tall = Qs,t / FS_E,t + W,   T ≤ Tall
  

  fs is the shaft friction of the chosen method, λt the ratio of the shaft friction in tension to that in compression (0.75 in sand, 1.0 in clay by default, after the usual finding that uplift in sand mobilises about two thirds to three quarters of the compression value), ζ a reduction for cyclic loading (1.0 by default; lower it where cyclic degradation is expected) and W the weight of the pile, buoyant below the water table. The factor of safety is reported on the friction alone, FS = Qs,t / (T − W); the weight is not factored. The base contributes nothing in tension.

• Uplift of the group: when the group as a whole is lifted (V < 0) it is also checked as one block, Tg ≤ Qs,block / FS_E,t + W_block, with the soil-on-soil shear round its outline (cu in clay, K0·σ′v·tan φ′ in sand, reduced by ζ) and the effective weight of the soil and the piles inside it.

The piles in tension are reported: their reinforcement has to carry T down the shaft and into the cap, which is a structural check outside this program.

7. Required length

The analysis is repeated for L from the shortest length tried to the foot of the profile, in the chosen step, every other input held as it is; the required length is the first that passes both the single-pile and the group check, and the seismic checks when the earthquake is on. The capacity against length is drawn from the same run.

8. Settlement

Single pile — Vesić (1977), as set out by Das

s1 = (Qwb + ξ·Qws)·L/(Ab·Ep)
s2 = qwb·D·(1 − ν²)·0.85/Es(tip)
s3 = Qws/(p·L)·D·(1 − ν²)·(2 + 0.35·√(L/D))/Es(shaft)

ξ = 0.5 for a uniform or parabolic friction, 0.67 triangular. The working load Q/n is shared between shaft and base in the proportion of Qs and Qb.

Group

• Equivalent raft (Terzaghi & Peck; Tomlinson): the load Q acts on Bg × Lg at z = head + (2/3)·L and spreads 2 : 1, Δσ = Q/[(Bg + Δz/2)(Lg + Δz/2)]. Below it, slices down to the foot of the profile or to where Δσ < 0.1·σ′v0. A clay with Cc consolidates: Cr/(1 + e0)·log(σ′f/σ′0) while σ′f ≤ σ′p = OCR·σ′0, Cr·log(σ′p/σ′0) + Cc·log(σ′f/σ′p) above; everything else compresses by Δσ·h/M, M = E(1 − ν)/((1 + ν)(1 − 2ν)). The piles' shortening above the raft, (Q/n)·(2/3)L/(Ab·Ep), is added.
• Vesić (1969): sg = s·√(Bg/D).
• Meyerhof (1976), sand: sg [mm] = 0.96·q·√Bg·I/N60, q = Q/(Bg·Lg), I = 1 − L/(8Bg) ≥ 0.5, N60 the mean within Bg below the tip.

9. Rock-socketed piles

The socket is analysed on its own: the load Q on one pile, the pile head at top, the rock surface at rock_depth, a socket of length Ls and diameter D. The friction of the overburden is ignored; its share of the pile's weight is not.

Unit side shear

qu in the correlations is min(qu,rock, f′c): the bond is no stronger than the weaker of the two materials.

correlation	fs [MPa]
Rosenberg & Journeaux (1976)	0.375·qu^0.515
Horvath & Kenney (1979)	0.21·qu^0.5
Meigh & Wolski (1979)	0.22·qu^0.6
Williams, Johnston & Donald (1980)	0.44·qu^0.36
Reynolds & Kaderabek (1980)	0.30·qu (weak rock)
Gupton & Logan (1984)	0.20·qu (weak rock)
Rowe & Armitage (1987)	0.45·qu^0.5
Carter & Kulhawy (1988)	0.20·qu^0.5 (= 0.63·pa·√(qu/pa))
Toh et al. (1989)	0.25·qu (weak rock)
Zhang & Einstein (1998)	0.40·qu^0.5 (smooth sockets; 0.8 for rough)
O'Neill & Reese (1999), AASHTO LRFD	0.65·αE·pa·(qu/pa)^0.5 ≤ 7.8·pa·(f′c/pa)^0.5
Kulhawy, Prakoso & Akbas (2005)	1.0·pa·(qu/pa)^0.5

The three linear rules were fitted to weak rock. Above the weak rock limit (5 MPa by default) they are still listed, marked, but left out of the statistics. αE is O'Neill & Reese's joint modification factor, 1.0 at Em/Ei = 1, 0.8 at 0.5, 0.7 at 0.3, 0.55 at 0.1 and 0.45 at 0.05 and below.

Unit base resistance

method	qb [MPa]
Coates (1967)	3·qu
Rowe & Armitage (1987)	2.7·qu
Carter & Kulhawy (1988)	[√s + √(m√s + s)]·qu, m and s of the Hoek–Brown mass (2002)
Zhang & Einstein (1998)	4.83·qu^0.51
AASHTO / O'Neill & Reese	2.5·qu (intact or tightly jointed rock below the base)
CFEM, Ladanyi & Roy (1971)	3·Ksp·d·qu, Ksp = (3 + c/D)/(10√(1 + 300δ/c)), d = 1 + 0.4·Ls/D ≤ 3

The CFEM rule gives an allowable pressure with a factor of about 3, so it is multiplied by 3 here; it holds for 0.05 < c/D < 2 and δ/c < 0.02. The design base resistance is none, the lowest, the mean, or one method. A base resistance above f′c is flagged.

Socket length

Qall(Ls) = π·D·Ls·fs/FSside + Ab·qb/FSbase − W(Ls) = Q

is solved by bisection for every correlation and for the design side shear (the mean, median, lowest or highest of the correlations in range, or one by name). The design length is at least (min Ls/D)·D. The socket at the length entered (or at the design length when 0 is entered) is checked and its settlement worked out.

Rock mass modulus

Em = Ei·(0.0231·RQD − 1.32), at least 0.15·Ei (Gardner 1987); or Em = Ei·[0.02 + (1 − D/2)/(1 + e^((60 + 15D − GSI)/11))] (Hoek & Diederichs 2006); or entered.

Elastic settlement

Randolph & Wroth (1978), for a compressible pile of radius r0 and length Ls in an elastic medium of shear modulus G = Em/(2(1 + ν)) along the socket and Gb below the base:

Pt/(G·r0·wt) = [4η/((1 − ν)ξ) + (2πρ/ζ)·(tanh μL/μL)·(L/r0)]
               / [1 + (1/(πλ))·(4η/((1 − ν)ξ))·(tanh μL/μL)·(L/r0)]

ξ = G/Gb, λ = Ec/G, ρ = 1, η = 1, ζ = ln(rm/r0), rm = {0.25 + ξ[2.5ρ(1 − ν) − 0.25]}·L (kept at 2·r0 or more), μL = √(2/(ζλ))·(L/r0). The share of the load reaching the base is (4η/((1 − ν)ξ))/[cosh μL·(4η/((1 − ν)ξ) + (2πρ/ζ)(tanh μL/μL)(L/r0))]. The side-only variant drops the base term. Vesić's three-part expression of §7 is given beside it, with Em for the soil. The elastic shortening of the pile through the overburden, Q·Lo/(Ab·Ec), is added to each.

10. Studies

As in the other SlipSurface programs: one-at-a-time sweeps, Latin hypercube or Monte Carlo sampling of any input by a range or by a normal, lognormal or uniform distribution; every sample is a whole pile analysis without the length search. The probability of failure of the single pile (Q/n > Qall), of the group (Q > Qg,all), of the settlement (s > sallow) and of the seismic checks (the largest seismic utilisation above 1) comes with Wilson's 95 % interval and the reliability index β = −Φ⁻¹(P).

References

API (1987, 2000) RP 2A, Recommended practice for planning, designing and constructing fixed offshore platforms. · Burland, J.B. (1973) Shaft friction of piles in clay. Ground Engineering 6(3). · Carter, J.P. & Kulhawy, F.H. (1988) Analysis and design of drilled shaft foundations socketed into rock, EPRI EL-5918. · Coates, D.F. (1967) Rock mechanics principles, Mines Branch Monograph 874. · Converse, F.J. (1962), Labarre formula, in Moorhouse & Sheehan. · Das, B.M. Principles of Foundation Engineering. · De Nicola, A. & Randolph, M.F. (1993) Tensile and compressive shaft capacity of piles in sand, JGE ASCE 119(12). · Feld, J. (1943) Discussion, Trans. ASCE 108. · Gardner, W.S. (1987) Design of drilled piers in the Atlantic Piedmont, ASCE GSP. · Gupton, C. & Logan, T. (1984) Design guidelines for drilled shafts in weak rocks of South Florida. · Hoek, E., Carranza-Torres, C. & Corkum, B. (2002) Hoek–Brown failure criterion, 2002 edition. · Hoek, E. & Diederichs, M.S. (2006) Empirical estimation of rock mass modulus, IJRMMS 43. · Horvath, R.G. & Kenney, T.C. (1979) Shaft resistance of rock socketed drilled piers, ASCE Symposium on Deep Foundations. · Janbu, N. (1976) Static bearing capacity of friction piles, Proc. 6th ECSMFE. · Kulhawy, F.H. & Phoon, K.K. (1993) Drilled shaft side resistance in clay soil to rock, ASCE GSP 38. · Kulhawy, F.H., Prakoso, W.A. & Akbas, S.O. (2005) Evaluation of capacity of rock foundation sockets, 40th US Symp. Rock Mech. · Ladanyi, B. & Roy, A. (1971) Some aspects of bearing capacity of rock mass, 7th Canadian Rock Mech. Symp.; Canadian Foundation Engineering Manual. · Meigh, A.C. & Wolski, W. (1979) Design parameters for weak rock, 7th ECSMFE. · Meyerhof, G.G. (1976) Bearing capacity and settlement of pile foundations, JGED ASCE 102(GT3). · O'Neill, M.W. & Reese, L.C. (1999) Drilled shafts: construction procedures and design methods, FHWA-IF-99-025; AASHTO LRFD Bridge Design Specifications §10.8. · Randolph, M.F. & Wroth, C.P. (1978) Analysis of deformation of vertically loaded piles, JGED ASCE 104(GT12). · Reynolds, R.T. & Kaderabek, T.J. (1980) Miami limestone foundation design and construction, ASCE. · Rosenberg, P. & Journeaux, N.L. (1976) Friction and end bearing tests on bedrock for high capacity socket design, Can. Geotech. J. 13. · Rowe, R.K. & Armitage, H.H. (1987) A design method for drilled piers in soft rock, Can. Geotech. J. 24. · Seiler, J.F. & Keeney, W.D. (1944) The efficiency of piles in groups, Wood Preserving News 22. · Skempton, A.W. (1951) The bearing capacity of clays, Building Research Congress. · Sladen, J.A. (1992) The adhesion factor: applications and limitations, Can. Geotech. J. 29. · Terzaghi, K. & Peck, R.B. (1967) Soil mechanics in engineering practice. · Toh, C.T. et al. (1989) Design parameters for bored piles in a weathered sedimentary formation, 12th ICSMFE. · Tomlinson, M.J. Pile design and construction practice. · Vesić, A.S. (1969, 1977) Experiments with instrumented pile groups in sand, ASTM STP 444; Design of pile foundations, NCHRP Synthesis 42. · Vijayvergiya, V.N. & Focht, J.A. (1972) A new way to predict capacity of piles in clay, OTC. · Williams, A.F., Johnston, I.W. & Donald, I.B. (1980) The design of socketed piles in weak rock, Int. Conf. Structural Foundations on Rock. · Zhang, L. & Einstein, H.H. (1998) End bearing capacity of drilled shafts in rock, JGGE ASCE 124(7).