calcsheet
RC beam shear and torsion
Verified against CSA A23.3-24, 2026-10-01
Combined shear and torsion design of a reinforced concrete beam, CSA A23.3-24 Cl. 11. Check a reinforced concrete beam for combined shear and torsion to CSA A23.3-24 Clause 11, simplified method. The calculator works through the concrete and steel shear resistances, the cracking torque that decides whether torsion has to be designed for, the crushing check for combined shear and torsion, the stirrups and spacing the two demands require together, and the longitudinal tension they add to the flexural steel. Full hand-calculation output for design records.
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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/rc-shear-torsion.pdf.
Checks
| Check | D/C | Utilisation | Result |
|---|---|---|---|
| Min stirrups provided\(\htmlClass{sym-A_v_min_per_s}{A_{v,\mathrm{min},\mathrm{per},s}} \leq \htmlClass{sym-total_stir_provided}{\mathrm{total}_{\mathrm{stir},\mathrm{provided}}} \quad \Rightarrow \quad \htmlClass{sym-A_v_min_per_s}{0.2625} \leq \htmlClass{sym-total_stir_provided}{2}\) | 0.13 | PASS | |
| Torsion reinf required\(\htmlClass{sym-T_0f}{T_{0f}} > 0.25 \cdot \htmlClass{sym-T_cr}{T_{\mathrm{cr}}} \quad \Rightarrow \quad \htmlClass{sym-T_0f}{10\ \mathrm{kN} \cdot \mathrm{m}} > 0.25 \cdot \htmlClass{sym-T_cr}{35.3\ \mathrm{kN} \cdot \mathrm{m}}\) | 0.88 | PASS | |
| Low shear\(\htmlClass{sym-shear_stress}{\mathrm{shear}_{\mathrm{stress}}} \leq \htmlClass{sym-threshold}{\mathrm{threshold}} \quad \Rightarrow \quad \htmlClass{sym-shear_stress}{1.362\ \mathrm{MPa}} \leq \htmlClass{sym-threshold}{2.031\ \mathrm{MPa}}\) | 0.67 | PASS | |
| Shear adequate\(\htmlClass{sym-V_f}{V_{f}} \leq \htmlClass{sym-V_r}{V_{r}} \quad \Rightarrow \quad \htmlClass{sym-V_f}{273\ \mathrm{kN}} \leq \htmlClass{sym-V_r}{673.2\ \mathrm{kN}}\) | 0.41 | PASS | |
| Torsion section adequate\(\htmlClass{sym-combined_stress}{\mathrm{combined}_{\mathrm{stress}}} \leq \htmlClass{sym-stress_limit}{\mathrm{stress}_{\mathrm{limit}}} \quad \Rightarrow \quad \htmlClass{sym-combined_stress}{1.441\ \mathrm{MPa}} \leq \htmlClass{sym-stress_limit}{4.062\ \mathrm{MPa}}\) | 0.35 | PASS | |
| Torsion stirrups adequate\(\htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}} \leq \htmlClass{sym-torsion_stir_provided}{\mathrm{torsion}_{\mathrm{stir},\mathrm{provided}}} \quad \Rightarrow \quad \htmlClass{sym-A_t_per_s}{0.07693} \leq \htmlClass{sym-torsion_stir_provided}{0.5}\) | 0.15 | PASS | |
| Total stirrups adequate\(\htmlClass{sym-total_stir_required}{\mathrm{total}_{\mathrm{stir},\mathrm{required}}} \leq \htmlClass{sym-total_stir_provided}{\mathrm{total}_{\mathrm{stir},\mathrm{provided}}} \quad \Rightarrow \quad \htmlClass{sym-total_stir_required}{0.7143} \leq \htmlClass{sym-total_stir_provided}{2}\) | 0.36 | PASS | |
| Spacing adequate\(\htmlClass{sym-s}{s} \leq \htmlClass{sym-S_max}{S_{\mathrm{max}}} \quad \Rightarrow \quad \htmlClass{sym-s}{200\ \mathrm{mm}} \leq \htmlClass{sym-S_max}{200.4\ \mathrm{mm}}\) | 1.00 | PASS | |
| Width spacing adequate\(\htmlClass{sym-s_w}{s_{w}} \leq \htmlClass{sym-s_w_max}{s_{w,\mathrm{max}}} \quad \Rightarrow \quad \htmlClass{sym-s_w}{86.23\ \mathrm{mm}} \leq \htmlClass{sym-s_w_max}{572.5\ \mathrm{mm}}\) | 0.15 | PASS |
Results
| Quantity | Description | Value | Unit |
|---|---|---|---|
| \(\htmlClass{sym-V_r}{V_{r}}\) | Factored shear resistance | 673.2 | \(\mathrm{kN}\) |
| \(\htmlClass{sym-S_max}{S_{\mathrm{max}}}\) | Maximum stirrup spacing | 200.4 | \(\mathrm{mm}\) |
| \(\htmlClass{sym-A_v_min_per_s}{A_{v,\mathrm{min},\mathrm{per},s}}\) | Minimum shear stirrup area per unit length, mm²/mm | 0.2625 | |
| \(\htmlClass{sym-A_v_per_s}{A_{v,\mathrm{per},s}}\) | Shear stirrup area per unit length required, mm²/mm | 0.5604 | |
| \(\htmlClass{sym-T_cr}{T_{\mathrm{cr}}}\) | Cracking torque | 35.3 | \(\mathrm{kN} \cdot \mathrm{m}\) |
| \(\htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}}\) | Torsion stirrup area per unit length required, mm²/mm | 0.07693 | |
| \(\htmlClass{sym-F_lt}{F_{\mathrm{lt}}}\) | Longitudinal tension from shear and torsion, add M_f / d_v | 199.3 | \(\mathrm{kN}\) |
Derivation
S.1
\[\begin{aligned}
\htmlClass{sym-lamb}{\lambda} &= 1
\end{aligned}\]
S.2
\[\begin{aligned}
\htmlClass{sym-phi_c}{\phi_{c}} &= 0.65 \quad \left(\text{Resistance factor, concrete | CSA A23.3 Cl. 8.4.2}\right)
\end{aligned}\]
S.3
\[\begin{aligned}
\htmlClass{sym-phi_s}{\phi_{s}} &= 0.85 \quad \left(\text{Resistance factor, reinforcing bars | CSA A23.3 Cl. 8.4.3 a)}\right)
\end{aligned}\]
S.4
\[\begin{aligned}
\htmlClass{sym-d_s}{d_{s}} &= 11.3\ \mathrm{mm} \quad \left(\text{Stirrup diameter | 10M}\right)
\end{aligned}\]
S.5
\[\begin{aligned}
\htmlClass{sym-d_b}{d_{b}} &= 25.2\ \mathrm{mm} \quad \left(\text{Longitudinal bar diameter | 25M}\right)
\end{aligned}\]
S.6
\[\begin{aligned}
\htmlClass{sym-A_sb}{A_{\mathrm{sb}}} &= 100\ \mathrm{mm}^{2} \quad \left(\text{Area of one stirrup leg | 10M}\right)
\end{aligned}\]
S.7
A23.3 Cl. 11.3.6.3, simplified method
\[\begin{aligned} \htmlClass{sym-theta}{\theta} &= \operatorname{radians}\left(35\right) \\ &= \operatorname{radians}\left(35\right) \\ &= 0.6109 \end{aligned}\]
S.8
\[\begin{aligned}
\htmlClass{sym-d}{d} &= \htmlClass{sym-h}{h} - \htmlClass{sym-c_c}{c_{c}} - \htmlClass{sym-d_s}{d_{s}} - \frac{\htmlClass{sym-d_b}{d_{b}}}{2} \\
&= \htmlClass{sym-h}{700\ \mathrm{mm}} - \htmlClass{sym-c_c}{40\ \mathrm{mm}} - \htmlClass{sym-d_s}{11.3\ \mathrm{mm}} - \frac{\htmlClass{sym-d_b}{25.2\ \mathrm{mm}}}{2} \\
&= 636.1\ \mathrm{mm}
\end{aligned}\]
S.9
A23.3 Cl. 3.2, effective shear depth
\[\begin{aligned} \htmlClass{sym-d_v}{d_{v}} &= \max\left(0.9 \cdot \htmlClass{sym-d}{d}, 0.72 \cdot \htmlClass{sym-h}{h}\right) \\ &= \max\left(0.9 \cdot \htmlClass{sym-d}{636.1\ \mathrm{mm}}, 0.72 \cdot \htmlClass{sym-h}{700\ \mathrm{mm}}\right) \\ &= 572.5\ \mathrm{mm} \end{aligned}\]
S.10
\[\begin{aligned}
\htmlClass{sym-x_0}{x_{0}} &= \htmlClass{sym-b}{b} - 2 \cdot \htmlClass{sym-c_c}{c_{c}} - \htmlClass{sym-d_s}{d_{s}} \\
&= \htmlClass{sym-b}{350\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-c_c}{40\ \mathrm{mm}} - \htmlClass{sym-d_s}{11.3\ \mathrm{mm}} \\
&= 258.7\ \mathrm{mm}
\end{aligned}\]
S.11
\[\begin{aligned}
\htmlClass{sym-y_0}{y_{0}} &= \htmlClass{sym-h}{h} - 2 \cdot \htmlClass{sym-c_c}{c_{c}} - \htmlClass{sym-d_s}{d_{s}} \\
&= \htmlClass{sym-h}{700\ \mathrm{mm}} - 2 \cdot \htmlClass{sym-c_c}{40\ \mathrm{mm}} - \htmlClass{sym-d_s}{11.3\ \mathrm{mm}} \\
&= 608.7\ \mathrm{mm}
\end{aligned}\]
S.12
A23.3 Cl. 3.2, p_h
\[\begin{aligned} \htmlClass{sym-P_h}{P_{h}} &= 2 \cdot \left(\htmlClass{sym-x_0}{x_{0}} + \htmlClass{sym-y_0}{y_{0}}\right) \\ &= 2 \cdot \left(\htmlClass{sym-x_0}{258.7\ \mathrm{mm}} + \htmlClass{sym-y_0}{608.7\ \mathrm{mm}}\right) \\ &= 1.735\ \mathrm{m} \end{aligned}\]
S.13
A23.3 Cl. 3.2, A_oh
\[\begin{aligned} \htmlClass{sym-A_0h}{A_{0h}} &= \htmlClass{sym-x_0}{x_{0}} \cdot \htmlClass{sym-y_0}{y_{0}} \\ &= \htmlClass{sym-x_0}{258.7\ \mathrm{mm}} \cdot \htmlClass{sym-y_0}{608.7\ \mathrm{mm}} \\ &= 157471\ \mathrm{mm}^{2} \end{aligned}\]
S.14
\[\begin{aligned}
\htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}} &= \frac{1}{\tan\left(\htmlClass{sym-theta}{\theta}\right)} \\
&= \frac{1}{\tan\left(\htmlClass{sym-theta}{0.6109}\right)} \\
&= 1.428
\end{aligned}\]
S.15
A23.3 Cl. 11.2.8.2, Eq. 11.1
\[\begin{aligned} \htmlClass{sym-A_v_min_per_s}{A_{v,\mathrm{min},\mathrm{per},s}} &= \frac{0.06 \cdot \sqrt{\htmlClass{sym-f_c}{f_{c}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{b}}{\htmlClass{sym-f_y}{f_{y}} \cdot 1\ \mathrm{mm}} \\ &= \frac{0.06 \cdot \sqrt{\htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{350\ \mathrm{mm}}}{\htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot 1\ \mathrm{mm}} \\ &= 0.2625 \end{aligned}\]
S.16
\[\begin{aligned}
\htmlClass{sym-total_stir_provided}{\mathrm{total}_{\mathrm{stir},\mathrm{provided}}} &= \frac{\htmlClass{sym-n}{n} \cdot \htmlClass{sym-A_sb}{A_{\mathrm{sb}}}}{\htmlClass{sym-s}{s} \cdot 1\ \mathrm{mm}} \\
&= \frac{\htmlClass{sym-n}{4} \cdot \htmlClass{sym-A_sb}{100\ \mathrm{mm}^{2}}}{\htmlClass{sym-s}{200\ \mathrm{mm}} \cdot 1\ \mathrm{mm}} \\
&= 2
\end{aligned}\]
Branch: \(\mathrm{total}_{\mathrm{stir},\mathrm{provided}} > 11 \cdot A_{v,\mathrm{min},\mathrm{per},s}\) did not hold
S.17
A23.3 Cl. 11.2.10.1, minimum web width within d
\[\begin{aligned} \htmlClass{sym-b_w}{b_{w}} &= 1 \cdot \htmlClass{sym-b}{b} \\ &= 1 \cdot \htmlClass{sym-b}{350\ \mathrm{mm}} \\ &= 350\ \mathrm{mm} \end{aligned}\]Branch: \(A_{v,\mathrm{min},\mathrm{per},s} \leq \mathrm{total}_{\mathrm{stir},\mathrm{provided}}\) held
Branch: \(f_{y} \leq 400\ \mathrm{MPa}\) held
S.18
\[\begin{aligned}
\htmlClass{sym-beta}{\beta} &= 0.18 \quad \left(\text{At least the minimum stirrups, f\_y up to 400 MPa | Cl. 11.3.6.3 a)}\right)
\end{aligned}\]
S.19
A23.3 Cl. 11.3.4, Eq. 11.6
\[\begin{aligned} \htmlClass{sym-V_c}{V_{c}} &= \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-lamb}{\lambda} \cdot \htmlClass{sym-beta}{\beta} \cdot \sqrt{\htmlClass{sym-f_c}{f_{c}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b_w}{b_{w}} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-lamb}{1} \cdot \htmlClass{sym-beta}{0.18} \cdot \sqrt{\htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b_w}{350\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}} \\ &= 117.2\ \mathrm{kN} \end{aligned}\]
S.20
\[\begin{aligned}
\htmlClass{sym-A_v}{A_{v}} &= \htmlClass{sym-n}{n} \cdot \htmlClass{sym-A_sb}{A_{\mathrm{sb}}} \\
&= \htmlClass{sym-n}{4} \cdot \htmlClass{sym-A_sb}{100\ \mathrm{mm}^{2}} \\
&= 400\ \mathrm{mm}^{2}
\end{aligned}\]
S.21
A23.3 Cl. 11.3.5.1, Eq. 11.7
\[\begin{aligned} \htmlClass{sym-V_s}{V_{s}} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_v}{A_{v}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \htmlClass{sym-d_v}{d_{v}} \cdot \htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}}}{\htmlClass{sym-s}{s}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_v}{400\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}} \cdot \htmlClass{sym-cot_theta}{1.428}}{\htmlClass{sym-s}{200\ \mathrm{mm}}} \\ &= 556\ \mathrm{kN} \end{aligned}\]
S.22
A23.3 Cl. 11.3.3, Eq. 11.5
\[\begin{aligned} \htmlClass{sym-V_rmax}{V_{\mathrm{rmax}}} &= 0.25 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \cdot \htmlClass{sym-b_w}{b_{w}} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= 0.25 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot \htmlClass{sym-b_w}{350\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}} \\ &= 814\ \mathrm{kN} \end{aligned}\]
S.23
A23.3 Cl. 11.3.3, Eqs. 11.4 and 11.5
\[\begin{aligned} \htmlClass{sym-V_r}{V_{r}} &= \min\left(\htmlClass{sym-V_c}{V_{c}} + \htmlClass{sym-V_s}{V_{s}}, \htmlClass{sym-V_rmax}{V_{\mathrm{rmax}}}\right) \\ &= \min\left(\htmlClass{sym-V_c}{117.2\ \mathrm{kN}} + \htmlClass{sym-V_s}{556\ \mathrm{kN}}, \htmlClass{sym-V_rmax}{814\ \mathrm{kN}}\right) \\ &= 673.2\ \mathrm{kN} \end{aligned}\]
S.24
\[\begin{aligned}
\htmlClass{sym-shear_utilization}{\mathrm{shear}_{\mathrm{utilization}}} &= \frac{\htmlClass{sym-V_f}{V_{f}}}{\htmlClass{sym-V_r}{V_{r}}} \\
&= \frac{\htmlClass{sym-V_f}{273\ \mathrm{kN}}}{\htmlClass{sym-V_r}{673.2\ \mathrm{kN}}} \\
&= 0.4055
\end{aligned}\]
S.25
A23.3 Cl. 3.2, A_c
\[\begin{aligned} \htmlClass{sym-A_c}{A_{c}} &= \htmlClass{sym-b}{b} \cdot \htmlClass{sym-h}{h} \\ &= \htmlClass{sym-b}{350\ \mathrm{mm}} \cdot \htmlClass{sym-h}{700\ \mathrm{mm}} \\ &= 245000\ \mathrm{mm}^{2} \end{aligned}\]
S.26
A23.3 Cl. 3.2, p_c
\[\begin{aligned} \htmlClass{sym-P_c}{P_{c}} &= 2 \cdot \left(\htmlClass{sym-b}{b} + \htmlClass{sym-h}{h}\right) \\ &= 2 \cdot \left(\htmlClass{sym-b}{350\ \mathrm{mm}} + \htmlClass{sym-h}{700\ \mathrm{mm}}\right) \\ &= 2.1\ \mathrm{m} \end{aligned}\]
S.27
A23.3 Cl. 11.2.9.1, Eq. 11.2
\[\begin{aligned} \htmlClass{sym-T_cr}{T_{\mathrm{cr}}} &= \frac{0.38 \cdot \htmlClass{sym-A_c}{A_{c}}^{2} \cdot \htmlClass{sym-lamb}{\lambda} \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \sqrt{\htmlClass{sym-f_c}{f_{c}} \cdot 1\ \mathrm{MPa}}}{\htmlClass{sym-P_c}{P_{c}}} \\ &= \frac{0.38 \cdot \left(\htmlClass{sym-A_c}{245000\ \mathrm{mm}^{2}}\right)^{2} \cdot \htmlClass{sym-lamb}{1} \cdot \htmlClass{sym-phi_c}{0.65} \cdot \sqrt{\htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}}}{\htmlClass{sym-P_c}{2.1\ \mathrm{m}}} \\ &= 35.3\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.28
A23.3 Cl. 11.3.3 and 11.3.5.1, Eqs. 11.4 and 11.7
\[\begin{aligned} \htmlClass{sym-A_v_per_s}{A_{v,\mathrm{per},s}} &= \max\left(\frac{\htmlClass{sym-V_f}{V_{f}} - \htmlClass{sym-V_c}{V_{c}}}{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \htmlClass{sym-d_v}{d_{v}} \cdot \htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}} \cdot 1\ \mathrm{mm}}, 0\right) \\ &= \max\left(\frac{\htmlClass{sym-V_f}{273\ \mathrm{kN}} - \htmlClass{sym-V_c}{117.2\ \mathrm{kN}}}{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}} \cdot \htmlClass{sym-cot_theta}{1.428} \cdot 1\ \mathrm{mm}}, 0\right) \\ &= 0.5604 \end{aligned}\]
S.29
A23.3 Cl. 11.3.10.3
\[\begin{aligned} \htmlClass{sym-A_0}{A_{0}} &= 0.85 \cdot \htmlClass{sym-A_0h}{A_{0h}} \\ &= 0.85 \cdot \htmlClass{sym-A_0h}{157471\ \mathrm{mm}^{2}} \\ &= 133850\ \mathrm{mm}^{2} \end{aligned}\]
S.30
A23.3 Cl. 11.3.10.3, Eq. 11.17
\[\begin{aligned} \htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}} &= \frac{\htmlClass{sym-T_0f}{T_{0f}}}{2 \cdot \htmlClass{sym-A_0}{A_{0}} \cdot \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}} \cdot 1\ \mathrm{mm}} \\ &= \frac{\htmlClass{sym-T_0f}{10\ \mathrm{kN} \cdot \mathrm{m}}}{2 \cdot \htmlClass{sym-A_0}{133850\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \htmlClass{sym-cot_theta}{1.428} \cdot 1\ \mathrm{mm}} \\ &= 0.07693 \end{aligned}\]
S.31
\[\begin{aligned}
\htmlClass{sym-torsion_stir_provided}{\mathrm{torsion}_{\mathrm{stir},\mathrm{provided}}} &= \frac{\htmlClass{sym-A_sb}{A_{\mathrm{sb}}}}{\htmlClass{sym-s}{s} \cdot 1\ \mathrm{mm}} \\
&= \frac{\htmlClass{sym-A_sb}{100\ \mathrm{mm}^{2}}}{\htmlClass{sym-s}{200\ \mathrm{mm}} \cdot 1\ \mathrm{mm}} \\
&= 0.5
\end{aligned}\]
S.32
\[\begin{aligned}
\htmlClass{sym-torsion_stir_utilization}{\mathrm{torsion}_{\mathrm{stir},\mathrm{utilization}}} &= \frac{\htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}}}{\htmlClass{sym-torsion_stir_provided}{\mathrm{torsion}_{\mathrm{stir},\mathrm{provided}}}} \\
&= \frac{\htmlClass{sym-A_t_per_s}{0.07693}}{\htmlClass{sym-torsion_stir_provided}{0.5}} \\
&= 0.1539
\end{aligned}\]
S.33
A23.3 Cl. 11.3.10.1
\[\begin{aligned} \htmlClass{sym-total_stir_required}{\mathrm{total}_{\mathrm{stir},\mathrm{required}}} &= \htmlClass{sym-A_v_per_s}{A_{v,\mathrm{per},s}} + 2 \cdot \htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}} \\ &= \htmlClass{sym-A_v_per_s}{0.5604} + 2 \cdot \htmlClass{sym-A_t_per_s}{0.07693} \\ &= 0.7143 \end{aligned}\]
S.34
\[\begin{aligned}
\htmlClass{sym-total_stir_utilization}{\mathrm{total}_{\mathrm{stir},\mathrm{utilization}}} &= \frac{\htmlClass{sym-total_stir_required}{\mathrm{total}_{\mathrm{stir},\mathrm{required}}}}{\htmlClass{sym-total_stir_provided}{\mathrm{total}_{\mathrm{stir},\mathrm{provided}}}} \\
&= \frac{\htmlClass{sym-total_stir_required}{0.7143}}{\htmlClass{sym-total_stir_provided}{2}} \\
&= 0.3571
\end{aligned}\]
S.35
\[\begin{aligned}
\htmlClass{sym-shear_stress}{\mathrm{shear}_{\mathrm{stress}}} &= \frac{\htmlClass{sym-V_f}{V_{f}}}{\htmlClass{sym-b_w}{b_{w}} \cdot \htmlClass{sym-d_v}{d_{v}}} \\
&= \frac{\htmlClass{sym-V_f}{273\ \mathrm{kN}}}{\htmlClass{sym-b_w}{350\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}}} \\
&= 1.362\ \mathrm{MPa}
\end{aligned}\]
S.36
A23.3 Cl. 11.3.8.3, as a stress on b_w d_v
\[\begin{aligned} \htmlClass{sym-threshold}{\mathrm{threshold}} &= 0.125 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-lamb}{\lambda} \cdot \htmlClass{sym-f_c}{f_{c}} \\ &= 0.125 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-lamb}{1} \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}} \\ &= 2.031\ \mathrm{MPa} \end{aligned}\]
S.37
A23.3 Cl. 11.3.8.3, half of Cl. 11.3.8.1
\[\begin{aligned} \htmlClass{sym-S_max}{S_{\mathrm{max}}} &= \min\left(300\ \mathrm{mm}, 0.35 \cdot \htmlClass{sym-d_v}{d_{v}}\right) \\ &= \min\left(300\ \mathrm{mm}, 0.35 \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}}\right) \\ &= 200.4\ \mathrm{mm} \end{aligned}\]
S.38
\[\begin{aligned}
\htmlClass{sym-s_w}{s_{w}} &= \frac{\htmlClass{sym-x_0}{x_{0}}}{\htmlClass{sym-n}{n} - 1} \\
&= \frac{\htmlClass{sym-x_0}{258.7\ \mathrm{mm}}}{\htmlClass{sym-n}{4} - 1} \\
&= 86.23\ \mathrm{mm}
\end{aligned}\]
S.39
A23.3 Cl. 11.3.8.4
\[\begin{aligned} \htmlClass{sym-s_w_max}{s_{w,\mathrm{max}}} &= \min\left(\htmlClass{sym-d_v}{d_{v}}, 600\ \mathrm{mm}\right) \\ &= \min\left(\htmlClass{sym-d_v}{572.5\ \mathrm{mm}}, 600\ \mathrm{mm}\right) \\ &= 572.5\ \mathrm{mm} \end{aligned}\]
S.40
A23.3 Cl. 11.3.10.4
\[\begin{aligned} \htmlClass{sym-alpha_T}{\alpha_{T}} &= 1 - \frac{0.005 \cdot \htmlClass{sym-f_c}{f_{c}}}{1\ \mathrm{MPa}} \\ &= 1 - \frac{0.005 \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}}}{1\ \mathrm{MPa}} \\ &= 0.875 \end{aligned}\]
S.41
A23.3 Cl. 11.3.10.4 b), torsion term
\[\begin{aligned} \htmlClass{sym-torsion_stress}{\mathrm{torsion}_{\mathrm{stress}}} &= \frac{\htmlClass{sym-T_0f}{T_{0f}} \cdot \htmlClass{sym-P_c}{P_{c}}}{0.85 \cdot \htmlClass{sym-alpha_T}{\alpha_{T}} \cdot \htmlClass{sym-A_c}{A_{c}}^{2}} \\ &= \frac{\htmlClass{sym-T_0f}{10\ \mathrm{kN} \cdot \mathrm{m}} \cdot \htmlClass{sym-P_c}{2.1\ \mathrm{m}}}{0.85 \cdot \htmlClass{sym-alpha_T}{0.875} \cdot \left(\htmlClass{sym-A_c}{245000\ \mathrm{mm}^{2}}\right)^{2}} \\ &= 470.4\ \mathrm{kPa} \end{aligned}\]
S.42
A23.3 Cl. 11.3.10.4 b)
\[\begin{aligned} \htmlClass{sym-combined_stress}{\mathrm{combined}_{\mathrm{stress}}} &= \sqrt{\htmlClass{sym-shear_stress}{\mathrm{shear}_{\mathrm{stress}}}^{2} + \htmlClass{sym-torsion_stress}{\mathrm{torsion}_{\mathrm{stress}}}^{2}} \\ &= \sqrt{\left(\htmlClass{sym-shear_stress}{1.362\ \mathrm{MPa}}\right)^{2} + \left(\htmlClass{sym-torsion_stress}{470.4\ \mathrm{kPa}}\right)^{2}} \\ &= 1.441\ \mathrm{MPa} \end{aligned}\]
S.43
A23.3 Cl. 11.3.10.4 b)
\[\begin{aligned} \htmlClass{sym-stress_limit}{\mathrm{stress}_{\mathrm{limit}}} &= 0.25 \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \\ &= 0.25 \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}} \\ &= 4.062\ \mathrm{MPa} \end{aligned}\]
S.44
\[\begin{aligned}
\htmlClass{sym-torsion_section_utilization}{\mathrm{torsion}_{\mathrm{section},\mathrm{utilization}}} &= \frac{\htmlClass{sym-combined_stress}{\mathrm{combined}_{\mathrm{stress}}}}{\htmlClass{sym-stress_limit}{\mathrm{stress}_{\mathrm{limit}}}} \\
&= \frac{\htmlClass{sym-combined_stress}{1.441\ \mathrm{MPa}}}{\htmlClass{sym-stress_limit}{4.062\ \mathrm{MPa}}} \\
&= 0.3548
\end{aligned}\]
S.45
A23.3 Cl. 11.3.10.1, shear share
\[\begin{aligned} \htmlClass{sym-A_vs_per_s}{A_{\mathrm{vs},\mathrm{per},s}} &= \max\left(\htmlClass{sym-total_stir_provided}{\mathrm{total}_{\mathrm{stir},\mathrm{provided}}} - 2 \cdot \htmlClass{sym-A_t_per_s}{A_{t,\mathrm{per},s}}, 0\right) \\ &= \max\left(\htmlClass{sym-total_stir_provided}{2} - 2 \cdot \htmlClass{sym-A_t_per_s}{0.07693}, 0\right) \\ &= 1.846 \end{aligned}\]
S.46
A23.3 Cl. 11.3.5.1, Eq. 11.7
\[\begin{aligned} \htmlClass{sym-V_s_shear}{V_{s,\mathrm{shear}}} &= \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_vs_per_s}{A_{\mathrm{vs},\mathrm{per},s}} \cdot 1\ \mathrm{mm} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \htmlClass{sym-d_v}{d_{v}} \cdot \htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}} \\ &= \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_vs_per_s}{1.846} \cdot 1\ \mathrm{mm} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \htmlClass{sym-d_v}{572.5\ \mathrm{mm}} \cdot \htmlClass{sym-cot_theta}{1.428} \\ &= 513.2\ \mathrm{kN} \end{aligned}\]
S.47
A23.3 Cl. 11.3.9.2, V_s not over V_f
\[\begin{aligned} \htmlClass{sym-V_s_lt}{V_{s,\mathrm{lt}}} &= \min\left(\htmlClass{sym-V_s_shear}{V_{s,\mathrm{shear}}}, \htmlClass{sym-V_f}{V_{f}}\right) \\ &= \min\left(\htmlClass{sym-V_s_shear}{513.2\ \mathrm{kN}}, \htmlClass{sym-V_f}{273\ \mathrm{kN}}\right) \\ &= 273\ \mathrm{kN} \end{aligned}\]
S.48
A23.3 Cl. 11.3.10.6, V_p = 0
\[\begin{aligned} \htmlClass{sym-shear_term}{\mathrm{shear}_{\mathrm{term}}} &= \htmlClass{sym-V_f}{V_{f}} - 0.5 \cdot \htmlClass{sym-V_s_lt}{V_{s,\mathrm{lt}}} \\ &= \htmlClass{sym-V_f}{273\ \mathrm{kN}} - 0.5 \cdot \htmlClass{sym-V_s_lt}{273\ \mathrm{kN}} \\ &= 136.5\ \mathrm{kN} \end{aligned}\]
S.49
A23.3 Cl. 11.3.10.6
\[\begin{aligned} \htmlClass{sym-torsion_term}{\mathrm{torsion}_{\mathrm{term}}} &= \frac{0.45 \cdot \htmlClass{sym-P_h}{P_{h}} \cdot \htmlClass{sym-T_0f}{T_{0f}}}{2 \cdot \htmlClass{sym-A_0}{A_{0}}} \\ &= \frac{0.45 \cdot \htmlClass{sym-P_h}{1.735\ \mathrm{m}} \cdot \htmlClass{sym-T_0f}{10\ \mathrm{kN} \cdot \mathrm{m}}}{2 \cdot \htmlClass{sym-A_0}{133850\ \mathrm{mm}^{2}}} \\ &= 29.16\ \mathrm{kN} \end{aligned}\]
S.50
A23.3 Eq. 11.14 with Cl. 11.3.10.6
\[\begin{aligned} \htmlClass{sym-F_lt}{F_{\mathrm{lt}}} &= \sqrt{\htmlClass{sym-shear_term}{\mathrm{shear}_{\mathrm{term}}}^{2} + \htmlClass{sym-torsion_term}{\mathrm{torsion}_{\mathrm{term}}}^{2}} \cdot \htmlClass{sym-cot_theta}{\mathrm{cot}_{\theta}} \\ &= \sqrt{\left(\htmlClass{sym-shear_term}{136.5\ \mathrm{kN}}\right)^{2} + \left(\htmlClass{sym-torsion_term}{29.16\ \mathrm{kN}}\right)^{2}} \cdot \htmlClass{sym-cot_theta}{1.428} \\ &= 199.3\ \mathrm{kN} \end{aligned}\]Questions
When does Cl. 11.2.9.1 require torsion reinforcement?
When the factored torsion exceeds a quarter of the cracking torque, T_0f > 0.25 T_cr, per Cl. 11.2.9.1. Below that the torsion may be neglected. This sheet still sizes A_t/s either way, which errs on the safe side.
Why is the stirrup demand A_v/s + 2 A_t/s?
Cl. 11.3.10.1 adds the transverse steel for shear to that for torsion. Torsion is resisted by a shear flow around the closed perimeter, so A_t/s is needed in each leg on that perimeter and both side legs carry it, hence the 2. Shear uses every leg crossing the crack, so the combined demand is compared with all n legs, while the torsion check on its own counts one outside leg.
What sets S_max, the maximum stirrup spacing?
Cl. 11.3.8.1 limits it to the lesser of 600 mm and 0.7 d_v. Cl. 11.3.8.3 halves that when the shear stress exceeds 0.125 lambda phi_c f'c or the torsion exceeds 0.25 T_cr.
Why is theta fixed at 35 degrees, and how is beta chosen?
The sheet uses the simplified method of Cl. 11.3.6.3, which fixes theta at 35 degrees and holds to f'c of 60 MPa. With at least the minimum stirrups of Eq. 11.1, beta is 0.18 for f_y up to 400 MPa and 0.4 / (1 + f_y / 320) above it. With less, beta comes from Eq. 11.9, which assumes coarse aggregate of at least 20 mm. Every bar is taken at the one yield strength f_y.