Retaining Wall and Footing: Method

Retaining Wall and Footing

What the tool does

Two classical geotechnical checks share one page. The retaining wall part is the conventional stability analysis of a cantilever or gravity wall per unit length (Das, Principles of Foundation Engineering, Sections 7.4–7.7): lateral earth pressure by Rankine or Coulomb, the weights and moment arms of the wall and the soil it carries, and the three factors of safety against overturning, sliding and bearing-capacity failure, with the base pressure distribution. The spread footing part is the ultimate bearing capacity of a strip, rectangular or circular footing by the four equations in general use (Terzaghi 1943, Meyerhof 1963, Vesić 1973 and the general equation as Das assembles it), with the corrections for shape, embedment, load inclination, eccentricity and a water table, the allowable pressure and load, and the immediate elastic settlement. The default inputs are Das's Example 7.1 (the wall) and Examples 3.1 / 3.2 (the footing), so the page reproduces the printed values on first load.

Conventions and units

The wall section is drawn with the toe at the left and the heel at the right, as in Das's Figure 7.11. The stem has a vertical back face; its front face slopes from the base width to the top width. The base width is B = toe + stem base width + heel. The active pressure acts on the vertical plane through the heel end (the Rankine "virtual back"), over the height H′ = base thickness + stem height + heel length × tan α, where α is the backfill slope. Moments are taken about the toe (point C). Forces are per unit length of wall. Unit weights are entered as plain numbers (kN/m³ in SI, lb/ft³ in USCS) because the unit dictionary has no force-per-volume type; pressures and cohesion go through the dictionary (kPa or lb/in² in the form; the results are written in kPa or lb/ft²). The unit weights do not convert on a unit-system switch: re-enter them.

Earth pressure

Rankine. For a level backfill K_a = (1 − sin φ′)/(1 + sin φ′) = tan²(45° − φ′/2). For a backfill sloping at α the Rankine coefficient is K_a = cos α (cos α − √(cos²α − cos²φ′)) / (cos α + √(cos²α − cos²φ′)), and the active force P_a = ½ K_a γ H′² acts at H′/3 above the base, inclined at α to the horizontal (Das Table 7.1). The passive coefficient of the soil in front of the toe is K_p = tan²(45° + φ′₂/2). Coulomb. With wall friction δ and a pressure plane battered at θ from the vertical (positive when the plane leans toward the toe as it rises, so the backfill overhangs it), K_a = cos²(φ′ − θ) / [cos²θ cos(δ + θ) (1 + √(sin(δ + φ′) sin(φ′ − α) / (cos(δ + θ) cos(θ − α))))²] and P_a is inclined at δ + θ to the normal of the plane (Das eq. 12.68). With δ = θ = 0 Coulomb's coefficient equals Rankine's. The tool verifies Coulomb's closed form against a trial-wedge maximisation in its test script.

Water, surcharge and cohesion. Below a water table standing h_w above the base the soil pressure is K_a times the effective vertical stress, built with the moist unit weight above the water table and the submerged unit weight γ_sat − γ_w below it, and the hydrostatic pressure γ_w (z − z_w) is added in full; the water force ½ γ_w h_w² acts at h_w/3 and is horizontal. A uniform surcharge q on the backfill adds K_a q H′ at mid-height and its weight on the heel to the resisting forces. A backfill cohesion reduces the active force by 2 c′ √K_a H′ (no tension crack is modelled, and a cohesive backfill is not recommended). The case is hydrostatic: seepage, drainage and the line-of-creep method of USACE EM 1110-2-2502 are not modelled; an optional uplift of γ_w h_w over the whole base (full head, no tail water) can be subtracted from the vertical forces as a conservative bound.

Wall stability

Overturning: FS = ΣM_R / M_o, where ΣM_R is the sum of the moments of the weights (stem, base, soil over the heel and its sloping triangle, surcharge on the heel, optionally the soil over the toe) and of the vertical component P_v of the active force about the toe, and M_o = P_h ȳ is the moment of the horizontal components about the toe. The passive resistance is not counted in overturning. Das requires 2 to 3; USACE EM 1110-2-2502 Table 4-1 instead requires the resultant within the middle third of the base (100 % of the base in compression) for usual loads on soil.

Sliding: FS = (ΣV tan δ_b + B k₂ c′₂ + P_p) / P_h, with the base friction angle δ_b = k₁ φ′₂ and the adhesion k₂ c′₂ (Das, k₁ = k₂ = ½ to ⅔; the second rule applies k₁ to tan φ′₂; the third takes an entered δ_b), and the Rankine passive resistance P_p = ½ K_p γ₂ D² + 2 c′₂ √K_p D of the soil in front of the toe when the box is checked (Das counts it; many codes neglect it because that soil may be removed). Das and USACE require 1.5.

Bearing: the resultant crosses the base at x̄ = (ΣM_R − M_o)/ΣV from the toe, e = B/2 − x̄. For e ≤ B/6 the whole base is in compression and q_toe, q_heel = (ΣV/B)(1 ± 6e/B); for e > B/6 the heel lifts off and q_toe = 2ΣV/(3(B/2 − e)) over the compressed length 3(B/2 − e). The ultimate bearing capacity of the base is computed as a strip footing of effective width B′ = B − 2e at the embedment D of the toe, under the load inclined at ψ = tan⁻¹(P_h/ΣV), by the bearing-capacity method chosen in the form (Das uses the general equation with the Meyerhof inclination factors); FS = q_u/q_toe, required 3. An entered allowable pressure from a geotechnical report is checked against the toe pressure as well.

Bearing capacity of a spread footing

The general equation is q_u = c′ N_c F_cs F_cd F_ci + q N_q F_qs F_qd F_qi + ½ γ B′ N_γ F_γs F_γd F_γi, where q is the effective overburden at the base, B′ the effective width and the F factors correct for shape (s), depth (d) and load inclination (i). N_q = e^{π tan φ′} tan²(45° + φ′/2) and N_c = (N_q − 1) cot φ′ (5.14 at φ′ = 0) are Prandtl–Reissner's and are shared by Meyerhof, Hansen and Vesić; the methods differ in N_γ and in the correction factors:

• Terzaghi (1943): 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), N_γ from Terzaghi's passive-wedge solution as evaluated numerically by Kumbhojkar (1993), the values printed as Das Table 3.1 (interpolated log-linearly between whole degrees). Shape coefficients 1.0 / 0.5 (strip), 1.3 / 0.4 (square), 1.3 / 0.3 (circle) on the c and γ terms; a rectangle uses 1 + 0.3 B/L and 0.5(1 − 0.2 B/L). No depth or inclination factors; the embedment enters only through q.
• Meyerhof (1963): N_γ = (N_q − 1) tan 1.4φ′; shape F_cs = 1 + 0.2 K_p B/L, F_qs = F_γs = 1 + 0.1 K_p B/L (φ′ > 10°); depth F_cd = 1 + 0.2 √K_p D_f/B, F_qd = F_γd = 1 + 0.1 √K_p D_f/B (φ′ > 10°); inclination F_ci = F_qi = (1 − δ/90°)², F_γi = (1 − δ/φ′)² (zero when δ ≥ φ′); K_p = tan²(45° + φ′/2). These are also the factors of the USACE CBEAR program printed in EM 1110-2-2502 Chapter 5 and Table 5-1.
• Vesić (1973): N_γ = 2 (N_q + 1) tan φ′ with De Beer's shape factors F_cs = 1 + (B/L)(N_q/N_c), F_qs = 1 + (B/L) tan φ′, F_γs = 1 − 0.4 B/L, Hansen's depth factors F_qd = 1 + 2 tan φ′ (1 − sin φ′)² k, F_cd = 1 + 0.4 k, F_γd = 1 with k = D_f/B for D_f/B ≤ 1 and k = tan⁻¹(D_f/B) otherwise, and Vesić's inclination factors F_qi = (1 − H/(V + B′L′c′ cot φ′))^m, F_γi = (…)^{m+1}, F_ci = F_qi − (1 − F_qi)/(N_c tan φ′), m = (2 + B/L)/(1 + B/L) for a load inclined in the B direction (H = V tan δ).
• General equation (Das): Vesić's N, De Beer's shape factors, Hansen's depth factors with F_cd = F_qd − (1 − F_qd)/(N_c tan φ′), and Meyerhof's inclination factors, exactly as Das Tables 3.3 and 3.4 assemble them. This is the method of Das's Examples 3.2 and 7.1.

All four are printed side by side; the chosen one gives the allowable pressure q_all = q_u/FS (gross), the net values q_net = q_u − q and q_all,net = q_net/FS, and the allowable load Q_all = q_all B′L′ (per unit length for a strip). With a vertical load V entered, the applied pressure V/(B′L′) gives the available factor of safety and the margin M.S. = q_all/q − 1.

Eccentricity and water. Meyerhof's effective area takes B′ = B − 2e_B and L′ = L − 2e_L, the smaller effective dimension being B′ in the formulas, and the ultimate load is q_u B′L′. The water table follows Das's three cases: above the base (case I) the overburden is the effective stress D₁γ + D₂(γ_sat − γ_w) and the γ term uses γ′; within a depth B below the base (case II) q = γ D_f and the γ term uses γ̄ = γ′ + (d/B)(γ − γ′); deeper than B (case III) there is no effect. A circular footing uses Terzaghi's circular coefficients and, for the other methods, the shape factors of a square of the same width.

Elastic settlement

S_e = q_o (α B′)(1 − μ_s²)/E_s · I_s I_f (Das eq. 5.33, after Steinbrenner 1934 and Bowles), with I_s = F₁ + (1 − 2μ_s)/(1 − μ_s) F₂, where F₁ and F₂ are Steinbrenner's influence factors for the corner of a uniformly loaded rectangle m′ = L/B wide on a layer n′ deep: 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)). The centre of the flexible footing uses α = 4, B′ = B/2, n′ = H/(B/2); the corner α = 1, B′ = B, n′ = H/B; a semi-infinite soil gives F₂ = 0 and the familiar values α I_s = 1.12 (centre of a square) and 0.56 (corner). A rigid footing settles about 0.93 times the flexible centre value (Das). I_f is Fox's (1948) embedment correction, read from the chart as reproduced by Bowles and Das and entered. The formula is exact for a homogeneous elastic layer on a rough rigid base in the sense that it equals the difference of the Boussinesq half-space displacements at the surface and at depth H; the tool's test script checks it against a direct numerical integration of the Boussinesq point-load displacement over the loaded area. The settlement is computed on the net applied pressure V/A − q.

Reading the results

The wall part prints the coefficients, the active-force components and their line of action, the table of areas, weights, arms and moments as Das lays it out, each factor of safety with its substituted equation and a margin M.S. = FS/FS_required − 1, the resultant location against B/6, the toe and heel pressures and the bearing capacity under the toe; Figure 1 draws the section with the active, water and passive pressure diagrams, the base pressure and the resultant. The footing part prints the water case, the effective area, the four methods in one table (with the chosen method in bold), the allowable pressure and load, the available factor of safety, the elastic settlement with its influence factors and a chart of the allowable pressure against the footing width at the same aspect ratio, embedment and eccentricity ratios.

Limits

Not covered: the structural design of the stem, toe, heel and footing (shear, moment, reinforcement), deep-seated (global) slope failure, settlement and tilt of the wall, seepage, drainage and the line-of-creep uplift of USACE EM 1110-2-2502, seismic earth pressure (Mononobe–Okabe), compaction-induced pressure, shear keys, local and punching shear (Terzaghi's reduced strength parameters), layered soils, footings on or near slopes, consolidation settlement of clays and Schmertmann's strain-influence settlement of sands. Vesić's inclination factors need the vertical load to form the adhesion term; without it the ratio 1 − H/V is used and the page says so.