Soldier-Pile (Berliner) Walls — MADEP CPT Interpreter

Scope and positioning

1. Analysis class and the hybrid idealisation

A soldier-pile wall is not a plane-strain wall. Above the excavation the lagging spans horizontally between the flanges and transfers the earth pressure of the whole tributary width s (centre-to-centre spacing) to the pile; below the excavation only the discrete pile of flange width b is in the ground and the passive resistance is three-dimensional. The engine therefore models the wall as a hybrid (embedded_model.hpp, kinds SoldierEffWidth and SoldierBrinchHansen):

Above the excavation: the retained-side ordinates (earth, variable surcharge, and water only when the lagging is declared watertight) are multiplied by the tributary width s. All quantities are per pile (kN/m of pile, kN, kNm).
Below the excavation — Model A, effective width: the active pressure acts on the flange width b and the passive resistance on b_eff = min(k·b, s) with the plane-strain K_p (EAB / Belgian guideline §5); k = 3 by default.
Below the excavation — Model B, Brinch Hansen: the net line resistance of one pile B·[e_w(z)]⁺ from the Brinch Hansen (1961) coefficients K_q(z/B), K_c(z/B), with the Andersen–Lodahl (2023) additional active term for the higher retained side, optionally capped by the continuous-wall tributary resistance s·p_net.
Never mixed: the hand calculation runs one of the two models; the PLAXIS Embedded Beam Row T_lat tables are always Brinch Hansen with B = flange width, never divided by the spacing and never 3b (Rekennota §5.7; course chapter §8.3).

Everything else — the four design branches, the over-excavation rule, the Blum embedment, the support reaction, the shear and moment diagrams and the anchor pull-out — is shared with the sheet-pile engine (embedded_branches.hpp, embedded_solver.hpp, embedded_wall.hpp) and documented in the retaining-wall chapter; this chapter repeats what the soldier-pile route needs and adds the pile-specific parts.

Outputs. Required embedment (GEO), M_Ed, V_Ed, T_Ed per pile (STR envelope), the factored lagging pressure, the lagging plate check, the EN 1993-1-1 checks of the H/I section (class, M_c,Rd, V_pl,Rd, M–V), a vertical-equilibrium screen, and the PLAXIS 2D Plate + Embedded Beam Row parameter set (EA, EI, w, ISF, T_skin, F_max, T_lat). It does not compute wall displacements (SSI/FE) and it does not verify the timber or the pile-to-lagging connection.

Design framework

2. Design branches (Belgian guideline)

Embedded walls follow the risk-class (RK) workflow of the Belgian embedded-wall guideline (BGGG/Buildwise 2022, “Richtlijnen EC7 beschoeiingen”) as implemented in embedded_branches.hpp. Every analysis runs four branches and reports each with its full intermediate set:

Branch	Actions	Strength	Excavation level	Role
DA1/2 (A2 + M2)	γ_G = 1.00 both sides; γ_Q of the risk class (1.10 RK1/RK2, 1.20 RK3; 1.30 generic ANB)	M2 of the risk class	design = nominal − Δa	governs the embedment (GEO)
DA1/1 (A1 + M1)	γ_G = 1.35 (RK2) on the retained-side actions, γ_Q = 1.50; passive resistance and front water at γ_G,fav = 1.00 (separate source, default) or at 1.35 (single source, EN 1997-1 §2.4.2(9)P)	M1	design = nominal − Δa	STR envelope; governs embedment only with the generic ANB sets
BGT + α_ver	γ = 1.0; α_ver = 1.10 on the variable surcharge	characteristic (M1)	nominal	section and support forces × 1.35 for STR
SLS	γ = 1.0	characteristic (M1)	nominal	serviceability reference

Factor sets (factors.hpp; γ never reduces the unit weight)

Generic NBN EN 1997-1 ANB: A1 = [1.35, 1.00, 1.50, 0]; A2 = [1.00, 1.00, 1.30, 0]; M2 = [1.25, 1.25, 1.40, 1.00]; K_FI = 0.90 / 1.00 / 1.10 (CC1/CC2/CC3) on the unfavourable actions — applied only with this scheme.
RK1: A1 = [1.20 / 1.30], A2 = [1.00 / 1.10], M2 = [1.10, 1.10, 1.25].
RK2 (default for embedded walls): A1 = [1.35 / 1.50], A2 = [1.00 / 1.10], M2 = [1.25, 1.25, 1.40].
RK3: A1 = [1.50 / 1.80], A2 = [1.00 / 1.20], M2 = [1.40, 1.40, 1.55].
Material override: Optional replacement of M2 in DA1/2, e.g. the SB260 value γ_φ = γ_c = 1.30 (γ_cu = 1.40) used in the Rekennota; also drives the “sensitivity” T_lat table.
STR envelope: M_Ed, V_Ed, T_Ed and the lagging pressure = max over DA1/2, DA1/1 and 1.35 × (BGT + α_ver) (guideline §3.5). SLS is reported, never enveloped.

Over-excavation Δa (overdiepte) — a ULS geometry assumption applied to the two ULS branches only; BGT + α_ver and SLS use the nominal excavation (guideline §3.3). h is the retained height for a cantilever, or the height below the lowest support for an anchored wall; “under water” means the front water table lies above the nominal excavation.

Belgian (default): Δa = 0.30 m (dry), Δa = min(0.1·h, 0.5 m) (under water)

EN 1997-1 §9.3.2.2: Δa = min(0.1·h, 0.5 m)

custom: Δa = user value (≥ 0); none: Δa = 0

toe elevation = nominal excavation − Δa − d_provided (the provided embedment counts below the design excavation)

Model B and DA1/1. The Brinch Hansen net coefficient cannot separate the active from the passive part below the excavation. In DA1/1 the γ_G = 1.35 factor is therefore applied to the retained-side load above the excavation and to the variable-surcharge part of the Andersen–Lodahl term; the net line resistance below the excavation carries γ_G,fav (1.00 separate-source, 1.35 single-source). The engine prints this as a note.

Pressure model

3. Ordinates, widths and water

Design strengths are formed at source per branch (φ′_d = atan(tan φ′_k/γ_φ), c′_d = c′_k/γ_c, c_u,d = c_u,k/γ_cu), then the per-layer coefficients: active Rankine on a vertical wall with δ = 0 (K_a, K_ac = 2√K_a), passive from the EN 1997-1 Annex C closed form with δ_p = (δ_p/φ′)·φ′_d (K_p, K_pc), and the Brinch Hansen constants of §5 at the design φ′. Undrained layers use K_a = K_p = 1, K_ac = K_pc = 2 on the total vertical stress with c_u, and the φ → 0 Brinch Hansen limits. The vertical effective stress on each side is precomputed on a 1 cm grid (stress_profile.hpp) so every ordinate query is O(1); σ′_v is continuous across layer boundaries while K and c are read from the layer present at each depth.

Soldier-pile inputs and defaults (wall-state.js / request-builder.js)

b, s: Flange width of the catalogue section (normal to the loading) and centre-to-centre spacing (default 1.00 m).
k: Effective-width factor, b_eff = min(k·b, s); default 3.
δ_p/φ′: Passive wall-friction ratio; default 0 for soldier piles (Rankine, as in the Rekennota; the Belgian guideline Table 4 caps a discontinuous wall at φ′_k/3 straight and φ′_k/2 curved), ⅔ for sheet piles.
Surcharge floor: Explicit, visible minimum variable surcharge (default 10 kPa, a practice value — not a Belgian requirement); q_var = max(q_user, floor).
Row cap: Model B cap by s·p_{net,continuous}; on by default.
Watertight lagging: Off by default (permeable lagging).

Retained (active) face ordinate at elevation el, per unit area, unfactored (activeOrdinate):

σ_ref = σ′_v (drained) σ_ref = σ′_v + u (undrained)

p_a = [K_a·σ_ref − K_ac·c]⁺

p_surch = K_a·q_var (variable, γ_Q)

u_back = γ_w·(WT_ret − el) below the retained water table (drained layers)

tension crack (p_a = 0, option on): u_crack = γ_w·(surface − el) above the phreatic line, else γ_w·(surface − WT_ret) (EN 1997-1 §9.6(5)P)

σ′_v on the retained side includes the permanent surcharge and, optionally, a retained berm or slope treated as an equivalent surcharge spread under 45° (Rekennota §7.3). This is an approximation, not EN 1997-1 Annex C sloping ground; Δh is the berm height, β its slope, γ the fill unit weight (top stratum by default), L = Δh/tan β:

q_berm(z) = γ·tan β·z/2 (z ≤ L)

q_berm(z) = γ·Δh·(1 − L/(2z)) (z > L)

Net terms handed to the solver, already multiplied by the acting width (positive pushes the pile toward the excavation):

above the excavation: p_earth·s, p_surch·s, u_back·s only if the lagging is watertight (else 0)

below the excavation: no water on either face (permeable lagging / flow around the pile)

Model A: p_earth·b, p_surch·b driving; (K_p·σ_ref,f + K_pc·c)·b_eff resisting

Model B: B·K_q^A·q_var driving (variable); T_lat,perm(z) resisting (§5)

factored drive = γ_G·(p_earth + u_back) + γ_Q·p_surch; factored resist = γ_G,resist·(p_resist + u_front)

Water assumption. With permeable lagging no water pressure is applied to the wall, which presumes free drainage through the lagging and a dewatered pit. When the retained water table lies above the design excavation the engine flags this as a binding execution condition (Rekennota §8.1). Tick “watertight lagging” to apply the hydrostatic thrust over the tributary width above the excavation.

Hand calculation A

4. Effective-width model (EAB / Belgian guideline §5)

The conventional Berliner-wall hand calculation. Below the excavation the active earth pressure (and the variable surcharge) act on the flange width b; the passive resistance acts on an effective width b_eff that accounts for the three-dimensional wedge in front of the pile, computed with the plane-strain passive coefficient:

b_eff = min(k·b, s) (k = 3 default)

e_a(z) = [K_a·σ_ref,b − K_ac·c_b]⁺·b + K_a·q_var·b

e_p(z) = [K_p·σ_ref,f + K_pc·c_f]⁺·b_eff (K_p, K_pc from EN 1997-1 Annex C with δ_p)

σ_ref,f is the front-side vertical stress from the design excavation level downward (a fresh stress profile starting at zero at the excavation). This is the model used for the Rekennota parity case in §9 (HEA180, k = 3, δ_p = 0). It is the default hand model in the app.

Hand calculation B and PLAXIS T_lat

5. Brinch Hansen (1961) net line resistance

brinch_hansen.hpp is the single source of truth for the coefficients — the hand calculation and the PLAXIS Embedded-Beam-Row tables both use it. The ultimate net pressure around a rigid pile of width B at depth z below the excavation is e(z) = q̄(z)·K_q(z/B) + c·K_c(z/B), with the depth-dependent coefficients interpolated rationally between the surface and the great-depth values (Brinch Hansen 1961, DGI Bulletin 12, pp. 5–9). Angles in radians; φ is the design friction angle of the front layer.

Surface coefficients (rough wall, translation):

P_q = exp[(π/2 + φ)·tan φ]·cos φ·tan(45° + φ/2)

K_q^A = exp[−(π/2 − φ)·tan φ]·cos φ·tan(45° − φ/2)

K_q⁰ = P_q − K_q^A

K_c⁰ = (P_q − 1)·cot φ

Great-depth coefficients (Jáky K₀, deep depth factor d_c^∞, bearing factors):

K₀ = 1 − sin φ

d_c^∞ = 1.58 + 4.09·tan⁴φ

N_q = exp(π·tan φ)·tan²(45° + φ/2)

N_c = (N_q − 1)·cot φ

K_c^∞ = N_c·d_c^∞

K_q^∞ = K_c^∞·K₀·tan φ

Interpolation parameters and depth-dependent coefficients (ξ = z/B ≥ 0):

a_q = K_q⁰ / (K_q^∞ − K_q⁰) · K₀·sin φ / sin(45° + φ/2) (0 if K_q^∞ ≤ K_q⁰)

a_c = K_c⁰ / (K_c^∞ − K_c⁰) · 2·sin(45° + φ/2) (0 if K_c^∞ ≤ K_c⁰)

K_q(ξ) = (K_q⁰ + K_q^∞·a_q·ξ) / (1 + a_q·ξ)

K_c(ξ) = (K_c⁰ + K_c^∞·a_c·ξ) / (1 + a_c·ξ)

φ → 0 limits (used for φ < 10⁻⁴ rad, i.e. undrained φ_u = 0 layers; Brinch Hansen 1961, course chapter §3.7):

P_q = K_q^A = 1, K_q⁰ = K_q^∞ = 0, a_q = 0 ⇒ K_q ≡ 0

K_c⁰ = 1 + π/2 = 2.5708, K₀ = 1, d_c^∞ = 1.58, N_q = 1, N_c = π + 2 = 5.1416

K_c^∞ = N_c·d_c^∞ = 8.1237, a_c = K_c⁰/(K_c^∞ − K_c⁰)·2·sin 45° = 0.6547

Net line resistance of one pile (brinchHansenOrdinate), z measured from the branch's design excavation, B = flange width b, σ_ref,f the front-side vertical stress (effective, or total for undrained layers), and Δq the retained-minus-front vertical-stress difference at the same elevation — layered, including the berm equivalent surcharge and the permanent surcharge — times the shallow active coefficient (Andersen & Lodahl 2023, eq. 2–3; course chapter eq. 5):

e_equal(z) = σ_ref,f·K_q(z/B) + c·K_c(z/B) (equal-level convention, Rekennota Table 5-7)

Δq = [σ_ref,b − σ_ref,f]⁺ (+ q_var in the T_lat tables)

e_w(z) = e_equal(z) − Δq·K_q^A (Andersen–Lodahl convention)

T_lat,perm(z) = B·[e_w(z)]⁺ [kN/m of pile]

p_net,cont = (K_p·σ_ref,f + K_pc·c_f) − [K_a·σ_ref,b − K_ac·c_b]⁺ (continuous wall, per unit width)

row cap: T_lat ≤ s·[p_net,cont]⁺ (the tributary strip cannot supply more than the continuous wall)

hand calculation, driving side: B·K_q^A·q_var (variable surcharge part of the A–L term, γ_Q)

Two conventions, one choice. The Rekennota Table 5-7 enters the equal-level value B·(σ′_vK_q + c′K_c) as the PLAXIS T_lat; the course chapter and Andersen & Lodahl (NUMGE 2023) define the Embedded-Beam-Row cap as B·[e_w]⁺ with the −(γ′H + p_b)·K_q^A term. For the Rekennota case the difference is ≈ 2.4 kN/m per pile over the first 2 m (5–45 %). The engine tabulates both columns side by side; the hand calculation uses the Andersen–Lodahl form, the PLAXIS export defaults to it, and the engineer must pick and justify (worklog review E-7). The positive-part operator, the row cap and the rule to recompute T_lat at φ_d are the chapter's implementation rules, not Brinch Hansen's.

Limit equilibrium

6. Blum embedment, support reaction and diagrams

The solver (embedded_solver.hpp) is idealisation-agnostic: it only sees the factored net terms of §3. The net pressure is integrated over 800 cells from the retained surface to the trial toe.

ODF(d) = |M_resist| / |M_drive| about the toe (cantilever) or about the anchor (anchored)

d₀: ODF(d₀) = 1 by bisection on [0.05 m, 40 m], 60 halvings (not bracketed within 40 m → flagged)

cantilever: d_design = 1.2·d₀ (Blum 1931; Rekennota §7.4 — the 20 % is part of the method, not an EC7 factor)

anchored: d_design = d₀ (free-earth support), T = H_drive − H_resist at d₀ (≥ 0)

GEO: d_provided ≥ d_required = max d_design over the branches that govern the embedment (DA1/2; also DA1/1 with the generic ANB sets)

Shear and moment on the provided pile by trapezoidal double integration on 600 cells (course manual eq. 21–22); the anchor reaction enters as a shear jump −T at its exact elevation:

V ← V + ½·(net_i−1 + net_i)·dz, M ← M + ½·(V_i−1 + V_i)·dz (anchor: V ← V − T)

M_max = largest |M| at a zero-shear crossing inside the model-valid depth (≤ H + d₀ + tol), else max |M|

beyond the last moment zero-crossing (free-earth closure) V and M are set to 0 — the tail diverges by construction

M_Ed = M_max·f_effect, V_Ed = V_max·f_effect, T_Ed = T·f_effect (f_effect = 1.35 for BGT + α_ver, else 1)

For an anchored soldier pile the EN 1537 pull-out check of the retaining-wall chapter is applied with the reaction converted to a per-metre value T/s before the per-anchor axial force (T/s)/cos α · s_anchor is formed. The HYD heave check and the wall vertical screening of continuous walls are not run for soldier piles; the pile's own vertical equilibrium is checked in §7.

Structural verifications

7. Lagging, EN 1993-1-1 section checks, vertical equilibrium

Lagging design pressure (laggingPressureAt, course/Rekennota §7.8) — the factored horizontal pressure on the deepest lagging board, at the branch's excavation level, enveloped over the non-SLS branches:

p_Ed = f_effect·[γ_G·(K_aσ_ref − K_acc)⁺ + γ_Q·K_a·q_var + γ_G·u (watertight only)] at z = design excavation of the branch

Lagging plate (steel-checks.js, checkLaggingPlate) — a steel plate of thickness t spanning horizontally between the flanges; span L = s (centre-to-centre, conservative, default) or the clear span s − b (min 0.05 m); per metre height; elastic by default:

M_Ed = p_Ed·L²/8 [kNm/m]

W_el = t²/6, W_pl = t²/4 [m³/m]

M_Rd = W·f_y/γ_M0 (W_el elastic default, W_pl plastic option), σ = M_Ed/W_el

δ = 5·p_k·L⁴ / (384·EI), EI = E·t³/12 (characteristic pressure p_k, informational)

H/I pile section (hSectionClass, checkHPile) — NBN EN 1993-1-1 with the ANB γ_M0 = 1.00; E = 210 000 N/mm², γ_steel = 78.5 kN/m³, f_y per EN 10025-2 (S235/S275/S355 for t ≤ 40 mm). Section properties come from the EN 10365 catalogue (90 HEA/HEB/HEM/IPE profiles; A_v,z is the catalogue shear area for η = 1.2).

ε = √(235/f_y)

flange: c/t = [(b − t_w − 2r)/2] / t_f ≤ 9ε (class 1), 10ε (2), 14ε (3), else 4 (Table 5.2, outstand in compression)

web: c/t = (h − 2t_f − 2r) / t_w ≤ 72ε (1), 83ε (2), 124ε (3), else 4 (Table 5.2, web in bending)

class = max(flange, web)

M_pl,Rd = W_pl,y·f_y/γ_M0, M_el,Rd = W_el,y·f_y/γ_M0; M_c,Rd = M_pl,Rd (class 1–2) or M_el,Rd (class 3) (§6.2.5)

V_pl,Rd = A_v,z·(f_y/√3)/γ_M0 (§6.2.6)

M–V: no reduction if V_Ed ≤ 0.5·V_pl,Rd; else ρ = (2V_Ed/V_pl,Rd − 1)², M_y,V,Rd = [W_pl,y − ρ·A_w²/(4t_w)]·f_y/γ_M0, A_w = (h − 2t_f)·t_w (§6.2.8(5))

N_pl,Rd = A·f_y/γ_M0 (reported)

The elastic bending check is always printed as a control row next to the plastic one. For continuous sheet piles the same module applies EN 1993-5 §5.2.2 (β_B·W_pl or W_el; shear area Σ webs per metre × t_w·(h − t_f) with the web inclination neglected — conservative; M–V interaction at 0.5·V_pl,Rd); an optional uniform corrosion loss is applied as a plain reduction factor on the thickness-driven properties.

Vertical equilibrium of one pile (checkVerticalEquilibrium, EN 1997-1 §9.7.5; Rekennota §7.10) — self-weight (plus the anchor down-drag T·tan α when present) against the β-method shaft resistance of §8 integrated over the embedment; base resistance is not credited:

G = w_pile·L_pile + γ_steel·t_lagging·H_lagging·s + V_extra

R_s = slope·d² / 2 (slope = dT_skin/dz′ of §8, uniform γ below the excavation)

pass: G ≤ R_s

PLAXIS 2D input

8. PLAXIS 2D (v24) parameter set

plaxis-parameters.js derives the values the engineer copies into the calculation note, following the hybrid model of Andersen & Lodahl (NUMGE 2023) and the Rekennota §5: a Plate from the pile head to the design excavation and a user-defined Embedded Beam Row below it. E = 210·10⁶ kN/m², γ_steel = 78.5 kN/m³ unless overridden. A_p, I_p are the catalogue properties of one pile.

Plate above the excavation (per metre of wall):

EA₁ = EA₂ = E·A_p/s [kN/m]

EI = E·I_p/s [kNm²/m] (lagging stiffness omitted: EI_lagging = E·t³/12 ≪ profile and no composite action)

w = γ_steel·(A_p/s + t_lagging) [kN/m/m] (the lagging self-weight IS included)

ν = 0 (wall of discrete elements; PLAXIS reference manual)

d_eq = √(12·I_p/A_p) (independent of s; computed by PLAXIS)

interfaces both sides, R_inter = tan δ / tan φ′ per layer (clamped to [0.01, 1]), Rayleigh damping 0 (static)

Embedded Beam Row below the excavation (properties of ONE pile — never divided by the spacing; PLAXIS smears the row itself):

L_spacing = s; A = A_p; I = I_p; E

γ_eff = γ_steel − γ_soil (the row occupies no soil volume)

D_eq = √(12·I/A)

ISF_RS = ISF_RN = 2.5·(L_spacing/D_eq)^−0.75, ISF_KF = 25·(L_spacing/D_eq)^−0.75 (PLAXIS 2D reference-manual defaults — a mismatch means A, I or L_spacing were mistyped)

continuity at the transition: EI/s of the plate = E·I_p/s of the row

Axial skin resistance, linear (β-method, Rekennota §5.5) — K = K₀ = 1 − sin φ′_k by default (the lower bound of the allowed installation methods: pre-augering with backfill), steel–soil friction δ = ⅔·φ′_k on the outer flange faces O_steel = 2b, soil–soil shear at φ′_k on the plug faces O_plug = 2h between the flanges; z′ from the design excavation:

T_skin(z′) = σ′_v(z′)·K·[O_steel·tan δ + O_plug·tan φ′_k] [kN/m]

slope = K·[…]·γ [kN/m per m] (uniform γ below the excavation — a simplification when layered)

T_{skin,start,max} = 1.0 kN/m (numerical floor at z′ = 0), T_skin,end,max = slope·d at z′ = d

R_s = slope·d²/2

Base resistance from the cone resistance near the toe (the least substantiated parameter of the set — a toe force approaching F_max in the results requires a separate pile calculation under NBN EN 1997-1 ANB):

F_max = α_b·q_c·A_b, A_b = b·h (plugged box area), α_b = 0.5 default

F_{max,unplugged} = q_c·A_steel (alternative, reported)

Lateral resistance, multilinear T_lat (buildTlatTable) — rows every 0.25 m from the top of the row (= the ULS design excavation), plus every layer boundary inside the embedment (two rows, ± 0.1 mm) and the toe. Each row carries z, σ′_v,f, Δq, K_q, K_c, K_q^A, the equal-level value, the Andersen–Lodahl value, the row cap and the adopted value; the table is built with γ_G = γ_Q = 1 and Δq including the variable surcharge (representative loads). Three sets are produced:

Characteristic (M1) — staged/SLS phases and the φ-c reduction.
Design (M2 of the risk class) — for an explicit DA1/2 plastic phase (T_lat recomputed at φ_d; the chapter's rule, prompted by the PLAXIS 2D T_lat strength-reduction release note).
Sensitivity — only when a material override is active (e.g. γ_φ = 1.30).

T_adopted(z) = min(T_AL, s·p_net,cont) (cap on) or T_AL; PLAXIS export: convention 'AL' (default) or 'equal', cap on/off

R_u = ∫ T_adopted dz, M_u = ∫ T_adopted·z dz, z̄ = M_u/R_u (trapezoidal, kN and kNm per pile about the row top)

For a continuous sheet pile the same module returns the Plate set per metre of wall (course manual §8.4; Bentley KB): EA₁ = E·A, EI = E·I, w = mass per m² × g, M_p = f_y·W_pl and N_p = f_y·A as numerical yield caps (not an EN 1993-5 verification), d_eq = √(12EI/EA), ν = 0 (no out-of-plane plate action for a corrugated section), EA₂ only from interlock tests (EA₁/20 is illustrative), “prevent punching” off.

Verification

9. Verified against

The native test suite (src/wasm/retaining/test_native.cpp) and the Node script scripts/verify_retaining_sections_plaxis.mjs reproduce the worked examples of the course chapter and of the MADEP Rekennota “beschoeiing berlinerwand HEA180” (h.o.h. 1.00 m, S235, lagging 10 mm). All checks pass within the stated tolerances.

Brinch Hansen constants, φ = 20.5° (course §7.3, 6 decimals): P_q = 2.776880, K_q^A = 0.412869, K_q⁰ = 2.364011, K_c⁰ = 4.752482, K₀ = 0.649793, d_c^∞ = 1.659923, N_c = 15.314396, K_c^∞ = 25.420725, K_q^∞ = 6.175902, a_q = 0.171761, a_c = 0.377861; K_q(z/B = 10) = 4.7732, K_q(14) = 5.0563.
φ = 25° (Rekennota Table 5-6): K_q⁰ = 3.2869, K_c⁰ = 5.6339, K_q^∞ = 9.8932, K_c^∞ = 36.7454, a_q = 0.14395, a_c = 0.30545. φ = 0: K_c⁰ = 2.5708, K_c^∞ = 8.1237, a_c = 0.6547, K_q ≡ 0.
Rekennota Blum, Model A (HEA180, b = 0.180 m, s = 1.00 m, k = 3, φ′_k = 25°, γ = 19.5 kN/m³, berm 1.577 m at 45°, δ_p = 0, Δa = 0.30 m, γ_φ = γ_c = 1.30): φ_red = 19.733°, K_a = 0.4952, K_p = 2.0195, design excavation 69.300, σ′_v,a at H_d = 55.46 kPa, t₀ = 3.539 m, D_req = 1.2·t₀ = 4.247 m against 4.484 m provided (UC 0.947); with RK2 γ_φ = 1.25: t₀ = 3.431 m. DA1/1 (separate source, passive at 1.00): t₀ = 3.346 m, M_Ed = 57.6 kNm/pile, V_Ed = 85.0 kN/pile, lagging p_Ed = 30.39 kPa.
T_lat tables (c′ = 0.5 kPa, equal-level rows): characteristic z = 1 m: K_q = 6.223, K_c = 25.210, T = 24.11 kN/m; z = 3 m: 86.56 kN/m; γ_φ = 1.30 at z = 1 m: 15.10 kN/m; the row cap reproduces K_pσ′_vs minus the active term; last row at the toe z = 4.484 m.
HEA180 section and EN 1993-1-1: A = 45.25 cm², I_y = 2510 cm⁴, W_el,y = 293.6, W_pl,y = 324.9 cm³, A_v,z = 14.47 cm²; flange c/t = 7.58, web c/t = 20.33 → class 1; M_pl,Rd = 76.35 kNm, M_el,Rd = 69.00 kNm, V_pl,Rd = 196.3 kN, N_pl,Rd = 1063 kN; at M_Ed = 57.64 kNm, V_Ed = 85.03 kN: UC 0.755 (plastic) / 0.835 (elastic) / 0.433 (shear), V/(0.5V_pl,Rd) = 0.866 → no M–V reduction.
PLAXIS Plate: EA = 9.503·10⁵ kN/m, EI = 5271 kNm²/m, d_eq = 0.2580 m, w = 1.140 kN/m/m (0.355 profile + 0.785 lagging), EI_lagging = 17.5 kNm²/m, ν = 0. EBR: γ_eff = 59.0 kN/m³, L/D_eq = 3.876, ISF_RS = ISF_RN = 0.905, ISF_KF = 9.050 (HEA240 at 1.5 m: 0.8360 / 8.361). T_skin: K₀ = 0.5774, T_skin/σ′_v = 0.15431 m, slope 3.009 kN/m per m, T_end = 13.49 kN/m, R_s = 30.25 kN. F_max (q_c = 3 MPa, α_b = 0.5): A_b = 0.03078 m², q_b = 1500 kPa, F_max = 46.2 kN (unplugged 13.6 kN).
Lagging 10 mm S235 (p_Ed = 30.39 kPa, p_k = 22.51 kPa): M_Ed = 3.798 kNm/m, σ = 227.9 N/mm², UC 0.970 elastic (0.646 plastic), δ ≈ 17 mm at L = 1.00 m; clear span 0.82 m: M_Ed = 2.554, UC 0.652, δ ≈ 7.6 mm; 12 mm plate: σ = 158.3, UC 0.674. Vertical: G = 3.73 kN (Rekennota 3.78 with g ≈ 10), R_s = 30.25 kN, UC 0.125.

The course manual's sheet-pile branches (§6.2 SLS, §6.3 BGT + α_ver, §6.4 DA1/2 with D = 3.568 m, T = 122.92 kN/m, M = 258.23 kNm/m) and its §4.3 cantilever illustration are reproduced by the same engine and are listed in the retaining-wall chapter.

Assumptions and limitations

10. Documented assumptions

All soldier-pile quantities are per pile; the per-metre values of the anchor check are obtained by dividing by s.
The effective-width factor k = 3 is the usual EAB / Belgian guideline value, not a code rule; Andersen & Lodahl note that d/B ≈ 3 behaving as a full wall is a single granular case (course §2.1).
Model B applies the row cap to the permanent part of the net resistance; the variable-surcharge part of the Andersen–Lodahl term is carried on the driving side with γ_Q. The positive-part operator, the row cap and the recomputation at φ_d are the course chapter's rules, not Brinch Hansen's.
No water pressure below the excavation on either face and none above it unless the lagging is watertight; pore pressures are hydrostatic, no seepage.
The berm/slope behind the wall is an equivalent surcharge averaged under a 45° spread (Rekennota §7.3) — conservative near the surface, not a rigorous sloping-ground earth-pressure solution.
Cantilever embedment is the Blum simplified method (d₀ × 1.2); reported as “Blum simplified”, not as an EC7 verification of the toe reaction. Wall displacement, passive mobilisation and SSI are not computed.
The β-method T_skin uses a uniform γ below the excavation and K = K₀ (no installation increase); F_max = α_b·q_c·A_b is a preliminary value.
The lagging check is for a steel plate in pure bending (M = pL²/8) with elastic W_el by default; timber lagging, the plate-to-flange bearing and the connection are not verified.
The soil profile is the interpreted CPT stratigraphy, vertically shifted to the wall datum (layers above the reference surface are cut off; a CPT ground level below the surface extends the uppermost layer upward, which is reported) with per-layer overrides of c′, φ′, γ, γ_sat, c_u and the drainage framework.
Guideline-specific values (Table 4 wall-friction limits, over-dig wording, risk-class factors) must be confirmed against the controlled BGGG/Buildwise 2022 text for a stamped design.

References

11. Reference basis

Brinch Hansen, J. (1961). The ultimate resistance of rigid piles against transversal forces. Danish Geotechnical Institute, Bulletin No. 12, pp. 5–9 — surface and great-depth coefficients K_q, K_c and the rational interpolation.
Andersen, K. & Lodahl, M. (2023). Soldier-pile walls in PLAXIS 2D: plate above / embedded beam row below the excavation. Proc. NUMGE 2023, doi 10.53243/NUMGE2023-25 — the retained-height active term and the hybrid idealisation.
BGGG / Buildwise (WTCB/CSTC) (2022). Richtlijnen EC7 beschoeiingen — Belgian embedded-wall guideline: risk classes RK1–RK3, DA1/2 embedment, BGT + α_ver structural route, over-excavation, wall-friction limits, effective width (§5).
EAB — Empfehlungen des Arbeitskreises “Baugruben” (DGGT): effective-width treatment of soldier piles below the excavation.
EN 1997-1:2004 & NBN EN 1997-1 ANB — Design Approach 1, §2.4.2(9)P single-source principle, §9.3.2.2 over-excavation, §9.6(5)P tension-crack water, §9.7.5 vertical equilibrium, Annex C passive coefficients, Table A.12 anchors.
NBN EN 1993-1-1 + ANB — Table 5.2 classification, §6.2.5 bending, §6.2.6 shear, §6.2.8 M–V interaction; γ_M0 = 1.00. NBN EN 1993-5 — sheet-pile section resistance (§5.2.2).
NBN EN 10365 — hot-rolled H/I dimensions; catalogue values from eurocodeapplied.com cross-checked against the ArcelorMittal sales programme V2023-5 (≤ 0.1 % on section properties). EN 10025-2 / EN 10248 — yield strengths.
Blum, H. (1931). Einspannungsverhältnisse bei Bohlwerken — equivalent-beam / free-earth embedment with the 20 % toe allowance.
Bentley Systems. PLAXIS 2D Reference Manual (v24) — Plate and Embedded Beam Row parameters, ISF defaults, d_eq, ν for discrete walls; PLAXIS Knowledge Base “Material datasets for plates: sheet pile wall in bending” (KB0110039).
MADEP. Rekennota beschoeiing berlinerwand HEA180 (v01) — the calculation-note format this route reproduces (§5 PLAXIS set, §5.7 “never mixed”, §7.3 berm, §7.4 Blum, §7.8 lagging, §7.10 vertical, §8.1 water).
Course texts: Brinch Hansen T_lat for soldier-pile walls (Rev. 1) and Sheet-pile retaining walls manual, EC7 / PLAXIS v24 (Rev. 1) — worked constants (φ = 20.5°) and the Belgian branch workflow, verified numerically (176/176 checks).

The coefficient set of §5 and the branch definitions of §2 are transcribed term-for-term from brinch_hansen.hpp and embedded_branches.hpp; the structural and PLAXIS formulas from steel-checks.js, section-properties.js and plaxis-parameters.js. Where a source could not be checked offline (PLAXIS release note on T_lat strength reduction; guideline Table 4) the worklog review says so.