Sheet Pile and Braced Excavation: Method

Sheet Pile and Braced Excavation

What the tool does

Three analyses of flexible earth-retaining walls by the classical limit-equilibrium methods of NAVFAC DM-7.02 (1986) and FHWA Geotechnical Engineering Circular No. 4 (1999): a cantilever sheet-pile wall, a sheet-pile wall with one tie (anchored bulkhead), and an excavation braced by struts. All results are per unit length of wall. The methods give embedment, anchor and strut forces and bending moments; they do not give deflections, and they do not replace a check of the overall stability of the wall and the soil mass.

Conventions

Depths are measured down from the top of the wall, which is the ground surface behind it. H is the retained height to the dredge line (or the base of the cut) and D the embedment below it. A pressure is positive when it pushes the wall toward the excavation. The shear is the running sum of the net pressure from the top (less the anchor pull below the tie) and the bending moment is its integral. Unit weights are total: the moist weight above the water level and the saturated weight below it; the effective vertical stress is the total stress less the pore pressure, and the unit weight of water is 62.4 lb/ft³ (9.81 kN/m³).

Earth pressures

Active pressure σ_a = K_A σ′_v − 2 c √K_A, not taken below zero; passive pressure σ_p = K_P σ′_v + 2 c √K_P (DM-7.02 Figure 2, used in its Figure 21). Rankine: K_A = tan²(45° − φ/2), K_P = tan²(45° + φ/2). Coulomb, vertical wall and level ground, wall friction δ: K_A = cos²φ / {cos δ [1 + √(sin(φ + δ) sin φ / cos δ)]²} and K_P with the minus sign in the bracket; the tool uses their horizontal components K cos δ. The plane-wedge passive coefficient is unsafe for wall friction above about φ/3; DM-7.02 reads K_P for larger wall friction from the log-spiral chart of its Figure 5, and such a value is entered in the layer (the default case enters 6.0, the value of Figure 21). A layer with φ = 0 is a clay analysed by total stress: K_A = K_P = 1 and the cohesion is the undrained shear strength, so the net resistance below the dredge line of a wall driven into clay is 4c − γ_E H, where γ_E H is the effective vertical stress at the dredge line (DM-7.02 Figure 25).

A uniform surcharge adds to the vertical stress. A line load Q_L at a distance x = mH gives, on the retained height, σ_H = (Q_L/H) 0.20 n / (0.16 + n²)² for m up to 0.4 and (Q_L/H) 1.28 m² n / (m² + n²)² beyond, n = z/H (DM-7.02 Figure 11, the Boussinesq stress modified by experiment). A strip load q between the distances a and a + b gives σ_H = (2q/π)(β − sin β cos 2α), the elastic stress doubled for an unyielding wall, β being the angle the strip subtends at the depth and α the angle of its centre line from the vertical; the 1.28 of the line-load formula is 4/π, the same doubling.

Water

Three options. Hydrostatic: the difference of the two hydrostatic pressures acts on the wall down to the toe (GEC 4 equation 17). Tapering: the same down to the dredge line, falling linearly to zero at the toe, which is how DM-7.02 Figure 21 draws its net water pressure. Seepage: the head difference is lost uniformly along the shortest path down the back of the sheeting and up the front (GEC 4 equations 15 and 16), so the pore pressure is below hydrostatic behind the wall and above it in front, the two are equal at the toe, and the effective stresses change accordingly; the embedment then depends on itself and is found by iteration.

Cantilever wall

Simplified method: active pressure behind and passive pressure in front act to the toe, and the embedment D_0 is where the moments of the net pressure about the toe vanish; the horizontal force left over is the passive reaction that develops behind the toe, and the embedment is increased to provide for it. Rigorous method (DM-7.02 Figure 23): the wall rotates about a point above the toe; above it the net pressure is active behind less passive in front, at the toe it is passive behind less active in front, and the change between the two is taken linear over a height Z. Horizontal equilibrium gives Z and moment equilibrium gives D; for a uniform sand without water Z = [K_P D² − K_A (H + D)²] / [(K_P − K_A)(H + 2D)], the expression printed in the figure. The tool integrates the net pressure numerically, so the same solution holds for two layers, clay, water and surface loads. DM-7.02 computes the pressures with no factor of safety and increases D by 20 to 40 %, which it equates with a factor of about 1.5 to 2; a factor dividing the passive pressure can be entered instead or as well. The maximum moment is at the depth of zero shear.

Anchored wall

Free earth support (DM-7.02 Figure 18): the toe is free to rotate. Moments about the anchor of the driving pressures and of the passive pressure divided by F_S (2 to 3 in coarse-grained, 1.5 to 2 in fine-grained soil) give the embedment, which is increased by 20 % for dredging and scour; the anchor pull is A_p = ΣP_A − P_P/F_S. When an embedment is entered the tool reports the factor of safety against toe failure, the ratio of the two moments, and shows the equilibrium state in which the passive pressure is divided by that factor. The maximum span moment may be reduced for the flexibility of the sheeting (Rowe): DM-7.02 Figure 19 gives M_design / M_max against ρ = (H + D)⁴ / (E I) in in²/lb per running foot of wall for penetration in compact and in very compact coarse-grained soil, and permits no reduction in fine-grained or loose soil. The two curves were read from the figure by eye (± 0.02).

Fixed earth support, equivalent beam: with deeper embedment the sheeting is restrained in the ground and its elastic line has a point of contraflexure near the dredge line. The tool places a hinge where the net pressure is zero, solves the upper beam (anchor to hinge) for the anchor pull and the hinge reaction, and finds the embedment at which the lower beam, loaded by the hinge reaction and the net passive pressure, is in moment equilibrium about its toe; the reaction C behind the toe is developed by the embedment increase. This is the textbook procedure attributed to Blum; no document held for this tool prints it, and it was checked by statics only.

Anchorage (DM-7.02 Figure 20): a continuous deadman reaching h below the surface resists ½ (K_P − K_A) γ h² per unit length when the tie is at or below its mid-height. It gives no resistance inside the active wedge of the wall, which rises at 45° + φ/2 from the estimated point of zero moment below the dredge line, and full resistance only behind the line rising at φ from that point.

Braced excavation

Strut loads are computed from apparent pressure envelopes, design diagrams that envelop measured strut loads. Terzaghi and Peck, as given in DM-7.02 Figure 26: sand, a uniform 0.65 K_A γ H; soft to medium clay (stability number γH/c above 6), K_A γ H rising from zero over the top quarter with K_A = 1 − m 4c/(γH), m = 1 or 0.4 F_SB over deep soft clay; stiff clay (below 4), 0.2 to 0.4 γH rising over the top quarter and falling over the bottom quarter; between 4 and 6 the larger of the two. FHWA (GEC 4 Figures 24 and 27): trapezoids that rise over two thirds of the distance from the top to the first support and fall over two thirds of the distance from the last support to the base, carrying 0.65 K_A γ H² in sand and with p = 0.2 to 0.4 γH in stiff to hard clay; for soft to medium clay Henkel's coefficient K_A = 1 − 4 s_u/(γH) + 2√2 (d/H)(1 − 5.14 s_ub/(γH)) (equation 12), valid for a stability number above 5.14.

Tributary area method: each support takes the diagram between the mid-points of the adjacent spans. Hinge method: the sheeting is hinged at every support but the top one and at the base. Wall moments: the cantilever above the top support, p l²/10 in the spans (GEC 4) and the simple-span moments of the hinge method. Stages: the excavation is carried a set distance below each strut level before the strut is placed; before the first strut the wall is a cantilever under the Rankine active pressure, afterwards the envelope of the stage depth is carried by the struts in place and the stage base (DM-7.02 Figure 27).

Base stability in clay: Terzaghi, F_S = N_c s_u / (γH + q − s_u H/B′) with B′ = B/√2 or the depth to a hard stratum (GEC 4 equations 34 to 36; the surcharge is added here to the weight of the block); Bjerrum and Eide, F_S = N_c s_u / (γH + q) with N_c from the chart for H/B and B/L (GEC 4 equation 37 and Figure 51, DM-7.02 Figure 28). When no N_c is entered the tool uses 5 (1 + 0.2 H/B)(1 + 0.2 B/L), H/B not above 2.5, a closed-form fit that falls about 3 % below the chart value in the DM-7.02 Figure 30 example. The unbalanced force on sheeting carried below an unstable base and the factor 2 N_γ (γ_2/γ_1) K_A tan φ of a cut in sand follow DM-7.02 Figure 28. Piping: critical gradient (γ_sat − γ_w)/γ_w against the gradient of a uniform head loss along the sheeting, or against the exit gradient of an entered flow net.

Checks

The default case is the anchored bulkhead of DM-7.02 Figure 21. The tool gives a factor of safety of 2.65 where the figure prints 2.61 and an anchor pull of 11.8 kips per foot by the figure's formula where it prints 12.37; the figure adds forces rounded to two or three figures, one of which (P_A1 = 1.55 kips) is larger than its own pressure diagram gives, and the anchor pull is a small difference of large numbers. The cantilever example of Figure 25 (D/H 0.69, moment ratio 0.33, chart readings) is reproduced as 0.688 and 0.327, the braced cut of Figure 30 and Design Example 1 of GEC 4 to the printed digits. Coulomb coefficients, the strip load, the cantilever solution in sand and twenty seeded two-layer walls are compared with an independent scipy implementation, and every solved diagram is in horizontal and moment equilibrium to 1e-9.

Limits

Two soil layers on the form (the library takes any number); level ground; one tie level; no surcharge in front of the wall; no earthquake loading; no sloping dredge line or berm. USACE EM 1110-2-2504, which applies factors to the soil strengths instead of the passive pressure, is cited but its procedure and examples are not implemented. Settlement beside a braced cut is not computed.