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CSA A23.3

Pad footing capacity

Bearing, punching, one-way shear and flexure of a concrete pad footing to CSA A23.3. Check a reinforced concrete pad footing under a column carrying axial load and moments about both axes. Soil bearing is gross, with the footing's own weight and the soil over it, linear at service and on Meyerhof's effective area at the factored level, and the resultant must stay inside the kern. The column bears on the footing concrete and on its own. Punching includes the unbalanced moment transfer, and one-way shear and flexure integrate the linear pressure in both directions. Minimum steel, neutral axis depth, bar spacing and development length of the bottom bars are checked, and the band steel of a rectangular footing and the minimum dowel area are reported.

Given

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\(D\) \(\mathrm{kN}\) 0-50,000
\(L\) \(\mathrm{kN}\) 0-50,000
\(M_{\mathrm{Dx}}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-20,000
\(M_{\mathrm{Lx}}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-20,000
\(M_{\mathrm{Dy}}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-20,000
\(M_{\mathrm{Ly}}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-20,000
\(q_{a}\) \(\mathrm{kPa}\) 25-2,000
\(q_{\mathrm{ult}}\) \(\mathrm{kPa}\) 25-3,000
\(b_{c}\) \(\mathrm{mm}\) 150-3,000
\(h_{c}\) \(\mathrm{mm}\) 150-3,000
\(B_{f}\) \(\mathrm{mm}\) 500-12,000
\(L_{f}\) \(\mathrm{mm}\) 500-12,000
\(t_{f}\) \(\mathrm{mm}\) 200-3,000
\(h_{s}\) \(\mathrm{mm}\) 0-5,000
\(\gamma_{s}\) \(\mathrm{kN/m³}\) 0-25
\(c_{c}\) \(\mathrm{mm}\) 25-150
\(f_{c}\) \(\mathrm{MPa}\) 20-80
\(f_{\mathrm{cc}}\) \(\mathrm{MPa}\) 20-80
\(f_{y}\) \(\mathrm{MPa}\) 300-500
\(\mathrm{layer}\)
\(\mathrm{bar}_{x}\)
\(n_{x}\) 2-200
\(\mathrm{bar}_{y}\)
\(n_{y}\) 2-200
Bf = 3,000 mm Lf = 3,600 mm lx = 1,275 mm ly = 1,350 mm x y
Plan, to scale: the column, the punching perimeter at d/2 from its faces, the one-way shear sections at d_v from them, and the kern with a dot at the factored resultant, each dashed, green where its check passes and red where it fails.
Bf = 3,000 mm tf = 1,400 mm bc = 450 mm Pu = 7,575 kN Asx = 10,500 mm², X bars Asy = 9,000 mm², Y bars dv = 1,170 mm qs = 698.7 kPa, gross que = 811.1 kPa, effective qu = 855.7 kPa, net
Section along x on the column's centre line, to scale: the X bars along the cut, the Y bars cut, the one-way shear sections d_v from the column faces, and the net factored bearing along the cut.

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This printed copy omits the derivation. The full report, with every step, is the PDF at https://calc.struct.work/calc/pad-footing-capacity.pdf.

Checks

Check D/C Result
Column fits x\(\htmlClass{sym-b_c}{b_{c}} < \htmlClass{sym-B_f}{B_{f}} \quad \Rightarrow \quad \htmlClass{sym-b_c}{450\ \mathrm{mm}} < \htmlClass{sym-B_f}{3000\ \mathrm{mm}}\) 0.15 PASS
Column fits y\(\htmlClass{sym-h_c}{h_{c}} < \htmlClass{sym-L_f}{L_{f}} \quad \Rightarrow \quad \htmlClass{sym-h_c}{900\ \mathrm{mm}} < \htmlClass{sym-L_f}{3600\ \mathrm{mm}}\) 0.25 PASS
Bearing service\(\htmlClass{sym-q_s}{q_{s}} \leq \htmlClass{sym-q_a}{q_{a}} \quad \Rightarrow \quad \htmlClass{sym-q_s}{698.7\ \mathrm{kPa}} \leq \htmlClass{sym-q_a}{950\ \mathrm{kPa}}\) 0.74 PASS
Contact service\(\htmlClass{sym-q_s_min}{q_{s,\mathrm{min}}} \geq 0\ \mathrm{Pa} \quad \Rightarrow \quad \htmlClass{sym-q_s_min}{464.2\ \mathrm{kPa}} \geq 0\ \mathrm{Pa}\) 0.00 PASS
Bearing factored\(\htmlClass{sym-q_ue}{q_{\mathrm{ue}}} \leq \htmlClass{sym-q_ult}{q_{\mathrm{ult}}} \quad \Rightarrow \quad \htmlClass{sym-q_ue}{811.1\ \mathrm{kPa}} \leq \htmlClass{sym-q_ult}{1400\ \mathrm{kPa}}\) 0.58 PASS
Contact factored\(\htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}} \geq 0\ \mathrm{Pa} \quad \Rightarrow \quad \htmlClass{sym-q_u_min}{547.1\ \mathrm{kPa}} \geq 0\ \mathrm{Pa}\) 0.00 PASS
Concrete bearing\(\htmlClass{sym-P_u}{P_{u}} \leq \htmlClass{sym-B_r}{B_{r}} \quad \Rightarrow \quad \htmlClass{sym-P_u}{7.575\ \mathrm{MN}} \leq \htmlClass{sym-B_r}{15.66\ \mathrm{MN}}\) 0.48 PASS
Bearing column\(\htmlClass{sym-P_u}{P_{u}} \leq \htmlClass{sym-B_rc}{B_{\mathrm{rc}}} \quad \Rightarrow \quad \htmlClass{sym-P_u}{7.575\ \mathrm{MN}} \leq \htmlClass{sym-B_rc}{7.832\ \mathrm{MN}}\) 0.97 PASS
Punching adequate\(\htmlClass{sym-v_u}{v_{u}} \leq \htmlClass{sym-v_c}{v_{c}} \quad \Rightarrow \quad \htmlClass{sym-v_u}{522.8\ \mathrm{kPa}} \leq \htmlClass{sym-v_c}{826\ \mathrm{kPa}}\) 0.63 PASS
Shear x adequate\(\htmlClass{sym-V_ux}{V_{\mathrm{ux}}} \leq \htmlClass{sym-V_cx}{V_{\mathrm{cx}}} \quad \Rightarrow \quad \htmlClass{sym-V_ux}{301.1\ \mathrm{kN}} \leq \htmlClass{sym-V_cx}{3.401\ \mathrm{MN}}\) 0.09 PASS
Shear y adequate\(\htmlClass{sym-V_uy}{V_{\mathrm{uy}}} \leq \htmlClass{sym-V_cy}{V_{\mathrm{cy}}} \quad \Rightarrow \quad \htmlClass{sym-V_uy}{408.4\ \mathrm{kN}} \leq \htmlClass{sym-V_cy}{2.834\ \mathrm{MN}}\) 0.14 PASS
Flexure x adequate\(\htmlClass{sym-M_fx}{M_{\mathrm{fx}}} \leq \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} \quad \Rightarrow \quad \htmlClass{sym-M_fx}{2.256\ \mathrm{MN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_rx}{4.588\ \mathrm{MN} \cdot \mathrm{m}}\) 0.49 PASS
Flexure y adequate\(\htmlClass{sym-M_fy}{M_{\mathrm{fy}}} \leq \htmlClass{sym-M_ry}{M_{\mathrm{ry}}} \quad \Rightarrow \quad \htmlClass{sym-M_fy}{2.034\ \mathrm{MN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_ry}{3.853\ \mathrm{MN} \cdot \mathrm{m}}\) 0.53 PASS
Minimum steel x\(\htmlClass{sym-A_sx}{A_{\mathrm{sx}}} \geq \htmlClass{sym-A_sx_min}{A_{\mathrm{sx},\mathrm{min}}} \quad \Rightarrow \quad \htmlClass{sym-A_sx}{10500\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_sx_min}{10080\ \mathrm{mm}^{2}}\) 0.96 PASS
Minimum steel y\(\htmlClass{sym-A_sy}{A_{\mathrm{sy}}} \geq \htmlClass{sym-A_sy_min}{A_{\mathrm{sy},\mathrm{min}}} \quad \Rightarrow \quad \htmlClass{sym-A_sy}{9000\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_sy_min}{8400\ \mathrm{mm}^{2}}\) 0.93 PASS
Steel yields x\(\frac{\htmlClass{sym-c_x}{c_{x}}}{\htmlClass{sym-d_x}{d_{x}}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{f_{y}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c_x}{61.94\ \mathrm{mm}}}{\htmlClass{sym-d_x}{1.312\ \mathrm{m}}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\) 0.07 PASS
Steel yields y\(\frac{\htmlClass{sym-c_y}{c_{y}}}{\htmlClass{sym-d_y}{d_{y}}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{f_{y}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c_y}{63.7\ \mathrm{mm}}}{\htmlClass{sym-d_y}{1.287\ \mathrm{m}}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\) 0.08 PASS
Spacing x\(\htmlClass{sym-s_x}{s_{x}} \leq \htmlClass{sym-s_max}{s_{\mathrm{max}}} \quad \Rightarrow \quad \htmlClass{sym-s_x}{172.5\ \mathrm{mm}} \leq \htmlClass{sym-s_max}{500\ \mathrm{mm}}\) 0.34 PASS
Spacing y\(\htmlClass{sym-s_y}{s_{y}} \leq \htmlClass{sym-s_max}{s_{\mathrm{max}}} \quad \Rightarrow \quad \htmlClass{sym-s_y}{167.6\ \mathrm{mm}} \leq \htmlClass{sym-s_max}{500\ \mathrm{mm}}\) 0.34 PASS
Development x\(\htmlClass{sym-l_dx}{l_{\mathrm{dx}}} \leq \htmlClass{sym-l_ax}{l_{\mathrm{ax}}} \quad \Rightarrow \quad \htmlClass{sym-l_dx}{766.7\ \mathrm{mm}} \leq \htmlClass{sym-l_ax}{1.2\ \mathrm{m}}\) 0.64 PASS
Development y\(\htmlClass{sym-l_dy}{l_{\mathrm{dy}}} \leq \htmlClass{sym-l_ay}{l_{\mathrm{ay}}} \quad \Rightarrow \quad \htmlClass{sym-l_dy}{766.7\ \mathrm{mm}} \leq \htmlClass{sym-l_ay}{1.275\ \mathrm{m}}\) 0.60 PASS

Results

Quantity Description Value Unit
\(\htmlClass{sym-q_s}{q_{s}}\) Service bearing pressure, gross, worst corner 698.7 \(\mathrm{kPa}\)
\(\htmlClass{sym-q_ue}{q_{\mathrm{ue}}}\) Factored bearing pressure, gross, on the effective area 811.1 \(\mathrm{kPa}\)
\(\htmlClass{sym-q_u}{q_{u}}\) Factored net pressure, worst corner 855.7 \(\mathrm{kPa}\)
\(\htmlClass{sym-v_u}{v_{u}}\) Factored punching shear stress 522.8 \(\mathrm{kPa}\)
\(\htmlClass{sym-v_c}{v_{c}}\) Factored punching shear resistance 826 \(\mathrm{kPa}\)
\(\htmlClass{sym-V_cx}{V_{\mathrm{cx}}}\) Factored one-way shear resistance, x 3.401 \(\mathrm{MN}\)
\(\htmlClass{sym-V_cy}{V_{\mathrm{cy}}}\) Factored one-way shear resistance, y 2.834 \(\mathrm{MN}\)
\(\htmlClass{sym-M_rx}{M_{\mathrm{rx}}}\) Factored flexural resistance, x 4.588 \(\mathrm{MN} \cdot \mathrm{m}\)
\(\htmlClass{sym-M_ry}{M_{\mathrm{ry}}}\) Factored flexural resistance, y 3.853 \(\mathrm{MN} \cdot \mathrm{m}\)
\(\htmlClass{sym-B_r}{B_{r}}\) Factored concrete bearing resistance, footing 15.66 \(\mathrm{MN}\)
\(\htmlClass{sym-B_rc}{B_{\mathrm{rc}}}\) Factored concrete bearing resistance, column 7.832governs \(\mathrm{MN}\)
\(\htmlClass{sym-A_dowel}{A_{\mathrm{dowel}}}\) Minimum dowel area across the interface 2025 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-l_dx}{l_{\mathrm{dx}}}\) Development length, X bars 766.7 \(\mathrm{mm}\)
\(\htmlClass{sym-l_dy}{l_{\mathrm{dy}}}\) Development length, Y bars 766.7 \(\mathrm{mm}\)
\(\htmlClass{sym-A_band}{A_{\mathrm{band}}}\) Short-direction steel within the central band 9545 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-w_band}{w_{\mathrm{band}}}\) Width of the central band 3 \(\mathrm{m}\)

Derivation

S.1 \[\begin{aligned} \htmlClass{sym-phi_c}{\phi_{c}} &= 0.65 \quad \left(\text{Resistance factor, concrete | CSA A23.3 Cl. 8.4.2}\right) \end{aligned}\]
S.2 \[\begin{aligned} \htmlClass{sym-phi_s}{\phi_{s}} &= 0.85 \quad \left(\text{Resistance factor, reinforcement | CSA A23.3 Cl. 8.4.3}\right) \end{aligned}\]
S.3 \[\begin{aligned} \htmlClass{sym-gamma_c}{\gamma_{c}} &= 24\ \mathrm{kN/m³} \quad \left(\text{Concrete unit weight}\right) \end{aligned}\]
S.4 \[\begin{aligned} \htmlClass{sym-P_s}{P_{s}} &= \htmlClass{sym-D}{D} + \htmlClass{sym-L}{L} \\ &= \htmlClass{sym-D}{4500\ \mathrm{kN}} + \htmlClass{sym-L}{1300\ \mathrm{kN}} \\ &= 5.8\ \mathrm{MN} \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-M_sx}{M_{\mathrm{sx}}} &= \htmlClass{sym-M_Dx}{M_{\mathrm{Dx}}} + \htmlClass{sym-M_Lx}{M_{\mathrm{Lx}}} \\ &= \htmlClass{sym-M_Dx}{200\ \mathrm{kN} \cdot \mathrm{m}} + \htmlClass{sym-M_Lx}{80\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 280\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.6 \[\begin{aligned} \htmlClass{sym-M_sy}{M_{\mathrm{sy}}} &= \htmlClass{sym-M_Dy}{M_{\mathrm{Dy}}} + \htmlClass{sym-M_Ly}{M_{\mathrm{Ly}}} \\ &= \htmlClass{sym-M_Dy}{300\ \mathrm{kN} \cdot \mathrm{m}} + \htmlClass{sym-M_Ly}{100\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 400\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.7

NBCC Table 4.1.3.2.A

\[\begin{aligned} \htmlClass{sym-P_u}{P_{u}} &= \max\left(1.4 \cdot \htmlClass{sym-D}{D}, 1.25 \cdot \htmlClass{sym-D}{D} + 1.5 \cdot \htmlClass{sym-L}{L}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-D}{4500\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-D}{4500\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-L}{1300\ \mathrm{kN}}\right) \\ &= 7.575\ \mathrm{MN} \end{aligned}\]
S.8 \[\begin{aligned} \htmlClass{sym-M_ux}{M_{\mathrm{ux}}} &= \max\left(1.4 \cdot \htmlClass{sym-M_Dx}{M_{\mathrm{Dx}}}, 1.25 \cdot \htmlClass{sym-M_Dx}{M_{\mathrm{Dx}}} + 1.5 \cdot \htmlClass{sym-M_Lx}{M_{\mathrm{Lx}}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-M_Dx}{200\ \mathrm{kN} \cdot \mathrm{m}}, 1.25 \cdot \htmlClass{sym-M_Dx}{200\ \mathrm{kN} \cdot \mathrm{m}} + 1.5 \cdot \htmlClass{sym-M_Lx}{80\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 370\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

Assumed: each at its own worst combination

S.9 \[\begin{aligned} \htmlClass{sym-M_uy}{M_{\mathrm{uy}}} &= \max\left(1.4 \cdot \htmlClass{sym-M_Dy}{M_{\mathrm{Dy}}}, 1.25 \cdot \htmlClass{sym-M_Dy}{M_{\mathrm{Dy}}} + 1.5 \cdot \htmlClass{sym-M_Ly}{M_{\mathrm{Ly}}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-M_Dy}{300\ \mathrm{kN} \cdot \mathrm{m}}, 1.25 \cdot \htmlClass{sym-M_Dy}{300\ \mathrm{kN} \cdot \mathrm{m}} + 1.5 \cdot \htmlClass{sym-M_Ly}{100\ \mathrm{kN} \cdot \mathrm{m}}\right) \\ &= 525\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

Assumed: dead and live moments act in the same sense

S.10 \[\begin{aligned} \htmlClass{sym-d_bx}{d_{\mathrm{bx}}} &= 25.2\ \mathrm{mm} \quad \left(\text{X bar diameter | 25M}\right) \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-d_by}{d_{\mathrm{by}}} &= 25.2\ \mathrm{mm} \quad \left(\text{Y bar diameter | 25M}\right) \end{aligned}\]
S.12 \[\begin{aligned} \htmlClass{sym-d}{d} &= \htmlClass{sym-t_f}{t_{f}} - \htmlClass{sym-c_c}{c_{c}} - \htmlClass{sym-d_bx}{d_{\mathrm{bx}}} \\ &= \htmlClass{sym-t_f}{1400\ \mathrm{mm}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} - \htmlClass{sym-d_bx}{25.2\ \mathrm{mm}} \\ &= 1.3\ \mathrm{m} \end{aligned}\]
S.13 \[\begin{aligned} \htmlClass{sym-d_x}{d_{x}} &= \htmlClass{sym-t_f}{t_{f}} - \htmlClass{sym-c_c}{c_{c}} - \frac{\htmlClass{sym-d_bx}{d_{\mathrm{bx}}}}{2} \\ &= \htmlClass{sym-t_f}{1400\ \mathrm{mm}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} - \frac{\htmlClass{sym-d_bx}{25.2\ \mathrm{mm}}}{2} \\ &= 1.312\ \mathrm{m} \end{aligned}\]
S.14 \[\begin{aligned} \htmlClass{sym-d_y}{d_{y}} &= \htmlClass{sym-t_f}{t_{f}} - \htmlClass{sym-c_c}{c_{c}} - \htmlClass{sym-d_bx}{d_{\mathrm{bx}}} - \frac{\htmlClass{sym-d_by}{d_{\mathrm{by}}}}{2} \\ &= \htmlClass{sym-t_f}{1400\ \mathrm{mm}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} - \htmlClass{sym-d_bx}{25.2\ \mathrm{mm}} - \frac{\htmlClass{sym-d_by}{25.2\ \mathrm{mm}}}{2} \\ &= 1.287\ \mathrm{m} \end{aligned}\]
S.15

CSA A23.3 Cl. 3.2

\[\begin{aligned} \htmlClass{sym-d_v}{d_{v}} &= \max\left(0.9 \cdot \htmlClass{sym-d}{d}, 0.72 \cdot \htmlClass{sym-t_f}{t_{f}}\right) \\ &= \max\left(0.9 \cdot \htmlClass{sym-d}{1.3\ \mathrm{m}}, 0.72 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}}\right) \\ &= 1.17\ \mathrm{m} \end{aligned}\]
S.16 \[\begin{aligned} \htmlClass{sym-l_x}{l_{x}} &= \frac{\htmlClass{sym-B_f}{B_{f}} - \htmlClass{sym-b_c}{b_{c}}}{2} \\ &= \frac{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} - \htmlClass{sym-b_c}{450\ \mathrm{mm}}}{2} \\ &= 1.275\ \mathrm{m} \end{aligned}\]
S.17 \[\begin{aligned} \htmlClass{sym-l_y}{l_{y}} &= \frac{\htmlClass{sym-L_f}{L_{f}} - \htmlClass{sym-h_c}{h_{c}}}{2} \\ &= \frac{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} - \htmlClass{sym-h_c}{900\ \mathrm{mm}}}{2} \\ &= 1.35\ \mathrm{m} \end{aligned}\]
S.18

CSA A23.3 Cl. 11.3.4

\[\begin{aligned} \htmlClass{sym-f_v}{f_{v}} &= \min\left(\sqrt{\htmlClass{sym-f_c}{f_{c}} \cdot 1\ \mathrm{MPa}}, 8\ \mathrm{MPa}\right) \\ &= \min\left(\sqrt{\htmlClass{sym-f_c}{35\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}}, 8\ \mathrm{MPa}\right) \\ &= 5.916\ \mathrm{MPa} \end{aligned}\]
S.19 \[\begin{aligned} \htmlClass{sym-A_f}{A_{f}} &= \htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}} \\ &= \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}} \\ &= 10.8\ \mathrm{m}^{2} \end{aligned}\]
S.20 \[\begin{aligned} \htmlClass{sym-W}{W} &= \left(\htmlClass{sym-gamma_s}{\gamma_{s}} \cdot \htmlClass{sym-h_s}{h_{s}} + \htmlClass{sym-gamma_c}{\gamma_{c}} \cdot \htmlClass{sym-t_f}{t_{f}}\right) \cdot \htmlClass{sym-A_f}{A_{f}} \\ &= \left(\htmlClass{sym-gamma_s}{18\ \mathrm{kN/m³}} \cdot \htmlClass{sym-h_s}{600\ \mathrm{mm}} + \htmlClass{sym-gamma_c}{24\ \mathrm{kN/m³}} \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}}\right) \cdot \htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}} \\ &= 479.5\ \mathrm{kN} \end{aligned}\]
S.21 \[\begin{aligned} \htmlClass{sym-q_s}{q_{s}} &= \frac{\htmlClass{sym-P_s}{P_{s}} + \htmlClass{sym-W}{W}}{\htmlClass{sym-A_f}{A_{f}}} + \frac{6 \cdot \htmlClass{sym-M_sy}{M_{\mathrm{sy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} + \frac{6 \cdot \htmlClass{sym-M_sx}{M_{\mathrm{sx}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_s}{5.8\ \mathrm{MN}} + \htmlClass{sym-W}{479.5\ \mathrm{kN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} + \frac{6 \cdot \htmlClass{sym-M_sy}{400\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} + \frac{6 \cdot \htmlClass{sym-M_sx}{280\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 698.7\ \mathrm{kPa} \end{aligned}\]
S.22 \[\begin{aligned} \htmlClass{sym-q_s_min}{q_{s,\mathrm{min}}} &= \frac{\htmlClass{sym-P_s}{P_{s}} + \htmlClass{sym-W}{W}}{\htmlClass{sym-A_f}{A_{f}}} - \frac{6 \cdot \htmlClass{sym-M_sy}{M_{\mathrm{sy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} - \frac{6 \cdot \htmlClass{sym-M_sx}{M_{\mathrm{sx}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_s}{5.8\ \mathrm{MN}} + \htmlClass{sym-W}{479.5\ \mathrm{kN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} - \frac{6 \cdot \htmlClass{sym-M_sy}{400\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} - \frac{6 \cdot \htmlClass{sym-M_sx}{280\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 464.2\ \mathrm{kPa} \end{aligned}\]
S.23

NBCC Table 4.1.3.2.A

\[\begin{aligned} \htmlClass{sym-P_ug}{P_{\mathrm{ug}}} &= \max\left(1.4 \cdot \left(\htmlClass{sym-D}{D} + \htmlClass{sym-W}{W}\right), 1.25 \cdot \left(\htmlClass{sym-D}{D} + \htmlClass{sym-W}{W}\right) + 1.5 \cdot \htmlClass{sym-L}{L}\right) \\ &= \max\left(1.4 \cdot \left(\htmlClass{sym-D}{4500\ \mathrm{kN}} + \htmlClass{sym-W}{479.5\ \mathrm{kN}}\right), 1.25 \cdot \left(\htmlClass{sym-D}{4500\ \mathrm{kN}} + \htmlClass{sym-W}{479.5\ \mathrm{kN}}\right) + 1.5 \cdot \htmlClass{sym-L}{1300\ \mathrm{kN}}\right) \\ &= 8.174\ \mathrm{MN} \end{aligned}\]
S.24 \[\begin{aligned} \htmlClass{sym-e_x}{e_{x}} &= \frac{\htmlClass{sym-M_uy}{M_{\mathrm{uy}}}}{\htmlClass{sym-P_ug}{P_{\mathrm{ug}}}} \\ &= \frac{\htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-P_ug}{8.174\ \mathrm{MN}}} \\ &= 64.22\ \mathrm{mm} \end{aligned}\]
S.25 \[\begin{aligned} \htmlClass{sym-e_y}{e_{y}} &= \frac{\htmlClass{sym-M_ux}{M_{\mathrm{ux}}}}{\htmlClass{sym-P_ug}{P_{\mathrm{ug}}}} \\ &= \frac{\htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-P_ug}{8.174\ \mathrm{MN}}} \\ &= 45.26\ \mathrm{mm} \end{aligned}\]
S.26 \[\begin{aligned} \htmlClass{sym-B_e}{B_{e}} &= \htmlClass{sym-B_f}{B_{f}} - 2 \cdot \htmlClass{sym-e_x}{e_{x}} \\ &= \htmlClass{sym-B_f}{3000\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-e_x}{64.22\ \mathrm{mm}} \\ &= 2.872\ \mathrm{m} \end{aligned}\]
S.27 \[\begin{aligned} \htmlClass{sym-L_e}{L_{e}} &= \htmlClass{sym-L_f}{L_{f}} - 2 \cdot \htmlClass{sym-e_y}{e_{y}} \\ &= \htmlClass{sym-L_f}{3600\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-e_y}{45.26\ \mathrm{mm}} \\ &= 3.509\ \mathrm{m} \end{aligned}\]
S.28 \[\begin{aligned} \htmlClass{sym-q_ue}{q_{\mathrm{ue}}} &= \frac{\htmlClass{sym-P_ug}{P_{\mathrm{ug}}}}{\htmlClass{sym-B_e}{B_{e}} \cdot \htmlClass{sym-L_e}{L_{e}}} \\ &= \frac{\htmlClass{sym-P_ug}{8.174\ \mathrm{MN}}}{\htmlClass{sym-B_e}{2.872\ \mathrm{m}} \cdot \htmlClass{sym-L_e}{3.509\ \mathrm{m}}} \\ &= 811.1\ \mathrm{kPa} \end{aligned}\]
S.29 \[\begin{aligned} \htmlClass{sym-q_ux}{q_{\mathrm{ux}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} + \frac{6 \cdot \htmlClass{sym-M_uy}{M_{\mathrm{uy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} + \frac{6 \cdot \htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} \\ &= 798.6\ \mathrm{kPa} \end{aligned}\]
S.30 \[\begin{aligned} \htmlClass{sym-q_ux_min}{q_{\mathrm{ux},\mathrm{min}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} - \frac{6 \cdot \htmlClass{sym-M_uy}{M_{\mathrm{uy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} - \frac{6 \cdot \htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} \\ &= 604.2\ \mathrm{kPa} \end{aligned}\]
S.31 \[\begin{aligned} \htmlClass{sym-q_uy}{q_{\mathrm{uy}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} + \frac{6 \cdot \htmlClass{sym-M_ux}{M_{\mathrm{ux}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} + \frac{6 \cdot \htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 758.5\ \mathrm{kPa} \end{aligned}\]
S.32 \[\begin{aligned} \htmlClass{sym-q_uy_min}{q_{\mathrm{uy},\mathrm{min}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} - \frac{6 \cdot \htmlClass{sym-M_ux}{M_{\mathrm{ux}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} - \frac{6 \cdot \htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 644.3\ \mathrm{kPa} \end{aligned}\]
S.33 \[\begin{aligned} \htmlClass{sym-q_u}{q_{u}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} + \frac{6 \cdot \htmlClass{sym-M_uy}{M_{\mathrm{uy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} + \frac{6 \cdot \htmlClass{sym-M_ux}{M_{\mathrm{ux}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} + \frac{6 \cdot \htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} + \frac{6 \cdot \htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 855.7\ \mathrm{kPa} \end{aligned}\]
S.34 \[\begin{aligned} \htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} - \frac{6 \cdot \htmlClass{sym-M_uy}{M_{\mathrm{uy}}}}{\htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-B_f}{B_{f}}^{2}} - \frac{6 \cdot \htmlClass{sym-M_ux}{M_{\mathrm{ux}}}}{\htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-L_f}{L_{f}}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} - \frac{6 \cdot \htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right)^{2}} - \frac{6 \cdot \htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)^{2}} \\ &= 547.1\ \mathrm{kPa} \end{aligned}\]
S.35 \[\begin{aligned} \htmlClass{sym-A_c}{A_{c}} &= \htmlClass{sym-b_c}{b_{c}} \cdot \htmlClass{sym-h_c}{h_{c}} \\ &= \htmlClass{sym-b_c}{450\ \mathrm{mm}} \cdot \htmlClass{sym-h_c}{900\ \mathrm{mm}} \\ &= 405000\ \mathrm{mm}^{2} \end{aligned}\]
S.36 \[\begin{aligned} \htmlClass{sym-A_2}{A_{2}} &= \min\left(\htmlClass{sym-b_c}{b_{c}} + 4 \cdot \htmlClass{sym-t_f}{t_{f}}, \htmlClass{sym-B_f}{B_{f}}\right) \cdot \min\left(\htmlClass{sym-h_c}{h_{c}} + 4 \cdot \htmlClass{sym-t_f}{t_{f}}, \htmlClass{sym-L_f}{L_{f}}\right) \\ &= \min\left(\htmlClass{sym-b_c}{450\ \mathrm{mm}} + 4 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}}, \htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right) \cdot \min\left(\htmlClass{sym-h_c}{900\ \mathrm{mm}} + 4 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}}, \htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right) \\ &= 10.8\ \mathrm{m}^{2} \end{aligned}\]
S.37

CSA A23.3 Cl. 10.8.1

\[\begin{aligned} \htmlClass{sym-B_r}{B_{r}} &= 0.85 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \cdot \htmlClass{sym-A_c}{A_{c}} \cdot \min\left(\sqrt{\frac{\htmlClass{sym-A_2}{A_{2}}}{\htmlClass{sym-A_c}{A_{c}}}}, 2\right) \\ &= 0.85 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-A_c}{405000\ \mathrm{mm}^{2}} \cdot \min\left(\sqrt{\frac{\htmlClass{sym-A_2}{10.8\ \mathrm{m}^{2}}}{\htmlClass{sym-A_c}{405000\ \mathrm{mm}^{2}}}}, 2\right) \\ &= 15.66\ \mathrm{MN} \end{aligned}\]
S.38

CSA A23.3 Cl. 10.8.1

\[\begin{aligned} \htmlClass{sym-B_rc}{B_{\mathrm{rc}}} &= 0.85 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cc}{f_{\mathrm{cc}}} \cdot \htmlClass{sym-A_c}{A_{c}} \\ &= 0.85 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cc}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-A_c}{405000\ \mathrm{mm}^{2}} \\ &= 7.832\ \mathrm{MN} \end{aligned}\]
S.39 \[\begin{aligned} \htmlClass{sym-B_min}{B_{\mathrm{min}}} &= \min\left(\htmlClass{sym-B_r}{B_{r}}, \htmlClass{sym-B_rc}{B_{\mathrm{rc}}}\right) \\ &= \min\left(\htmlClass{sym-B_r}{15.66\ \mathrm{MN}}, \htmlClass{sym-B_rc}{7.832\ \mathrm{MN}}\right) \\ &= 7.832\ \mathrm{MN} \end{aligned}\]
S.40

CSA A23.3 Cl. 15.9.2

\[\begin{aligned} \htmlClass{sym-A_dowel}{A_{\mathrm{dowel}}} &= \max\left(0.005 \cdot \htmlClass{sym-A_c}{A_{c}}, \frac{\htmlClass{sym-P_u}{P_{u}} - \htmlClass{sym-B_min}{B_{\mathrm{min}}}}{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-f_y}{f_{y}}}\right) \\ &= \max\left(0.005 \cdot \htmlClass{sym-A_c}{405000\ \mathrm{mm}^{2}}, \frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}} - \htmlClass{sym-B_min}{7.832\ \mathrm{MN}}}{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\right) \\ &= 2025\ \mathrm{mm}^{2} \end{aligned}\]
S.41 \[\begin{aligned} \htmlClass{sym-b_1}{b_{1}} &= \min\left(\htmlClass{sym-b_c}{b_{c}} + \htmlClass{sym-d}{d}, \htmlClass{sym-B_f}{B_{f}}\right) \\ &= \min\left(\htmlClass{sym-b_c}{450\ \mathrm{mm}} + \htmlClass{sym-d}{1.3\ \mathrm{m}}, \htmlClass{sym-B_f}{3000\ \mathrm{mm}}\right) \\ &= 1.75\ \mathrm{m} \end{aligned}\]
S.42 \[\begin{aligned} \htmlClass{sym-b_2}{b_{2}} &= \min\left(\htmlClass{sym-h_c}{h_{c}} + \htmlClass{sym-d}{d}, \htmlClass{sym-L_f}{L_{f}}\right) \\ &= \min\left(\htmlClass{sym-h_c}{900\ \mathrm{mm}} + \htmlClass{sym-d}{1.3\ \mathrm{m}}, \htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right) \\ &= 2.2\ \mathrm{m} \end{aligned}\]
S.43

CSA A23.3 Cl. 13.3.3

\[\begin{aligned} \htmlClass{sym-b_o}{b_{o}} &= 2 \cdot \left(\htmlClass{sym-b_1}{b_{1}} + \htmlClass{sym-b_2}{b_{2}}\right) \\ &= 2 \cdot \left(\htmlClass{sym-b_1}{1.75\ \mathrm{m}} + \htmlClass{sym-b_2}{2.2\ \mathrm{m}}\right) \\ &= 7.899\ \mathrm{m} \end{aligned}\]
S.44 \[\begin{aligned} \htmlClass{sym-v_d}{v_{d}} &= \frac{\frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-A_f}{A_{f}}} \cdot \left(\htmlClass{sym-A_f}{A_{f}} - \htmlClass{sym-b_1}{b_{1}} \cdot \htmlClass{sym-b_2}{b_{2}}\right)}{\htmlClass{sym-b_o}{b_{o}} \cdot \htmlClass{sym-d}{d}} \\ &= \frac{\frac{\htmlClass{sym-P_u}{7.575\ \mathrm{MN}}}{\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}}} \cdot \left(\htmlClass{sym-A_f}{10.8\ \mathrm{m}^{2}} - \htmlClass{sym-b_1}{1.75\ \mathrm{m}} \cdot \htmlClass{sym-b_2}{2.2\ \mathrm{m}}\right)}{\htmlClass{sym-b_o}{7.899\ \mathrm{m}} \cdot \htmlClass{sym-d}{1.3\ \mathrm{m}}} \\ &= 474.8\ \mathrm{kPa} \end{aligned}\]
S.45

CSA A23.3 Cl. 13.3.5.3

\[\begin{aligned} \htmlClass{sym-gamma_vy}{\gamma_{\mathrm{vy}}} &= 1 - \frac{1}{1 + \frac{2}{3} \cdot \sqrt{\frac{\htmlClass{sym-b_1}{b_{1}}}{\htmlClass{sym-b_2}{b_{2}}}}} \\ &= 1 - \frac{1}{1 + \frac{2}{3} \cdot \sqrt{\frac{\htmlClass{sym-b_1}{1.75\ \mathrm{m}}}{\htmlClass{sym-b_2}{2.2\ \mathrm{m}}}}} \\ &= 0.3729 \end{aligned}\]
S.46

CSA A23.3 Cl. 13.3.5.5

\[\begin{aligned} \htmlClass{sym-J_y}{J_{y}} &= \frac{\htmlClass{sym-b_1}{b_{1}} \cdot \htmlClass{sym-d}{d}^{3}}{6} + \frac{\htmlClass{sym-d}{d} \cdot \htmlClass{sym-b_1}{b_{1}}^{3}}{6} + \frac{\htmlClass{sym-b_2}{b_{2}} \cdot \htmlClass{sym-d}{d} \cdot \htmlClass{sym-b_1}{b_{1}}^{2}}{2} \\ &= \frac{\htmlClass{sym-b_1}{1.75\ \mathrm{m}} \cdot \left(\htmlClass{sym-d}{1.3\ \mathrm{m}}\right)^{3}}{6} + \frac{\htmlClass{sym-d}{1.3\ \mathrm{m}} \cdot \left(\htmlClass{sym-b_1}{1.75\ \mathrm{m}}\right)^{3}}{6} + \frac{\htmlClass{sym-b_2}{2.2\ \mathrm{m}} \cdot \htmlClass{sym-d}{1.3\ \mathrm{m}} \cdot \left(\htmlClass{sym-b_1}{1.75\ \mathrm{m}}\right)^{2}}{2} \\ &= 6.178\ \mathrm{m}^{4} \end{aligned}\]
S.47 \[\begin{aligned} \htmlClass{sym-v_my}{v_{\mathrm{my}}} &= \frac{\htmlClass{sym-gamma_vy}{\gamma_{\mathrm{vy}}} \cdot \htmlClass{sym-M_uy}{M_{\mathrm{uy}}} \cdot \frac{\htmlClass{sym-b_1}{b_{1}}}{2}}{\htmlClass{sym-J_y}{J_{y}}} \\ &= \frac{\htmlClass{sym-gamma_vy}{0.3729} \cdot \htmlClass{sym-M_uy}{525\ \mathrm{kN} \cdot \mathrm{m}} \cdot \frac{\htmlClass{sym-b_1}{1.75\ \mathrm{m}}}{2}}{\htmlClass{sym-J_y}{6.178\ \mathrm{m}^{4}}} \\ &= 27.72\ \mathrm{kPa} \end{aligned}\]
S.48

CSA A23.3 Cl. 13.3.5.3

\[\begin{aligned} \htmlClass{sym-gamma_vx}{\gamma_{\mathrm{vx}}} &= 1 - \frac{1}{1 + \frac{2}{3} \cdot \sqrt{\frac{\htmlClass{sym-b_2}{b_{2}}}{\htmlClass{sym-b_1}{b_{1}}}}} \\ &= 1 - \frac{1}{1 + \frac{2}{3} \cdot \sqrt{\frac{\htmlClass{sym-b_2}{2.2\ \mathrm{m}}}{\htmlClass{sym-b_1}{1.75\ \mathrm{m}}}}} \\ &= 0.4278 \end{aligned}\]
S.49

CSA A23.3 Cl. 13.3.5.5

\[\begin{aligned} \htmlClass{sym-J_x}{J_{x}} &= \frac{\htmlClass{sym-b_2}{b_{2}} \cdot \htmlClass{sym-d}{d}^{3}}{6} + \frac{\htmlClass{sym-d}{d} \cdot \htmlClass{sym-b_2}{b_{2}}^{3}}{6} + \frac{\htmlClass{sym-b_1}{b_{1}} \cdot \htmlClass{sym-d}{d} \cdot \htmlClass{sym-b_2}{b_{2}}^{2}}{2} \\ &= \frac{\htmlClass{sym-b_2}{2.2\ \mathrm{m}} \cdot \left(\htmlClass{sym-d}{1.3\ \mathrm{m}}\right)^{3}}{6} + \frac{\htmlClass{sym-d}{1.3\ \mathrm{m}} \cdot \left(\htmlClass{sym-b_2}{2.2\ \mathrm{m}}\right)^{3}}{6} + \frac{\htmlClass{sym-b_1}{1.75\ \mathrm{m}} \cdot \htmlClass{sym-d}{1.3\ \mathrm{m}} \cdot \left(\htmlClass{sym-b_2}{2.2\ \mathrm{m}}\right)^{2}}{2} \\ &= 8.614\ \mathrm{m}^{4} \end{aligned}\]
S.50 \[\begin{aligned} \htmlClass{sym-v_mx}{v_{\mathrm{mx}}} &= \frac{\htmlClass{sym-gamma_vx}{\gamma_{\mathrm{vx}}} \cdot \htmlClass{sym-M_ux}{M_{\mathrm{ux}}} \cdot \frac{\htmlClass{sym-b_2}{b_{2}}}{2}}{\htmlClass{sym-J_x}{J_{x}}} \\ &= \frac{\htmlClass{sym-gamma_vx}{0.4278} \cdot \htmlClass{sym-M_ux}{370\ \mathrm{kN} \cdot \mathrm{m}} \cdot \frac{\htmlClass{sym-b_2}{2.2\ \mathrm{m}}}{2}}{\htmlClass{sym-J_x}{8.614\ \mathrm{m}^{4}}} \\ &= 20.21\ \mathrm{kPa} \end{aligned}\]
S.51 \[\begin{aligned} \htmlClass{sym-v_u}{v_{u}} &= \htmlClass{sym-v_d}{v_{d}} + \htmlClass{sym-v_my}{v_{\mathrm{my}}} + \htmlClass{sym-v_mx}{v_{\mathrm{mx}}} \\ &= \htmlClass{sym-v_d}{474.8\ \mathrm{kPa}} + \htmlClass{sym-v_my}{27.72\ \mathrm{kPa}} + \htmlClass{sym-v_mx}{20.21\ \mathrm{kPa}} \\ &= 522.8\ \mathrm{kPa} \end{aligned}\]
S.52

CSA A23.3 Cl. 13.3.4.3

\[\begin{aligned} \htmlClass{sym-lambda_s}{\lambda_{s}} &= \min\left(1, \frac{1.3\ \mathrm{m}}{1\ \mathrm{m} + \htmlClass{sym-d}{d}}\right) \\ &= \min\left(1, \frac{1.3\ \mathrm{m}}{1\ \mathrm{m} + \htmlClass{sym-d}{1.3\ \mathrm{m}}}\right) \\ &= 0.5653 \end{aligned}\]
S.53 \[\begin{aligned} \htmlClass{sym-beta_c}{\beta_{c}} &= \frac{\max\left(\htmlClass{sym-b_c}{b_{c}}, \htmlClass{sym-h_c}{h_{c}}\right)}{\min\left(\htmlClass{sym-b_c}{b_{c}}, \htmlClass{sym-h_c}{h_{c}}\right)} \\ &= \frac{\max\left(\htmlClass{sym-b_c}{450\ \mathrm{mm}}, \htmlClass{sym-h_c}{900\ \mathrm{mm}}\right)}{\min\left(\htmlClass{sym-b_c}{450\ \mathrm{mm}}, \htmlClass{sym-h_c}{900\ \mathrm{mm}}\right)} \\ &= 2 \end{aligned}\]
S.54

CSA A23.3 Cl. 13.3.4.1

\[\begin{aligned} \htmlClass{sym-alpha}{\alpha} &= \min\left(0.19 \cdot \left(1 + \frac{2}{\htmlClass{sym-beta_c}{\beta_{c}}}\right), \frac{4 \cdot \htmlClass{sym-d}{d}}{\htmlClass{sym-b_o}{b_{o}}} + 0.19, 0.38\right) \\ &= \min\left(0.19 \cdot \left(1 + \frac{2}{\htmlClass{sym-beta_c}{2}}\right), \frac{4 \cdot \htmlClass{sym-d}{1.3\ \mathrm{m}}}{\htmlClass{sym-b_o}{7.899\ \mathrm{m}}} + 0.19, 0.38\right) \\ &= 0.38 \end{aligned}\]
S.55 \[\begin{aligned} \htmlClass{sym-v_c}{v_{c}} &= \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-lambda_s}{\lambda_{s}} \cdot \htmlClass{sym-alpha}{\alpha} \cdot \htmlClass{sym-f_v}{f_{v}} \\ &= \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-lambda_s}{0.5653} \cdot \htmlClass{sym-alpha}{0.38} \cdot \htmlClass{sym-f_v}{5.916\ \mathrm{MPa}} \\ &= 826\ \mathrm{kPa} \end{aligned}\]

Branch: \(l_{x} \leq 2 \cdot d_{v}\) held

S.56 \[\begin{aligned} \htmlClass{sym-beta_x}{\beta_{x}} &= 0.21 \quad \left(\text{Shallow section or short cantilever | CSA A23.3 Cl. 11.3.6.2}\right) \end{aligned}\]
S.57 \[\begin{aligned} \htmlClass{sym-x_vx}{x_{\mathrm{vx}}} &= \max\left(\htmlClass{sym-l_x}{l_{x}} - \htmlClass{sym-d_v}{d_{v}}, 0\ \mathrm{m}\right) \\ &= \max\left(\htmlClass{sym-l_x}{1.275\ \mathrm{m}} - \htmlClass{sym-d_v}{1.17\ \mathrm{m}}, 0\ \mathrm{m}\right) \\ &= 105.2\ \mathrm{mm} \end{aligned}\]
S.58 \[\begin{aligned} \htmlClass{sym-q_vx}{q_{\mathrm{vx}}} &= \htmlClass{sym-q_ux}{q_{\mathrm{ux}}} - \frac{\left(\htmlClass{sym-q_ux}{q_{\mathrm{ux}}} - \htmlClass{sym-q_ux_min}{q_{\mathrm{ux},\mathrm{min}}}\right) \cdot \htmlClass{sym-x_vx}{x_{\mathrm{vx}}}}{\htmlClass{sym-B_f}{B_{f}}} \\ &= \htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}} - \htmlClass{sym-q_ux_min}{604.2\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-x_vx}{105.2\ \mathrm{mm}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}}} \\ &= 791.8\ \mathrm{kPa} \end{aligned}\]
S.59 \[\begin{aligned} \htmlClass{sym-V_ux}{V_{\mathrm{ux}}} &= \frac{\htmlClass{sym-q_ux}{q_{\mathrm{ux}}} + \htmlClass{sym-q_vx}{q_{\mathrm{vx}}}}{2} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-x_vx}{x_{\mathrm{vx}}} \\ &= \frac{\htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}} + \htmlClass{sym-q_vx}{791.8\ \mathrm{kPa}}}{2} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \htmlClass{sym-x_vx}{105.2\ \mathrm{mm}} \\ &= 301.1\ \mathrm{kN} \end{aligned}\]
S.60

CSA A23.3 Cl. 11.3.4

\[\begin{aligned} \htmlClass{sym-V_cx}{V_{\mathrm{cx}}} &= \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-beta_x}{\beta_{x}} \cdot \htmlClass{sym-f_v}{f_{v}} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-beta_x}{0.21} \cdot \htmlClass{sym-f_v}{5.916\ \mathrm{MPa}} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{1.17\ \mathrm{m}} \\ &= 3.401\ \mathrm{MN} \end{aligned}\]

Branch: \(l_{y} \leq 2 \cdot d_{v}\) held

S.61 \[\begin{aligned} \htmlClass{sym-beta_y}{\beta_{y}} &= 0.21 \quad \left(\text{Shallow section or short cantilever | CSA A23.3 Cl. 11.3.6.2}\right) \end{aligned}\]
S.62 \[\begin{aligned} \htmlClass{sym-x_vy}{x_{\mathrm{vy}}} &= \max\left(\htmlClass{sym-l_y}{l_{y}} - \htmlClass{sym-d_v}{d_{v}}, 0\ \mathrm{m}\right) \\ &= \max\left(\htmlClass{sym-l_y}{1.35\ \mathrm{m}} - \htmlClass{sym-d_v}{1.17\ \mathrm{m}}, 0\ \mathrm{m}\right) \\ &= 180.2\ \mathrm{mm} \end{aligned}\]
S.63 \[\begin{aligned} \htmlClass{sym-q_vy}{q_{\mathrm{vy}}} &= \htmlClass{sym-q_uy}{q_{\mathrm{uy}}} - \frac{\left(\htmlClass{sym-q_uy}{q_{\mathrm{uy}}} - \htmlClass{sym-q_uy_min}{q_{\mathrm{uy},\mathrm{min}}}\right) \cdot \htmlClass{sym-x_vy}{x_{\mathrm{vy}}}}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}} - \htmlClass{sym-q_uy_min}{644.3\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-x_vy}{180.2\ \mathrm{mm}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}}} \\ &= 752.8\ \mathrm{kPa} \end{aligned}\]
S.64 \[\begin{aligned} \htmlClass{sym-V_uy}{V_{\mathrm{uy}}} &= \frac{\htmlClass{sym-q_uy}{q_{\mathrm{uy}}} + \htmlClass{sym-q_vy}{q_{\mathrm{vy}}}}{2} \cdot \htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-x_vy}{x_{\mathrm{vy}}} \\ &= \frac{\htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}} + \htmlClass{sym-q_vy}{752.8\ \mathrm{kPa}}}{2} \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \htmlClass{sym-x_vy}{180.2\ \mathrm{mm}} \\ &= 408.4\ \mathrm{kN} \end{aligned}\]
S.65

CSA A23.3 Cl. 11.3.4

\[\begin{aligned} \htmlClass{sym-V_cy}{V_{\mathrm{cy}}} &= \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-beta_y}{\beta_{y}} \cdot \htmlClass{sym-f_v}{f_{v}} \cdot \htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-beta_y}{0.21} \cdot \htmlClass{sym-f_v}{5.916\ \mathrm{MPa}} \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{1.17\ \mathrm{m}} \\ &= 2.834\ \mathrm{MN} \end{aligned}\]
S.66

CSA A23.3 Cl. 10.1.7

\[\begin{aligned} \htmlClass{sym-alpha_1}{\alpha_{1}} &= \max\left(0.67, 0.85 - \frac{0.0015 \cdot \htmlClass{sym-f_c}{f_{c}}}{1\ \mathrm{MPa}}\right) \\ &= \max\left(0.67, 0.85 - \frac{0.0015 \cdot \htmlClass{sym-f_c}{35\ \mathrm{MPa}}}{1\ \mathrm{MPa}}\right) \\ &= 0.7975 \end{aligned}\]
S.67

CSA A23.3 Cl. 10.1.7

\[\begin{aligned} \htmlClass{sym-beta_1}{\beta_{1}} &= \max\left(0.67, 0.97 - \frac{0.0025 \cdot \htmlClass{sym-f_c}{f_{c}}}{1\ \mathrm{MPa}}\right) \\ &= \max\left(0.67, 0.97 - \frac{0.0025 \cdot \htmlClass{sym-f_c}{35\ \mathrm{MPa}}}{1\ \mathrm{MPa}}\right) \\ &= 0.8825 \end{aligned}\]
S.68 \[\begin{aligned} \htmlClass{sym-q_fx}{q_{\mathrm{fx}}} &= \htmlClass{sym-q_ux}{q_{\mathrm{ux}}} - \frac{\left(\htmlClass{sym-q_ux}{q_{\mathrm{ux}}} - \htmlClass{sym-q_ux_min}{q_{\mathrm{ux},\mathrm{min}}}\right) \cdot \htmlClass{sym-l_x}{l_{x}}}{\htmlClass{sym-B_f}{B_{f}}} \\ &= \htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}} - \htmlClass{sym-q_ux_min}{604.2\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-l_x}{1.275\ \mathrm{m}}}{\htmlClass{sym-B_f}{3000\ \mathrm{mm}}} \\ &= 716\ \mathrm{kPa} \end{aligned}\]
S.69 \[\begin{aligned} \htmlClass{sym-M_fx}{M_{\mathrm{fx}}} &= \frac{\htmlClass{sym-q_fx}{q_{\mathrm{fx}}} + 2 \cdot \htmlClass{sym-q_ux}{q_{\mathrm{ux}}}}{6} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-l_x}{l_{x}}^{2} \\ &= \frac{\htmlClass{sym-q_fx}{716\ \mathrm{kPa}} + 2 \cdot \htmlClass{sym-q_ux}{798.6\ \mathrm{kPa}}}{6} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}} \cdot \left(\htmlClass{sym-l_x}{1.275\ \mathrm{m}}\right)^{2} \\ &= 2.256\ \mathrm{MN} \cdot \mathrm{m} \end{aligned}\]
S.70 \[\begin{aligned} \htmlClass{sym-A_sx}{A_{\mathrm{sx}}} &= \htmlClass{sym-n_x}{n_{x}} \cdot 500\ \mathrm{mm}^{2} \\ &= \htmlClass{sym-n_x}{21} \cdot 500\ \mathrm{mm}^{2} \\ &= 10500\ \mathrm{mm}^{2} \end{aligned}\]
S.71 \[\begin{aligned} \htmlClass{sym-s_x}{s_{x}} &= \frac{\htmlClass{sym-L_f}{L_{f}} - 2 \cdot \htmlClass{sym-c_c}{c_{c}}}{\htmlClass{sym-n_x}{n_{x}} - 1} \\ &= \frac{\htmlClass{sym-L_f}{3600\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-c_c}{75\ \mathrm{mm}}}{\htmlClass{sym-n_x}{21} - 1} \\ &= 172.5\ \mathrm{mm} \end{aligned}\]
S.72 \[\begin{aligned} \htmlClass{sym-a_x}{a_{x}} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_sx}{A_{\mathrm{sx}}} \cdot \htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-alpha_1}{\alpha_{1}} \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \cdot \htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_sx}{10500\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-alpha_1}{0.7975} \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}}} \\ &= 54.66\ \mathrm{mm} \end{aligned}\]
S.73

CSA A23.3 Cl. 10.1

\[\begin{aligned} \htmlClass{sym-M_rx}{M_{\mathrm{rx}}} &= \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_sx}{A_{\mathrm{sx}}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \left(\htmlClass{sym-d_x}{d_{x}} - \frac{\htmlClass{sym-a_x}{a_{x}}}{2}\right) \\ &= \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_sx}{10500\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \left(\htmlClass{sym-d_x}{1.312\ \mathrm{m}} - \frac{\htmlClass{sym-a_x}{54.66\ \mathrm{mm}}}{2}\right) \\ &= 4.588\ \mathrm{MN} \cdot \mathrm{m} \end{aligned}\]
S.74 \[\begin{aligned} \htmlClass{sym-q_fy}{q_{\mathrm{fy}}} &= \htmlClass{sym-q_uy}{q_{\mathrm{uy}}} - \frac{\left(\htmlClass{sym-q_uy}{q_{\mathrm{uy}}} - \htmlClass{sym-q_uy_min}{q_{\mathrm{uy},\mathrm{min}}}\right) \cdot \htmlClass{sym-l_y}{l_{y}}}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}} - \htmlClass{sym-q_uy_min}{644.3\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-l_y}{1.35\ \mathrm{m}}}{\htmlClass{sym-L_f}{3600\ \mathrm{mm}}} \\ &= 715.7\ \mathrm{kPa} \end{aligned}\]
S.75 \[\begin{aligned} \htmlClass{sym-M_fy}{M_{\mathrm{fy}}} &= \frac{\htmlClass{sym-q_fy}{q_{\mathrm{fy}}} + 2 \cdot \htmlClass{sym-q_uy}{q_{\mathrm{uy}}}}{6} \cdot \htmlClass{sym-B_f}{B_{f}} \cdot \htmlClass{sym-l_y}{l_{y}}^{2} \\ &= \frac{\htmlClass{sym-q_fy}{715.7\ \mathrm{kPa}} + 2 \cdot \htmlClass{sym-q_uy}{758.5\ \mathrm{kPa}}}{6} \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \cdot \left(\htmlClass{sym-l_y}{1.35\ \mathrm{m}}\right)^{2} \\ &= 2.034\ \mathrm{MN} \cdot \mathrm{m} \end{aligned}\]
S.76 \[\begin{aligned} \htmlClass{sym-A_sy}{A_{\mathrm{sy}}} &= \htmlClass{sym-n_y}{n_{y}} \cdot 500\ \mathrm{mm}^{2} \\ &= \htmlClass{sym-n_y}{18} \cdot 500\ \mathrm{mm}^{2} \\ &= 9000\ \mathrm{mm}^{2} \end{aligned}\]
S.77 \[\begin{aligned} \htmlClass{sym-s_y}{s_{y}} &= \frac{\htmlClass{sym-B_f}{B_{f}} - 2 \cdot \htmlClass{sym-c_c}{c_{c}}}{\htmlClass{sym-n_y}{n_{y}} - 1} \\ &= \frac{\htmlClass{sym-B_f}{3000\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-c_c}{75\ \mathrm{mm}}}{\htmlClass{sym-n_y}{18} - 1} \\ &= 167.6\ \mathrm{mm} \end{aligned}\]
S.78 \[\begin{aligned} \htmlClass{sym-a_y}{a_{y}} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_sy}{A_{\mathrm{sy}}} \cdot \htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-alpha_1}{\alpha_{1}} \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \cdot \htmlClass{sym-B_f}{B_{f}}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_sy}{9000\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-alpha_1}{0.7975} \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}}} \\ &= 56.22\ \mathrm{mm} \end{aligned}\]
S.79

CSA A23.3 Cl. 10.1

\[\begin{aligned} \htmlClass{sym-M_ry}{M_{\mathrm{ry}}} &= \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_sy}{A_{\mathrm{sy}}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \left(\htmlClass{sym-d_y}{d_{y}} - \frac{\htmlClass{sym-a_y}{a_{y}}}{2}\right) \\ &= \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_sy}{9000\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \left(\htmlClass{sym-d_y}{1.287\ \mathrm{m}} - \frac{\htmlClass{sym-a_y}{56.22\ \mathrm{mm}}}{2}\right) \\ &= 3.853\ \mathrm{MN} \cdot \mathrm{m} \end{aligned}\]
S.80

CSA A23.3 Cl. 7.8.1

\[\begin{aligned} \htmlClass{sym-A_sx_min}{A_{\mathrm{sx},\mathrm{min}}} &= 0.002 \cdot \htmlClass{sym-t_f}{t_{f}} \cdot \htmlClass{sym-L_f}{L_{f}} \\ &= 0.002 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}} \cdot \htmlClass{sym-L_f}{3600\ \mathrm{mm}} \\ &= 10080\ \mathrm{mm}^{2} \end{aligned}\]
S.81

CSA A23.3 Cl. 7.8.1

\[\begin{aligned} \htmlClass{sym-A_sy_min}{A_{\mathrm{sy},\mathrm{min}}} &= 0.002 \cdot \htmlClass{sym-t_f}{t_{f}} \cdot \htmlClass{sym-B_f}{B_{f}} \\ &= 0.002 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}} \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \\ &= 8400\ \mathrm{mm}^{2} \end{aligned}\]
S.82 \[\begin{aligned} \htmlClass{sym-c_x}{c_{x}} &= \frac{\htmlClass{sym-a_x}{a_{x}}}{\htmlClass{sym-beta_1}{\beta_{1}}} \\ &= \frac{\htmlClass{sym-a_x}{54.66\ \mathrm{mm}}}{\htmlClass{sym-beta_1}{0.8825}} \\ &= 61.94\ \mathrm{mm} \end{aligned}\]
S.83 \[\begin{aligned} \htmlClass{sym-c_y}{c_{y}} &= \frac{\htmlClass{sym-a_y}{a_{y}}}{\htmlClass{sym-beta_1}{\beta_{1}}} \\ &= \frac{\htmlClass{sym-a_y}{56.22\ \mathrm{mm}}}{\htmlClass{sym-beta_1}{0.8825}} \\ &= 63.7\ \mathrm{mm} \end{aligned}\]
S.84

CSA A23.3 Cl. 13.10.4

\[\begin{aligned} \htmlClass{sym-s_max}{s_{\mathrm{max}}} &= \min\left(3 \cdot \htmlClass{sym-t_f}{t_{f}}, 500\ \mathrm{mm}\right) \\ &= \min\left(3 \cdot \htmlClass{sym-t_f}{1400\ \mathrm{mm}}, 500\ \mathrm{mm}\right) \\ &= 500\ \mathrm{mm} \end{aligned}\]
S.85 \[\begin{aligned} \htmlClass{sym-k_1}{k_{1}} &= 1 \quad \left(\text{Bar location factor, bottom bars | Cl. 12.2.4.a}\right) \end{aligned}\]
S.86 \[\begin{aligned} \htmlClass{sym-k_2}{k_{2}} &= 1 \quad \left(\text{Coating factor, uncoated | Cl. 12.2.4.b}\right) \end{aligned}\]
S.87 \[\begin{aligned} \htmlClass{sym-k_3}{k_{3}} &= 1 \quad \left(\text{Concrete density factor, normal density | Cl. 12.2.4.c}\right) \end{aligned}\]
S.88 \[\begin{aligned} \htmlClass{sym-k_4x}{k_{4x}} &= 1 \quad \left(\text{Bar size factor, X bars | Cl. 12.2.4.d}\right) \end{aligned}\]

Branch: \(c_{c} \geq d_{\mathrm{bx}}\ \text{and}\ s_{x} - d_{\mathrm{bx}} \geq 2 \cdot d_{\mathrm{bx}}\) held

S.89 \[\begin{aligned} \htmlClass{sym-k_x}{k_{x}} &= 0.45 \quad \left(\text{Slab, cover and clear spacing | CSA A23.3 Cl. 12.2.3}\right) \end{aligned}\]
S.90

CSA A23.3 Cl. 12.2.3

\[\begin{aligned} \htmlClass{sym-l_dx}{l_{\mathrm{dx}}} &= \max\left(\frac{\htmlClass{sym-k_x}{k_{x}} \cdot \htmlClass{sym-k_1}{k_{1}} \cdot \htmlClass{sym-k_2}{k_{2}} \cdot \htmlClass{sym-k_3}{k_{3}} \cdot \htmlClass{sym-k_4x}{k_{4x}} \cdot \htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-f_v}{f_{v}}} \cdot \htmlClass{sym-d_bx}{d_{\mathrm{bx}}}, 300\ \mathrm{mm}\right) \\ &= \max\left(\frac{\htmlClass{sym-k_x}{0.45} \cdot \htmlClass{sym-k_1}{1} \cdot \htmlClass{sym-k_2}{1} \cdot \htmlClass{sym-k_3}{1} \cdot \htmlClass{sym-k_4x}{1} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-f_v}{5.916\ \mathrm{MPa}}} \cdot \htmlClass{sym-d_bx}{25.2\ \mathrm{mm}}, 300\ \mathrm{mm}\right) \\ &= 766.7\ \mathrm{mm} \end{aligned}\]
S.91 \[\begin{aligned} \htmlClass{sym-l_ax}{l_{\mathrm{ax}}} &= \htmlClass{sym-l_x}{l_{x}} - \htmlClass{sym-c_c}{c_{c}} \\ &= \htmlClass{sym-l_x}{1.275\ \mathrm{m}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} \\ &= 1.2\ \mathrm{m} \end{aligned}\]
S.92 \[\begin{aligned} \htmlClass{sym-k_4y}{k_{4y}} &= 1 \quad \left(\text{Bar size factor, Y bars | Cl. 12.2.4.d}\right) \end{aligned}\]

Branch: \(c_{c} \geq d_{\mathrm{by}}\ \text{and}\ s_{y} - d_{\mathrm{by}} \geq 2 \cdot d_{\mathrm{by}}\) held

S.93 \[\begin{aligned} \htmlClass{sym-k_y}{k_{y}} &= 0.45 \quad \left(\text{Slab, cover and clear spacing | CSA A23.3 Cl. 12.2.3}\right) \end{aligned}\]
S.94

CSA A23.3 Cl. 12.2.3

\[\begin{aligned} \htmlClass{sym-l_dy}{l_{\mathrm{dy}}} &= \max\left(\frac{\htmlClass{sym-k_y}{k_{y}} \cdot \htmlClass{sym-k_1}{k_{1}} \cdot \htmlClass{sym-k_2}{k_{2}} \cdot \htmlClass{sym-k_3}{k_{3}} \cdot \htmlClass{sym-k_4y}{k_{4y}} \cdot \htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-f_v}{f_{v}}} \cdot \htmlClass{sym-d_by}{d_{\mathrm{by}}}, 300\ \mathrm{mm}\right) \\ &= \max\left(\frac{\htmlClass{sym-k_y}{0.45} \cdot \htmlClass{sym-k_1}{1} \cdot \htmlClass{sym-k_2}{1} \cdot \htmlClass{sym-k_3}{1} \cdot \htmlClass{sym-k_4y}{1} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-f_v}{5.916\ \mathrm{MPa}}} \cdot \htmlClass{sym-d_by}{25.2\ \mathrm{mm}}, 300\ \mathrm{mm}\right) \\ &= 766.7\ \mathrm{mm} \end{aligned}\]
S.95 \[\begin{aligned} \htmlClass{sym-l_ay}{l_{\mathrm{ay}}} &= \htmlClass{sym-l_y}{l_{y}} - \htmlClass{sym-c_c}{c_{c}} \\ &= \htmlClass{sym-l_y}{1.35\ \mathrm{m}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} \\ &= 1.275\ \mathrm{m} \end{aligned}\]
S.96 \[\begin{aligned} \htmlClass{sym-beta_f}{\beta_{f}} &= \frac{\max\left(\htmlClass{sym-B_f}{B_{f}}, \htmlClass{sym-L_f}{L_{f}}\right)}{\min\left(\htmlClass{sym-B_f}{B_{f}}, \htmlClass{sym-L_f}{L_{f}}\right)} \\ &= \frac{\max\left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}, \htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)}{\min\left(\htmlClass{sym-B_f}{3000\ \mathrm{mm}}, \htmlClass{sym-L_f}{3600\ \mathrm{mm}}\right)} \\ &= 1.2 \end{aligned}\]
S.97

CSA A23.3 Cl. 15.4.4

\[\begin{aligned} \htmlClass{sym-r_band}{r_{\mathrm{band}}} &= \frac{2}{\htmlClass{sym-beta_f}{\beta_{f}} + 1} \\ &= \frac{2}{\htmlClass{sym-beta_f}{1.2} + 1} \\ &= 0.9091 \end{aligned}\]

Branch: \(B_{f} > L_{f}\) did not hold

Branch: \(L_{f} > B_{f}\) held

S.98 \[\begin{aligned} \htmlClass{sym-A_band}{A_{\mathrm{band}}} &= \htmlClass{sym-r_band}{r_{\mathrm{band}}} \cdot \htmlClass{sym-A_sx}{A_{\mathrm{sx}}} \\ &= \htmlClass{sym-r_band}{0.9091} \cdot \htmlClass{sym-A_sx}{10500\ \mathrm{mm}^{2}} \\ &= 9545\ \mathrm{mm}^{2} \end{aligned}\]
S.99 \[\begin{aligned} \htmlClass{sym-w_band}{w_{\mathrm{band}}} &= 1 \cdot \htmlClass{sym-B_f}{B_{f}} \\ &= 1 \cdot \htmlClass{sym-B_f}{3000\ \mathrm{mm}} \\ &= 3\ \mathrm{m} \end{aligned}\]

Questions

Are q_a and q_ult gross or net, and is the footing's weight included?

Gross, at the underside of the footing. The concrete at 24 kN/m³ over t_f and the soil over it, gamma_s h_s, are added to the column load in the soil bearing checks; they are uniform, so they cause no shear or moment in the footing, and the structural checks use the net pressure from the column alone. If your geotechnical report gives net values, add gamma_s (h_s + t_f) to them.

Why does bearing_factored use B_e and L_e rather than the whole footing?

At the ultimate limit state the soil check uses Meyerhof's effective area: the footing is shrunk by twice the eccentricity each way, B_e = B_f - 2 e_x and L_e = L_f - 2 e_y, and the factored load is spread uniformly over what is left. The service check and the structural checks keep the linear pressure over the whole footing, which is only valid with the resultant inside the kern; contact_service and contact_factored check that, and partial contact is not analysed.

Where do the short-direction bars go under Cl. 15.4.4?

In a rectangular footing CSA A23.3 Cl. 15.4.4 puts the fraction 2 / (beta + 1) of the short-direction steel in a band as wide as the short side, centred on the column, and the rest outside it. The sheet reports that area and the band width. It takes the bars as evenly spaced, so place them to the band yourself.

What does a column bearing FAIL under Cl. 10.8.1 mean for the dowels?

That the column load is more than bearing alone can carry across the interface. CSA A23.3 Cl. 15.9 lets reinforcement carry the rest, and the minimum dowel area reported is then that excess over phi_s f_y, never less than 0.005 of the column area. Tension in the dowels from the column moment is not designed here.

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