Pile Capacity and Lateral Load: Method

Pile Capacity and Lateral Load

What the tool does

The tool takes a layered soil profile (thickness, unit weight, φ′ or s_u, N_60 per layer, water table) and one pile or drilled shaft (section, embedded length, head depth below the ground surface) and computes the static axial resistance by one of the FHWA, NAVFAC, USACE or GEC 10 methods, the resistance against depth, the design resistance in the LRFD or ASD format, the uplift resistance, the group resistance with its efficiency, block-failure and group-uplift checks, the neutral plane and drag force, the settlement of the single pile, the lateral capacity and deflection by three methods, and the nominal resistance that the common driving formulas would report for an observed set. The default inputs are the HP 12x74 of the FHWA GEC 12 (2016) Volume III North Abutment example (three sand layers, water table at 15 ft, head 5 ft below grade), whose Nordlund shaft resistance the page reproduces to the printed digits at every depth of Table D-6.

Conventions and units

Depths of the soil layers and the water table are measured from the ground surface; the pile head (bottom of the cap or of the excavation) sits D_cap below it and the embedded length L runs from the head to the toe. The overburden above the head counts in the effective stress but contributes no shaft resistance. Unit weights are total (moist) above the water table and saturated below it; the effective stress is σ′_v = Σ γ_i h_i − γ_w h_w with γ_w = 62.4 pcf (9.81 kN/m³). Lengths, forces, pressures, unit weights, moments and the subgrade moduli go through the unit dictionary (ft / m, kip / kN, psf / kPa, pcf / kN/m³, kip-ft / kN-m, pci / kN/m³); angles, blow counts and factors are plain numbers. The displaced volume of the Nordlund method is quoted in ft³/ft because the charts are drawn in those units.

Axial resistance of the single pile

R_n = R_s + R_p with the shaft resistance summed over segments no longer than 1 ft (0.5 m) that never cross a layer boundary or the water table, each at the effective stress of its midpoint, and the toe resistance from the layer at the toe (GEC 12 Eqs. 7-3 to 7-5).

• FHWA (Nordlund / α-method). Sand: f_s = K_δ C_F σ′_d sin δ (Eq. 7-8 for a uniform pile; for a tapered pile the K_δ of Figures 7-10 to 7-13 at the taper angle ω). δ/φ is read from Figure 7-9 for the pile type (pipe a, timber b, precast concrete c, Raymond step-taper d, uniform taper e, H-pile f, tapered Monotube g) at the displaced volume V; K_δ from Tables 7-6 / 7-7 (linear in φ, log-linear in V, GEC 12 Step 3) for ω = 0 or from the digitized figures for ω > 0; C_F from Figure 7-14 at φ and δ/φ. Toe: q_p = α_t N′_q σ′_p with α_t and N′_q from Figure 7-16 (α_t at the D/b of the pile), σ′_p limited to 3 ksf, and q_p ≤ q_L from Figure 7-15 (Meyerhof 1976). Clay: f_s = c_a from Tomlinson's Figure 7-17 for the pile surface (smooth steel or concrete / timber / corrugated) and D/b, with D the distance from the ground surface to the bottom of the clay layer or the toe; beyond the chart's 0.25 to 3.45 ksf the API RP 2A factor α = 0.5 ψ^−0.5 (ψ ≤ 1) or 0.5 ψ^−0.25 is used (Section 7.2.1.3.3). Toe in clay: 9 s_u. The seven curves of Figures 7-9, 7-10 to 7-13, 7-14, 7-15, 7-16 and 7-17 were digitized from the figure images of the FHWA PDF (government works); the digitizer, its specifications and the overlays that verify every curve are filed in References/Pile-Capacity/checks.
• Effective stress β-method. f_s = β σ′_v, q_p = N_t σ′_p (Eqs. 7-19 / 7-20); β and N_t from GEC 12 Table 7-9 (Fellenius 2014) by soil type and φ′ unless entered per layer. No limiting values.
• NAVFAC DM-7.02. Figure 1: Q_ult = P_T N_q A_T + Σ K_HC p_o tan δ S Δd with p_o held at its value at 20 B below the pile head (Note 1), K_HC by pile type (H-pile 0.5 to 1.0, displacement 1.0 to 1.5, the tool uses 0.75 and 1.25), δ 20° for steel and ¾ φ for concrete and timber, N_q from the driven-pile row of the figure's table. Clay (Figure 2): adhesion from the recommended-values table by consistency and pile material, N_c from the embedment ratio.
• USACE EM 1110-2-2906. Section 4-3: f_s = K σ′_v tan δ with σ′_v constant below the critical depth D_c (10 B loose, 15 B medium, 20 B dense; the tool uses 15 B), K = 1.0 (Table 4-4) and δ = ¾ φ for steel, 0.9 φ for concrete (Table 4-3), q = σ′_v N_q with Terzaghi's N_q; clay f_s = α c with α from the Figure 4-5a shape (1.0 below 0.5 ksf falling to 0.5 at 2 ksf) and q = 9 c.
• Meyerhof SPT. FHWA NHI-05-042 Section 9.7.1.1a (Meyerhof 1976): f_s = 2 N′ kPa (displacement piles) or N′ kPa (H-piles), ≤ 100 kPa; toe q_p = 400 N′_o + (40 N′_B − 40 N′_o) D_B/b ≤ 400 N′_B kPa when a weaker stratum overlies the bearing stratum, otherwise 40 N′_B D_B/b ≤ 400 N′_B kPa. The entered N_60 is taken as the overburden-corrected N′ of the method. Clay layers by the α-method.
• GEC 10 drilled shafts. Cohesionless: β = K_o tan φ′ with K_o = (1 − sin φ′) OCR^sin φ′ ≤ K_p and OCR = σ′_p/σ′_v, σ′_p = 0.47 N_60^0.6 p_a (sands) (Eqs. 10-8 to 10-13), β held at its value at 7.5 ft above that depth; base q_BN = 0.6 N_60 tsf ≤ 30 tsf (Eq. 10-14). Cohesive: α = 0.30 + 0.17/(s_u/p_a) (Eq. 10-18, the entered s_u taken as the CIUC value), top 5 ft neglected, base N*_c s_u with N*_c from Table 10-2 and the short-shaft reduction of Eq. 10-20.

Design format. LRFD: R_r = φ_stat R_n with the AASHTO resistance factors printed in GEC 12 Table 7-1 (Nordlund 0.45, α-method 0.35, Schmertmann 0.50, block failure 0.60, uplift 0.35 / 0.25 / 0.20, group uplift 0.50, lateral 1.0) or the field-verification factors of Table 7-2 (static load test 0.75 to 0.80, dynamic testing 0.65, wave equation 0.50, Gates 0.40, Engineering News 0.10); the entered loads are factored. ASD: Q_all = R_n/FS with FS 3.5 for a static analysis alone, 2.75 wave equation, 2.25 dynamic testing, 2.00 static load test (FHWA 2006) or 3 / 2 for permanent / temporary loads (NAVFAC); the entered loads are service loads. The margin boxes name the format and factor.

Uplift, groups, downdrag, settlement

The nominal uplift resistance of a single pile is its shaft resistance (GEC 12 Eq. 7-38). The group resistance is η_g n R_n with η_g = 1 in sand and, in clay with s_u < 2 ksf and the cap clear of the ground, 0.65 at 2.5 b rising linearly to 1.0 at 6 b (Section 7.2.2); with the toe in clay the block failure R_ng = 2 D (B + Z) s_u1 + B Z N_c s_u2, N_c = 5 (1 + 0.2 B/Z)(1 + 0.2 D/B) ≤ 9 (Eqs. 7-36 / 7-37) is checked and the smaller governs. Group uplift is the lesser of the piles' resistances and the block: in sand the effective weight of the soil within planes spreading at 1H:4V from the toe (Figure 7-33, integrated over the profile) plus the cap; in clay 2 D (B + Z) s_u plus the block weight (Eq. 7-41).

Downdrag. Neutral-plane mode (Section 7.3.5.7, Goudreault and Fellenius 1994): the depth where the unfactored permanent load plus the negative shaft resistance above it equals the positive shaft resistance below plus the mobilized fraction of the toe resistance, found on the segment resistances; the drag force and the maximum pile load at that depth are reported as structural loads (Siegel et al. 2013), not subtracted from the geotechnical resistance. Entered-depth mode (FHWA 2006 Appendix F.6): the shaft resistance over the settling depth is negative and the net resistance below it is compared with the load.

Settlement. Vesić's (1977) semi-empirical method as given in EM 1110-2-2906 Section 4-4a: w = (Q_p + α_s Q_s) L/(A E_p) + C_p Q_p/(b q_p) + C_s Q_s/(L q_p) with C_s = (0.93 + 0.16 √(L/b)) C_p and C_p from Table 4-7; and the elastic method (Vesić 1977 as Das gives it): s_2 = q_wp b (1 − μ²) I_wp/E_s and s_3 = (Q_s/(p L))(b/E_s)(1 − μ²) I_ws, I_ws = 2 + 0.35 √(L/b). The permanent load is split between toe and shaft in proportion to the nominal resistances.

Lateral load

Broms (1964a, b). Ultimate lateral load from the limiting soil pressure: 9 s_u b below 1.5 b in clay, 3 K_p γ′ b z in sand, with the short-pile (rigid rotation), long-pile (plastic hinge at M_y) and fixed-head intermediate cases; the governing case is the smaller load and the page prints both. The soil is the surface layer at the pile head, s_u averaged over 8 b and φ′, γ′ over the embedded length (the FHWA 2006 Appendix F.4.1 practice). Broms' head-deflection closed forms (long pile y_0 = 2 H β (1 + β e)/(k_h b) or 2.40 H/(n_h^0.6 (E I)^0.4) + 1.60 M/(n_h^0.4 (E I)^0.6); short pile 4 H (1 + 1.5 e/L)/(k_h b L) or 18 H (1 + 1.33 e/L)/(n_h L²); fixed head H β/(k_h b), 0.93 H/(n_h^0.6 (E I)^0.4), H/(k_h b L), 2 H/(n_h L²)) are printed beside the numerical solution.

Beam on an elastic foundation. E_p I_p y′′′′ + E_s(z) y = 0 with E_s = n_h z (Reese and Matlock 1956; NAVFAC DM-7.02 Section 7 and Figures 11 / 12) or a constant k_h b (Hetenyi 1946), solved by central finite differences on 240 segments with fictitious nodes: at the head E I y′′′ = H and either E I y′′ = H e (free) or y′ = 0 (fixed); at the toe y′′ = y′′′ = 0. The sign convention is Hetenyi's (y positive in the direction of H, M = E I y′′). For a long pile (L/T ≥ 5) the solution reproduces the chart coefficients A_y = 2.435, B_y = 1.623, A_m = 0.772 at z = 1.3 T, and 0.93 / −0.93 for a fixed head; the page prints the solution's own coefficients and the charts of y(z) and M(z). The test script checks the solver against scipy's solve_bvp and against Hetenyi's semi-infinite closed form.

Characteristic load method. Evans and Duncan (1982), Duncan, Evans and Ooi (1994): P_c = 7.34 b² (E_p R_I)(s_u/(E_p R_I))^0.683 (clay) or 1.57 b² (E_p R_I)(γ′ b φ′ K_p/(E_p R_I))^0.57 (sand, φ′ in degrees), M_c = 3.86 b³ (E_p R_I)(s_u/(E_p R_I))^0.46 or 1.33 b³ (E_p R_I)(γ′ b φ′ K_p/(E_p R_I))^0.40, with E_p R_I the stiffness of a solid circular section of the same E I. The fixed-head deflection and moment are the regressions of the Evans and Duncan curves as printed in Ooi (1991) Eqs. A1 to A4; the free-head deflection is the Duncan et al. (1994) fit (cited; about four times the fixed-head value) and its maximum moment comes from the characteristic length T = (y_t E_p I_p/(2.43 H))^(1/3) with 0.772 H T (Matlock and Reese). The soil is averaged over 8 b below the head. The fits hold for H/P_c below about 0.1; the page says when the load exceeds it. The p-y method itself (LPILE, FB-Pier) is the FHWA recommended analysis for major projects and is described, not computed.

Driving formulas

From the ram weight, stroke and observed set at the end of driving: FHWA modified Gates R = 1.75 √E_d log₁₀(10 N_b) − 100 (kips; φ 0.40), AASHTO Engineering News R = 12 E_d/(s + 0.1) (E_d in ft-kips; φ 0.10), WSDOT R = 6.6 F_eff E_d ln(10 N_b) (φ 0.55 redundant, 0.45 non-redundant) and MnDOT MPF12 R = 40 √(W h/1000) log₁₀(10/s) (φ 0.50 pipe and concrete, 0.60 H-piles), GEC 12 Vol. II Eqs. 13-2 to 13-7, plus NAVFAC's Table 4 allowable 2 W h/(s + 0.1). The default hammer and set are the Table 13-4 case history (ICE I-46, 49 blows/ft), which the page reproduces: 2,872 / 710 / 950 / 585 kips against a static load test of 510 kips. The wave equation is cited only.

Checks and sources

The test script reproduces the GEC 12 (2016) Volume III Table D-6 shaft resistances of the HP 12x74 at the North Abutment to 0.1 kip at every tabulated depth and its toe resistances within the chart-reading differences of the DrivenPiles program (N′_q at 40° read 155 here against about 190 in the program), Table D-100 (α-method, South Abutment) within 3 %, the FHWA NHI-05-042 (2006) Appendix F hand calculations by the Nordlund, effective-stress, Meyerhof SPT and α-methods (R_s 898, 537, 418, 1,648 kN; toe 940, 1,294, 854, 182 kN) and its Broms example (long fixed-head pile, 219 kN from the chart against 201 kN from the closed form; deflection 16 mm), the GEC 10 Illustrative Examples 10-1 to 10-3 (β 0.97 / 0.98, 837 and 2,248 psf, α 0.44, 1,131 psf, base 453.6 kips), the resistance-factor and group rules, the Table 13-4 driving formulas, and the lateral solver against Hetenyi, Reese–Matlock and scipy. Sources, page renders, the digitized figures with their overlays and the numpy check are in References/Pile-Capacity.

Limits

Not covered: scour and liquefaction, seismic downdrag, time effects (setup, relaxation), pile structural resistance and driving stresses, the p-y analysis, group lateral p-multipliers and batter piles, piles driven to rock, CPT methods, Janbu and equivalent-footing group settlement, the special-stratigraphy adhesion cases of GEC 12 Figure 7-18, and the Nordlund method's calibration limits (piles above about 24 in). A tapered pile uses one taper angle over its whole length. Chart values carry the ±1 to 2 % reading accuracy of the digitizing; the Nordlund δ/φ chart ends at 1.9 ft³/ft and α_t at D/b = 45, beyond which the end values are used and the page says so.