Stage 6 retaining-wall design.

Scope and positioning

1. Analysis class and intended use

The application performs ultimate and serviceability limit-state verification of retaining structures on a per-metre-run (plane-strain) section. Five wall types are covered:

• RC cantilever (L / T-shaped) — stem, toe, heel and backfill; GEO sliding and bearing, eccentricity, EQU overturning, SLS, plus structural design forces.
• Gravity / mass — resists by self-weight; the same GEO/EQU/SLS checks with active thrust on the inclined back face (Coulomb).
• Embedded sheet pile (cantilever) — free-earth / Blum simplified embedment, maximum bending moment.
• Anchored / propped sheet pile — free-earth-support embedment, anchor/prop force, maximum bending moment, and an EN 1537 ground-anchor pull-out verification.
• Soldier-pile (Berliner) wall — H-piles with lagging: Blum embedment with effective widths or Brinch Hansen line resistance, lagging plate and EN 1993-1-1 checks, PLAXIS Plate + Embedded Beam Row parameter set.

Design framework

2. Belgian Eurocode 7 — Design Approach 1 and the embedded-wall guideline

Belgium adopts Design Approach 1 (DA1) for the GEO/STR ultimate limit states of retaining structures (NBN EN 1997-1 ANB:2022). For embedded walls the application follows the workflow of the Belgian embedded-wall guideline (BGGG/Buildwise 2022, Richtlijnen EC7 — beschoeiingen) with its risk-class partial factors; for gravity and RC cantilever walls the generic ANB sets apply. The partial-factor scheme is an explicit, visible input.

Design excavation. The over-excavation Δa (overdiepte) is a ULS geometry assumption applied to the DA1 branches only: Belgian guideline §3.3 — +0.30 m for a dry excavation, min(0.1·h, 0.5 m) under water (default); EN 1997-1 §9.3.2.2 — 10 % of h, ≤ 0.5 m; a custom value; or none where controlled execution is justified. The BGT and SLS branches use the nominal level. h is the retained height (cantilever) or the height below the lowest support (anchored).

Design strengths are formed at source (tan φ′_d = tan φ′_k/γ_φ, c′_d = c′_k/γ_c, c_u,d = c_u,k/γ_cu) and every earth-pressure coefficient is recomputed from φ′_d — never K_p/γ. The single-source principle (EN 1997-1 §2.4.2(9)P) is honoured within each branch; the split between retained-side actions and favourable passive resistance in DA1/1 is a documented, selectable interpretation.

Earth pressure

3. Earth-pressure model

Pressures are computed on effective stress; water is added separately so each action type can be factored independently. A fine downward integration marches from the ground surface, accumulating the effective vertical stress σ′_v and the permanent-earth, variable-surcharge and water resultants. The soil profile is subdivided layer-by-layer: σ′_v is continuous across every boundary while K and the cohesion are read from the layer present at each depth, so the pressure diagram steps correctly at each interface. The same machinery runs on both faces of an embedded wall and on the front (passive) soil and resistance-side overburden of a gravity/cantilever wall, so a layered CPT profile is honoured throughout. The retained fill behind a gravity/cantilever wall is the single engineered backfill you define; the in-situ CPT layers drive the foundation, the passive toe and the overburden.

Method selection is automatic — there is no method menu. Active pressure uses Coulomb on the real inclined back face (mass-gravity wall, with δ and θ from the drawn geometry) or Rankine on the vertical virtual plane through the rear of the heel (RC cantilever / embedded wall). Passive resistance always uses the EN 1997-1:2004 Annex C closed form; the planar Coulomb-passive coefficient and the older lookup table were removed.

Gravity & cantilever

4. GEO / EQU / SLS verification

For an RC cantilever, active thrust is taken on a vertical virtual plane through the rear of the heel (Rankine), with the soil wedge over the heel counted as gravity mass; for a mass gravity wall, Coulomb pressure acts on the real back face. Each check applies its own favourable / unfavourable action roles per EC7 and is run for both DA1 combinations.

• Sliding (GEO): H_d ≤ (V_d·tan δ_b,d + c_a·B′)/γ_R;h + R_p,d; driving uses unfavourable factors, the resisting vertical load uses the favourable factor, and the base uplift is single-sourced with the water thrust. Passive resistance at the toe (R_p,d) is the EN Annex C value at M-factored strength divided by R1 = 1.0 — no lumped mobilisation factor. The unplanned over-dig Δa = min(0.10·H, 0.5 m) automatically removes the top band (scour / future excavation), and the front soil subdivides per CPT layer; the toe passive can be switched off for a fully conservative design (see §8). The small vertical drag R_p,v = R_p,h·tan δ_p is reported.
• Bearing (GEO): V_d ≤ B′·q_Rd, effective width B′ = B − 2e (Meyerhof), single-source γ_G on all permanent actions. Three selectable routes (see §7): EN 1997-1 Annex D (drained N_q/N_c/N_γ + load-inclination factors) — the default — with optional Brinch-Hansen/Vesić depth factors for the buried toe, or the De Beer / NF P94-261 CPT-direct method (q_net = k_c·q_ce). The inclined, eccentric load on a shallow footing is what governs bearing here.
• Eccentricity: e ≤ B/3 (ULS) and the SLS middle-third e ≤ B/6 with the bearing-pressure distribution reported.
• Overturning (EQU): M_dst,d ≤ M_stb,d with the 1.10 / 0.90 split — reported chiefly for rock/pinned bases, as toppling of a soil-founded wall is normally captured by the bearing/eccentricity check.
• Uplift / flotation (UPL): the base hydrostatic uplift U must not exceed the favourable dead weight — γ_G,dst·U ≤ γ_G,stb·G (1.00 on uplift, 0.90 on weight). Reported for every wall; it governs only a deeply submerged, light section.

Embedded walls

5. Sheet-pile and soldier-pile walls

The embedded engine (v2) runs the four Belgian design branches of §2 on one of three idealisations: a continuous wall (sheet pile, cantilever or singly anchored — all quantities per metre of wall), a soldier-pile wall with effective widths (lagging transfers the tributary width s above the excavation; below it active on the flange width b and passive on b_eff = min(k·b, s), EAB / guideline §5), or a soldier-pile wall with the Brinch Hansen (1961) net line resistance B·[e_w(z)]⁺ with the Andersen–Lodahl (2023) retained-height term and the s·p_net row cap. The soldier-pile methods are detailed on the soldier-pile page.

• Cantilever (Blum): moments about the trial toe give the free-earth depth d₀ (ODF = M_resist/M_drive = 1 by bisection); design embedment 1.2·d₀. The 20 % is part of Blum's method (Rekennota §7.4), not an EC7 verification of the toe reaction.
• Anchored / propped: free-earth support — moments about the anchor give d₀; horizontal equilibrium at d₀ gives the support reaction T of each branch (course manual §4.4, eq. 19–20).
• Diagrams: V(z) and M(z) by trapezoidal double integration of the factored net pressure on the provided pile, the support reaction as an exact shear jump; M_max at the governing zero-shear crossing inside the free-earth closure; pressure ordinates (retained side, resistance, water) reported per branch.
• Envelope: d_required from DA1/2 (and DA1/1 with the generic sets); M_Ed, V_Ed, T_Ed = max over DA1/2, DA1/1 and 1.35 × (BGT + α_ver).
• Actions: variable surcharge (γ_Q, α_ver, with a visible practice floor), permanent surcharge (γ_G), a retained berm/slope treated as an equivalent permanent surcharge averaged under a 45° spread (Rekennota §7.3 — an approximation, not Annex C sloping ground), hydrostatic water per face, water-filled tension crack (EN 1997-1 9.6(5)P).
• Soil profile: the CPT layer model with a vertical shift relative to the wall datum (top layer extended upward or cut off) and per-layer overrides of γ, γ_sat, φ′, c′, c_u and the drainage framework — every override is flagged in the results and the note.
• Wall friction: active Rankine (δ = 0); passive EN 1997-1 Annex C with δ_p = ratio × φ′_d per layer (sheet pile default ⅔, soldier pile default 0). Belgian caps shown in the UI: steel sheet piles ⅔φ′_k (straight) / φ′_k − 2.5° and ≤ 30° (curved); Berliner walls φ′_k/3 (straight) / φ′_k/2 (curved).
• Structural checks: sheet piles to NBN EN 1993-5 (elastic W_el by default, W_pl with β_B for class 1–2; shear on the webs with the inclination neglected; optional uniform corrosion loss); H-piles to NBN EN 1993-1-1 (class per Table 5.2, M_pl,Rd/M_el,Rd, V_pl,Rd, M–V interaction §6.2.8); lagging plate M = p·L²/8 with W_el = t²/6; vertical equilibrium of a soldier pile. Sections from verified catalogues (EN 10365 H-profiles, ArcelorMittal 2024 sheet piles).
• Ground-anchor pull-out (EN 1537): R_a,k = π·Ø·L_fixed·τ, R_a,d = R_a,k/γ_a (γ_a = 1.1, EN 1997-1 Table A.12); T_d per anchor = T_Ed·s/cos(angle). The calculated bond must be proven by acceptance testing.
• HYD heave / piping (§10.3): γ_G,dst·γ_wΔh ≤ γ_G,stb·γ′·d with the full head dissipated over the downstream embedment (continuous walls).
• PLAXIS 2D input: Plate (EA, EI, w, M_p, N_p, d_eq, ν = 0, R_inter per layer) for sheet piles; Plate above + Embedded Beam Row below the design excavation for soldier piles (A, I, E of one pile, γ_eff, ISF defaults, linear T_skin, F_max, multilinear T_lat per pile from Brinch Hansen with B = b).
• Drivability and vibration: non-normative estimators on the same CPT trace — see the drivability and vibration pages.

Structural forces

6. Structural design forces

Per the agreed scope the application reports design forces only, as an envelope of the DA1 combinations. For gravity/cantilever walls it returns M_Ed and V_Ed at the stem base, toe (front face of stem) and heel (rear face of stem), from the factored earth pressure, surcharge, soil and bearing-pressure distributions. For embedded walls it returns the maximum bending moment, the anchor/prop force and the EN 1537 pull-out utilisation. Section sizing, reinforcement and crack-width design to EN 1992 / EN 1993-5 — with the Belgian National-Annex values (α_cc = 0.85, γ_c = 1.5, γ_s = 1.15, the +25 % plate-shear NDP, cover/exposure and the 0.3 mm crack limit) — remain the engineer’s responsibility.

Equation reference

7. Implemented equations

Each topic below is given twice: first as clean, typeset formulas in the project house style, then as the literal as-evaluated block (subscripts written inline — K_p, sigma'_v, phi'_d — reading as the calculation the engine actually performs). Both were transcribed term-for-term from the C++ source and cross-checked against it; where a historical inline block disagreed with the code, the code wins and the typeset formula is authoritative.

Design values and partial-factor sets (γ never reduces unit weight).

Design (material set M) strength reductions — γ never reduces unit weight (γ_γ = 1.00).

Partial-factor sets — actions [γ_G,unfav, γ_G,fav, γ_Q,unfav, γ_Q,fav], materials [γ_φ, γ_c, γ_cu, γ_γ], resistance [γ_R;v, γ_R;h, γ_R;e].

Verification combinations — both DA1 sets run and the worst governs; the EQU/HYD/UPL action triple is [γ_G,dst, γ_G,stb, γ_Q,dst].

Optional Buildwise risk classes (project-confirmed only; default is the EN/ANB sets above). Each overrides A1, A2 as [γ_G,unfav / γ_Q,unfav] and M2 as [γ_φ, γ_c, γ_cu].

Design (material set M):   phi'_d = atan( tan phi'_k / g_phi )    c'_d = c'_k / g_c    cu_d = cu_k / g_cu

Action sets    [g_G,unfav · g_G,fav · g_Q,unfav · g_Q,fav]
   A1 = [1.35 · 1.00 · 1.50 · 0]         A2 = [1.00 · 1.00 · 1.30 · 0]
Material sets  [g_phi · g_c · g_cu · g_gamma]
   M1 = [1.00 · 1.00 · 1.00 · 1.00]      M2 = [1.25 · 1.25 · 1.40 · 1.00]
Resistance     R1 = [g_R;v · g_R;h · g_R;e] = [1.00 · 1.00 · 1.00]
K_FI = 0.90 (CC1) | 1.00 (CC2) | 1.10 (CC3)   — multiplies the UNFAVOURABLE actions

DA1-C1 = A1 + M1 + R1          DA1-C2 = A2 + M2 + R1     (both run; worst governs)
EQU = [1.10 · 0.90 · 1.50] + M2 + R1   (stabilising/destabilising split; K_FI = 1)
HYD = [1.35 · 0.90 · 1.50] + M1 + R1   UPL = [1.00 · 0.90 · 1.50] + M2 + R1

Optional Buildwise risk classes (project-confirmed only; default = EN/ANB above):
   RK1: A1[1.20/1.30]  A2[1.00/1.10]  M2[1.10, 1.10, 1.25]
   RK2: 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]

Earth-pressure coefficients — active by geometry, passive always EN Annex C.

At-rest (Jáky), with the over-consolidation form capped at the passive limit.

Rankine — active by geometry, level backfill.

Rankine — sloping backfill slope β (engine caps |β| at φ′ − 0.1°).

Coulomb active — wall friction δ, wall batter θ from vertical, backfill slope β.

EN 1997-1:2004 Annex C passive (Brinch-Hansen log-spiral); passive wall friction δ recomputed per layer.

At rest:   K0 = 1 − sin phi'        K0,OC = min( (1−sin phi')·OCR^(sin phi') ,  (1+sin phi')/(1−sin phi') )

Rankine, level:   Ka = (1−sin phi')/(1+sin phi')        Kp = (1+sin phi')/(1−sin phi')
Rankine, slope b  (|b| capped at phi'−0.1°):   r = sqrt( cos²b − cos²phi' )
   Ka = cos b·(cos b − r)/(cos b + r)          Kp = cos b·(cos b + r)/(cos b − r)

Coulomb active (wall friction d, back batter t, slope b):
   Ka = cos²(phi'−t) / [ cos²t · cos(d+t) · [ 1 + sqrt( sin(d+phi')·sin(phi'−b) / (cos(d+t)·cos(b−t)) ) ]² ]

EN 1997-1:2004 Annex C passive (log spiral); d_p = (2/3)·phi'_d recomputed per layer:
   sb = sin b / sin phi'      sd = sin d / sin phi'           (each clamped to [−1, 1])
   m_t = ½( arccos(−sb) − phi' − b )                          (C.3)
   m_w = ½( arccos( sd) − phi' − d )                          (C.4)
   v   = m_t + b − m_w − t                                    (C.5)
   K_n = [1 + sin phi'·sin(2 m_w + phi')] / [1 − sin phi'·sin(2 m_t + phi')] · exp(2 v · tan phi')   (C.6)
   K_p  = K_n·cos b·cos(b−t)     (C.9)      K_q = K_n·cos²b     (C.7)
   K_pc = (K_n − 1)·cot phi'     (C.8)   →   equals 2·sqrt(K_p) when d = b = t = 0

Effective-stress pressure integration (per cell, midpoint march, layered).

Downward cell march (surface → base): resolution, effective unit weight, and the running effective vertical stress.

Lateral pressure ordinate at the cell midpoint — drained on effective σ′_v, undrained on total σ_v.

Tension crack and ponded crack-water (active face only).

Resultant on a face and projection onto the analysed (virtual or battered) plane.

N = max(n_steps, 50) cells,  dz = (surface − base)/N,  g_w = 9.81 kN/m³
   g_eff = g_sat − g_w  below the water table, else g_moist
   sigma'_v,mid = sigma'_v + ½ g_eff dz        sigma'_v += g_eff dz     (continuous across layers)

Drained ordinates (design strength):
   active:   sigma'_a = Ka·sigma'_v,mid − 2·sqrt(Ka)·c'      (if sigma'_a < 0 ⇒ 0; first non-tension cell sets z0)
   passive:  sigma'_p = Kp·sigma'_v,mid + K_pc·c'            (cohesion ADDS; K_pc from C.8)
Undrained (phi_u = 0):   sigma = (sigma'_v,mid + u) ∓ 2·cu          u = g_w·(WT − el)
Surcharge:  d_sigma_h = K·q          Water:  u = g_w·(WT − el)
Tension crack (level):  z0 = 2c'/(g·sqrt(Ka))
   crack water (drained, dry-capped):  P = ½ g_w · z_c²  at  (2/3) z_c,   z_c = min(z0, depth to WT)
Resultant on a face:    N = Σ sigma·dz        lever  zbar = (Σ sigma·dz·depth) / N

Active thrust on the analysed plane:
   virtual-plane height  H_v = top-of-wall + heel·tan b
   back batter from drawn taper  t = atan( (t_b − t_t)/H_stem );   d = k_a·phi'_d  (0 on the Rankine plane)
   H_soil = N·cos(incl)   V_soil = N·sin(incl)   y = H_v − zbar     (water/crack horizontal, no cosine)

Gravity / cantilever — free body, bearing, and the GEO/EQU/SLS/UPL checks.

Geometry, resultant position and effective base width.

Permanent vertical free-body weights and lever arms about the toe tip (characteristic; γ is never factored).

Base hydrostatic uplift on the underside (permanent water action, single-source with the lateral water thrust; γ_w = 9.81 kN/m³).

Annex D bearing resistance (strip, shape factors = 1; φ′_d floored at 1°; design strengths). The depth factors d_c, d_q, d_γ are the optional Brinch-Hansen / Vesić refinement detailed by the bearing family (d_γ = 1, and d = 1 when disabled).

Design partial factors and design strengths used in the checks.

Sliding (GEO) — favourable weights, uplift single-sourced with the water thrust at γ_G.

Bearing (GEO) — single-source γ_G on all permanent actions including uplift.

Eccentricity (ULS) — evaluated on the bearing-case loads.

EQU static equilibrium — overturning about the toe (γ_G,dst = 1.10, γ_G,stb = 0.90, γ_Q,dst = 1.50; no K_FI).

SLS bearing-pressure distribution (characteristic strength, set M1; unfactored actions).

UPL flotation (γ_G,dst = 1.00 on uplift, γ_G,stb = 0.90 on favourable weight; side/base friction neglected).

B = toe + t_b + heel        top-of-wall = baseThk + H_stem        B' = max(B − 2e, 0.05 B)
W_base = g_c·B·baseThk @ B/2     W_stem = g_c·½(t_t+t_b)·H_stem @ toe+(t_b²+t_b t_t+t_t²)/(3(t_b+t_t))
W_heelsoil = heel·(g_moist·h_above + g_sat·h_below) @ heel mid   (split at retained WT)
W_wedge = ½ g_moist·heel²·tan b @ toe+t_b+(2/3)heel   (sloping fill)
Base uplift:  u_F = g_w·max(WT_front,0),  u_B = g_w·max(WT_ret,0)
   U = ½(u_F + u_B)·B    at   x = B·(u_F + 2u_B)/(3(u_F + u_B))

Bearing factors (Annex D, strip; phi' floored at 1°):
   N_q = e^(pi·tan phi')·tan²(45° + phi'/2)    N_c = (N_q − 1)/tan phi'    N_g = 2(N_q − 1)·tan phi'
   base = 1 − H/(V + B'·c'/tan phi');   i_q = base²,  i_g = base³,  i_c = i_q − (1−i_q)/(N_c·tan phi')
   q_Rd = c'·N_c·i_c·d_c + q'·N_q·i_q·d_q + ½ g'·B'·N_g·i_g·d_g    (d_* = Brinch-Hansen/Vesic depth factors, default-on, d_g=1, =1 when off)
        (undrained: q_Rd = (pi+2)·cu·i_c·d_c + q',  i_c = ½(1+sqrt(1−H/(B'·cu))),  d_c = 1 + 0.4·k)

Checks   (g_G = g_G,unfav·K_FI,  g_Q = g_Q,unfav·K_FI,  g_Gf = g_G,fav):
 Sliding (GEO):  H_d = g_G·H_soil + g_Q·H_surch + g_G·(H_w + H_crack)
   V_res = g_Gf·(ΣW + V_soil) − g_G·U;   d_b = (d_b/phi')·phi'_d
   R_d = (V_res·tan d_b + c_a·B')/g_R;h + R_p     [undrained: R_d = min(cu_d·B'/g_R;h, 0.4·V_res) + R_p]   pass: H_d ≤ R_d
 Bearing (GEO, single-source g_G incl. U):  V_d = g_G·(ΣW + V_soil − U);  q_Ed = V_d/B';  R_d = B'·q_Rd/g_R;v;  pass: V_d ≤ R_d
 Eccentricity:  e ≤ B/3
 EQU:  M_dst = g_G,dst·(H_soil·y + H_w·y + H_crack·y + U·x) + g_Q,dst·H_surch·y ≤ M_stb = g_G,stb·(ΣW·x + V_soil·B)   (1.10/0.90)
 SLS (characteristic, M1):  e ≤ B/6;   q_max,min = (V/B)(1 ± 6e/B)   [if e ≤ B/6, else q_max = 2V/(3(B/2−e)), q_min = 0]
 UPL flotation:  g_G,dst·U ≤ g_G,stb·(ΣW + V_soil)   (1.00/0.90)

Bearing resistance — three selectable routes. The default is EN 1997-1 Annex D with the depth-factor refinement; the De Beer / NF P94-261 CPT-direct method is selectable when a CPT profile is present.

Route 1 — EN 1997-1 Annex D, analytical c′–φ′ (strip; shape and base factors = 1; inside DA1, R1 = 1.0). φ′_d floored at 1°.

Annex D — undrained branch (founding stratum total-stress, φ = 0).

Route 2 — Brinch-Hansen / Vesic depth factors (opt-in refinement, default on; credits the soil shear above the buried toe; d_γ = 1 always). D = founding depth below the design front level.

Route 3 — De Beer / NF P94-261 CPT-direct (shallow strip toe; selected when a CPT q_c profile is present, else reverts to Annex D). q_cm = mean q_c over the influence zone; q_cc is the peak-clipped cone resistance.

NF P94-261 Tableau E.2.3 — penetrometric bearing factor k_c (strip, B/L = 0; founding stratum auto-classified).

(1) EN 1997-1 ANNEX D — analytical c-φ (informative sample method; inside DA1, R1 = 1.0)
   N_q = e^(pi·tan phi'_d)·tan²(45° + phi'_d/2)     N_c = (N_q − 1)·cot phi'_d     N_gamma = 2(N_q − 1)·tan phi'_d
   strip shape factors s = 1, horizontal base b = 1.   Load inclination (m = 2, strip):
      base = 1 − H_d/(V_d + B'·c'_d·cot phi'_d);   i_q = base²,   i_gamma = base³,   i_c = i_q − (1−i_q)/(N_c·tan phi'_d)
   q_Rd = c'_d·N_c·i_c·d_c + q'·N_q·i_q·d_q + ½·gamma'·B'·N_gamma·i_gamma·d_gamma
   B' = B − 2e (Meyerhof effective width); q' and the N_gamma soil use the LAYERED front profile; R_d = B'·q_Rd / gamma_R;v.
   Annex D itself omits depth factors (d = 1) — so the inclined, eccentric load on a shallow footing governs.

(2) DEPTH FACTORS (Brinch-Hansen 1970 / Vesic 1973; opt-in refinement, default ON)
   D = founding depth below the design front level;   k = D/B'  if D/B' ≤ 1,   else   arctan(D/B')  [radians]
   d_q = 1 + 2·tan phi'_d·(1 − sin phi'_d)²·k       d_c = d_q − (1 − d_q)/(N_c·tan phi'_d)       d_gamma = 1  (always)
   undrained (phi = 0):   d_c = 1 + 0.4·k   on the (pi + 2)·c_u term.
   Credits the soil shear above the buried toe; switching it off reverts to the conservative pure-Annex-D value.

(3) DE BEER / NF P94-261 — CPT-direct (Fascicule 62-V; De Beer shallow-footing lineage; a DIRECT method)
   q_ce = mean q_c over [founding, founding − 1.5 B'], with the De Beer peak clip:  q_c > 1.3·q_cm  ⇒  clip to 1.3·q_cm
   D_e  = (1/q_ce)·∫₀^D min(q_c, 1.3·q_cm) dz     (equivalent embedment; SAME De Beer clip as q_ce; weak surface cover ⇒ D_e < D)
   k_c  = min[ k_c0 + (a + b·D_e/B')·(1 − exp(−c·D_e/B')) ,  k_c,max ]      (unconditional cap; strip footing, B/L = 0)
   i_delta = [ clamp(1 − H_d/V_d, 0, 1) ]²        (load-inclination surrogate, clamped then squared)
   q_net = k_c·q_ce·i_delta;      R_d = B'·( q_net/(gamma_R;v·gamma_R;d) + q0 ),      gamma_R;v·gamma_R;d = 1.4 · 1.2
   NF P94-261 Tableau E.2.3 (strip):    sand & gravel   k_c0 = 0.09   (a 0.04, b 0.006, c 2.0, k_c,max 0.141)
                                         clay & silt     k_c0 = 0.27   (a 0.07, b 0.007, c 1.3, k_c,max 0.348)
   A shallow toe mobilises only k_c ≪ 1 of q_c. DISTINCT from the deep-pile BGGG-GBMS De Beer method (q_b ≈ q_c) in the
   pile module — surface-shear (shallow toe) vs deep-punching (pile base). Reverts to Annex D if no q_c is supplied.

Parameters. φ′_d, c′_d — design (M-factored) friction and cohesion of the founding stratum; q′ — effective overburden at founding from the layered front soil above the design dig level; γ′ — buoyant unit weight of the N_γ failure wedge (submerged when either water table reaches founding); B′ = B − 2e — the Meyerhof effective width; H_d, V_d — the factored horizontal and vertical actions; q_c — the CPT cone resistance; q_ce — the De Beer equivalent cone resistance near founding; D_e — the equivalent embedment; k_c — the NF penetrometric bearing factor (largest in clay, smallest in sand, because q_ce is small in clay and large in sand).

The two frameworks differ and are not combined. Annex D is the analytical method inside Design Approach 1 — soil strength is M-factored at source and the resistance factor is R1 = 1.0. The De Beer / NF P94-261 route is a direct method: the resistance is the measured q_c, divided by the NF resistance + model factors (γ_R;v·γ_R;d = 1.4·1.2). Both descend from Egide De Beer's work, but the shallow toe (q_net = k_c·q_ce, k_c ≪ 1, a general/surface shear mechanism) and the deep pile base (q_b ≈ q_c via the BGGG-GBMS scale transformation used in the pile module, a deep-punching mechanism) are physically different problems and therefore correctly different formulas — they should not be conflated. For dense sand the De Beer route typically returns a higher capacity than the φ-capped Annex D method; for soft or low-q_c ground the Annex D c-φ method is the more reliable choice.

Modelling notes (De Beer route — verified against NF P94-261). A handful of implementation choices are worth stating explicitly so the route is transparent and auditable:

• Effective width. The equivalent-resistance window depth (1.5·B′) and the De Beer embedment ratio D_e/B′ both use the Meyerhof effective width B′ = B − 2e, not the gross width. Using B′ keeps the q_ce-averaging depth and the k_c penetration ratio consistent with the eccentric contact area, and the seating resistance is likewise R₀ = q₀·B′ (per metre run). This is conservative for an eccentric base.
• Per-layer q_c. q_ce is the De Beer-clipped mean of the measured q_c over the window — each stratum contributes its own q_c (parsed per layer), so a layered profile is integrated cell-by-cell rather than with a single representative value. The clip (q_c > 1.3·q_cm → 1.3·q_cm, NF C.2.2) is applied inside both the q_ce mean and the D_e = (1/q_ce)∫q_c dz integral, so a thin stiff lens cannot inflate either quantity.
• Load inclination. NF P94-261 reduces the CPT bearing term with an inclination factor i_δ. We use the conservative surrogate i_δ = (1 − H_d/V_d)², clamped to [0, 1], which collapses to 1 for a vertical resultant and degrades faster than the φ-dependent Annex D form — a deliberately safe choice when no founding-level φ is invoked by the direct method.
• Sand vs clay k_c. The k_c coefficient set is selected by soil class at founding level: φ′ ≥ 25° (or a granular classification) takes the sand row, otherwise the clay row. Because k_c is larger in clay but q_ce is far larger in sand, the product q_net = k_c·q_ce still ranks sand above clay.
• CPT depth caveat. The direct route is only as deep as the CPT sounding. If the founding level or the 1.5·B′ window extends below the deepest q_c reading the mean is taken over the available data only — extend the sounding past the influence zone for a fully governed result.

Embedded walls — net pressure, embedment, heave, anchor force.

Unplanned over-excavation (EN 1997-1 §9.3.2.2): the engine lowers the front level by Δa before any pressures are built. Cantilever walls use 10% of the retained height; anchored walls 10% of the height below the lowest support (both capped at 0.5 m). With a dry front (no standing water in the pit) the Belgian dry-excavation floor of 0.30 m applies on top of the percentage rule.

Net horizontal pressure per cell — back active over the full retained height, front passive only below the (over-dug) excavation level; permanent earth and water by γ_G, variable surcharge by γ_Q.

Limiting-equilibrium embedment — moments about the rotation point (toe for a cantilever, anchor/prop for a propped wall), with the over-design factor driven to unity by bisection.

Design embedment — Blum 20% addition for a cantilever, none when propped; governed by the worst of the two DA1 combinations. The moment/shear diagrams and M_max are computed on the provided pile (a diagram cannot be longer than the structure it acts on); embedment adequacy is reported separately as the d_provided ≥ d_required check, with the moment-equilibrium ODF at the provided depth alongside.

Hydraulic heave / piping (HYD, EN 1997-1 §10.3) at the toe — destabilising excess pore pressure versus stabilising buoyant effective stress, the full differential head dissipated over the embedment.

Anchor / prop force — free-earth horizontal equilibrium of the factored thrusts at the equilibrium (FES) depth, per combination.

Engine over-dig:  excavationEl ← excavationEl − max(da, 0),   da = min(0.10·H_ret, 0.5 m)
Net pressure per cell (back active / front passive; passive only below excavation):
   drive  = g_G·(p_a + u_back) + g_Q·p_surch        p_a = Ka·sigma'_v − 2·sqrt(Ka)·c'
   resist = g_G·(p_resist + u_front)                p_resist = Kp·sigma'_v + K_pc·c'   (Annex C)
Moment about the rotation point (toe = cantilever, anchor = propped):  M = Σ (drive − resist)·dz·(el − ref)
Over-design factor:  ODF = |M_resist| / |M_drive| ≥ 1   →   required d by bisection on [0.10, 40] m
   cantilever:  d_design = 1.2·d_req (Blum)        propped:  d_design = d_req
   governing d = max over [C1, C2];   analysis depth = max(d_required, d_provided)
HYD heave/piping:  E_d = 1.35·g_w·dh ≤ R_d = 0.90·g'·d        i_exit = dh/d,   i_crit = g'/g_w
Anchor/prop force (free-earth horizontal equilibrium at the FES depth):  T = H_drive − H_resist

Structural design forces and the anchor / wall-vertical checks.

Gross contact pressure for slab design — base uplift is NOT subtracted (handled by superposition), so V_d carries γ_G on all permanent vertical load. e is signed, positive toward the toe.

For |e| ≤ B/6 the bearing diagram is linear across the full base (q_toe at x = 0, q_heel at x = B); beyond B/6 it is triangular over the contact length a, with the peak at the loaded edge.

Stem design forces — active backfill thrust over the stem height H_s, moment taken about the stem base. N is the soil active-thrust resultant; z_i is each resultant's depth below the top of the stem.

Toe slab — net upward bearing pressure relieved by the favourable toe self-weight, moment about the stem front. x runs from the toe tip (x = 0) to the stem front (x = L_toe); γ_conc = concrete unit weight, t_base = base-slab thickness.

Heel slab — downward soil + slab + surcharge load minus the upward bearing pressure, moment about the stem back x_b. The retained soil column is split at the water table z_w into a moist part above and a saturated part below; y_top = top-of-wall elevation.

Embedded wall — factored bending moment by double integration of the net pressure down the wall; the structural maximum is the governing (largest-|M|) zero-shear crossing per Blum — each V = 0 crossing is a local moment extremum, and for anchored walls the first crossing is not necessarily the largest. The cohesion c′ enters through p_a and p_resist.

Ground-anchor pull-out (EN 1537) — calculated grout-body bond resistance, checked per anchor at the design axial force. T = anchor force per metre run, α = inclination from horizontal, s = horizontal spacing, Ø = grout-body diameter, L_fixed = fixed (bond) length, τ = characteristic bond stress.

Wall vertical equilibrium (screening) — the inclined-anchor down-drag must be carried by the embedded shaft friction over both faces (Jaky K₀ = 1 − sin φ′, δ = ⅔ φ′; base resistance and wall self-weight neglected). σ′_v is the front-soil effective vertical stress from the excavation level down.

Contact pressure for slab design (gross — base uplift NOT subtracted, by superposition):
   V_d = g_G·(ΣW + V_soil),   e signed (+ toward toe)
   |e| ≤ B/6:  q(x) = q_mean·(1 ± 6e/B) (linear)    |e| beyond B/6:  triangular over a = 3(B/2−|e|), peak q_pk = 2V_d/a
Stem (about base):  M = g_G·N·(H_s − zbar) + g_Q·M_surch + g_G·M_water     V = g_G·N + g_Q·N_surch + g_G·(N_w + N_crack)
Toe  (about stem front):  M = integral over [0,toe] of [ q(x) − g_Gf·g_c·baseThk ]·(toe − x) dx     (slab self-weight = favourable relief)
Heel (about stem back):   M = integral over the heel of [ (g_G·g_soil·h(x) + g_G·g_c·baseThk + g_Q·q) − q(x) ]·(x − x_b) dx
   h(x) = top-of-wall + (x − x_b)·tan b − baseThk     (carries the sloping wedge), split at the WT
Embedded M_max:  double integration  V ← V + net·dz,  M ← M + V·dz,  anchor adds  V ← V − T;
   M_max at the first zero-shear crossing (Blum), else max|M|

Ground-anchor pull-out (EN 1537):  R_ak = pi·D·L_fixed·tau;   R_ad = R_ak/g_a   (g_a = 1.1, EN Table A.12)
   axial per anchor  T_axial = T/cos(alpha)·s;   down-drag  V = T·tan(alpha);   pass: T_axial ≤ R_ad
Wall vertical equilibrium (screening; base resistance neglected):
   R_v = Σ 2·(1 − sin phi')·sigma'_v·tan((2/3) phi')·dz   over the embedded length, both faces;   pass: V ≤ R_v

Inputs and workflow

8. Inputs and interactive section

Foundation / in-situ parameters default from the active CPT layer model (γ, γ_sat, φ′, c′, c_u) and are editable; the backfill behind a gravity/cantilever wall is a separate user-defined material; the front soil supplies the passive resistance. The section is drawn live — the CPT soil layers, the wall, water, surcharge, the active/passive pressure and bending-moment diagrams, and the ground anchor are all to scale, and the geometry can be edited by dragging the handles directly on the drawing. The earth-pressure method is fixed by the geometry (no method menu). Editable settings are the active, base and passive wall- friction ratios (δ/φ′), the consequence class, the surcharge, and the full anchor configuration. The unplanned over-dig Δa is applied automatically (EN 1997-1 §9.3.2.2) and drawn on the section. Every verification reports the governing DA1 combination, the demand, the resistance and the utilisation.

Assumptions and limitations

9. Documented assumptions

• Built on EN 1997-1:2004 + NBN EN 1997-1 ANB (DA1). Factor sets are configurable for the 2nd-generation EN 1997-1:2024 verification cases.
• Embedded walls default to the Belgian embedded-wall guideline RK2 sets (DA1/2 γ_Q = 1.10); RK1, RK3 and the generic NBN EN 1997-1 ANB sets (γ_Q = 1.30 with K_FI) are selectable. Gravity walls use the generic sets.
• R1 = 1.0 applies to the wall body. Ground anchors are verified here for pull-out only (R_a,k = π·Ø·L_fixed·τ, R_a,d = R_a,k/γ_a): γ_a = 1.1 is the EN 1997-1 Table A.12 anchor factor (temporary and permanent) and is EN-conforming — the ~1.5–1.6 range is the pile R4 set and does not apply to anchors. The real caveat is that the calculated π·Ø·L·τ bond is untested and must be proven by EN 1537 acceptance testing (apply a model factor / reduce τ until then). The tendon/steel design and the free-length adequacy remain the engineer's responsibility. The inclined anchor's downward component V = ΣT·tan(angle) is carried into a screening wall vertical-equilibrium check (embedded shaft friction only, base neglected) that must be confirmed against the real section. Axially-loaded piles are out of scope.
• Passive resistance at the toe is computed from the EN 1997-1 Annex C closed form at M-factored strength / R1 = 1.0 — no lumped mobilisation factor. The unplanned over-dig Δa = min(0.10·H, 0.5 m) removes the top band (scour/future excavation), the front soil and overburden subdivide per CPT layer, and the value is the rigorous log-spiral coefficient for all δ (planar Coulomb passive was removed). Full passive needs large movement — verify SLS displacement separately, or disable the toe passive for a fully conservative design.
• Brinch-Hansen/Vesić depth factors are applied to the Annex D bearing check by default (Annex D itself omits them; untick “Bearing depth factors” for the strict, more conservative Annex-D form). The cantilever uses the virtual-plane Rankine idealisation; for a short heel (heel below H_v/tan(45°+φ′_d/2)) the active wedge is interrupted by the stem, so the engine flags it at runtime and recommends a Coulomb-on-stem cross-check.
• Deliberate conservative simplifications (all err safe): the base uplift uses a linear head toe→heel (no flow net); the resisting front-water thrust and the shear-key passive block are neglected; the passive vertical drag R_p,v = R_p,h·tan δ_p is reported but not credited to vertical equilibrium; and the unplanned over-dig removes the top band of front soil so the passive starts at zero effective stress at the design dig level.
• Embedded-wall embedment uses the free-earth / Blum limit-equilibrium method; SLS wall displacement and passive mobilisation are not computed (a nonlinear beam-on-springs model is a planned extension; PLAXIS 2D with the exported parameter set is the reference for movements).
• No Belgian-codified minimum backfill surcharge exists. The minimum variable surcharge of embedded walls is an explicit, visible input (default 10 kPa, a practice value) — nothing is floored silently.
• The in-situ / foundation profile is the full layered CPT stratigraphy — σ′_v continuous, K and cohesion per layer — shifted to the wall datum where the sounding was pushed from another level and with per-layer overrides (c′ in particular) that are flagged everywhere. The retained fill behind a gravity/cantilever wall is a single engineered backfill material.
• Hydraulic limit states are verified: HYD heave/piping for embedded walls (conservative exit gradient — confirm with a flow net for stratified ground) and UPL flotation for gravity walls. SLS is limited to the bearing-resultant middle-third / eccentricity; settlement, tilt and embedded-wall deflection (SSI/FE) are not yet automated and remain the engineer's responsibility.
• The partial factors and equations were verified against the EN/ANB recommended values and multiple independent sources; for a stamped design every National-Annex value must still be confirmed against the controlled standards (NBN EN 1997-1 ANB, NBN EN 1992-1-1 ANB:2010, Buildwise).

References

10. Reference basis

• EN 1997-1:2004 (Eurocode 7) & NBN EN 1997-1 ANB — geotechnical design; Design Approach 1, partial-factor sets A1/A2, M1/M2, R1, and the Belgian National Annex values.
• EN 1990 & NBN EN 1990 ANB — basis of structural design; action combinations 6.10 / 6.10a-b, ψ factors, consequence-class K_FI.
• EN 1992-1-1 & NBN EN 1992-1-1 ANB:2010 — concrete design parameters referenced for structural-force interpretation.
• EN 1993-5 — design of steel sheet piles (section resistance, referenced for the structural force context).
• Bond, A. & Harris, A. Decoding Eurocode 7. Taylor & Francis, 2008 — worked-example basis for the DA1 retaining-wall verification recipes.
• EN 1997-1:2004 Annex C (Brinch-Hansen log-spiral) — the closed-form active/passive coefficients (eqs C.3–C.9) the engine evaluates for passive resistance. Kérisel, J. & Absi, E. Tables for the Calculation of Passive Pressure… Gauthier-Villars — retained only as an independent cross-check of the closed form.
• Coulomb (1776), Rankine (1857) — classical earth-pressure theory; Blum (1931) — embedded-wall equivalent-beam method.
• CIRIA C760 — guidance on embedded retaining walls (over-dig, passive mobilisation, hydraulic checks) used for context.
• Normalisatiecommissie NBN E25007 / Buildwise (WTCB–SECO), March 2022 — Richtlijnen voor de toepassing van de Eurocode 7 in België volgens NBN EN 1997-1 ANB: het grondmechanische ontwerp van ingebedde kerende constructies (beschoeiingen) — risk classes, partial factors, over-excavation, BGT + α_ver route, φ-c reduction ≥ 1.25, wall-friction limits, effective width of discontinuous walls.
• Blum, H. (1931) — Einspannungsverhältnisse bei Bohlwerken; Brinch Hansen, J. (1961) — The ultimate resistance of rigid piles against transversal forces, DGI Bulletin 12; Andersen, F. & Lodahl, M.R. (2023) — Modelling of soldier pile walls in Plaxis 2D, NUMGE 2023, doi 10.53243/NUMGE2023-25.
• NBN EN 1993-1-1:2005 + ANB:2018; NBN EN 1993-5:2007 + ANB:2011; NBN EN 10365:2017; ArcelorMittal Sheet Piling General Catalogue 2024 — steel section checks and catalogues.
• Bentley Systems (2024) — PLAXIS 2D Reference Manual V24 (Plate, Embedded Beam Row, interface stiffness factors); PLAXIS Knowledge Base articles “Material datasets for plates: sheet pile wall in bending” (KB0110039) and “End bearing of plates” (KB0110231).
• MADEP course material (2026) — Steel sheet-pile retaining walls: manual calculation, EC7 design and PLAXIS 2D v24; Brinch Hansen lateral resistance for soldier-pile walls — worked examples used as verification fixtures.

The partial-factor tables (§2), earth-pressure equations (§3) and verification recipes (§4–§6) on this page are the complete methodology the application implements; the assumptions and limitations are listed in full in §8. For a stamped design, confirm each National-Annex value against the controlled NBN EN 1997-1 ANB, NBN EN 1992-1-1 ANB and Buildwise texts.