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CSA S304-14 Cl. 7.14, 11.2 and 11.3; NBCC 2020 Table 4.1.3.2.-A

Masonry lintel

Verified against CSA S304-14, NBC 2020 Table 4.1.3.2.-A and CMHA TEK 17-01D, 2026-10-12

Reinforced block or brick lintel: arching, load dispersion, flexure, shear and end bearing. Design check of a simply supported reinforced concrete block or brick lintel over an opening. The wall above is carried with or without arching, a concentrated load is dispersed through the masonry, and the lintel is checked for flexure, shear, reinforcement limits and bearing at its ends, with every step shown.

Given

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Opening and lintel
\(L_{o}\) \(\mathrm{mm}\) 400-8,000
\(l_{b}\) \(\mathrm{mm}\) 50-1,000
\(b\) \(\mathrm{mm}\) 90-400
\(h\) \(\mathrm{mm}\) 150-2,400
\(d\) \(\mathrm{mm}\) 100-2,350
\(w_{\mathrm{lintel}}\) \(\mathrm{kPa}\) 0-10
Reinforcement
\(\mathrm{layout}\)
\(A_{s}\) \(\mathrm{mm}^{2}\) 50-5,000
\({A'}_{s}\) \(\mathrm{mm}^{2}\) 0-5,000
\({d'}\) \(\mathrm{mm}\) 20-400
\(f_{y}\) \(\mathrm{MPa}\) 300-400
Masonry
\(\mathrm{unit}_{\mathrm{type}}\)
\(f_{\mathrm{unit}}\) \(\mathrm{MPa}\) 10-90
\(\mathrm{mortar}\)
\(h_{k}\) \(\mathrm{mm}\) 0-600
\(\mathrm{density}\)
Wall above
\(h_{\mathrm{wall}}\) \(\mathrm{mm}\) 0-20,000
\(w_{\mathrm{wall}}\) \(\mathrm{kPa}\) 0-10
\(\mathrm{arching}\)
\(\mathrm{wall}_{\mathrm{masonry}}\)
Loads
\(w_{D}\) \(\mathrm{kN/m}\) 0-200
\(w_{L}\) \(\mathrm{kN/m}\) 0-200
\(w_{S}\) \(\mathrm{kN/m}\) 0-200
\(P_{D}\) \(\mathrm{kN}\) 0-1,000
\(P_{L}\) \(\mathrm{kN}\) 0-1,000
\(P_{S}\) \(\mathrm{kN}\) 0-1,000
\(b_{p}\) \(\mathrm{mm}\) 10-2,000
\(h_{p}\) \(\mathrm{mm}\) 0-20,000
Stirrups
\(\mathrm{stirrups}\)
\(A_{v}\) \(\mathrm{mm}^{2}\) 10-1,000
\(s\) \(\mathrm{mm}\) 50-1,200
Support
\(\mathrm{support}_{\mathrm{masonry}}\)
ha = 1,500 mm Pf = 149.8 kN lp = 1,220 mm h = 1,000 mm Lo = 2,800 mm
The lintel in its wall: opening, bearings, bars, arching triangle and the concentrated load's spread.

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/masonry-lintel.pdf.

Checks

Flexure

Flexure ok\(\htmlClass{sym-M_f}{M_{f}} \leq \htmlClass{sym-M_r}{M_{r}} \quad \Rightarrow \quad \htmlClass{sym-M_f}{98\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r}{142.5\ \mathrm{kN} \cdot \mathrm{m}}\) 0.69 PASS
Tension steel yields\(\frac{\htmlClass{sym-c}{c}}{\htmlClass{sym-d}{d}} \leq \frac{600\ \mathrm{MPa}}{600\ \mathrm{MPa} + \htmlClass{sym-f_y}{f_{y}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c}{467.8\ \mathrm{mm}}}{\htmlClass{sym-d}{840\ \mathrm{mm}}} \leq \frac{600\ \mathrm{MPa}}{600\ \mathrm{MPa} + \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\) 0.93 PASS
Min steel ok\(\htmlClass{sym-rho}{\rho} \geq \htmlClass{sym-rho_min}{\rho_{\mathrm{min}}} \quad \Rightarrow \quad \htmlClass{sym-rho}{0.003759} \geq \htmlClass{sym-rho_min}{0.002}\) 0.53 PASS
Not deep beam\(\frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h}{h}} \geq 2 \quad \Rightarrow \quad \frac{\htmlClass{sym-L}{3\ \mathrm{m}}}{\htmlClass{sym-h}{1000\ \mathrm{mm}}} \geq 2\) 0.67 PASS
Lateral support ok\(\min\left(30 \cdot \htmlClass{sym-b}{b}, \frac{120 \cdot \htmlClass{sym-b}{b}^{2}}{\htmlClass{sym-d}{d}}\right) \geq \htmlClass{sym-L}{L} \quad \Rightarrow \quad \min\left(30 \cdot \htmlClass{sym-b}{190\ \mathrm{mm}}, \frac{120 \cdot \left(\htmlClass{sym-b}{190\ \mathrm{mm}}\right)^{2}}{\htmlClass{sym-d}{840\ \mathrm{mm}}}\right) \geq \htmlClass{sym-L}{3\ \mathrm{m}}\) 0.58 PASS

Shear

Shear ok\(\htmlClass{sym-V_f}{V_{f}} \leq \htmlClass{sym-V_r}{V_{r}} \quad \Rightarrow \quad \htmlClass{sym-V_f}{80.07\ \mathrm{kN}} \leq \htmlClass{sym-V_r}{113.9\ \mathrm{kN}}\) 0.70 PASS
Stirrup spacing ok\(\htmlClass{sym-s}{s} \leq \htmlClass{sym-s_max}{s_{\mathrm{max}}} \quad \Rightarrow \quad \htmlClass{sym-s}{400\ \mathrm{mm}} \leq \htmlClass{sym-s_max}{420\ \mathrm{mm}}\) 0.95 PASS
Stirrup area ok\(\htmlClass{sym-A_v}{A_{v}} \geq \htmlClass{sym-A_v_min}{A_{v,\mathrm{min}}} \quad \Rightarrow \quad \htmlClass{sym-A_v}{100\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_v_min}{66.5\ \mathrm{mm}^{2}}\) 0.67 PASS

Bearing

Bearing ok\(\htmlClass{sym-R_f}{R_{f}} \leq \htmlClass{sym-B_r}{B_{r}} \quad \Rightarrow \quad \htmlClass{sym-R_f}{85.22\ \mathrm{kN}} \leq \htmlClass{sym-B_r}{85.5\ \mathrm{kN}}\) 1.00 PASS
Min bearing ok\(\htmlClass{sym-l_b}{l_{b}} \geq 200\ \mathrm{mm} \quad \Rightarrow \quad \htmlClass{sym-l_b}{200\ \mathrm{mm}} \geq 200\ \mathrm{mm}\) 1.00 PASS

Geometry

Tension bars in section\(\htmlClass{sym-d}{d} < \htmlClass{sym-h}{h} \quad \Rightarrow \quad \htmlClass{sym-d}{840\ \mathrm{mm}} < \htmlClass{sym-h}{1000\ \mathrm{mm}}\) 0.84 PASS
Compression bars above\(\htmlClass{sym-d_prime}{{d'}} < \htmlClass{sym-d}{d} \quad \Rightarrow \quad \htmlClass{sym-d_prime}{50\ \mathrm{mm}} < \htmlClass{sym-d}{840\ \mathrm{mm}}\) 0.06 PASS
Load bearing on span\(\htmlClass{sym-b_p}{b_{p}} \leq \htmlClass{sym-L}{L} \quad \Rightarrow \quad \htmlClass{sym-b_p}{500\ \mathrm{mm}} \leq \htmlClass{sym-L}{3\ \mathrm{m}}\) 0.17 PASS

Results

\(\htmlClass{sym-M_f}{M_{f}}\) Factored moment at midspan 98 \(\mathrm{kN} \cdot \mathrm{m}\)
\(\htmlClass{sym-M_r}{M_{r}}\) Factored moment resistance 142.5 \(\mathrm{kN} \cdot \mathrm{m}\)
\(\htmlClass{sym-V_f}{V_{f}}\) Factored shear at the critical section near the support 80.07 \(\mathrm{kN}\)
\(\htmlClass{sym-V_r}{V_{r}}\) Factored shear resistance 113.9 \(\mathrm{kN}\)
\(\htmlClass{sym-R_f}{R_{f}}\) Factored reaction at each end 85.22 \(\mathrm{kN}\)
\(\htmlClass{sym-B_r}{B_{r}}\) Factored bearing resistance at each end 85.5 \(\mathrm{kN}\)

Derivation

S.1 \[\begin{aligned} \htmlClass{sym-L}{L} &= \htmlClass{sym-L_o}{L_{o}} + \htmlClass{sym-l_b}{l_{b}} \\ &= \htmlClass{sym-L_o}{2800\ \mathrm{mm}} + \htmlClass{sym-l_b}{200\ \mathrm{mm}} \\ &= 3\ \mathrm{m} \end{aligned}\]

Assumed: span between the centres of the bearings, the larger of the usual definitions

S.2

Cl. 11.3.4.2

\[\begin{aligned} \htmlClass{sym-d_v}{d_{v}} &= \max\left(0.9 \cdot \htmlClass{sym-d}{d}, 0.72 \cdot \htmlClass{sym-h}{h}\right) \\ &= \max\left(0.9 \cdot \htmlClass{sym-d}{840\ \mathrm{mm}}, 0.72 \cdot \htmlClass{sym-h}{1000\ \mathrm{mm}}\right) \\ &= 756\ \mathrm{mm} \end{aligned}\]
S.3

Cl. 11.3.4.2, d_v past the support face, which is l_b / 2 from the bearing centre

\[\begin{aligned} \htmlClass{sym-x_v}{x_{v}} &= \min\left(\frac{\htmlClass{sym-l_b}{l_{b}}}{2} + \htmlClass{sym-d_v}{d_{v}}, \frac{\htmlClass{sym-L}{L}}{2}\right) \\ &= \min\left(\frac{\htmlClass{sym-l_b}{200\ \mathrm{mm}}}{2} + \htmlClass{sym-d_v}{756\ \mathrm{mm}}, \frac{\htmlClass{sym-L}{3\ \mathrm{m}}}{2}\right) \\ &= 856\ \mathrm{mm} \end{aligned}\]
S.4

Cl. 7.14.2, 2.5:1

\[\begin{aligned} \htmlClass{sym-l_p}{l_{p}} &= \min\left(\htmlClass{sym-b_p}{b_{p}} + \frac{2 \cdot \htmlClass{sym-h_p}{h_{p}}}{2.5}, \htmlClass{sym-L}{L}\right) \\ &= \min\left(\htmlClass{sym-b_p}{500\ \mathrm{mm}} + \frac{2 \cdot \htmlClass{sym-h_p}{900\ \mathrm{mm}}}{2.5}, \htmlClass{sym-L}{3\ \mathrm{m}}\right) \\ &= 1.22\ \mathrm{m} \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-m_p}{m_{p}} &= \frac{\htmlClass{sym-L}{L}}{4} - \frac{\htmlClass{sym-l_p}{l_{p}}}{8} \\ &= \frac{\htmlClass{sym-L}{3\ \mathrm{m}}}{4} - \frac{\htmlClass{sym-l_p}{1.22\ \mathrm{m}}}{8} \\ &= 597.5\ \mathrm{mm} \end{aligned}\]
S.6 \[\begin{aligned} \htmlClass{sym-w_s}{w_{s}} &= \htmlClass{sym-w_lintel}{w_{\mathrm{lintel}}} \cdot \htmlClass{sym-h}{h} \\ &= \htmlClass{sym-w_lintel}{3.91\ \mathrm{kPa}} \cdot \htmlClass{sym-h}{1000\ \mathrm{mm}} \\ &= 3.91\ \mathrm{kN/m} \end{aligned}\]
S.7

CMHA TEK 17-01D, 45 degree arching triangle on the span

\[\begin{aligned} \htmlClass{sym-h_a}{h_{a}} &= \frac{\htmlClass{sym-L}{L}}{2} \\ &= \frac{\htmlClass{sym-L}{3\ \mathrm{m}}}{2} \\ &= 1.5\ \mathrm{m} \end{aligned}\]
S.8

CMHA TEK 17-01D, 203 mm of masonry above the apex

\[\begin{aligned} \htmlClass{sym-h_arch}{h_{\mathrm{arch}}} &= \htmlClass{sym-h_a}{h_{a}} + 203\ \mathrm{mm} \\ &= \htmlClass{sym-h_a}{1.5\ \mathrm{m}} + 203\ \mathrm{mm} \\ &= 1.703\ \mathrm{m} \end{aligned}\]

Branch: \(\mathrm{arching} = \mathrm{YES}\ \text{and}\ h_{\mathrm{wall}} \geq h_{\mathrm{arch}}\) held

S.9 \[\begin{aligned} \htmlClass{sym-w_t}{w_{t}} &= \htmlClass{sym-w_wall}{w_{\mathrm{wall}}} \cdot \htmlClass{sym-h_a}{h_{a}} \\ &= \htmlClass{sym-w_wall}{2.11\ \mathrm{kPa}} \cdot \htmlClass{sym-h_a}{1.5\ \mathrm{m}} \\ &= 3.165\ \mathrm{kN/m} \end{aligned}\]
S.10 \[\begin{aligned} \htmlClass{sym-M_D}{M_{D}} &= \frac{\htmlClass{sym-w_s}{w_{s}} \cdot \htmlClass{sym-L}{L}^{2}}{8} + \frac{\htmlClass{sym-w_t}{w_{t}} \cdot \htmlClass{sym-L}{L}^{2}}{12} + \htmlClass{sym-P_D}{P_{D}} \cdot \htmlClass{sym-m_p}{m_{p}} \\ &= \frac{\htmlClass{sym-w_s}{3.91\ \mathrm{kN/m}} \cdot \left(\htmlClass{sym-L}{3\ \mathrm{m}}\right)^{2}}{8} + \frac{\htmlClass{sym-w_t}{3.165\ \mathrm{kN/m}} \cdot \left(\htmlClass{sym-L}{3\ \mathrm{m}}\right)^{2}}{12} + \htmlClass{sym-P_D}{81\ \mathrm{kN}} \cdot \htmlClass{sym-m_p}{597.5\ \mathrm{mm}} \\ &= 55.17\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-V_D}{V_{D}} &= \htmlClass{sym-w_s}{w_{s}} \cdot \left(\frac{\htmlClass{sym-L}{L}}{2} - \htmlClass{sym-x_v}{x_{v}}\right) + \frac{\htmlClass{sym-w_t}{w_{t}} \cdot \htmlClass{sym-L}{L}}{4} - \frac{\htmlClass{sym-w_t}{w_{t}} \cdot \htmlClass{sym-x_v}{x_{v}}^{2}}{\htmlClass{sym-L}{L}} + \frac{\htmlClass{sym-P_D}{P_{D}}}{2} \\ &= \htmlClass{sym-w_s}{3.91\ \mathrm{kN/m}} \cdot \left(\frac{\htmlClass{sym-L}{3\ \mathrm{m}}}{2} - \htmlClass{sym-x_v}{856\ \mathrm{mm}}\right) + \frac{\htmlClass{sym-w_t}{3.165\ \mathrm{kN/m}} \cdot \htmlClass{sym-L}{3\ \mathrm{m}}}{4} - \frac{\htmlClass{sym-w_t}{3.165\ \mathrm{kN/m}} \cdot \left(\htmlClass{sym-x_v}{856\ \mathrm{mm}}\right)^{2}}{\htmlClass{sym-L}{3\ \mathrm{m}}} + \frac{\htmlClass{sym-P_D}{81\ \mathrm{kN}}}{2} \\ &= 44.62\ \mathrm{kN} \end{aligned}\]
S.12 \[\begin{aligned} \htmlClass{sym-R_D}{R_{D}} &= \frac{\htmlClass{sym-w_s}{w_{s}} \cdot \htmlClass{sym-L}{L}}{2} + \frac{\htmlClass{sym-w_t}{w_{t}} \cdot \htmlClass{sym-L}{L}}{4} + \frac{\htmlClass{sym-P_D}{P_{D}}}{2} \\ &= \frac{\htmlClass{sym-w_s}{3.91\ \mathrm{kN/m}} \cdot \htmlClass{sym-L}{3\ \mathrm{m}}}{2} + \frac{\htmlClass{sym-w_t}{3.165\ \mathrm{kN/m}} \cdot \htmlClass{sym-L}{3\ \mathrm{m}}}{4} + \frac{\htmlClass{sym-P_D}{81\ \mathrm{kN}}}{2} \\ &= 48.74\ \mathrm{kN} \end{aligned}\]
S.13 \[\begin{aligned} \htmlClass{sym-M_L}{M_{L}} &= \htmlClass{sym-P_L}{P_{L}} \cdot \htmlClass{sym-m_p}{m_{p}} \\ &= \htmlClass{sym-P_L}{0\ \mathrm{kN}} \cdot \htmlClass{sym-m_p}{597.5\ \mathrm{mm}} \\ &= 0\ \mathrm{N} \cdot \mathrm{m} \end{aligned}\]
S.14 \[\begin{aligned} \htmlClass{sym-V_L}{V_{L}} &= \frac{\htmlClass{sym-P_L}{P_{L}}}{2} \\ &= \frac{\htmlClass{sym-P_L}{0\ \mathrm{kN}}}{2} \\ &= 0\ \mathrm{N} \end{aligned}\]
S.15 \[\begin{aligned} \htmlClass{sym-R_L}{R_{L}} &= \frac{\htmlClass{sym-P_L}{P_{L}}}{2} \\ &= \frac{\htmlClass{sym-P_L}{0\ \mathrm{kN}}}{2} \\ &= 0\ \mathrm{N} \end{aligned}\]
S.16 \[\begin{aligned} \htmlClass{sym-M_S}{M_{S}} &= \htmlClass{sym-P_S}{P_{S}} \cdot \htmlClass{sym-m_p}{m_{p}} \\ &= \htmlClass{sym-P_S}{32.4\ \mathrm{kN}} \cdot \htmlClass{sym-m_p}{597.5\ \mathrm{mm}} \\ &= 19.36\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.17 \[\begin{aligned} \htmlClass{sym-V_S}{V_{S}} &= \frac{\htmlClass{sym-P_S}{P_{S}}}{2} \\ &= \frac{\htmlClass{sym-P_S}{32.4\ \mathrm{kN}}}{2} \\ &= 16.2\ \mathrm{kN} \end{aligned}\]
S.18 \[\begin{aligned} \htmlClass{sym-R_S}{R_{S}} &= \frac{\htmlClass{sym-P_S}{P_{S}}}{2} \\ &= \frac{\htmlClass{sym-P_S}{32.4\ \mathrm{kN}}}{2} \\ &= 16.2\ \mathrm{kN} \end{aligned}\]
S.19

NBCC Table 4.1.3.2.-A

\[\begin{aligned} \htmlClass{sym-M_f}{M_{f}} &= \max\left(1.4 \cdot \htmlClass{sym-M_D}{M_{D}}, 1.25 \cdot \htmlClass{sym-M_D}{M_{D}} + 1.5 \cdot \htmlClass{sym-M_L}{M_{L}} + 1 \cdot \htmlClass{sym-M_S}{M_{S}}, 1.25 \cdot \htmlClass{sym-M_D}{M_{D}} + 1.5 \cdot \htmlClass{sym-M_S}{M_{S}} + 1 \cdot \htmlClass{sym-M_L}{M_{L}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-M_D}{55.17\ \mathrm{kN} \cdot \mathrm{m}}, 1.25 \cdot \htmlClass{sym-M_D}{55.17\ \mathrm{kN} \cdot \mathrm{m}} + 1.5 \cdot \htmlClass{sym-M_L}{0\ \mathrm{N} \cdot \mathrm{m}} + 1 \cdot \htmlClass{sym-M_S}{19.36\ \mathrm{kN} \cdot \mathrm{m}}, 1.25 \cdot \htmlClass{sym-M_D}{55.17\ \mathrm{kN} \cdot \mathrm{m}} + 1.5 \cdot \htmlClass{sym-M_S}{19.36\ \mathrm{kN} \cdot \mathrm{m}} + 1 \cdot \htmlClass{sym-M_L}{0\ \mathrm{N} \cdot \mathrm{m}}\right) \\ &= 98\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.20

NBCC Table 4.1.3.2.-A

\[\begin{aligned} \htmlClass{sym-V_f}{V_{f}} &= \max\left(1.4 \cdot \htmlClass{sym-V_D}{V_{D}}, 1.25 \cdot \htmlClass{sym-V_D}{V_{D}} + 1.5 \cdot \htmlClass{sym-V_L}{V_{L}} + 1 \cdot \htmlClass{sym-V_S}{V_{S}}, 1.25 \cdot \htmlClass{sym-V_D}{V_{D}} + 1.5 \cdot \htmlClass{sym-V_S}{V_{S}} + 1 \cdot \htmlClass{sym-V_L}{V_{L}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-V_D}{44.62\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-V_D}{44.62\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-V_L}{0\ \mathrm{N}} + 1 \cdot \htmlClass{sym-V_S}{16.2\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-V_D}{44.62\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-V_S}{16.2\ \mathrm{kN}} + 1 \cdot \htmlClass{sym-V_L}{0\ \mathrm{N}}\right) \\ &= 80.07\ \mathrm{kN} \end{aligned}\]
S.21

NBCC Table 4.1.3.2.-A

\[\begin{aligned} \htmlClass{sym-R_f}{R_{f}} &= \max\left(1.4 \cdot \htmlClass{sym-R_D}{R_{D}}, 1.25 \cdot \htmlClass{sym-R_D}{R_{D}} + 1.5 \cdot \htmlClass{sym-R_L}{R_{L}} + 1 \cdot \htmlClass{sym-R_S}{R_{S}}, 1.25 \cdot \htmlClass{sym-R_D}{R_{D}} + 1.5 \cdot \htmlClass{sym-R_S}{R_{S}} + 1 \cdot \htmlClass{sym-R_L}{R_{L}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-R_D}{48.74\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-R_D}{48.74\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-R_L}{0\ \mathrm{N}} + 1 \cdot \htmlClass{sym-R_S}{16.2\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-R_D}{48.74\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-R_S}{16.2\ \mathrm{kN}} + 1 \cdot \htmlClass{sym-R_L}{0\ \mathrm{N}}\right) \\ &= 85.22\ \mathrm{kN} \end{aligned}\]
S.22

NBCC Table 4.1.3.2.-A

\[\begin{aligned} \htmlClass{sym-P_f}{P_{f}} &= \max\left(1.4 \cdot \htmlClass{sym-P_D}{P_{D}}, 1.25 \cdot \htmlClass{sym-P_D}{P_{D}} + 1.5 \cdot \htmlClass{sym-P_L}{P_{L}} + 1 \cdot \htmlClass{sym-P_S}{P_{S}}, 1.25 \cdot \htmlClass{sym-P_D}{P_{D}} + 1.5 \cdot \htmlClass{sym-P_S}{P_{S}} + 1 \cdot \htmlClass{sym-P_L}{P_{L}}\right) \\ &= \max\left(1.4 \cdot \htmlClass{sym-P_D}{81\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-P_D}{81\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-P_L}{0\ \mathrm{kN}} + 1 \cdot \htmlClass{sym-P_S}{32.4\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-P_D}{81\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-P_S}{32.4\ \mathrm{kN}} + 1 \cdot \htmlClass{sym-P_L}{0\ \mathrm{kN}}\right) \\ &= 149.8\ \mathrm{kN} \end{aligned}\]
S.23 \[\begin{aligned} \htmlClass{sym-mortar_column}{\mathrm{mortar}_{\mathrm{column}}} &= 0 \end{aligned}\]
S.24 \[\begin{aligned} \htmlClass{sym-f_prime_m}{{f'}_{m}} &= 7.5\ \mathrm{MPa} \quad \left(\text{15 MPa block, Type S, grouted | Table 4}\right) \end{aligned}\]
S.25 \[\begin{aligned} \htmlClass{sym-f_prime_m_s}{{f'}_{m,s}} &= 7.5\ \mathrm{MPa} \quad \left(\text{15 MPa block, Type S, grouted | Table 4}\right) \end{aligned}\]
S.26 \[\begin{aligned} \htmlClass{sym-phi_m}{\phi_{m}} &= 0.6 \quad \left(\text{Masonry | Cl. 4.3.2.1}\right) \end{aligned}\]
S.27 \[\begin{aligned} \htmlClass{sym-phi_s}{\phi_{s}} &= 0.85 \quad \left(\text{Reinforcement | Cl. 4.3.2.2}\right) \end{aligned}\]
S.28 \[\begin{aligned} \htmlClass{sym-a_trial}{a_{\mathrm{trial}}} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \left(\htmlClass{sym-A_s}{A_{s}} - \htmlClass{sym-A_s_prime}{{A'}_{s}}\right) \cdot \htmlClass{sym-f_y}{f_{y}}}{0.85 \cdot \htmlClass{sym-phi_m}{\phi_{m}} \cdot 0.7 \cdot \htmlClass{sym-f_prime_m}{{f'}_{m}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \left(\htmlClass{sym-A_s}{600\ \mathrm{mm}^{2}} - \htmlClass{sym-A_s_prime}{200\ \mathrm{mm}^{2}}\right) \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{0.85 \cdot \htmlClass{sym-phi_m}{0.6} \cdot 0.7 \cdot \htmlClass{sym-f_prime_m}{7.5\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}}} \\ &= 267.3\ \mathrm{mm} \end{aligned}\]

Branch: \(a_{\mathrm{trial}} \leq h_{k}\) did not hold

S.29 \[\begin{aligned} \htmlClass{sym-chi}{\chi} &= 0.5 \quad \left(\text{Stress block reaches the webs | Cl. 11.2.1.6}\right) \end{aligned}\]

Branch: \({f'}_{m} \leq 20\ \mathrm{MPa}\) held

S.30 \[\begin{aligned} \htmlClass{sym-beta_1}{\beta_{1}} &= 0.8 \quad \left(\text{Cl. 11.2.1.6}\right) \end{aligned}\]
S.31 \[\begin{aligned} \htmlClass{sym-E_s}{E_{s}} &= 200\ \mathrm{GPa} \end{aligned}\]
S.32 \[\begin{aligned} \htmlClass{sym-eps_m}{\epsilon_{m}} &= 0.003 \quad \left(\text{Cl. 11.2.1.2}\right) \end{aligned}\]
S.33 \[\begin{aligned} \htmlClass{sym-eps_y}{\epsilon_{y}} &= \frac{\htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-E_s}{E_{s}}} \\ &= \frac{\htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-E_s}{200\ \mathrm{GPa}}} \\ &= 0.002 \end{aligned}\]

Branch: \(A_{s} > {A'}_{s}\) held

S.34

Cl. 11.2.1.6

\[\begin{aligned} \htmlClass{sym-a_y}{a_{y}} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \left(\htmlClass{sym-A_s}{A_{s}} - \htmlClass{sym-A_s_prime}{{A'}_{s}}\right) \cdot \htmlClass{sym-f_y}{f_{y}}}{0.85 \cdot \htmlClass{sym-phi_m}{\phi_{m}} \cdot \htmlClass{sym-chi}{\chi} \cdot \htmlClass{sym-f_prime_m}{{f'}_{m}} \cdot \htmlClass{sym-b}{b}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \left(\htmlClass{sym-A_s}{600\ \mathrm{mm}^{2}} - \htmlClass{sym-A_s_prime}{200\ \mathrm{mm}^{2}}\right) \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{0.85 \cdot \htmlClass{sym-phi_m}{0.6} \cdot \htmlClass{sym-chi}{0.5} \cdot \htmlClass{sym-f_prime_m}{7.5\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}}} \\ &= 374.3\ \mathrm{mm} \end{aligned}\]
S.35 \[\begin{aligned} \htmlClass{sym-c_y}{c_{y}} &= \frac{\htmlClass{sym-a_y}{a_{y}}}{\htmlClass{sym-beta_1}{\beta_{1}}} \\ &= \frac{\htmlClass{sym-a_y}{374.3\ \mathrm{mm}}}{\htmlClass{sym-beta_1}{0.8}} \\ &= 467.8\ \mathrm{mm} \end{aligned}\]
S.36 \[\begin{aligned} \htmlClass{sym-eps_s_prime}{{\epsilon'}_{s}} &= \frac{\htmlClass{sym-eps_m}{\epsilon_{m}} \cdot \left(\htmlClass{sym-c_y}{c_{y}} - \htmlClass{sym-d_prime}{{d'}}\right)}{\htmlClass{sym-c_y}{c_{y}}} \\ &= \frac{\htmlClass{sym-eps_m}{0.003} \cdot \left(\htmlClass{sym-c_y}{467.8\ \mathrm{mm}} - \htmlClass{sym-d_prime}{50\ \mathrm{mm}}\right)}{\htmlClass{sym-c_y}{467.8\ \mathrm{mm}}} \\ &= 0.002679 \end{aligned}\]

Branch: \(A_{s} > {A'}_{s}\ \text{and}\ {\epsilon'}_{s} \geq \epsilon_{y}\) held

S.37 \[\begin{aligned} \htmlClass{sym-c}{c} &= 1 \cdot \htmlClass{sym-c_y}{c_{y}} \\ &= 1 \cdot \htmlClass{sym-c_y}{467.8\ \mathrm{mm}} \\ &= 467.8\ \mathrm{mm} \end{aligned}\]
S.38

Cl. 11.2.1.3

\[\begin{aligned} \htmlClass{sym-f_s_prime}{{f'}_{s}} &= 1 \cdot \htmlClass{sym-f_y}{f_{y}} \\ &= 1 \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \\ &= 400\ \mathrm{MPa} \end{aligned}\]
S.39 \[\begin{aligned} \htmlClass{sym-a}{a} &= \htmlClass{sym-beta_1}{\beta_{1}} \cdot \htmlClass{sym-c}{c} \\ &= \htmlClass{sym-beta_1}{0.8} \cdot \htmlClass{sym-c}{467.8\ \mathrm{mm}} \\ &= 374.3\ \mathrm{mm} \end{aligned}\]
S.40

Cl. 11.2.1.6

\[\begin{aligned} \htmlClass{sym-C_m}{C_{m}} &= 0.85 \cdot \htmlClass{sym-phi_m}{\phi_{m}} \cdot \htmlClass{sym-chi}{\chi} \cdot \htmlClass{sym-f_prime_m}{{f'}_{m}} \cdot \htmlClass{sym-a}{a} \cdot \htmlClass{sym-b}{b} \\ &= 0.85 \cdot \htmlClass{sym-phi_m}{0.6} \cdot \htmlClass{sym-chi}{0.5} \cdot \htmlClass{sym-f_prime_m}{7.5\ \mathrm{MPa}} \cdot \htmlClass{sym-a}{374.3\ \mathrm{mm}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}} \\ &= 136\ \mathrm{kN} \end{aligned}\]
S.41 \[\begin{aligned} \htmlClass{sym-C_s}{C_{s}} &= \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_s_prime}{{A'}_{s}} \cdot \htmlClass{sym-f_s_prime}{{f'}_{s}} \\ &= \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_s_prime}{200\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_s_prime}{400\ \mathrm{MPa}} \\ &= 68\ \mathrm{kN} \end{aligned}\]
S.42 \[\begin{aligned} \htmlClass{sym-M_r}{M_{r}} &= \htmlClass{sym-C_m}{C_{m}} \cdot \left(\htmlClass{sym-d}{d} - \frac{\htmlClass{sym-a}{a}}{2}\right) + \htmlClass{sym-C_s}{C_{s}} \cdot \left(\htmlClass{sym-d}{d} - \htmlClass{sym-d_prime}{{d'}}\right) \\ &= \htmlClass{sym-C_m}{136\ \mathrm{kN}} \cdot \left(\htmlClass{sym-d}{840\ \mathrm{mm}} - \frac{\htmlClass{sym-a}{374.3\ \mathrm{mm}}}{2}\right) + \htmlClass{sym-C_s}{68\ \mathrm{kN}} \cdot \left(\htmlClass{sym-d}{840\ \mathrm{mm}} - \htmlClass{sym-d_prime}{50\ \mathrm{mm}}\right) \\ &= 142.5\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

Branch: \(a_{\mathrm{trial}} \leq h_{k}\) did not hold

S.43 \[\begin{aligned} \htmlClass{sym-rho}{\rho} &= \frac{\htmlClass{sym-A_s}{A_{s}}}{\htmlClass{sym-b}{b} \cdot \htmlClass{sym-d}{d}} \\ &= \frac{\htmlClass{sym-A_s}{600\ \mathrm{mm}^{2}}}{\htmlClass{sym-b}{190\ \mathrm{mm}} \cdot \htmlClass{sym-d}{840\ \mathrm{mm}}} \\ &= 0.003759 \end{aligned}\]
S.44

Cl. 11.2.3.1

\[\begin{aligned} \htmlClass{sym-rho_min}{\rho_{\mathrm{min}}} &= \frac{800\ \mathrm{kPa}}{\htmlClass{sym-f_y}{f_{y}}} \\ &= \frac{800\ \mathrm{kPa}}{\htmlClass{sym-f_y}{400\ \mathrm{MPa}}} \\ &= 0.002 \end{aligned}\]

Branch: \(\rho \geq \rho_{\mathrm{min}}\) held

S.45 \[\begin{aligned} \htmlClass{sym-lamb}{\lambda} &= 1 \quad \left(\text{Over 2000 kg/m3 | Cl. 11.3.3}\right) \end{aligned}\]
S.46 \[\begin{aligned} \htmlClass{sym-K_b}{K_{b}} &= 1 \quad \left(\text{Grouted hollow or semi-solid block, grouted hollow clay | Cl. 11.3.4.3}\right) \end{aligned}\]
S.47

Cl. 11.3.4.6, simplified method

\[\begin{aligned} \htmlClass{sym-theta_1}{\theta_{1}} &= \operatorname{radians}\left(42\right) \\ &= \operatorname{radians}\left(42\right) \\ &= 0.733 \end{aligned}\]

Branch: \(h \leq 800\ \mathrm{mm}\) did not hold

S.48 \[\begin{aligned} \htmlClass{sym-V_lim_factor}{V_{\mathrm{lim},\mathrm{factor}}} &= 0.5 \quad \left(\text{h\_b over 800 mm | Cl. 11.3.4.8.1}\right) \end{aligned}\]
S.49

Cl. 11.3.4.8.2

\[\begin{aligned} \htmlClass{sym-A_v_min}{A_{v,\mathrm{min}}} &= \frac{350\ \mathrm{kPa} \cdot \htmlClass{sym-b}{b} \cdot \htmlClass{sym-s}{s}}{\htmlClass{sym-f_y}{f_{y}}} \\ &= \frac{350\ \mathrm{kPa} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}} \cdot \htmlClass{sym-s}{400\ \mathrm{mm}}}{\htmlClass{sym-f_y}{400\ \mathrm{MPa}}} \\ &= 66.5\ \mathrm{mm}^{2} \end{aligned}\]

Branch: \(A_{v} \geq A_{v,\mathrm{min}}\) held

S.50 \[\begin{aligned} \htmlClass{sym-beta}{\beta} &= 0.18 \quad \left(\text{Minimum stirrups | Cl. 11.3.4.6 a)}\right) \end{aligned}\]
S.51

Cl. 11.3.4.3

\[\begin{aligned} \htmlClass{sym-V_m}{V_{m}} &= \htmlClass{sym-phi_m}{\phi_{m}} \cdot \htmlClass{sym-lamb}{\lambda} \cdot \htmlClass{sym-K_b}{K_{b}} \cdot \htmlClass{sym-beta}{\beta} \cdot \sqrt{\htmlClass{sym-f_prime_m}{{f'}_{m}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{b} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= \htmlClass{sym-phi_m}{0.6} \cdot \htmlClass{sym-lamb}{1} \cdot \htmlClass{sym-K_b}{1} \cdot \htmlClass{sym-beta}{0.18} \cdot \sqrt{\htmlClass{sym-f_prime_m}{7.5\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{756\ \mathrm{mm}} \\ &= 42.48\ \mathrm{kN} \end{aligned}\]
S.52

Cl. 11.3.4.4

\[\begin{aligned} \htmlClass{sym-V_s}{V_{s}} &= \min\left(\frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_v}{A_{v}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \htmlClass{sym-d_v}{d_{v}} \cdot \cot\left(\htmlClass{sym-theta_1}{\theta_{1}}\right)}{\htmlClass{sym-s}{s}}, 0.36 \cdot \htmlClass{sym-phi_m}{\phi_{m}} \cdot \sqrt{\htmlClass{sym-f_prime_m}{{f'}_{m}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{b} \cdot \htmlClass{sym-d_v}{d_{v}}\right) \\ &= \min\left(\frac{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_v}{100\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \htmlClass{sym-d_v}{756\ \mathrm{mm}} \cdot \cot\left(\htmlClass{sym-theta_1}{0.733}\right)}{\htmlClass{sym-s}{400\ \mathrm{mm}}}, 0.36 \cdot \htmlClass{sym-phi_m}{0.6} \cdot \sqrt{\htmlClass{sym-f_prime_m}{7.5\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}} \cdot \htmlClass{sym-d_v}{756\ \mathrm{mm}}\right) \\ &= 71.37\ \mathrm{kN} \end{aligned}\]
S.53

Cl. 11.3.4.1

\[\begin{aligned} \htmlClass{sym-V_r}{V_{r}} &= \htmlClass{sym-V_m}{V_{m}} + \htmlClass{sym-V_s}{V_{s}} \\ &= \htmlClass{sym-V_m}{42.48\ \mathrm{kN}} + \htmlClass{sym-V_s}{71.37\ \mathrm{kN}} \\ &= 113.9\ \mathrm{kN} \end{aligned}\]

Branch: \(V_{f} > V_{\mathrm{lim},\mathrm{factor}} \cdot V_{m}\) held

S.54

Cl. 11.3.4.9

\[\begin{aligned} \htmlClass{sym-s_lim}{s_{\mathrm{lim}}} &= \min\left(\frac{\htmlClass{sym-d}{d}}{2}, 600\ \mathrm{mm}\right) \\ &= \min\left(\frac{\htmlClass{sym-d}{840\ \mathrm{mm}}}{2}, 600\ \mathrm{mm}\right) \\ &= 420\ \mathrm{mm} \end{aligned}\]

Branch: \(d \geq 300\ \mathrm{mm}\) held

S.55

Cl. 11.3.4.9, need not be under 200 mm

\[\begin{aligned} \htmlClass{sym-s_max}{s_{\mathrm{max}}} &= \max\left(\htmlClass{sym-s_lim}{s_{\mathrm{lim}}}, 200\ \mathrm{mm}\right) \\ &= \max\left(\htmlClass{sym-s_lim}{420\ \mathrm{mm}}, 200\ \mathrm{mm}\right) \\ &= 420\ \mathrm{mm} \end{aligned}\]
S.56 \[\begin{aligned} \htmlClass{sym-A_bp}{A_{\mathrm{bp}}} &= \htmlClass{sym-l_b}{l_{b}} \cdot \htmlClass{sym-b}{b} \\ &= \htmlClass{sym-l_b}{200\ \mathrm{mm}} \cdot \htmlClass{sym-b}{190\ \mathrm{mm}} \\ &= 38000\ \mathrm{mm}^{2} \end{aligned}\]
S.57 \[\begin{aligned} \htmlClass{sym-k_dist}{k_{\mathrm{dist}}} &= 0.5 \quad \left(\text{Triangular | Cl. 7.14.1.1}\right) \end{aligned}\]
S.58 \[\begin{aligned} \htmlClass{sym-K_1}{K_{1}} &= 1 \quad \left(\text{Lower bound | Cl. 7.14.3}\right) \end{aligned}\]
S.59

Cl. 7.14.3

\[\begin{aligned} \htmlClass{sym-B_r}{B_{r}} &= \htmlClass{sym-k_dist}{k_{\mathrm{dist}}} \cdot \htmlClass{sym-K_1}{K_{1}} \cdot \htmlClass{sym-A_bp}{A_{\mathrm{bp}}} \cdot \htmlClass{sym-phi_m}{\phi_{m}} \cdot \htmlClass{sym-f_prime_m_s}{{f'}_{m,s}} \\ &= \htmlClass{sym-k_dist}{0.5} \cdot \htmlClass{sym-K_1}{1} \cdot \htmlClass{sym-A_bp}{38000\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-phi_m}{0.6} \cdot \htmlClass{sym-f_prime_m_s}{7.5\ \mathrm{MPa}} \\ &= 85.5\ \mathrm{kN} \end{aligned}\]

Questions

When does the calc take arching over the lintel?

When arching is chosen and the masonry above the lintel reaches 203 mm past the apex h_a of a 45 degree triangle on the span. The uniform floor or roof load is taken at the top of that masonry, so a floor any lower means no arching and the lintel carries the full wall and floor load. S304-14 has no arching clause; the rule is CMHA TEK 17-01D's, taken conservatively: the concentrated load is still carried in full, whatever its height.

Why does chi reduce M_r under S304-14 Cl. 11.2.1.6?

The compression in a lintel runs normal to the head faces of the units, so the stress block is 0.85 phi_m chi f'_m. chi is 0.7 while the stress block depth a stays inside the horizontally continuous grout of lintel or knockout units, h_k, and 0.5 once it reaches the webs below.

Which beta does V_m use under Cl. 11.3.4.6?

0.18 when the stirrups reach the minimum area of Cl. 11.3.4.8.2, 0.35 b_w s / f_y, and 230 / (1000 + 1.4 d_v) otherwise, d_v in millimetres. Without stirrups, Cl. 11.3.4.8.1 also limits V_f to V_m, or to half of V_m for a lintel deeper than 800 mm.

What does the calc leave out of S304-14, and where is it conservative?

It does not check deflection (Cl. 11.4, required once the clear span passes 10 d), crack control and intermediate bars (Cl. 11.2.6), or bar development. A deep lintel, span under twice its depth, is refused by the not deep beam check of Cl. 11.2.7.1 rather than designed. End bearing takes K_1 at 1.0, the floor of Cl. 7.14.3, which gives no credit for the masonry around the bearing, and f'_m above the top row of Table 4 stays at that row.

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