| phi_c | \(\displaystyle 0.65\) | | A23.3-24 Cl. 8.4.2, phi_c = 0.65 | 2026-10-07 |
| phi_s | \(\displaystyle 0.85\) | | A23.3-24 Cl. 8.4.3 a), phi_s = 0.85 | 2026-10-07 |
| gamma_c | \(\displaystyle 23.5\ \mathrm{kN/m³}\) | | 2400 kg/m3 times g = 23.54 kN/m3, taken as 23.5; not in ref/, design assumption, owner decision 2026-10-09 | 2026-10-09 |
| phi_gu_sl | \(\displaystyle 0.8\) | | CFEM 5th ed. Table 6.2, Retaining systems, base sliding, typical understanding 0.80 | 2026-10-07 |
| phi_gu_p | \(\displaystyle 0.5\) | | CFEM 5th ed. Table 6.2, Shallow foundations, passive resistance, typical understanding 0.50 | 2026-10-07 |
| alpha_H | \(\displaystyle 1.5\) | | NBC 2020 Div B Art. 4.1.3.2.(4), load factor 1.5 for lateral earth pressure H, checked in ref/ | 2026-10-08 |
| alpha_L | \(\displaystyle 1.5\) | | NBC 2020 Div B Table 4.1.3.2.-A case 2, principal live load factor 1.5L, checked in ref/; treating the surcharge as L is the calc's classification | 2026-10-08 |
| alpha_D | \(\displaystyle 1.25\) | | NBC 2020 Div B Table 4.1.3.2.-A cases 2 to 4, 1.25D with note (3) to Art. 4.1.3.2.(8), soil handled by alpha_Ds, checked in ref/ e6a079e | 2026-10-09 |
| alpha_Dc | \(\displaystyle 0.9\) | | NBC 2020 Div B Art. 4.1.3.2.(5), counteracting dead load 0.9D in cases 2 to 4 when D resists sliding, checked in ref/ | 2026-10-08 |
| alpha_Ds | \(\displaystyle \min\left(1.5, \max\left(1.25, 1 + \frac{600\ \mathrm{mm}}{h_{s}}\right)\right)\) | NBC 2020 Div B Art. 4.1.3.2.(8), soil dead load 1.5, or 1 + 0.6/h_s but not below 1.25; assume: h_s is the soil depth | NBC 2020 Div B Art. 4.1.3.2.(8), 1.5 on soil D, or 1 + 0.6/h_s above 1.2 m but not below 1.25, checked in ref/; h_s taken as the retained height | 2026-10-08 |
| alpha_E | \(\displaystyle 1\) | | NBC 2020 Div B Table 4.1.3.2.-A case 5, 1.0D + 1.0E (0.5L companion), and Art. 4.1.3.2.(5) 1.0D counteracting, checked in ref/; the factor on static earth pressure in case 5 is not settled by the text | 2026-10-08 |
| H | \(\displaystyle h_{s} + t_{b}\) | | Geometry; CFEM 5th ed. Fig. 20.1 H, wall height, at the base underside, checked in ref/ e6a079e | 2026-10-09 |
| B | \(\displaystyle L_{\mathrm{toe}} + t_{\mathrm{bot}} + L_{\mathrm{heel}}\) | | Geometry; CFEM 5th ed. Fig. 20.10 B, width of wall foundation, checked in ref/ e6a079e | 2026-10-09 |
| phi_r | \(\displaystyle \operatorname{radians}\left(\phi\right)\) | | Unit conversion of CFEM Fig. 20.1 phi' | 2026-10-09 |
| beta_r | \(\displaystyle \operatorname{radians}\left(\beta\right)\) | | Unit conversion of CFEM Fig. 20.1 beta | 2026-10-09 |
| delta_r | \(\displaystyle \operatorname{radians}\left(\delta\right)\) | | Unit conversion of CFEM Cl. 20.2.6 wall friction delta | 2026-10-09 |
| delta_br | \(\displaystyle \operatorname{radians}\left(\delta_{b}\right)\) | | Unit conversion; base friction delta of CFEM 5th ed. Fig. 20.10, F = (W + P_v) tan delta, checked in ref/ e6a079e | 2026-10-09 |
| H_v | \(\displaystyle H + L_{\mathrm{heel}} \cdot \tan\left(\beta_{r}\right)\) | | Geometry of the vertical virtual back, CFEM 5th ed. Fig. 20.1 with alpha = 90 deg; the virtual back is the calc's modelling, checked in ref/ e6a079e | 2026-10-09 |
| K_a | \(\displaystyle \frac{\cos\left(\phi_{r}\right)^{2}}{\left(1 + \sqrt{\frac{\sin\left(\delta_{r} + \phi_{r}\right) \cdot \sin\left(\phi_{r} - \beta_{r}\right)}{\cos\left(\delta_{r}\right) \cdot \cos\left(\beta_{r}\right)}}\right)^{2}}\) | CFEM 20.2.2, Fig. 20.1, Coulomb with a vertical back | CFEM 5th ed. Cl. 20.2.2, Fig. 20.1 Coulomb K_a (horizontal), alpha = 90 deg reduces to the typeset form; checked on the figure image too | 2026-10-07 |
| K_p | \(\displaystyle \tan\left(\operatorname{radians}\left(45\right) + \frac{\phi_{r}}{2}\right)^{2}\) | CFEM 20.2.3, Fig. 20.3, Rankine | CFEM 5th ed. Cl. 20.2.3, Fig. 20.3 Rankine, beta = 0 gives tan^2(45 + phi/2); wall friction ignored for passive | 2026-10-07 |
| P_h | \(\displaystyle 0.5 \cdot K_{a} \cdot \gamma \cdot H_{v}^{2}\) | CFEM 20.7.1, active thrust on the virtual back | CFEM 5th ed. Cl. 20.7.1 (active pressure for free-standing walls), Fig. 20.1 P = 1/2 gamma H^2 K_a horizontal; the virtual back H_v is the calc's modelling, not in CFEM | 2026-10-07 |
| P_v | \(\displaystyle P_{h} \cdot \tan\left(\delta_{r}\right)\) | | CFEM 5th ed. Fig. 20.1 note, P_a in direction of delta = P_h / cos(delta + i), i = 0 for a vertical back, checked in ref/ e6a079e | 2026-10-09 |
| y_a | \(\displaystyle \frac{H_{v}}{3}\) | | CFEM 5th ed. Fig. 20.1 resultant at H/3, linear K_a sigma'_z (Cl. 20.2.2), taken on H_v, checked in ref/ e6a079e | 2026-10-09 |
| P_q | \(\displaystyle K_{a} \cdot q \cdot H_{v}\) | CFEM 20.4, eq. 20.7 | CFEM 5th ed. Cl. 20.4, sigma_hq = q K with K = K_a; text limits this to a surcharge under 30% of the active force (the calc tests it); equation printed as 20.7, a duplicate number | 2026-10-07 |
| y_q | \(\displaystyle \frac{H_{v}}{2}\) | | Statics; CFEM 5th ed. Cl. 20.4 sigma_hq = qK uniform with depth, resultant at mid-height, checked in ref/ e6a079e | 2026-10-09 |
| W_base | \(\displaystyle \gamma_{c} \cdot B \cdot t_{b}\) | | Statics, rectangle; CFEM 5th ed. Fig. 20.10 W includes the wall | 2026-10-09 |
| x_base | \(\displaystyle \frac{B}{2}\) | | Statics, centroid from the toe tip (CFEM Fig. 20.10 moments about toe) | 2026-10-09 |
| W_rect | \(\displaystyle \gamma_{c} \cdot t_{\mathrm{top}} \cdot h_{s}\) | | Statics, stem rectangle; CFEM 5th ed. Fig. 20.10 W includes the wall | 2026-10-09 |
| x_rect | \(\displaystyle L_{\mathrm{toe}} + t_{\mathrm{bot}} - t_{\mathrm{top}} + \frac{t_{\mathrm{top}}}{2}\) | | Statics, centroid from the toe tip; back face at B - L_heel | 2026-10-09 |
| W_wedge | \(\displaystyle \frac{\gamma_{c} \cdot \left(t_{\mathrm{bot}} - t_{\mathrm{top}}\right) \cdot h_{s}}{2}\) | | Statics, front batter triangle; CFEM 5th ed. Fig. 20.10 W includes the wall | 2026-10-09 |
| x_wedge | \(\displaystyle L_{\mathrm{toe}} + \frac{2 \cdot \left(t_{\mathrm{bot}} - t_{\mathrm{top}}\right)}{3}\) | | Statics, triangle centroid L_toe + 2(t_bot - t_top)/3 from the toe tip | 2026-10-09 |
| W_soil | \(\displaystyle \gamma \cdot L_{\mathrm{heel}} \cdot h_{s}\) | | Statics; CFEM 5th ed. Fig. 20.10, W includes soil above footing for cantilever walls, checked in ref/ e6a079e | 2026-10-09 |
| x_soil | \(\displaystyle L_{\mathrm{toe}} + t_{\mathrm{bot}} + \frac{L_{\mathrm{heel}}}{2}\) | | Statics, centroid B - L_heel/2 from the toe tip | 2026-10-09 |
| W_slope | \(\displaystyle \frac{\gamma \cdot L_{\mathrm{heel}} \cdot L_{\mathrm{heel}} \cdot \tan\left(\beta_{r}\right)}{2}\) | | Statics, backslope triangle inside the virtual back; CFEM Fig. 20.10 soil above footing | 2026-10-09 |
| x_slope | \(\displaystyle L_{\mathrm{toe}} + t_{\mathrm{bot}} + \frac{2 \cdot L_{\mathrm{heel}}}{3}\) | | Statics, triangle centroid B - L_heel/3 from the toe tip | 2026-10-09 |
| W_q | \(\displaystyle q \cdot L_{\mathrm{heel}}\) | | Statics, q over the heel; its use in V_s and M_Rs is a separate owner decision | 2026-10-09 |
| x_q | \(\displaystyle 1 \cdot x_{\mathrm{soil}}\) | | Statics, uniform load centroid at the heel midpoint | 2026-10-09 |
| W | \(\displaystyle W_{\mathrm{base}} + W_{\mathrm{rect}} + W_{\mathrm{wedge}} + W_{\mathrm{soil}} + W_{\mathrm{slope}}\) | | CFEM 5th ed. Fig. 20.10, W = wall and soil above footing for cantilever walls, checked in ref/ e6a079e | 2026-10-09 |
| M_W | \(\displaystyle W_{\mathrm{base}} \cdot x_{\mathrm{base}} + W_{\mathrm{rect}} \cdot x_{\mathrm{rect}} + W_{\mathrm{wedge}} \cdot x_{\mathrm{wedge}} + W_{\mathrm{soil}} \cdot x_{\mathrm{soil}} + W_{\mathrm{slope}} \cdot x_{\mathrm{slope}}\) | | CFEM 5th ed. Fig. 20.10, moments about toe, W a, checked in ref/ e6a079e | 2026-10-09 |
| V_s | \(\displaystyle W + P_{v}\) | CFEM Fig. 20.10, W is the wall and the soil above the footing, no surcharge | CFEM 5th ed. Fig. 20.10, W + P_v with W the wall and soil above the footing, no surcharge, owner decision 2026-10-09, checked in ref/ e6a079e | 2026-10-09 |
| M_Rs | \(\displaystyle M_{W} + P_{v} \cdot B\) | | CFEM 5th ed. Fig. 20.10, moments about toe W a + P_v e, no surcharge, owner decision 2026-10-09, checked in ref/ e6a079e | 2026-10-09 |
| M_Os | \(\displaystyle P_{h} \cdot y_{a} + P_{q} \cdot y_{q}\) | | CFEM 5th ed. Fig. 20.10, P_H b about the toe, with P_q part of the active force (Cl. 20.4), checked in ref/ e6a079e | 2026-10-09 |
| x_s | \(\displaystyle \frac{M_{\mathrm{Rs}} - M_{\mathrm{Os}}}{V_{s}}\) | CFEM Fig. 20.10, location of the resultant | CFEM 5th ed. Fig. 20.10, location of resultant d = (W a + P_v e - P_H b)/(W + P_v), moments about the toe | 2026-10-07 |
| e_s | \(\displaystyle \frac{B}{2} - x_{s}\) | | CFEM 5th ed. Cl. 10.3.6 eccentricity from the centroid, positive toward the toe, checked in ref/ e6a079e | 2026-10-09 |
| e_abs_s | \(\displaystyle \sqrt{e_{s}^{2}}\) | | CFEM 5th ed. Cl. 10.3.6 Eq. 10.12 uses the magnitude of e, checked in ref/ e6a079e | 2026-10-09 |
| z_bot | \(\displaystyle D_{f} + d_{\mathrm{key}}\) | | Geometry, depth below grade in front of the toe (CFEM Cl. 20.7.3.3 depth below ground surface) | 2026-10-09 |
| z_top | \(\displaystyle \min\left(d_{\mathrm{frost}}, z_{\mathrm{bot}}\right)\) | | CFEM 5th ed. Cl. 20.7.3.3 and Fig. 20.10, passive only below frost depth, checked in ref/ e6a079e | 2026-10-09 |
| P_p | \(\displaystyle 0.5 \cdot K_{p} \cdot \gamma \cdot \left(z_{\mathrm{bot}}^{2} - z_{\mathrm{top}}^{2}\right)\) | CFEM 20.7.3.3, passive below frost depth only | CFEM 5th ed. Cl. 20.7.3.3 and Fig. 20.10 (P_p below frost depth only), Fig. 20.3 P_p = 1/2 gamma H^2 K_p horizontal | 2026-10-07 |
| F_sl | \(\displaystyle \left(W + P_{v}\right) \cdot \tan\left(\delta_{\mathrm{br}}\right)\) | CFEM Fig. 20.10, F = (W + P_v) tan delta, without the surcharge weight | CFEM 5th ed. Fig. 20.10, F = (W + P_v) tan delta | 2026-10-07 |
| F_s | \(\displaystyle \frac{F_{\mathrm{sl}} + P_{p}}{P_{h} + P_{q}}\) | CFEM Fig. 20.10 | CFEM 5th ed. Fig. 20.10, F_s = ((W + P_v) tan delta + P_p)/P_H; the calc adds the surcharge thrust P_q to P_H | 2026-10-07 |
| FS_lim | \(\displaystyle 1.5\) | | CFEM 5th ed. Fig. 20.10, 1.5 without P_p, 2.0 with P_p (the calc switches on P_p; 1.5 is the no-passive default) | 2026-10-07 |
| B_s | \(\displaystyle \max\left(B - 2 \cdot e_{\mathrm{abs},s}, 1\ \mu \mathrm{m}\right)\) | CFEM 10.3.6, B' = B - 2e; assume: clamped, see docstring | CFEM 5th ed. Cl. 10.3.6, Eq. 10.12, B' = B - 2e; the 1 um clamp is the calc's own assumption | 2026-10-07 |
| q_s | \(\displaystyle \frac{V_{s}}{B_{s}}\) | CFEM 20.7.3.4 | CFEM 5th ed. Cl. 20.7.3.4 and Cl. 10.3.6 Eq. 10.12 B' = B - 2e; V/B' as the applied pressure is the effective-width method, not printed, checked in ref/ e6a079e | 2026-10-09 |
| P_hu | \(\displaystyle \alpha_{H} \cdot P_{h} + \alpha_{L} \cdot P_{q}\) | | NBC 2020 Div B Art. 4.1.3.2.(4) H at 1.5 and Table 4.1.3.2.-A case 2 1.5L, checked in ref/ e6a079e | 2026-10-09 |
| R_u | \(\displaystyle \phi_{\mathrm{gu},\mathrm{sl}} \cdot \left(\alpha_{\mathrm{Dc}} \cdot W + \alpha_{H} \cdot P_{v}\right) \cdot \tan\left(\delta_{\mathrm{br}}\right) + \phi_{\mathrm{gu},p} \cdot P_{p}\) | CFEM Table 6.2, base sliding 0.80 and passive 0.50 | CFEM 5th ed. Table 6.2, Retaining systems base sliding 0.80 (Analysis, typical) and Shallow foundations passive resistance 0.50 (typical), checked in ref/; load factors alpha_Dc and alpha_H settled by owner decision 2026-10-08: the active thrust is one load, so P_v keeps alpha_H 1.5 (see Reviewer decisions) | 2026-10-08 |
| V_u | \(\displaystyle \alpha_{D} \cdot \left(W_{\mathrm{base}} + W_{\mathrm{rect}} + W_{\mathrm{wedge}}\right) + \alpha_{\mathrm{Ds}} \cdot \left(W_{\mathrm{soil}} + W_{\mathrm{slope}}\right) + \alpha_{L} \cdot W_{q} + \alpha_{H} \cdot P_{v}\) | | NBC 2020 Div B Table 4.1.3.2.-A case 2 1.25D and 1.5L, Art. 4.1.3.2.(4) H 1.5, Art. 4.1.3.2.(8) soil, checked in ref/ e6a079e | 2026-10-09 |
| M_Ru | \(\displaystyle \alpha_{D} \cdot \left(W_{\mathrm{base}} \cdot x_{\mathrm{base}} + W_{\mathrm{rect}} \cdot x_{\mathrm{rect}} + W_{\mathrm{wedge}} \cdot x_{\mathrm{wedge}}\right) + \alpha_{\mathrm{Ds}} \cdot \left(W_{\mathrm{soil}} \cdot x_{\mathrm{soil}} + W_{\mathrm{slope}} \cdot x_{\mathrm{slope}}\right) + \alpha_{L} \cdot W_{q} \cdot x_{q} + \alpha_{H} \cdot P_{v} \cdot B\) | | NBC 2020 Div B Table 4.1.3.2.-A case 2, adding case for bearing and member pressures (1.25D, soil per Art. 4.1.3.2.(8), 1.5L, 1.5H); overturning is checked on M_Ruo, checked in ref/ e6a079e | 2026-10-09 |
| M_Ou | \(\displaystyle \alpha_{H} \cdot P_{h} \cdot y_{a} + \alpha_{L} \cdot P_{q} \cdot y_{q}\) | | NBC 2020 Div B Art. 4.1.3.2.(4) H at 1.5 and Table 4.1.3.2.-A case 2 1.5L, checked in ref/ e6a079e | 2026-10-09 |
| x_u | \(\displaystyle \frac{M_{\mathrm{Ru}} - M_{\mathrm{Ou}}}{V_{u}}\) | | Statics, moment balance about the toe (CFEM Fig. 20.10 frame) | 2026-10-09 |
| e_u | \(\displaystyle \frac{B}{2} - x_{u}\) | | CFEM 5th ed. Cl. 10.3.6 eccentricity from the centroid, positive toward the toe, checked in ref/ e6a079e | 2026-10-09 |
| e_abs_u | \(\displaystyle \sqrt{e_{u}^{2}}\) | | CFEM 5th ed. Cl. 10.3.6 Eq. 10.12 uses the magnitude of e, checked in ref/ e6a079e | 2026-10-09 |
| B_u | \(\displaystyle \max\left(B - 2 \cdot e_{\mathrm{abs},u}, 1\ \mu \mathrm{m}\right)\) | CFEM 10.3.6; assume: clamped, see docstring | CFEM 5th ed. Cl. 10.3.6, Eq. 10.12, B' = B - 2e; the 1 um clamp is the calc's own assumption | 2026-10-07 |
| q_u | \(\displaystyle \frac{V_{u}}{B_{u}}\) | CFEM 20.7.3.4, Chapter 10 with the eccentricity | CFEM 5th ed. Cl. 20.7.3.4 and Cl. 10.3.6 Eq. 10.12 B' = B - 2e; V/B' as the applied pressure is the effective-width method, not printed, checked in ref/ e6a079e | 2026-10-09 |
| V_uo | \(\displaystyle \alpha_{\mathrm{Dc}} \cdot W + \alpha_{H} \cdot P_{v}\) | NBC 2020 Div B Art. 4.1.3.2.(5), 0.9D resisting overturning | NBC 2020 Div B Art. 4.1.3.2.(5), counteracting 0.9D where dead load resists overturning; surcharge weight left out as counteracting live load, H at 1.5 per Art. 4.1.3.2.(4), checked in ref/ e6a079e | 2026-10-09 |
| M_Ruo | \(\displaystyle \alpha_{\mathrm{Dc}} \cdot M_{W} + \alpha_{H} \cdot P_{v} \cdot B\) | | NBC 2020 Div B Art. 4.1.3.2.(5), counteracting 0.9D where dead load resists overturning, moments about the toe as in CFEM Fig. 20.10, checked in ref/ e6a079e | 2026-10-09 |
| x_uo | \(\displaystyle \frac{M_{\mathrm{Ruo}} - M_{\mathrm{Ou}}}{V_{\mathrm{uo}}}\) | | Statics, moment balance about the toe (CFEM Fig. 20.10 frame) | 2026-10-09 |
| e_uo | \(\displaystyle \frac{B}{2} - x_{\mathrm{uo}}\) | | CFEM 5th ed. Cl. 10.3.6 eccentricity from the centroid, positive toward the toe, checked in ref/ e6a079e | 2026-10-09 |
| e_abs_uo | \(\displaystyle \sqrt{e_{\mathrm{uo}}^{2}}\) | | CFEM 5th ed. Cl. 10.3.6 Eq. 10.12 uses the magnitude of e, checked in ref/ e6a079e | 2026-10-09 |
| B_uo | \(\displaystyle \max\left(B - 2 \cdot e_{\mathrm{abs},\mathrm{uo}}, 1\ \mu \mathrm{m}\right)\) | CFEM 10.3.6; assume: clamped, see docstring | CFEM 5th ed. Cl. 10.3.6, Eq. 10.12, B' = B - 2e; the 1 um clamp is the calc's own assumption, checked in ref/ e6a079e | 2026-10-09 |
| q_uo | \(\displaystyle \frac{V_{\mathrm{uo}}}{B_{\mathrm{uo}}}\) | CFEM 20.7.3.4, Chapter 10 with the eccentricity | CFEM 5th ed. Cl. 20.7.3.4 and Cl. 10.3.6 Eq. 10.12 B' = B - 2e; V/B' as the applied pressure is the effective-width method, not printed, checked in ref/ e6a079e | 2026-10-09 |
| alpha_1 | \(\displaystyle \max\left(0.67, 0.85 - \frac{0.0015 \cdot f_{c}}{1\ \mathrm{MPa}}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 c), Eq. 10.1 | 2026-10-07 |
| beta_1 | \(\displaystyle \max\left(0.67, 0.97 - \frac{0.0025 \cdot f_{c}}{1\ \mathrm{MPa}}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 c), Eq. 10.2 | 2026-10-07 |
| f_v | \(\displaystyle \min\left(\sqrt{f_{c} \cdot 1\ \mathrm{MPa}}, 8\ \mathrm{MPa}\right)\) | CSA A23.3 Cl. 11.3.4 | A23.3-24 Cl. 11.3.4, Eq. 11.6, sqrt(f'c) capped at 8 MPa, lambda = 1 | 2026-10-07 |
| k_beta | \(\displaystyle 230\ \mathrm{mm}\) | | A23.3-24 Cl. 11.3.6.3 b), Eq. 11.9a and 11.9b | 2026-10-07 |
| b_1 | \(\displaystyle 1\ \mathrm{m}\) | | Convention, per-metre strip | 2026-10-09 |
| cd_lim | \(\displaystyle \frac{560\ \mathrm{MPa}}{700\ \mathrm{MPa} + f_{y}}\) | CSA A23.3 Cl. 10.5.2 | A23.3-24 Cl. 10.5.2, Eq. 10.5 | 2026-10-07 |
| f_s | \(\displaystyle 0.6 \cdot f_{y}\) | CSA A23.3 Cl. 10.6.1, in lieu of computing the service stress | A23.3-24 Cl. 10.6.1, f_s = 0.6 f_y | 2026-10-07 |
| z_lim | \(\displaystyle 25\ \mathrm{MN/m}\) | | A23.3-24 Cl. 10.6.1, exterior exposure | 2026-10-07 |
| V_a_stem | \(\displaystyle 0.5 \cdot K_{a} \cdot \gamma \cdot h_{s}^{2}\) | | CFEM 5th ed. Cl. 20.2.2, Fig. 20.1, 1/2 K_a gamma H^2 horizontal on the stem height, checked in ref/ e6a079e | 2026-10-09 |
| V_q_stem | \(\displaystyle K_{a} \cdot q \cdot h_{s}\) | | CFEM 5th ed. Cl. 20.4, sigma_hq = qK over the stem height, checked in ref/ e6a079e | 2026-10-09 |
| M_s_stem | \(\displaystyle \left(\frac{V_{a,\mathrm{stem}} \cdot h_{s}}{3} + \frac{V_{q,\mathrm{stem}} \cdot h_{s}}{2}\right) \cdot b_{1}\) | | Statics; CFEM Fig. 20.1 H/3 and the uniform surcharge at H/2 | 2026-10-09 |
| V_f_stem | \(\displaystyle \left(\alpha_{H} \cdot V_{a,\mathrm{stem}} + \alpha_{L} \cdot V_{q,\mathrm{stem}}\right) \cdot b_{1}\) | | NBC 2020 Div B Art. 4.1.3.2.(4) H at 1.5 and Table 4.1.3.2.-A case 2 1.5L, checked in ref/ e6a079e | 2026-10-09 |
| M_f_stem | \(\displaystyle \left(\frac{\alpha_{H} \cdot V_{a,\mathrm{stem}} \cdot h_{s}}{3} + \frac{\alpha_{L} \cdot V_{q,\mathrm{stem}} \cdot h_{s}}{2}\right) \cdot b_{1}\) | | NBC 2020 Div B Art. 4.1.3.2.(4) and Table 4.1.3.2.-A case 2 on the M_s_stem arms, checked in ref/ e6a079e | 2026-10-09 |
| d_b_stem | \(\displaystyle 16\ \mathrm{mm}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_b_stem | \(\displaystyle 200\ \mathrm{mm}^{2}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_s_stem | \(\displaystyle \frac{A_{b,\mathrm{stem}} \cdot b_{1}}{s_{\mathrm{stem}}}\) | | A23.3-24 Cl. 3.2 A_s, area per metre from bar area and spacing | 2026-10-09 |
| d_stem | \(\displaystyle t_{\mathrm{bot}} - c_{\mathrm{formed}} - \frac{d_{b,\mathrm{stem}}}{2}\) | | A23.3-24 Cl. 3.2 d; cover c_formed is the designer's input to suit the Annex A Table 17 exposure class, owner decision 2026-10-09, checked in ref/ e6a079e | 2026-10-09 |
| a_stem | \(\displaystyle \frac{\phi_{s} \cdot A_{s,\mathrm{stem}} \cdot f_{y}}{\alpha_{1} \cdot \phi_{c} \cdot f_{c} \cdot b_{1}}\) | | A23.3-24 Cl. 10.1.7 a), T = phi_s A_s f_y | 2026-10-07 |
| c_stem | \(\displaystyle \frac{a_{\mathrm{stem}}}{\beta_{1}}\) | | A23.3-24 Cl. 10.1.7 a), a = beta_1 c | 2026-10-07 |
| M_r_stem | \(\displaystyle \phi_{s} \cdot A_{s,\mathrm{stem}} \cdot f_{y} \cdot \left(d_{\mathrm{stem}} - \frac{a_{\mathrm{stem}}}{2}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 a) | 2026-10-07 |
| d_v_stem | \(\displaystyle \max\left(0.9 \cdot d_{\mathrm{stem}}, 0.72 \cdot t_{\mathrm{bot}}\right)\) | CSA A23.3 Cl. 3.2 | A23.3-24 Cl. 3.2, d_v | 2026-10-07 |
| beta_stem | \(\displaystyle \frac{k_{\beta}}{1\ \mathrm{m} + d_{v,\mathrm{stem}}}\) | CSA A23.3 Cl. 11.3.6.3 b), Eq. 11.9 | A23.3-24 Cl. 11.3.6.3 b), Eq. 11.9 | 2026-10-07 |
| V_c_stem | \(\displaystyle \phi_{c} \cdot \beta_{\mathrm{stem}} \cdot f_{v} \cdot b_{1} \cdot d_{v,\mathrm{stem}}\) | CSA A23.3 Cl. 11.3.4 | A23.3-24 Cl. 11.3.4, Eq. 11.6 | 2026-10-07 |
| A_s_min_stem | \(\displaystyle 0\ \mathrm{m}^{2}\) | | A23.3-24 Cl. 14.1.8.2.5, waiver per Cl. 10.5.1.3 | 2026-10-07 |
| s_max_stem | \(\displaystyle \min\left(3 \cdot t_{\mathrm{bot}}, 500\ \mathrm{mm}\right)\) | CSA A23.3 Cl. 7.4.1.2, 14.1.8.2.3 | A23.3-24 Cl. 7.4.1.2, 14.1.8.2.3 | 2026-10-07 |
| d_c_stem | \(\displaystyle \min\left(c_{\mathrm{formed}}, 50\ \mathrm{mm}\right) + \frac{d_{b,\mathrm{stem}}}{2}\) | CSA A23.3 Cl. 10.6.1, cover capped at 50 mm | A23.3-24 Cl. 10.6.1, cover capped at 50 mm | 2026-10-07 |
| A_c_stem | \(\displaystyle 2 \cdot d_{c,\mathrm{stem}} \cdot s_{\mathrm{stem}}\) | | A23.3-24 Cl. 3.2, A per bar | 2026-10-07 |
| z_stem | \(\displaystyle f_{s} \cdot \left(d_{c,\mathrm{stem}} \cdot A_{c,\mathrm{stem}}\right)^{\frac{1}{3}}\) | CSA A23.3 Cl. 10.6.1, 14.1.8.2.6 | A23.3-24 Cl. 10.6.1, Eq. 10.6, 14.1.8.2.6 | 2026-10-07 |
| q_ut | \(\displaystyle \frac{V_{u}}{B} \cdot \left(1 + \frac{6 \cdot e_{u}}{B}\right)\) | | Statics, linear contact pressure V/B (1 + 6e/B); CFEM Cl. 20.7.3.4 middle third, not printed | 2026-10-09 |
| q_uh | \(\displaystyle \frac{V_{u}}{B} \cdot \left(1 - \frac{6 \cdot e_{u}}{B}\right)\) | | Statics, linear contact pressure V/B (1 - 6e/B); CFEM Cl. 20.7.3.4 middle third, not printed | 2026-10-09 |
| g_u | \(\displaystyle \frac{q_{\mathrm{ut}} - q_{\mathrm{uh}}}{B}\) | | Statics, gradient of the linear contact pressure | 2026-10-09 |
| w_b | \(\displaystyle \alpha_{\mathrm{Dc}} \cdot \gamma_{c} \cdot t_{b}\) | NBC 2020 Div B Art. 4.1.3.2.(5), base self-weight offsetting the toe pressure | NBC 2020 Div B Art. 4.1.3.2.(5), counteracting 0.9D where dead load resists overturning: the base weight offsets the upward toe pressure, checked in ref/ e6a079e | 2026-10-09 |
| q_ft | \(\displaystyle q_{\mathrm{ut}} - g_{u} \cdot L_{\mathrm{toe}}\) | | Statics, pressure at the wall face, the section of A23.3-24 Cl. 15.4.3 a), checked in ref/ e6a079e | 2026-10-09 |
| M_f_toe | \(\displaystyle \frac{L_{\mathrm{toe}}^{2}}{6} \cdot \left(2 \cdot \left(q_{\mathrm{ut}} - w_{b}\right) + q_{\mathrm{ft}} - w_{b}\right) \cdot b_{1}\) | CSA A23.3 Cl. 15.4.3 a), moment at the wall face | A23.3-24 Cl. 15.4.3 a), face of wall | 2026-10-07 |
| d_b_toe | \(\displaystyle 16\ \mathrm{mm}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_b_toe | \(\displaystyle 200\ \mathrm{mm}^{2}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_s_toe | \(\displaystyle \frac{A_{b,\mathrm{toe}} \cdot b_{1}}{s_{\mathrm{toe}}}\) | | A23.3-24 Cl. 3.2 A_s, area per metre from bar area and spacing | 2026-10-09 |
| d_toe | \(\displaystyle t_{b} - c_{\mathrm{cast}} - \frac{d_{b,\mathrm{toe}}}{2}\) | | A23.3-24 Cl. 3.2 d; Annex A Table 17, 75 mm cast against earth, checked in ref/ e6a079e | 2026-10-09 |
| a_toe | \(\displaystyle \frac{\phi_{s} \cdot A_{s,\mathrm{toe}} \cdot f_{y}}{\alpha_{1} \cdot \phi_{c} \cdot f_{c} \cdot b_{1}}\) | | A23.3-24 Cl. 10.1.7 a), T = phi_s A_s f_y | 2026-10-07 |
| c_toe | \(\displaystyle \frac{a_{\mathrm{toe}}}{\beta_{1}}\) | | A23.3-24 Cl. 10.1.7 a), a = beta_1 c | 2026-10-07 |
| M_r_toe | \(\displaystyle \phi_{s} \cdot A_{s,\mathrm{toe}} \cdot f_{y} \cdot \left(d_{\mathrm{toe}} - \frac{a_{\mathrm{toe}}}{2}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 a) | 2026-10-07 |
| d_v_toe | \(\displaystyle \max\left(0.9 \cdot d_{\mathrm{toe}}, 0.72 \cdot t_{b}\right)\) | CSA A23.3 Cl. 3.2 | A23.3-24 Cl. 3.2, d_v | 2026-10-07 |
| beta_toe | \(\displaystyle \frac{k_{\beta}}{1\ \mathrm{m} + d_{v,\mathrm{toe}}}\) | CSA A23.3 Cl. 11.3.6.3 b), Eq. 11.9 | A23.3-24 Cl. 11.3.6.3 b), Eq. 11.9 | 2026-10-07 |
| V_c_toe | \(\displaystyle \phi_{c} \cdot \beta_{\mathrm{toe}} \cdot f_{v} \cdot b_{1} \cdot d_{v,\mathrm{toe}}\) | CSA A23.3 Cl. 11.3.4 | A23.3-24 Cl. 11.3.4, Eq. 11.6 | 2026-10-07 |
| x_v_toe | \(\displaystyle \max\left(L_{\mathrm{toe}} - d_{v,\mathrm{toe}}, 0\ \mathrm{m}\right)\) | CSA A23.3 Cl. 15.5.2 and 11.3.2, section d_v from the wall face | A23.3-24 Cl. 15.5.2 and 11.3.2, d_v from the face | 2026-10-07 |
| q_vt | \(\displaystyle q_{\mathrm{ut}} - g_{u} \cdot x_{v,\mathrm{toe}}\) | | Statics, pressure at d_v from the face, A23.3-24 Cl. 15.5.2 and 11.3.2, checked in ref/ e6a079e | 2026-10-09 |
| V_f_toe | \(\displaystyle \frac{q_{\mathrm{ut}} - w_{b} + q_{\mathrm{vt}} - w_{b}}{2} \cdot x_{v,\mathrm{toe}} \cdot b_{1}\) | | A23.3-24 Cl. 15.5.2, 13.3.2.1 and 11.3.2, net trapezoid between tip and the d_v section, checked in ref/ e6a079e | 2026-10-09 |
| A_s_min_toe | \(\displaystyle 0.002 \cdot t_{b} \cdot b_{1}\) | CSA A23.3 Cl. 7.8.1 | A23.3-24 Cl. 7.8.1, 0.002 A_g | 2026-10-07 |
| s_max_toe | \(\displaystyle \min\left(3 \cdot t_{b}, 500\ \mathrm{mm}\right)\) | CSA A23.3 Cl. 7.4.1.2 | A23.3-24 Cl. 7.4.1.2 | 2026-10-07 |
| d_c_toe | \(\displaystyle \min\left(c_{\mathrm{cast}}, 50\ \mathrm{mm}\right) + \frac{d_{b,\mathrm{toe}}}{2}\) | CSA A23.3 Cl. 10.6.1 | A23.3-24 Cl. 10.6.1, cover capped at 50 mm | 2026-10-07 |
| A_c_toe | \(\displaystyle 2 \cdot d_{c,\mathrm{toe}} \cdot s_{\mathrm{toe}}\) | | A23.3-24 Cl. 3.2, A per bar | 2026-10-07 |
| z_toe | \(\displaystyle f_{s} \cdot \left(d_{c,\mathrm{toe}} \cdot A_{c,\mathrm{toe}}\right)^{\frac{1}{3}}\) | CSA A23.3 Cl. 10.6.1 | A23.3-24 Cl. 10.6.1, Eq. 10.6 | 2026-10-07 |
| w_d | \(\displaystyle \alpha_{\mathrm{Ds}} \cdot \gamma \cdot \left(h_{s} + \frac{L_{\mathrm{heel}} \cdot \tan\left(\beta_{r}\right)}{2}\right) + \alpha_{D} \cdot \gamma_{c} \cdot t_{b} + \alpha_{L} \cdot q\) | | NBC 2020 Div B Table 4.1.3.2.-A case 2 1.25D and 1.5L, Art. 4.1.3.2.(8) soil at alpha_Ds, checked in ref/ e6a079e | 2026-10-09 |
| x_f | \(\displaystyle L_{\mathrm{toe}} + t_{\mathrm{bot}}\) | | Geometry, back face of the stem from the toe tip | 2026-10-09 |
| q_fh | \(\displaystyle q_{\mathrm{ut}} - g_{u} \cdot x_{f}\) | | Statics, pressure at the back face, the section of A23.3-24 Cl. 15.4.3 a), checked in ref/ e6a079e | 2026-10-09 |
| n_face | \(\displaystyle w_{d} - q_{\mathrm{fh}}\) | | Statics, net downward load at the back face | 2026-10-09 |
| n_tip | \(\displaystyle w_{d} - q_{\mathrm{uh}}\) | | Statics, net downward load at the heel tip | 2026-10-09 |
| M_f_heel | \(\displaystyle \frac{L_{\mathrm{heel}}^{2}}{6} \cdot \left(2 \cdot n_{\mathrm{tip}} + n_{\mathrm{face}}\right) \cdot b_{1}\) | CSA A23.3 Cl. 15.4.3 a) | A23.3-24 Cl. 15.4.3 a), face of wall | 2026-10-07 |
| d_b_heel | \(\displaystyle 16\ \mathrm{mm}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_b_heel | \(\displaystyle 200\ \mathrm{mm}^{2}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_s_heel | \(\displaystyle \frac{A_{b,\mathrm{heel}} \cdot b_{1}}{s_{\mathrm{heel}}}\) | | A23.3-24 Cl. 3.2 A_s, area per metre from bar area and spacing | 2026-10-09 |
| d_heel | \(\displaystyle t_{b} - c_{\mathrm{formed}} - \frac{d_{b,\mathrm{heel}}}{2}\) | | A23.3-24 Cl. 3.2 d; cover c_formed is the designer's input to suit the Annex A Table 17 exposure class, owner decision 2026-10-09, checked in ref/ e6a079e | 2026-10-09 |
| a_heel | \(\displaystyle \frac{\phi_{s} \cdot A_{s,\mathrm{heel}} \cdot f_{y}}{\alpha_{1} \cdot \phi_{c} \cdot f_{c} \cdot b_{1}}\) | | A23.3-24 Cl. 10.1.7 a), T = phi_s A_s f_y | 2026-10-07 |
| c_heel | \(\displaystyle \frac{a_{\mathrm{heel}}}{\beta_{1}}\) | | A23.3-24 Cl. 10.1.7 a), a = beta_1 c | 2026-10-07 |
| M_r_heel | \(\displaystyle \phi_{s} \cdot A_{s,\mathrm{heel}} \cdot f_{y} \cdot \left(d_{\mathrm{heel}} - \frac{a_{\mathrm{heel}}}{2}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 a) | 2026-10-07 |
| d_v_heel | \(\displaystyle \max\left(0.9 \cdot d_{\mathrm{heel}}, 0.72 \cdot t_{b}\right)\) | CSA A23.3 Cl. 3.2 | A23.3-24 Cl. 3.2, d_v | 2026-10-07 |
| beta_heel | \(\displaystyle \frac{k_{\beta}}{1\ \mathrm{m} + d_{v,\mathrm{heel}}}\) | CSA A23.3 Cl. 11.3.6.3 b), Eq. 11.9 | A23.3-24 Cl. 11.3.6.3 b), Eq. 11.9 | 2026-10-07 |
| V_c_heel | \(\displaystyle \phi_{c} \cdot \beta_{\mathrm{heel}} \cdot f_{v} \cdot b_{1} \cdot d_{v,\mathrm{heel}}\) | CSA A23.3 Cl. 11.3.4 | A23.3-24 Cl. 11.3.4, Eq. 11.6 | 2026-10-07 |
| V_f_heel | \(\displaystyle \frac{n_{\mathrm{face}} + n_{\mathrm{tip}}}{2} \cdot L_{\mathrm{heel}} \cdot b_{1}\) | CSA A23.3 Cl. 15.5.2, 11.3.2 a), section at the stem face | A23.3-24 Cl. 15.5.2, 11.3.2 a), section at the stem face | 2026-10-07 |
| A_s_min_heel | \(\displaystyle 0.002 \cdot t_{b} \cdot b_{1}\) | CSA A23.3 Cl. 7.8.1 | A23.3-24 Cl. 7.8.1, 0.002 A_g | 2026-10-07 |
| s_max_heel | \(\displaystyle \min\left(3 \cdot t_{b}, 500\ \mathrm{mm}\right)\) | CSA A23.3 Cl. 7.4.1.2 | A23.3-24 Cl. 7.4.1.2 | 2026-10-07 |
| d_c_heel | \(\displaystyle \min\left(c_{\mathrm{formed}}, 50\ \mathrm{mm}\right) + \frac{d_{b,\mathrm{heel}}}{2}\) | CSA A23.3 Cl. 10.6.1 | A23.3-24 Cl. 10.6.1, cover capped at 50 mm | 2026-10-07 |
| A_c_heel | \(\displaystyle 2 \cdot d_{c,\mathrm{heel}} \cdot s_{\mathrm{heel}}\) | | A23.3-24 Cl. 3.2, A per bar | 2026-10-07 |
| z_heel | \(\displaystyle f_{s} \cdot \left(d_{c,\mathrm{heel}} \cdot A_{c,\mathrm{heel}}\right)^{\frac{1}{3}}\) | CSA A23.3 Cl. 10.6.1 | A23.3-24 Cl. 10.6.1, Eq. 10.6 | 2026-10-07 |
| z_k | \(\displaystyle \max\left(z_{\mathrm{top}}, D_{f}\right)\) | | CFEM 5th ed. Cl. 20.7.3.3, no passive above frost depth; key top at D_f, checked in ref/ e6a079e | 2026-10-09 |
| h_k | \(\displaystyle z_{\mathrm{bot}} - z_{k}\) | | Geometry, loaded height of the key face | 2026-10-09 |
| p_k1 | \(\displaystyle K_{p} \cdot \gamma \cdot z_{k}\) | | CFEM 5th ed. Cl. 20.2.3, Fig. 20.3 Rankine K_p gamma z, checked in ref/ e6a079e | 2026-10-09 |
| p_k2 | \(\displaystyle K_{p} \cdot \gamma \cdot z_{\mathrm{bot}}\) | | CFEM 5th ed. Cl. 20.2.3, Fig. 20.3 Rankine K_p gamma z, checked in ref/ e6a079e | 2026-10-09 |
| P_key | \(\displaystyle \frac{p_{k1} + p_{k2}}{2} \cdot h_{k}\) | CFEM 20.2.3, passive on the key face | CFEM 5th ed. Cl. 20.2.3, Fig. 20.3 horizontal K_p, pressure K_p gamma z; trapezoid area is statics | 2026-10-07 |
| M_f_key | \(\displaystyle \left(\frac{h_{k}^{2}}{6} \cdot \left(2 \cdot p_{k2} + p_{k1}\right) + \left(z_{k} - D_{f}\right) \cdot P_{\mathrm{key}}\right) \cdot b_{1}\) | | Statics, moment of the trapezoidal passive load about the key root at D_f (CFEM 5th ed. Cl. 20.2.3 K_p gamma z, Cl. 20.7.3.3 below frost) | 2026-10-09 |
| V_f_key | \(\displaystyle P_{\mathrm{key}} \cdot b_{1}\) | | Statics, full passive resultant at the key root | 2026-10-09 |
| d_b_key | \(\displaystyle 16\ \mathrm{mm}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_b_key | \(\displaystyle 200\ \mathrm{mm}^{2}\) | | Nominal CSA G30.18 bar size from calcsheet.rebar, not in ref/; accepted as industry norm, owner decision 2026-10-09 | 2026-10-09 |
| A_s_key | \(\displaystyle \frac{A_{b,\mathrm{key}} \cdot b_{1}}{s_{\mathrm{key}}}\) | | A23.3-24 Cl. 3.2 A_s, area per metre from bar area and spacing | 2026-10-09 |
| d_key_eff | \(\displaystyle t_{\mathrm{key}} - c_{\mathrm{cast}} - \frac{d_{b,\mathrm{key}}}{2}\) | | A23.3-24 Cl. 3.2 d, front-face bars in tension under passive; Annex A Table 17, 75 mm cast against earth, checked in ref/ e6a079e | 2026-10-09 |
| a_key | \(\displaystyle \frac{\phi_{s} \cdot A_{s,\mathrm{key}} \cdot f_{y}}{\alpha_{1} \cdot \phi_{c} \cdot f_{c} \cdot b_{1}}\) | | A23.3-24 Cl. 10.1.7 a), T = phi_s A_s f_y | 2026-10-07 |
| M_r_key | \(\displaystyle \phi_{s} \cdot A_{s,\mathrm{key}} \cdot f_{y} \cdot \left(d_{\mathrm{key},\mathrm{eff}} - \frac{a_{\mathrm{key}}}{2}\right)\) | CSA A23.3 Cl. 10.1.7 | A23.3-24 Cl. 10.1.7 a) | 2026-10-07 |
| d_v_key | \(\displaystyle \max\left(0.9 \cdot d_{\mathrm{key},\mathrm{eff}}, 0.72 \cdot t_{\mathrm{key}}\right)\) | CSA A23.3 Cl. 3.2 | A23.3-24 Cl. 3.2, d_v | 2026-10-07 |
| beta_key | \(\displaystyle \frac{k_{\beta}}{1\ \mathrm{m} + d_{v,\mathrm{key}}}\) | CSA A23.3 Cl. 11.3.6.3 b), Eq. 11.9 | A23.3-24 Cl. 11.3.6.3 b), Eq. 11.9 | 2026-10-07 |
| V_c_key | \(\displaystyle \phi_{c} \cdot \beta_{\mathrm{key}} \cdot f_{v} \cdot b_{1} \cdot d_{v,\mathrm{key}}\) | CSA A23.3 Cl. 11.3.4 | A23.3-24 Cl. 11.3.4, Eq. 11.6 | 2026-10-07 |
| k_h | \(\displaystyle 1 \cdot \mathrm{PGA}\) | CFEM 18.7.1.1, eq. 18.22, k_h = a_h / g | CFEM 5th ed. Cl. 18.7.1.1, Eq. 18.22, k_h = a_h/g; a_h taken as the surface PGA input, with no F(PGA) amplification | 2026-10-07 |
| k_v | \(\displaystyle \frac{2}{3} \cdot k_{h}\) | CFEM 18.7.1.1, absent a site-specific assessment | CFEM 5th ed. Cl. 18.7.1.1, k_v = 2/3 k_h absent a site-specific assessment | 2026-10-07 |
| psi | \(\displaystyle \arctan\left(\frac{k_{h}}{1 - k_{v}}\right)\) | CFEM 18.7.1.1, eq. 18.21 | CFEM 5th ed. Cl. 18.7.1.1, psi = atan[k_h/(1 - k_v)] defined under Eq. 18.21 (also Eq. 20.11); calc raises when phi - beta < psi as Cl. 18.7.1.3 requires | 2026-10-07 |
| K_AE | \(\displaystyle \frac{\cos\left(\phi_{r} - \psi\right)^{2}}{\cos\left(\psi\right) \cdot \cos\left(\delta_{r} + \psi\right) \cdot \left(1 + \sqrt{\frac{\sin\left(\delta_{r} + \phi_{r}\right) \cdot \sin\left(\phi_{r} - \beta_{r} - \psi\right)}{\cos\left(\delta_{r} + \psi\right) \cdot \cos\left(\beta_{r}\right)}}\right)^{2}}\) | CFEM 18.7.1.1, eq. 18.21, theta = 0 | CFEM 5th ed. Cl. 18.7.1.1, Eq. 18.21 with theta = 0 | 2026-10-07 |
| K_AE_h | \(\displaystyle K_{\mathrm{AE}} \cdot \cos\left(\delta_{r}\right)\) | | CFEM 5th ed. Cl. 18.7.1.1 Eq. 18.21 and 18.25 (along the thrust) with the Fig. 20.1 note, horizontal = K_AE cos(delta), checked in ref/ e6a079e | 2026-10-09 |
| P_AE | \(\displaystyle 0.5 \cdot K_{\mathrm{AE}} \cdot \gamma \cdot H_{v}^{2} \cdot \left(1 - k_{v}\right)\) | CFEM 18.7.1.1, eq. 18.20 | CFEM 5th ed. Cl. 18.7.1.1, Eq. 18.20 | 2026-10-07 |
| Delta_P_AE | \(\displaystyle P_{\mathrm{AE}} - \frac{P_{h}}{\cos\left(\delta_{r}\right)}\) | CFEM 18.7.1.1, eq. 18.23 | CFEM 5th ed. Cl. 18.7.1.1, Eq. 18.23 with P_A = P_h/cos(delta) from Eq. 18.24 and 18.25 at theta = 0 | 2026-10-07 |
| Delta_P_h | \(\displaystyle \Delta_{P,\mathrm{AE}} \cdot \cos\left(\delta_{r}\right)\) | | CFEM 5th ed. Cl. 18.7.1.1 Eq. 18.23 and 18.24, increment along delta, horizontal part, checked in ref/ e6a079e | 2026-10-09 |
| Delta_P_v | \(\displaystyle \Delta_{P,\mathrm{AE}} \cdot \sin\left(\delta_{r}\right)\) | | Statics; CFEM 5th ed. Fig. 20.1 note, thrust along delta, vertical part, checked in ref/ e6a079e | 2026-10-09 |
| y_E | \(\displaystyle 0.6 \cdot H_{v}\) | CFEM 18.7.1.1, eq. 18.26, Seed and Whitman | CFEM 5th ed. Cl. 18.7.1.1 Eq. 18.26, Seed and Whitman 0.6H for the dynamic increment (the 0.6H sentence of Cl. 20.5 was removed by the 2026 errata), checked in ref/ e6a079e | 2026-10-09 |
| P_I | \(\displaystyle k_{h} \cdot W\) | CFEM 18.7.5.3, inertia of the wall and the soil over the heel, which M-O leaves out | CFEM 5th ed. Cl. 18.7.5.3, M-O ignores the wall's inertia; the soil over the heel sits inside the vertical virtual back and moves with the wall, so the M-O wedge of Fig. 18.34 starts behind it and leaves its inertia out, checked in ref/ e6a079e | 2026-10-09 |
| y_base | \(\displaystyle \frac{t_{b}}{2}\) | | Statics, centroid height above the base underside | 2026-10-09 |
| y_rect | \(\displaystyle t_{b} + \frac{h_{s}}{2}\) | | Statics, centroid height above the base underside | 2026-10-09 |
| y_wedge | \(\displaystyle t_{b} + \frac{h_{s}}{3}\) | | Statics, triangle centroid h_s/3 above the base top | 2026-10-09 |
| y_soil | \(\displaystyle t_{b} + \frac{h_{s}}{2}\) | | Statics, centroid height above the base underside | 2026-10-09 |
| y_slope | \(\displaystyle t_{b} + h_{s} + \frac{L_{\mathrm{heel}} \cdot \tan\left(\beta_{r}\right)}{3}\) | | Statics, triangle centroid L tan(beta)/3 above the retained surface | 2026-10-09 |
| M_I | \(\displaystyle k_{h} \cdot \left(W_{\mathrm{base}} \cdot y_{\mathrm{base}} + W_{\mathrm{rect}} \cdot y_{\mathrm{rect}} + W_{\mathrm{wedge}} \cdot y_{\mathrm{wedge}} + W_{\mathrm{soil}} \cdot y_{\mathrm{soil}} + W_{\mathrm{slope}} \cdot y_{\mathrm{slope}}\right)\) | | Statics, k_h W at each centroid above the base underside; CFEM 5th ed. Cl. 18.7.5.3, the soil over the heel sits inside the vertical virtual back and moves with the wall, so the M-O wedge of Fig. 18.34 starts behind it and leaves its inertia out, checked in ref/ e6a079e | 2026-10-09 |
| V_E | \(\displaystyle \alpha_{E} \cdot \left(W + 0.5 \cdot W_{q} + P_{v} + \Delta_{P,v}\right)\) | NBCC Table 4.1.3.2.-A, case 5 | NBC 2020 Div B Table 4.1.3.2.-A case 5; static earth pressure at 1.0 per owner decision 2026-10-08, checked in ref/ | 2026-10-08 |
| M_RE | \(\displaystyle \alpha_{E} \cdot \left(M_{W} + 0.5 \cdot W_{q} \cdot x_{q} + \left(P_{v} + \Delta_{P,v}\right) \cdot B\right)\) | | CFEM 5th ed. Fig. 20.10 moments about toe; NBC 2020 Table 4.1.3.2.-A case 5 with 0.5L and Art. 4.1.3.2.(5) 1.0D, checked in ref/ e6a079e | 2026-10-09 |
| M_OE | \(\displaystyle \alpha_{E} \cdot \left(P_{h} \cdot y_{a} + 0.5 \cdot P_{q} \cdot y_{q} + \Delta_{P,h} \cdot y_{E} + M_{I}\right)\) | | CFEM 5th ed. Fig. 20.10 moments about toe, Eq. 18.26 increment at 0.6H, plus M_I; NBC 2020 Table 4.1.3.2.-A case 5 with 0.5L, checked in ref/ e6a079e | 2026-10-09 |
| x_E | \(\displaystyle \frac{M_{\mathrm{RE}} - M_{\mathrm{OE}}}{V_{E}}\) | | Statics, moment balance about the toe (CFEM Fig. 20.10 frame) | 2026-10-09 |
| e_E | \(\displaystyle \frac{B}{2} - x_{E}\) | | CFEM 5th ed. Cl. 10.3.6 eccentricity from the centroid, checked in ref/ e6a079e | 2026-10-09 |
| e_abs_E | \(\displaystyle \sqrt{e_{E}^{2}}\) | | CFEM 5th ed. Cl. 10.3.6 Eq. 10.12 uses the magnitude of e, checked in ref/ e6a079e | 2026-10-09 |
| B_E | \(\displaystyle \max\left(B - 2 \cdot e_{\mathrm{abs},E}, 1\ \mu \mathrm{m}\right)\) | CFEM 10.3.6; assume: clamped, see docstring | CFEM 5th ed. Cl. 10.3.6, Eq. 10.12, B' = B - 2e; the 1 um clamp is the calc's own assumption | 2026-10-07 |
| q_E | \(\displaystyle \frac{V_{E}}{B_{E}}\) | CFEM 20.7.3.4 | CFEM 5th ed. Cl. 20.7.3.4 and Cl. 10.3.6 Eq. 10.12 B' = B - 2e; V/B' as the applied pressure is the effective-width method, not printed, checked in ref/ e6a079e | 2026-10-09 |
| P_hE | \(\displaystyle \alpha_{E} \cdot \left(P_{h} + 0.5 \cdot P_{q} + \Delta_{P,h} + P_{I}\right)\) | | CFEM 5th ed. Cl. 18.7.1.1 and 18.7.5.3; NBC 2020 Table 4.1.3.2.-A case 5 with 0.5L, checked in ref/ e6a079e | 2026-10-09 |
| R_E | \(\displaystyle \phi_{\mathrm{gu},\mathrm{sl}} \cdot \alpha_{E} \cdot \left(W + P_{v} + \Delta_{P,v}\right) \cdot \tan\left(\delta_{\mathrm{br}}\right)\) | CFEM Table 6.2, base sliding 0.80, without the surcharge weight | CFEM 5th ed. Table 6.2, Retaining systems base sliding 0.80 (Analysis, typical), checked in ref/; alpha_E is NBCC, not checked here | 2026-10-08 |
| P_AE_stem | \(\displaystyle 0.5 \cdot K_{\mathrm{AE}} \cdot \gamma \cdot h_{s}^{2} \cdot \left(1 - k_{v}\right) \cdot \cos\left(\delta_{r}\right)\) | CFEM 18.7.1.1, eq. 18.20 | CFEM 5th ed. Cl. 18.7.1.1, Eq. 18.20 at H = h_s; the cos(delta) horizontal projection follows Fig. 18.34 (delta from the wall normal) and is not written in the text | 2026-10-07 |
| Delta_stem | \(\displaystyle P_{\mathrm{AE},\mathrm{stem}} - V_{a,\mathrm{stem}}\) | | CFEM 5th ed. Cl. 18.7.1.1 Eq. 18.20, increment over the stem height, checked in ref/ e6a079e | 2026-10-09 |
| P_I_stem | \(\displaystyle k_{h} \cdot \left(W_{\mathrm{rect}} + W_{\mathrm{wedge}}\right)\) | | CFEM 5th ed. Cl. 18.7.5.3, M-O ignores wall inertia, k_h on the stem, checked in ref/ e6a079e | 2026-10-09 |
| M_I_stem | \(\displaystyle k_{h} \cdot \left(\frac{W_{\mathrm{rect}} \cdot h_{s}}{2} + \frac{W_{\mathrm{wedge}} \cdot h_{s}}{3}\right)\) | | Statics, stem inertia at the centroids above the base top | 2026-10-09 |
| V_E_stem | \(\displaystyle \alpha_{E} \cdot \left(V_{a,\mathrm{stem}} + 0.5 \cdot V_{q,\mathrm{stem}} + \Delta_{\mathrm{stem}} + P_{I,\mathrm{stem}}\right) \cdot b_{1}\) | | NBC 2020 Table 4.1.3.2.-A case 5 with 0.5L; CFEM 5th ed. Cl. 18.7.1.1, checked in ref/ e6a079e | 2026-10-09 |
| M_E_stem | \(\displaystyle \alpha_{E} \cdot \left(\frac{V_{a,\mathrm{stem}} \cdot h_{s}}{3} + \frac{0.5 \cdot V_{q,\mathrm{stem}} \cdot h_{s}}{2} + \Delta_{\mathrm{stem}} \cdot 0.6 \cdot h_{s} + M_{I,\mathrm{stem}}\right) \cdot b_{1}\) | CFEM 18.7.1.1, increment at 0.6 H | CFEM 5th ed. Cl. 18.7.1.1 (0.6H above base, Eq. 18.26), checked in ref/ e6a079e | 2026-10-09 |
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