Bearing Capacity and Settlement: Method

Bearing Capacity and Settlement

What the tool does

The tool checks a shallow footing the way a geotechnical report does: the ultimate bearing capacity of the soil under the footing by the four classical equations in current use (Terzaghi 1943, Meyerhof 1963, Hansen 1970, Vesić 1973), each with its own bearing-capacity factors and its own corrections for shape, embedment and load inclination as tabulated in the U.S. Army Corps of Engineers manual Bearing Capacity of Soils (EM 1110-1-1905, 1992); the allowable pressure as the ultimate capacity divided by the code factor of safety; the applied pressure from the column load and moments on Meyerhof's effective area; and the settlement, immediate and consolidation, against the tolerable value. The checks are printed as plain requirements (applied ≤ allowable, settlement ≤ allowable): civil practice puts the factor of safety on the resistance, and the tool carries no load-level radio or margin boxes.

Conventions and units

B is the footing width (the smaller plan dimension, or the diameter of a circular footing), L the length, D_f the depth of the base below the ground surface. A strip footing carries its load per unit length: enter Q and T per metre or per foot, the force unit standing for force per unit length. Moments M_B and M_L shift the load by e_B = M_B/Q in the B direction and e_L = M_L/Q in the L direction; a horizontal load T acts parallel to B. Unit weights are plain numbers (kN/m³ in SI, lb/ft³ in USCS) because the unit dictionary has no force-per-volume type, and they do not convert on a unit-system switch; pressures, cohesion, moduli, lengths, forces and moments go through the dictionary (results are written in kPa or lb/ft²). The tool is SI by default, as civil practice is.

The general bearing-capacity equation

EM 1110-1-1905 eq. 4-1 writes every method in one form,

q_u = c N_c ζ_c + σ′_D N_q ζ_q + ½ B′ γ′_H N_γ ζ_γ,

where c is the cohesion (or the undrained strength with φ = 0), σ′_D the effective overburden pressure at the base, B′ = B − 2e_B the effective width, γ′_H the effective unit weight of the soil in the failure zone and ζ = ζ_s ζ_i ζ_d the product of the shape (with eccentricity), inclination and depth corrections of the method. The net ultimate capacity is q′_u = q_u − σ′_D (eq. 4-2), the pressure the footing may add to the overburden it replaced.

Bearing-capacity factors. Meyerhof, Hansen and Vesić share Prandtl–Reissner's N_q = e^{π tan φ} tan²(45° + φ/2) and N_c = (N_q − 1) cot φ (5.14 at φ = 0) and differ in N_γ, the factor that depends on the assumed shape of the failure zone: Meyerhof (N_q − 1) tan 1.4φ, Hansen 1.5 (N_q − 1) tan φ, Vesić 2 (N_q + 1) tan φ. Terzaghi's factors come from his own wedge with a rough base: N_q = a²/[2 cos²(45° + φ/2)] with a = e^{(0.75π − φ/2) tan φ}, N_c = (N_q − 1) cot φ (5.7 at φ = 0), and N_γ as tabulated in EM 1110-1-1905 Table 4-1 (Bowles' values back-figured from Terzaghi's charts), which the tool interpolates log-linearly between the tabulated angles. Vesić's factors are also those of AASHTO (1996) and FHWA GEC 6 Table 5-1. The four N_γ values differ by a factor of about 1.5 between Hansen (least) and Terzaghi (greatest) at φ = 30°, which is why EM 1110-1-1905 asks for two or more methods on every design; the tool prints all four and lets the chosen one govern.

Correction factors. Terzaghi (Table 4-2) corrects only for shape: ζ_c, ζ_γ = 1.0, 1.0 for a strip, 1.3, 0.8 for a square, 1.3, 0.6 for a circle (the tool takes a rectangle linear in B/L); the embedment enters only through σ′_D. Meyerhof (Table 4-3): ζ_cs = 1 + 0.2 N_φ B′/L′, ζ_qs = ζ_γs = 1 + 0.1 N_φ B′/L′ for φ > 10° (1.0 at φ = 0, linear between); ζ_cd = 1 + 0.2 √N_φ D_f/B, ζ_qd = ζ_γd = 1 + 0.1 √N_φ D_f/B (φ > 10°); ζ_ci = ζ_qi = (1 − θ/90°)², ζ_γi = (1 − θ/φ)² with θ = tan⁻¹(T/Q), zero when θ ≥ φ; N_φ = tan²(45° + φ/2). Hansen (Table 4-5) and Vesić (Table 4-6) share De Beer's shape factors ζ_cs = 1 + (N_q/N_c) B′/L′, ζ_qs = 1 + (B′/L′) tan φ, ζ_γs = 1 − 0.4 B′/L′ (1.0 for a strip) and Hansen's depth factors ζ_cd = 1 + 0.4k, ζ_qd = 1 + 2 tan φ (1 − sin φ)² k, ζ_γd = 1 with k = D_f/B for D_f/B ≤ 1 and k = tan⁻¹(D_f/B) in radians otherwise (the full width B, as the manual's Example 4-5c uses it). Their inclination factors differ: Hansen ζ_qi = [1 − 0.5T/(Q + A′c cot φ)]⁵ and ζ_γi = [1 − 0.7T/(Q + A′c cot φ)]⁵; Vesić ζ_qi = [1 − T/(Q + A′c cot φ)]^m and ζ_γi = [1 − T/(Q + A′c cot φ)]^m+1 with m = (2 + B′/L′)/(1 + B′/L′) for a load inclined in the B direction; both take ζ_ci = ζ_qi − (1 − ζ_qi)/(N_q − 1). The base adhesion is taken equal to c, and T is capped at the sliding resistance Q tan φ + A′c of the base, as the tables note. At φ = 0 Hansen and Vesić use their additive form q_u = 5.14 c (1 + s′_c + d′_c − i′_c) + σ′_D with s′_c = 0.2 B′/L′, d′_c = 0.4k and i′_c = 0.5 − 0.5 √(1 − T/(A′c)) (Hansen) or m T/(A′ c N_c) (Vesić), the φ = 0 rows of the two tables.

Eccentricity and inclination together. EM 1110-1-1905 notes under each table that the eccentricity (shape) and inclination corrections are not to be used simultaneously, and FHWA GEC 6 (Section 5.2.3.7) omits the inclination factors altogether and checks sliding separately, because applying every correction at once can leave an allowable pressure of a few percent of the uncorrected ultimate. The tool applies the inclination factors when the box is checked and prints a note when both corrections are active, so the analyst can follow either rule.

Effective area, contact pressure and the water table

Meyerhof's effective area (EM eqs 4-4) reduces the footing to B′ = B − 2e_B and L′ = L − 2e_L, the smaller effective dimension being B′ in the equations; the applied pressure compared with the allowable is q = Q/(B′L′), and the ultimate load is q_u B′L′. The tool also prints the linear contact pressure on the full base, q_max,min = (Q/A)(1 ± 6e_B/B ± 6e_L/L), and the middle-third check 6e_B/B + 6e_L/L ≤ 1 (e ≤ B/6 for a strip), which GEC 6 uses as its overturning criterion. A circular footing is analysed for a concentric load only.

Two rules for the ground water are offered. EM 1110-1-1905 paragraph 1-5d (eqs 1-6, 1-7): with the failure depth H = B tan(45° + φ/2) below the base, water deeper than D_f + H has no effect; between the base and D_f + H the unit weight of the N_γ term is γ′_H = γ_sub + [(D_w − D_f)/H] γ_w; above the base γ′_H = γ_sub and the overburden is the effective stress. FHWA GEC 6 eqs 5-9 and 5-10 (AASHTO 1996 Table 5-3) instead multiply the N_γ term by C_Wγ = 0.5 + 0.5 D_w/(1.5B + D_f) and the surcharge term by C_Wq = 0.5 + 0.5 D_w/D_f, each at most 1.0, with the moist unit weight in both. The saturated unit weight defaults to the moist one, as the manual's examples assume.

Allowable pressure and the factor of safety

q_all = q_u/FS (gross) and q_all,net = q′_u/FS. The default FS = 3.0 is the value of EM 1110-1-1905 Table 1-2 for footings (3.5 for highway bridges and light industrial buildings, 4 for railway bridges) and of FHWA GEC 6 Section 5.2.6 for bridge foundations (2.5 to 3.5 typical). The requirement printed is q = Q/A′ ≤ q_all, with the factor of safety available q_u/q beside the required value, and the allowable load Q_all = q_all A′. The chart of q_u against the footing width keeps the aspect ratio, the depth, the load and the moments fixed, so the eccentricity stays absolute and the effective width grows with B.

Immediate settlement

Elastic influence factors. S = Δp B′ (1 − ν²) I / E_s, the form of EM 1110-1-1904 eq. 3-18 (Perloff), with Δp the net pressure on the effective area (the applied pressure less the overburden the excavation removed) and I from Steinbrenner's (1934) solution for the corner of a uniformly loaded rectangle L′/B′ = m′ on an elastic layer n′ = H/B′ deep over a rigid base: I_s = F₁ + (1 − 2ν)/(1 − ν) F₂, F₁ = (A₀ + A₁)/π, F₂ = (n′/2π) tan⁻¹ A₂, with A₀ = m′ ln[(1 + √(m′² + 1)) √(m′² + n′²) / (m′ (1 + √(m′² + n′² + 1)))], A₁ = ln[(m′ + √(m′² + 1)) √(1 + n′²) / (m′ + √(m′² + n′² + 1))] and A₂ = m′/(n′ √(m′² + n′² + 1)); on a half-space F₂ = 0. The centre of the flexible footing is the sum of four corners of B′/2 × L′/2 (I = 2 I_s: 1.12 for a square, 1.52 for L/B = 2, 2.53 for L/B = 10, the values of EM 1110-1-1904 Table 3-3a), the corner one of B′ × L′. A rigid footing settles about 0.99/1.12 of the flexible centre value (Table 3-3a: rigid square 0.99, flexible centre 1.12) and a rigid circle π/4 of it (0.79 against 1.00); the tool applies the square ratio to every rectangle. A strip is run as a rectangle 100 B long, since the half-space factor of an infinite strip is unbounded; a rigid-base depth gives it a finite value.

Schmertmann's strain influence. For sands and gravels, S = C₁ C₂ Δp Σ (I_z/E_s) Δz over layers 0.2 B′ thick (Schmertmann, Hartman and Brown 1978; EM 1110-1-1904 eqs 3-8 and 3-9, Figure 3-4): I_z rises linearly from 0.1 at the base (0.2 in plane strain) to the peak I_zp = 0.5 + 0.1 √(Δp/σ′_Izp) at a depth B′/2 (axisymmetric, L/B = 1) or B′ (plane strain, L/B ≥ 10) and falls to zero at 2B′ or 4B′, all three linear in L/B between 1 and 10; σ′_Izp is the effective overburden at the peak depth. C₁ = 1 − 0.5 σ′_od/Δp ≥ 0.5 credits the strain relief of the embedment and C₂ = 1 + 0.2 log₁₀(t/0.1 yr) the creep, 1.0 at the end of construction. The tool integrates the piecewise-linear I_z exactly over each layer (the manual's worked sheet reads layer averages off the diagram), stops the sum at a rigid base when one is entered, and takes one modulus for the whole depth; the manual's own sheet (Figure 3-6: 10 ft footing, E_s = 175 tsf, q = 2 tsf) is reproduced to 1 %.

Consolidation settlement

For a clay layer below the footing the primary consolidation is summed over sublayers (FHWA GEC 6 eqs 5-20 and 5-22; EM 1110-1-1904 eq. 3-23): S_c = Σ H_i/(1 + e₀) [C_r log(σ′_p/σ′_vo) + C_c log(σ′_vf/σ′_p)], where σ′_vo is the initial effective stress at the sublayer midpoint (built from the unit weights above and in the clay, with the water subtracted below the water table), σ′_vf = σ′_vo + Δσ the final stress, and σ′_p the preconsolidation pressure: a normally consolidated sublayer uses the C_c term alone, an overconsolidated one that stays below σ′_p the C_r term alone, and one that crosses σ′_p both. A blank C_r is taken as C_c/10, the usual estimate GEC 6 quotes. The stress increase Δσ at each midpoint comes from the net pressure by the 2:1 spread Δσ = Δp B′L′/((B′ + z)(L′ + z)) (GEC 6 eq. 5-17) or by Boussinesq under the centre, four corners of Newmark's formula (EM 1110-1-1904 Table C-1), the exact strip and circle forms for those shapes. The chart of both distributions against depth shows how far they differ: Boussinesq is higher close to the base and the two agree within about 15 % from one to four widths down, which is why the 2:1 method is accepted for settlement work. The same routine in its strain form (C_cε = 1/C′) reproduces the Hough settlements of GEC 6 Example B-1.

Typical moduli

The soil-type select fills E_s with the geometric mean of the range printed in EM 1110-1-1904 Table D-3 (very soft clay 5–50 tsf, soft clay 50–200, medium clay 200–500, stiff and silty clay 500–1,000, sandy clay 250–2,000, clay shale 1,000–2,000, loose sand 100–250, dense sand 250–1,000, dense sand and gravel 1,000–2,000, silty sand 250–2,000; 1 tsf = 95.76 kPa) and ν with 0.40 for clays (the manual's overall value, paragraph D-4), 0.35 for sandy clay and clay shale and 0.30 for sands (unsaturated soils 0.25 to 0.40). These are starting values: the manual's own remark is that the variation of the modulus at a site matters more than the choice of Poisson's ratio, and a site value should replace them.

Reading the results

The page prints the footing, soil and loads, the eccentricity and effective area with the middle-third check and the applied pressure, the overburden and water-table rule, Figure 1 (the section with the failure depth, the Prandtl wedge, the water table and the loads), the table of N factors and the table of the four methods with every correction factor and the three terms of each, the governing method's equation with the substituted numbers, the allowable pressure and the bearing requirement, Figure 2 (q_u of the four methods against the footing width with the allowable curve and the applied pressure), the immediate settlement by the chosen method with the other method's value in one line, the consolidation table by sublayer, Figure 3 (stress increase with depth by both distributions) and the total-settlement requirement.

Limits

Not covered: footings on or near slopes and tilted bases (Hansen's and Vesić's ground and base factors, GEC 6 Figure 5-7), layered bearing soils and punching through a strong layer over a weak one (EM 1110-1-1905 paragraph 4-2g), the local-shear strength reduction (GEC 6 eqs 5-12 and 5-13), Vesić's compressibility factors, secondary compression and the time rate of consolidation, Hough's and D'Appolonia's SPT methods, pressuremeter methods, the structural design of the footing and the sliding check itself (compare T with Q tan δ + A′c_a). The circular footing is concentric only. The settlement methods take one modulus for the whole depth; a layered profile is approximated by the equivalent modulus or by running the clay layer through the consolidation part.