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CSA A23.3-24 Cl. 11.6

RC bracket and corbel

Verified against CSA A23.3-24, 2026-10-01

Strut-and-tie design of a reinforced concrete bracket or corbel to CSA A23.3-24. Design reinforced concrete brackets and corbels using the strut-and-tie method per CSA A23.3-24 Clause 11.6. The calculator models the internal load path with concrete compression struts and steel tension ties, then determines required reinforcement and checks node bearing stresses. Input corbel geometry, applied loads, and material properties for a complete design check.

Given

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\(b\) \(\mathrm{mm}\) 100-2,000
\(h\) \(\mathrm{mm}\) 100-2,000
\(w_{2}\) \(\mathrm{mm}\) 100-2,000
\(a_{2}\) \(\mathrm{mm}\) 20-200
\(f_{y}\) \(\mathrm{MPa}\) 200-700
\({f'}_{c}\) \(\mathrm{MPa}\) 20-80
\(h_{1}\) \(\mathrm{mm}\) 50-1,000
\(h_{2}\) \(\mathrm{mm}\) 0-1,000
\(w_{1}\) \(\mathrm{mm}\) 50-1,000
\(w_{4}\) \(\mathrm{mm}\) 50-1,000
\(w_{3}\) \(\mathrm{mm}\) 20-1,000
\(a_{1}\) \(\mathrm{mm}\) 20-200
\(L_{p}\) \(\mathrm{mm}\) 50-1,000
\(b_{p}\) \(\mathrm{mm}\) 25-500
\(A_{s1}\) \(\mathrm{mm}^{2}\) 100-20,000
\(A_{s2}\) \(\mathrm{mm}^{2}\) 100-20,000
\(V_{f}\) \(\mathrm{kN}\) 10-558
\(N_{f}\) \(\mathrm{kN}\) 0-5,000
\(E_{s}\) \(\mathrm{MPa}\) 150,000-250,000
\(\phi_{c}\) 0.5-1
w1 = 300 mm h1 = 250 mm h2 = 250 mm h = 400 mm
Corbel profile as entered, with the dimensions that fix it.
Vf = 300 kN Nf,design = 60 kN A B C D θ1 = 60.7° PAC = 344 kN PAB = 228 kN w3 = 200 mm LBD = 455 mm
Corbel elevation with the strut-and-tie model. Struts dashed, ties solid, drawn at the angles the calc computed.

Title block

Optional. Printed under the title of the PDF.

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

Checks

Check D/C Result
Depth ok\(\htmlClass{sym-w3}{w_{3}} \leq \htmlClass{sym-L_BD}{L_{\mathrm{BD}}} \quad \Rightarrow \quad \htmlClass{sym-w3}{200\ \mathrm{mm}} \leq \htmlClass{sym-L_BD}{455\ \mathrm{mm}}\) 0.44 PASS
Edge depth ok\(\htmlClass{sym-h_avg}{h_{\mathrm{avg}}} \leq \htmlClass{sym-h12}{h_{12}} \quad \Rightarrow \quad \htmlClass{sym-h_avg}{250\ \mathrm{mm}} \leq \htmlClass{sym-h12}{302.1\ \mathrm{mm}}\) 0.83 PASS
Strut CE ok\(0.75 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \leq \htmlClass{sym-f_cu_CE}{f_{\mathrm{cu},\mathrm{CE}}} \quad \Rightarrow \quad 0.75 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \leq \htmlClass{sym-f_cu_CE}{26.25\ \mathrm{MPa}}\) 1.00 PASS
Shear ok\(\htmlClass{sym-V_f}{V_{f}} \leq \htmlClass{sym-V_r}{V_{r}} \quad \Rightarrow \quad \htmlClass{sym-V_f}{300\ \mathrm{kN}} \leq \htmlClass{sym-V_r}{815.2\ \mathrm{kN}}\) 0.37 PASS
Plate ok\(\htmlClass{sym-b_p_req}{b_{p,\mathrm{req}}} \leq \htmlClass{sym-b_p}{b_{p}} \quad \Rightarrow \quad \htmlClass{sym-b_p_req}{58.61\ \mathrm{mm}} \leq \htmlClass{sym-b_p}{75\ \mathrm{mm}}\) 0.78 PASS
Steel ok\(\htmlClass{sym-A_s1_req}{A_{s1,\mathrm{req}}} \leq \htmlClass{sym-A_s1}{A_{s1}} \quad \Rightarrow \quad \htmlClass{sym-A_s1_req}{671.2\ \mathrm{mm}^{2}} \leq \htmlClass{sym-A_s1}{900\ \mathrm{mm}^{2}}\) 0.75 PASS

Results

Quantity Description Value Unit
\(\htmlClass{sym-b_p_req}{b_{p,\mathrm{req}}}\) Bearing plate width required 58.61 \(\mathrm{mm}\)
\(\htmlClass{sym-V_r}{V_{r}}\) Corbel shear resistance 815.2 \(\mathrm{kN}\)
\(\htmlClass{sym-A_s1_req}{A_{s1,\mathrm{req}}}\) Tie AB steel area required 671.2 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-A_s2_req}{A_{s2,\mathrm{req}}}\) Tie CD steel area required 176.5 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-A_s3_req}{A_{s3,\mathrm{req}}}\) Tie BD steel area required 1053 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-A_s4_req}{A_{s4,\mathrm{req}}}\) Tie DF steel area required 1053 \(\mathrm{mm}^{2}\)
\(\htmlClass{sym-w_CE}{w_{\mathrm{CE}}}\) Strut CE width required at the node 110.2 \(\mathrm{mm}\)

Derivation

S.1

A23.3 Cl. 11.6.4

\[\begin{aligned} \htmlClass{sym-N_f_min}{N_{f,\mathrm{min}}} &= 0.2 \cdot \htmlClass{sym-V_f}{V_{f}} \\ &= 0.2 \cdot \htmlClass{sym-V_f}{300\ \mathrm{kN}} \\ &= 60\ \mathrm{kN} \end{aligned}\]
S.2

A23.3 Cl. 11.6.4

\[\begin{aligned} \htmlClass{sym-N_f_design}{N_{f,\mathrm{design}}} &= \max\left(\htmlClass{sym-N_f}{N_{f}}, \htmlClass{sym-N_f_min}{N_{f,\mathrm{min}}}\right) \\ &= \max\left(\htmlClass{sym-N_f}{0\ \mathrm{kN}}, \htmlClass{sym-N_f_min}{60\ \mathrm{kN}}\right) \\ &= 60\ \mathrm{kN} \end{aligned}\]
S.3

A23.3 Cl. 11.4.4.1 b), 11.4.4.2 a)

\[\begin{aligned} \htmlClass{sym-b_p_req}{b_{p,\mathrm{req}}} &= \frac{\htmlClass{sym-V_f}{V_{f}}}{\htmlClass{sym-L_p}{L_{p}} \cdot 0.75 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}} \\ &= \frac{\htmlClass{sym-V_f}{300\ \mathrm{kN}}}{\htmlClass{sym-L_p}{300\ \mathrm{mm}} \cdot 0.75 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}} \\ &= 58.61\ \mathrm{mm} \end{aligned}\]
S.4 \[\begin{aligned} \htmlClass{sym-L_BD}{L_{\mathrm{BD}}} &= \htmlClass{sym-h1}{h_{1}} + \htmlClass{sym-h2}{h_{2}} - \htmlClass{sym-a1}{a_{1}} \\ &= \htmlClass{sym-h1}{250\ \mathrm{mm}} + \htmlClass{sym-h2}{250\ \mathrm{mm}} - \htmlClass{sym-a1}{45\ \mathrm{mm}} \\ &= 455\ \mathrm{mm} \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-x_p}{x_{p}} &= \htmlClass{sym-w3}{w_{3}} + \frac{\htmlClass{sym-b_p}{b_{p}}}{2} \\ &= \htmlClass{sym-w3}{200\ \mathrm{mm}} + \frac{\htmlClass{sym-b_p}{75\ \mathrm{mm}}}{2} \\ &= 237.5\ \mathrm{mm} \end{aligned}\]

Branch: \(w_{1} \leq x_{p}\) did not hold

Branch: \(x_{p} \leq w_{4}\) held

S.6 \[\begin{aligned} \htmlClass{sym-h12}{h_{12}} &= \htmlClass{sym-h1}{h_{1}} + \frac{\htmlClass{sym-h2}{h_{2}} \cdot \left(\htmlClass{sym-w4}{w_{4}} - \htmlClass{sym-x_p}{x_{p}}\right)}{\htmlClass{sym-w4}{w_{4}}} \\ &= \htmlClass{sym-h1}{250\ \mathrm{mm}} + \frac{\htmlClass{sym-h2}{250\ \mathrm{mm}} \cdot \left(\htmlClass{sym-w4}{300\ \mathrm{mm}} - \htmlClass{sym-x_p}{237.5\ \mathrm{mm}}\right)}{\htmlClass{sym-w4}{300\ \mathrm{mm}}} \\ &= 302.1\ \mathrm{mm} \end{aligned}\]
S.7 \[\begin{aligned} \htmlClass{sym-h_avg}{h_{\mathrm{avg}}} &= \frac{\htmlClass{sym-h1}{h_{1}} + \htmlClass{sym-h2}{h_{2}}}{2} \\ &= \frac{\htmlClass{sym-h1}{250\ \mathrm{mm}} + \htmlClass{sym-h2}{250\ \mathrm{mm}}}{2} \\ &= 250\ \mathrm{mm} \end{aligned}\]
S.8 \[\begin{aligned} \htmlClass{sym-height_ratio}{\mathrm{height}_{\mathrm{ratio}}} &= \frac{\htmlClass{sym-h_avg}{h_{\mathrm{avg}}}}{\htmlClass{sym-h12}{h_{12}}} \\ &= \frac{\htmlClass{sym-h_avg}{250\ \mathrm{mm}}}{\htmlClass{sym-h12}{302.1\ \mathrm{mm}}} \\ &= 0.8276 \end{aligned}\]
S.9

A23.3 Cl. 3.2, effective shear depth

\[\begin{aligned} \htmlClass{sym-d_v}{d_{v}} &= \max\left(0.9 \cdot \htmlClass{sym-L_BD}{L_{\mathrm{BD}}}, 0.72 \cdot \left(\htmlClass{sym-h1}{h_{1}} + \htmlClass{sym-h2}{h_{2}}\right)\right) \\ &= \max\left(0.9 \cdot \htmlClass{sym-L_BD}{455\ \mathrm{mm}}, 0.72 \cdot \left(\htmlClass{sym-h1}{250\ \mathrm{mm}} + \htmlClass{sym-h2}{250\ \mathrm{mm}}\right)\right) \\ &= 409.5\ \mathrm{mm} \end{aligned}\]
S.10

A23.3 Cl. 11.3.3, V_r,max with V_p = 0

\[\begin{aligned} \htmlClass{sym-V_r}{V_{r}} &= 0.25 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \cdot \htmlClass{sym-b}{b} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= 0.25 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{409.5\ \mathrm{mm}} \\ &= 815.2\ \mathrm{kN} \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-L_AB}{L_{\mathrm{AB}}} &= \htmlClass{sym-w3}{w_{3}} + \htmlClass{sym-h}{h} - \htmlClass{sym-a2}{a_{2}} \\ &= \htmlClass{sym-w3}{200\ \mathrm{mm}} + \htmlClass{sym-h}{400\ \mathrm{mm}} - \htmlClass{sym-a2}{55\ \mathrm{mm}} \\ &= 545\ \mathrm{mm} \end{aligned}\]
S.12

A23.3 Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-g1}{g_{1}} &= \frac{0.75 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \cdot \htmlClass{sym-b}{b}}{2} \\ &= \frac{0.75 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}}{2} \\ &= 2.986\ \mathrm{MN/m} \end{aligned}\]
S.13

A23.3 Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-g2}{g_{2}} &= \left(-0.75\right) \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \cdot \htmlClass{sym-b}{b} \cdot \left(\htmlClass{sym-h}{h} - \htmlClass{sym-a2}{a_{2}}\right) \\ &= \left(-0.75\right) \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}} \cdot \left(\htmlClass{sym-h}{400\ \mathrm{mm}} - \htmlClass{sym-a2}{55\ \mathrm{mm}}\right) \\ &= -2.06\ \mathrm{MN} \end{aligned}\]
S.14 \[\begin{aligned} \htmlClass{sym-g3}{g_{3}} &= \htmlClass{sym-N_f_design}{N_{f,\mathrm{design}}} \cdot \htmlClass{sym-L_BD}{L_{\mathrm{BD}}} + \htmlClass{sym-V_f}{V_{f}} \cdot \htmlClass{sym-L_AB}{L_{\mathrm{AB}}} \\ &= \htmlClass{sym-N_f_design}{60\ \mathrm{kN}} \cdot \htmlClass{sym-L_BD}{455\ \mathrm{mm}} + \htmlClass{sym-V_f}{300\ \mathrm{kN}} \cdot \htmlClass{sym-L_AB}{545\ \mathrm{mm}} \\ &= 190.8\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.15 \[\begin{aligned} \htmlClass{sym-discriminant}{\mathrm{discriminant}} &= \htmlClass{sym-g2}{g_{2}}^{2} - 4 \cdot \htmlClass{sym-g1}{g_{1}} \cdot \htmlClass{sym-g3}{g_{3}} \\ &= \left(\htmlClass{sym-g2}{-2.06\ \mathrm{MN}}\right)^{2} - 4 \cdot \htmlClass{sym-g1}{2.986\ \mathrm{MN/m}} \cdot \htmlClass{sym-g3}{190.8\ \mathrm{kN} \cdot \mathrm{m}} \\ &= 1.966\ \mathrm{MN}^{2} \end{aligned}\]
S.16 \[\begin{aligned} \htmlClass{sym-a_node}{a_{\mathrm{node}}} &= \frac{\left(-\htmlClass{sym-g2}{g_{2}}\right) - \sqrt{\htmlClass{sym-discriminant}{\mathrm{discriminant}}}}{2 \cdot \htmlClass{sym-g1}{g_{1}}} \\ &= \frac{\left(-\left(\htmlClass{sym-g2}{-2.06\ \mathrm{MN}}\right)\right) - \sqrt{\htmlClass{sym-discriminant}{1.966\ \mathrm{MN}^{2}}}}{2 \cdot \htmlClass{sym-g1}{2.986\ \mathrm{MN/m}}} \\ &= 110.2\ \mathrm{mm} \end{aligned}\]
S.17 \[\begin{aligned} \htmlClass{sym-L_CD}{L_{\mathrm{CD}}} &= \htmlClass{sym-h}{h} - \htmlClass{sym-a2}{a_{2}} - \frac{\htmlClass{sym-a_node}{a_{\mathrm{node}}}}{2} \\ &= \htmlClass{sym-h}{400\ \mathrm{mm}} - \htmlClass{sym-a2}{55\ \mathrm{mm}} - \frac{\htmlClass{sym-a_node}{110.2\ \mathrm{mm}}}{2} \\ &= 289.9\ \mathrm{mm} \end{aligned}\]
S.18 \[\begin{aligned} \htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}} &= \arctan\left(\frac{\htmlClass{sym-L_BD}{L_{\mathrm{BD}}}}{\htmlClass{sym-L_AB}{L_{\mathrm{AB}}} - \htmlClass{sym-L_CD}{L_{\mathrm{CD}}}}\right) \\ &= \arctan\left(\frac{\htmlClass{sym-L_BD}{455\ \mathrm{mm}}}{\htmlClass{sym-L_AB}{545\ \mathrm{mm}} - \htmlClass{sym-L_CD}{289.9\ \mathrm{mm}}}\right) \\ &= 1.06 \end{aligned}\]
S.19 \[\begin{aligned} \htmlClass{sym-theta1}{\theta_{1}} &= \operatorname{degrees}\left(\htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}}\right) \\ &= \operatorname{degrees}\left(\htmlClass{sym-theta1_rad}{1.06}\right) \\ &= 60.72 \end{aligned}\]
S.20 \[\begin{aligned} \htmlClass{sym-P_AC}{P_{\mathrm{AC}}} &= \frac{\htmlClass{sym-V_f}{V_{f}}}{\sin\left(\htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}}\right)} \\ &= \frac{\htmlClass{sym-V_f}{300\ \mathrm{kN}}}{\sin\left(\htmlClass{sym-theta1_rad}{1.06}\right)} \\ &= 343.9\ \mathrm{kN} \end{aligned}\]
S.21 \[\begin{aligned} \htmlClass{sym-P_AB}{P_{\mathrm{AB}}} &= \frac{\htmlClass{sym-V_f}{V_{f}}}{\tan\left(\htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}}\right)} + \htmlClass{sym-N_f_design}{N_{f,\mathrm{design}}} \\ &= \frac{\htmlClass{sym-V_f}{300\ \mathrm{kN}}}{\tan\left(\htmlClass{sym-theta1_rad}{1.06}\right)} + \htmlClass{sym-N_f_design}{60\ \mathrm{kN}} \\ &= 228.2\ \mathrm{kN} \end{aligned}\]
S.22 \[\begin{aligned} \htmlClass{sym-theta2_rad}{\theta_{2,\mathrm{rad}}} &= \arctan\left(\frac{\htmlClass{sym-L_CD}{L_{\mathrm{CD}}}}{\htmlClass{sym-L_BD}{L_{\mathrm{BD}}}}\right) \\ &= \arctan\left(\frac{\htmlClass{sym-L_CD}{289.9\ \mathrm{mm}}}{\htmlClass{sym-L_BD}{455\ \mathrm{mm}}}\right) \\ &= 0.5673 \end{aligned}\]
S.23 \[\begin{aligned} \htmlClass{sym-theta2}{\theta_{2}} &= \operatorname{degrees}\left(\htmlClass{sym-theta2_rad}{\theta_{2,\mathrm{rad}}}\right) \\ &= \operatorname{degrees}\left(\htmlClass{sym-theta2_rad}{0.5673}\right) \\ &= 32.5 \end{aligned}\]
S.24 \[\begin{aligned} \htmlClass{sym-P_BC}{P_{\mathrm{BC}}} &= \frac{\htmlClass{sym-P_AB}{P_{\mathrm{AB}}}}{\sin\left(\htmlClass{sym-theta2_rad}{\theta_{2,\mathrm{rad}}}\right)} \\ &= \frac{\htmlClass{sym-P_AB}{228.2\ \mathrm{kN}}}{\sin\left(\htmlClass{sym-theta2_rad}{0.5673}\right)} \\ &= 424.7\ \mathrm{kN} \end{aligned}\]
S.25 \[\begin{aligned} \htmlClass{sym-P_BD}{P_{\mathrm{BD}}} &= \frac{\htmlClass{sym-P_AB}{P_{\mathrm{AB}}}}{\tan\left(\htmlClass{sym-theta2_rad}{\theta_{2,\mathrm{rad}}}\right)} \\ &= \frac{\htmlClass{sym-P_AB}{228.2\ \mathrm{kN}}}{\tan\left(\htmlClass{sym-theta2_rad}{0.5673}\right)} \\ &= 358.2\ \mathrm{kN} \end{aligned}\]
S.26 \[\begin{aligned} \htmlClass{sym-P_CD}{P_{\mathrm{CD}}} &= \htmlClass{sym-P_AB}{P_{\mathrm{AB}}} - \htmlClass{sym-P_AC}{P_{\mathrm{AC}}} \cdot \cos\left(\htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}}\right) \\ &= \htmlClass{sym-P_AB}{228.2\ \mathrm{kN}} - \htmlClass{sym-P_AC}{343.9\ \mathrm{kN}} \cdot \cos\left(\htmlClass{sym-theta1_rad}{1.06}\right) \\ &= 60\ \mathrm{kN} \end{aligned}\]
S.27 \[\begin{aligned} \htmlClass{sym-P_CE}{P_{\mathrm{CE}}} &= \htmlClass{sym-V_f}{V_{f}} + \htmlClass{sym-P_BC}{P_{\mathrm{BC}}} \cdot \cos\left(\htmlClass{sym-theta2_rad}{\theta_{2,\mathrm{rad}}}\right) \\ &= \htmlClass{sym-V_f}{300\ \mathrm{kN}} + \htmlClass{sym-P_BC}{424.7\ \mathrm{kN}} \cdot \cos\left(\htmlClass{sym-theta2_rad}{0.5673}\right) \\ &= 658.2\ \mathrm{kN} \end{aligned}\]
S.28 \[\begin{aligned} \htmlClass{sym-theta4}{\theta_{4}} &= 90 - \htmlClass{sym-theta2}{\theta_{2}} \\ &= 90 - \htmlClass{sym-theta2}{32.5} \\ &= 57.5 \end{aligned}\]
S.29 \[\begin{aligned} \htmlClass{sym-theta4_rad}{\theta_{4,\mathrm{rad}}} &= \operatorname{radians}\left(\htmlClass{sym-theta4}{\theta_{4}}\right) \\ &= \operatorname{radians}\left(\htmlClass{sym-theta4}{57.5}\right) \\ &= 1.004 \end{aligned}\]
S.30 \[\begin{aligned} \htmlClass{sym-P_DF}{P_{\mathrm{DF}}} &= 1 \cdot \htmlClass{sym-P_BD}{P_{\mathrm{BD}}} \\ &= 1 \cdot \htmlClass{sym-P_BD}{358.2\ \mathrm{kN}} \\ &= 358.2\ \mathrm{kN} \end{aligned}\]
S.31

A23.3 Cl. 11.4.3.1, 11.6.6

\[\begin{aligned} \htmlClass{sym-A_s1_req}{A_{s1,\mathrm{req}}} &= \max\left(\frac{\htmlClass{sym-P_AB}{P_{\mathrm{AB}}}}{0.85 \cdot \htmlClass{sym-f_y}{f_{y}}}, \frac{0.04 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \cdot \htmlClass{sym-b}{b} \cdot \htmlClass{sym-L_BD}{L_{\mathrm{BD}}}}{\htmlClass{sym-f_y}{f_{y}}}\right) \\ &= \max\left(\frac{\htmlClass{sym-P_AB}{228.2\ \mathrm{kN}}}{0.85 \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}, \frac{0.04 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}} \cdot \htmlClass{sym-L_BD}{455\ \mathrm{mm}}}{\htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\right) \\ &= 671.2\ \mathrm{mm}^{2} \end{aligned}\]
S.32

A23.3 Cl. 11.4.3.1

\[\begin{aligned} \htmlClass{sym-A_s2_req}{A_{s2,\mathrm{req}}} &= \frac{\htmlClass{sym-P_CD}{P_{\mathrm{CD}}}}{0.85 \cdot \htmlClass{sym-f_y}{f_{y}}} \\ &= \frac{\htmlClass{sym-P_CD}{60\ \mathrm{kN}}}{0.85 \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}} \\ &= 176.5\ \mathrm{mm}^{2} \end{aligned}\]
S.33

A23.3 Cl. 11.4.3.1

\[\begin{aligned} \htmlClass{sym-A_s3_req}{A_{s3,\mathrm{req}}} &= \frac{\htmlClass{sym-P_BD}{P_{\mathrm{BD}}}}{0.85 \cdot \htmlClass{sym-f_y}{f_{y}}} \\ &= \frac{\htmlClass{sym-P_BD}{358.2\ \mathrm{kN}}}{0.85 \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}} \\ &= 1053\ \mathrm{mm}^{2} \end{aligned}\]
S.34

A23.3 Cl. 11.4.3.1

\[\begin{aligned} \htmlClass{sym-A_s4_req}{A_{s4,\mathrm{req}}} &= \frac{\htmlClass{sym-P_DF}{P_{\mathrm{DF}}}}{0.85 \cdot \htmlClass{sym-f_y}{f_{y}}} \\ &= \frac{\htmlClass{sym-P_DF}{358.2\ \mathrm{kN}}}{0.85 \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}} \\ &= 1053\ \mathrm{mm}^{2} \end{aligned}\]
S.35 \[\begin{aligned} \htmlClass{sym-eps_s_A}{\epsilon_{s,A}} &= \frac{\htmlClass{sym-P_AB}{P_{\mathrm{AB}}}}{\htmlClass{sym-A_s1}{A_{s1}} \cdot \htmlClass{sym-E_s}{E_{s}}} \\ &= \frac{\htmlClass{sym-P_AB}{228.2\ \mathrm{kN}}}{\htmlClass{sym-A_s1}{900\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-E_s}{200000\ \mathrm{MPa}}} \\ &= 0.001268 \end{aligned}\]
S.36 \[\begin{aligned} \htmlClass{sym-cot_theta1}{\mathrm{cot}_{\theta 1}} &= \cot\left(\htmlClass{sym-theta1_rad}{\theta_{1,\mathrm{rad}}}\right) \\ &= \cot\left(\htmlClass{sym-theta1_rad}{1.06}\right) \\ &= 0.5607 \end{aligned}\]
S.37

A23.3 Cl. 11.4.2.3, Eq. 11.23

\[\begin{aligned} \htmlClass{sym-eps_1_A}{\epsilon_{1,A}} &= \htmlClass{sym-eps_s_A}{\epsilon_{s,A}} + \left(\htmlClass{sym-eps_s_A}{\epsilon_{s,A}} + 0.002\right) \cdot \htmlClass{sym-cot_theta1}{\mathrm{cot}_{\theta 1}}^{2} \\ &= \htmlClass{sym-eps_s_A}{0.001268} + \left(\htmlClass{sym-eps_s_A}{0.001268} + 0.002\right) \cdot \htmlClass{sym-cot_theta1}{0.5607}^{2} \\ &= 0.002295 \end{aligned}\]
S.38

A23.3 Eq. 11.22, Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-f_cu_A}{f_{\mathrm{cu},A}} &= \min\left(\frac{\htmlClass{sym-f_prime_c}{{f'}_{c}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_A}{\epsilon_{1,A}}}, 0.85 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}\right) \\ &= \min\left(\frac{\htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_A}{0.002295}}, 0.85 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}\right) \\ &= 26.25\ \mathrm{MPa} \end{aligned}\]
S.39

A23.3 Cl. 11.4.2.1

\[\begin{aligned} \htmlClass{sym-w_AC_A}{w_{\mathrm{AC},A}} &= \frac{\htmlClass{sym-P_AC}{P_{\mathrm{AC}}}}{\htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cu_A}{f_{\mathrm{cu},A}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_AC}{343.9\ \mathrm{kN}}}{\htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cu_A}{26.25\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 57.59\ \mathrm{mm} \end{aligned}\]
S.40 \[\begin{aligned} \htmlClass{sym-eps_s_C}{\epsilon_{s,C}} &= \frac{\htmlClass{sym-P_CD}{P_{\mathrm{CD}}}}{\htmlClass{sym-A_s2}{A_{s2}} \cdot \htmlClass{sym-E_s}{E_{s}}} \\ &= \frac{\htmlClass{sym-P_CD}{60\ \mathrm{kN}}}{\htmlClass{sym-A_s2}{200\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-E_s}{200000\ \mathrm{MPa}}} \\ &= 0.0015 \end{aligned}\]
S.41

A23.3 Cl. 11.4.2.3, Eq. 11.23

\[\begin{aligned} \htmlClass{sym-eps_1_C}{\epsilon_{1,C}} &= \htmlClass{sym-eps_s_C}{\epsilon_{s,C}} + \left(\htmlClass{sym-eps_s_C}{\epsilon_{s,C}} + 0.002\right) \cdot \htmlClass{sym-cot_theta1}{\mathrm{cot}_{\theta 1}}^{2} \\ &= \htmlClass{sym-eps_s_C}{0.0015} + \left(\htmlClass{sym-eps_s_C}{0.0015} + 0.002\right) \cdot \htmlClass{sym-cot_theta1}{0.5607}^{2} \\ &= 0.0026 \end{aligned}\]
S.42

A23.3 Eq. 11.22, Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-f_cu_C}{f_{\mathrm{cu},C}} &= \min\left(\frac{\htmlClass{sym-f_prime_c}{{f'}_{c}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_C}{\epsilon_{1,C}}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}\right) \\ &= \min\left(\frac{\htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_C}{0.0026}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}\right) \\ &= 26.25\ \mathrm{MPa} \end{aligned}\]
S.43

A23.3 Cl. 11.4.2.1

\[\begin{aligned} \htmlClass{sym-w_AC_C}{w_{\mathrm{AC},C}} &= \frac{\htmlClass{sym-P_AC}{P_{\mathrm{AC}}}}{\htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cu_C}{f_{\mathrm{cu},C}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_AC}{343.9\ \mathrm{kN}}}{\htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cu_C}{26.25\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 57.59\ \mathrm{mm} \end{aligned}\]
S.44 \[\begin{aligned} \htmlClass{sym-theta_min}{\theta_{\mathrm{min}}} &= \min\left(\htmlClass{sym-theta2}{\theta_{2}}, 90 - \htmlClass{sym-theta2}{\theta_{2}}\right) \\ &= \min\left(\htmlClass{sym-theta2}{32.5}, 90 - \htmlClass{sym-theta2}{32.5}\right) \\ &= 32.5 \end{aligned}\]
S.45 \[\begin{aligned} \htmlClass{sym-theta_min_rad}{\theta_{\mathrm{min},\mathrm{rad}}} &= \operatorname{radians}\left(\htmlClass{sym-theta_min}{\theta_{\mathrm{min}}}\right) \\ &= \operatorname{radians}\left(\htmlClass{sym-theta_min}{32.5}\right) \\ &= 0.5673 \end{aligned}\]
S.46 \[\begin{aligned} \htmlClass{sym-cot_theta_min}{\mathrm{cot}_{\theta,\mathrm{min}}} &= \cot\left(\htmlClass{sym-theta_min_rad}{\theta_{\mathrm{min},\mathrm{rad}}}\right) \\ &= \cot\left(\htmlClass{sym-theta_min_rad}{0.5673}\right) \\ &= 1.57 \end{aligned}\]

Branch: \(\theta_{2} \leq 45\) held

S.47 \[\begin{aligned} \htmlClass{sym-eps_s_B}{\epsilon_{s,B}} &= \frac{\htmlClass{sym-P_BD}{P_{\mathrm{BD}}}}{\htmlClass{sym-A_s3_req}{A_{s3,\mathrm{req}}} \cdot \htmlClass{sym-E_s}{E_{s}}} \\ &= \frac{\htmlClass{sym-P_BD}{358.2\ \mathrm{kN}}}{\htmlClass{sym-A_s3_req}{1053\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-E_s}{200000\ \mathrm{MPa}}} \\ &= 0.0017 \end{aligned}\]

Assumed: tie BD provided at A_s3_req

S.48

A23.3 Cl. 11.4.2.3, Eq. 11.23

\[\begin{aligned} \htmlClass{sym-eps_1_B}{\epsilon_{1,B}} &= \htmlClass{sym-eps_s_B}{\epsilon_{s,B}} + \left(\htmlClass{sym-eps_s_B}{\epsilon_{s,B}} + 0.002\right) \cdot \htmlClass{sym-cot_theta_min}{\mathrm{cot}_{\theta,\mathrm{min}}}^{2} \\ &= \htmlClass{sym-eps_s_B}{0.0017} + \left(\htmlClass{sym-eps_s_B}{0.0017} + 0.002\right) \cdot \htmlClass{sym-cot_theta_min}{1.57}^{2} \\ &= 0.01081 \end{aligned}\]
S.49

A23.3 Eq. 11.22, Cl. 11.4.4.1 c)

\[\begin{aligned} \htmlClass{sym-f_cu_B}{f_{\mathrm{cu},B}} &= \min\left(\frac{\htmlClass{sym-f_prime_c}{{f'}_{c}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_B}{\epsilon_{1,B}}}, 0.65 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}\right) \\ &= \min\left(\frac{\htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_B}{0.01081}}, 0.65 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}\right) \\ &= 13.27\ \mathrm{MPa} \end{aligned}\]
S.50

A23.3 Cl. 11.4.2.1

\[\begin{aligned} \htmlClass{sym-w_BC_B}{w_{\mathrm{BC},B}} &= \frac{\htmlClass{sym-P_BC}{P_{\mathrm{BC}}}}{\htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cu_B}{f_{\mathrm{cu},B}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_BC}{424.7\ \mathrm{kN}}}{\htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cu_B}{13.27\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 140.7\ \mathrm{mm} \end{aligned}\]
S.51 \[\begin{aligned} \htmlClass{sym-eps_s_BC}{\epsilon_{s,\mathrm{BC}}} &= \frac{\htmlClass{sym-P_CD}{P_{\mathrm{CD}}}}{\htmlClass{sym-A_s2}{A_{s2}} \cdot \htmlClass{sym-E_s}{E_{s}}} \\ &= \frac{\htmlClass{sym-P_CD}{60\ \mathrm{kN}}}{\htmlClass{sym-A_s2}{200\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-E_s}{200000\ \mathrm{MPa}}} \\ &= 0.0015 \end{aligned}\]
S.52 \[\begin{aligned} \htmlClass{sym-cot_theta4}{\mathrm{cot}_{\theta 4}} &= \cot\left(\htmlClass{sym-theta4_rad}{\theta_{4,\mathrm{rad}}}\right) \\ &= \cot\left(\htmlClass{sym-theta4_rad}{1.004}\right) \\ &= 0.6371 \end{aligned}\]
S.53

A23.3 Cl. 11.4.2.3, Eq. 11.23

\[\begin{aligned} \htmlClass{sym-eps_1_BC}{\epsilon_{1,\mathrm{BC}}} &= \htmlClass{sym-eps_s_BC}{\epsilon_{s,\mathrm{BC}}} + \left(\htmlClass{sym-eps_s_BC}{\epsilon_{s,\mathrm{BC}}} + 0.002\right) \cdot \htmlClass{sym-cot_theta4}{\mathrm{cot}_{\theta 4}}^{2} \\ &= \htmlClass{sym-eps_s_BC}{0.0015} + \left(\htmlClass{sym-eps_s_BC}{0.0015} + 0.002\right) \cdot \htmlClass{sym-cot_theta4}{0.6371}^{2} \\ &= 0.002921 \end{aligned}\]
S.54

A23.3 Eq. 11.22, Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-f_cu_BC}{f_{\mathrm{cu},\mathrm{BC}}} &= \min\left(\frac{\htmlClass{sym-f_prime_c}{{f'}_{c}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_BC}{\epsilon_{1,\mathrm{BC}}}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}\right) \\ &= \min\left(\frac{\htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}}{0.8 + 170 \cdot \htmlClass{sym-eps_1_BC}{0.002921}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}\right) \\ &= 26.25\ \mathrm{MPa} \end{aligned}\]
S.55

A23.3 Cl. 11.4.2.1

\[\begin{aligned} \htmlClass{sym-w_BC_C}{w_{\mathrm{BC},C}} &= \frac{\htmlClass{sym-P_BC}{P_{\mathrm{BC}}}}{\htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cu_BC}{f_{\mathrm{cu},\mathrm{BC}}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_BC}{424.7\ \mathrm{kN}}}{\htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cu_BC}{26.25\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 71.12\ \mathrm{mm} \end{aligned}\]
S.56

A23.3 Eq. 11.22, Cl. 11.4.4.1 b)

\[\begin{aligned} \htmlClass{sym-f_cu_CE}{f_{\mathrm{cu},\mathrm{CE}}} &= \min\left(\frac{\htmlClass{sym-f_prime_c}{{f'}_{c}}}{0.8 + 170 \cdot \htmlClass{sym-eps_s_C}{\epsilon_{s,C}}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}}\right) \\ &= \min\left(\frac{\htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}}{0.8 + 170 \cdot \htmlClass{sym-eps_s_C}{0.0015}}, 0.75 \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}}\right) \\ &= 26.25\ \mathrm{MPa} \end{aligned}\]
S.57

A23.3 Cl. 11.4.2.1

\[\begin{aligned} \htmlClass{sym-w_CE}{w_{\mathrm{CE}}} &= \frac{\htmlClass{sym-P_CE}{P_{\mathrm{CE}}}}{\htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_cu_CE}{f_{\mathrm{cu},\mathrm{CE}}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_CE}{658.2\ \mathrm{kN}}}{\htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_cu_CE}{26.25\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 110.2\ \mathrm{mm} \end{aligned}\]
S.58 \[\begin{aligned} \htmlClass{sym-w_CE_check_ratio}{w_{\mathrm{CE},\mathrm{check},\mathrm{ratio}}} &= \frac{\htmlClass{sym-w_CE}{w_{\mathrm{CE}}}}{\htmlClass{sym-a_node}{a_{\mathrm{node}}}} \\ &= \frac{\htmlClass{sym-w_CE}{110.2\ \mathrm{mm}}}{\htmlClass{sym-a_node}{110.2\ \mathrm{mm}}} \\ &= 1 \end{aligned}\]
S.59

A23.3 Cl. 11.6.5

\[\begin{aligned} \htmlClass{sym-min_tie_area}{\mathrm{min}_{\mathrm{tie},\mathrm{area}}} &= 0.5 \cdot \htmlClass{sym-A_s1}{A_{s1}} \\ &= 0.5 \cdot \htmlClass{sym-A_s1}{900\ \mathrm{mm}^{2}} \\ &= 450\ \mathrm{mm}^{2} \end{aligned}\]
S.60

A23.3 Cl. 11.6.5

\[\begin{aligned} \htmlClass{sym-tie_zone_height}{\mathrm{tie}_{\mathrm{zone},\mathrm{height}}} &= \frac{2}{3} \cdot \htmlClass{sym-L_BD}{L_{\mathrm{BD}}} \\ &= \frac{2}{3} \cdot \htmlClass{sym-L_BD}{455\ \mathrm{mm}} \\ &= 303.3\ \mathrm{mm} \end{aligned}\]
S.61

A23.3 Cl. 11.4.4.1 b), 11.4.4.2 b)

\[\begin{aligned} \htmlClass{sym-h_AB}{h_{\mathrm{AB}}} &= \frac{\htmlClass{sym-P_AB}{P_{\mathrm{AB}}}}{0.75 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_prime_c}{{f'}_{c}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-P_AB}{228.2\ \mathrm{kN}}}{0.75 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_prime_c}{35\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}} \\ &= 38.21\ \mathrm{mm} \end{aligned}\]

Questions

Why is N_f never taken below 0.2 V_f?

CSA A23.3 requires a bracket or corbel to be designed for a horizontal tension of at least 0.2 V_f acting with the vertical load, because restrained shrinkage, creep and temperature pull on the bearing even when no horizontal load is specified. N_f_design is the larger of your N_f and that minimum, and it adds directly to the top tie force P_AB.

Why does A_s1_req have a second, minimum term?

The top tie is the larger of the strut-and-tie force P_AB over 0.85 f_y and a minimum of 0.04 f'c / f_y times b times the depth L_BD. The minimum governs on a lightly loaded corbel, where the tie force alone would size a tie that yields as soon as the concrete cracks. The 0.85 in the first term is the steel resistance factor.

What limits V_f in the strut-and-tie model?

The depth of the node at the column face, a_node, is the root of a quadratic in the strut-and-tie equilibrium. Past a certain V_f the quadratic has no real root: no depth of concrete at the face can supply the compression the tie needs, so there is no strut-and-tie solution and the calc refuses rather than print one. At the declared geometry that happens near 558 kN; a narrower b, a weaker f'c or a shallower h lowers the limit.

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