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CSA A23.3

Strip footing capacity

Bearing, one-way shear and flexure of a concrete strip footing to CSA A23.3. Check a reinforced concrete strip footing under a wall. Service and factored bearing pressures are compared with the soil's resistances, and the footing's one-way shear and flexural resistance with the factored shear and moment at the wall face.

Given

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\(B\) \(\mathrm{mm}\) 300-6,000
\(h\) \(\mathrm{mm}\) 150-2,000
\(c_{c}\) \(\mathrm{mm}\) 25-150
\(t_{w}\) \(\mathrm{mm}\) 100-2,000
\(L_{f}\) \(\mathrm{m}\) 0.3-20
\(\mathrm{wall}\)
\(\mathrm{bar}\)
\(s\) \(\mathrm{mm}\) 75-600
\(D\) \(\mathrm{kN}\) 0-10,000
\(L\) \(\mathrm{kN}\) 0-10,000
\(M_{D}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-5,000
\(M_{L}\) \(\mathrm{kN} \cdot \mathrm{m}\) 0-5,000
\(q_{a}\) \(\mathrm{kPa}\) 25-2,000
\(q_{\mathrm{ult}}\) \(\mathrm{kPa}\) 25-3,000
\(f_{c}\) \(\mathrm{MPa}\) 20-80
\(f_{y}\) \(\mathrm{MPa}\) 300-500
\(\lambda_{c}\) 0.75-1
B = 900 mm h = 450 mm tw = 300 mm Pu = 237.5 kN/m As = 1,000 mm² dv = 330 mm qs = 194.4 kPa qu = 263.9 kPa
Section through the footing, to scale: the wall, the bottom bars, and the shear section one shear depth from the wall face.

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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/strip-footing-capacity.pdf.

Checks

Check D/C Result
Bearing service\(\htmlClass{sym-q_s}{q_{s}} \leq \htmlClass{sym-q_a}{q_{a}} \quad \Rightarrow \quad \htmlClass{sym-q_s}{194.4\ \mathrm{kPa}} \leq \htmlClass{sym-q_a}{200\ \mathrm{kPa}}\) 0.97 PASS
Bearing factored\(\htmlClass{sym-q_u}{q_{u}} \leq \htmlClass{sym-q_ult}{q_{\mathrm{ult}}} \quad \Rightarrow \quad \htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} \leq \htmlClass{sym-q_ult}{300\ \mathrm{kPa}}\) 0.88 PASS
Contact service\(\htmlClass{sym-q_s_min}{q_{s,\mathrm{min}}} \geq 0\ \mathrm{Pa} \quad \Rightarrow \quad \htmlClass{sym-q_s_min}{194.4\ \mathrm{kPa}} \geq 0\ \mathrm{Pa}\) 0.00 PASS
Contact factored\(\htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}} \geq 0\ \mathrm{Pa} \quad \Rightarrow \quad \htmlClass{sym-q_u_min}{263.9\ \mathrm{kPa}} \geq 0\ \mathrm{Pa}\) 0.00 PASS
Shear adequate\(\htmlClass{sym-V_u}{V_{u}} \leq \htmlClass{sym-V_c}{V_{c}} \quad \Rightarrow \quad \htmlClass{sym-V_u}{71.17\ \mathrm{kN}} \leq \htmlClass{sym-V_c}{225.4\ \mathrm{kN}}\) 0.32 PASS
Minimum steel\(\htmlClass{sym-A_s}{A_{s}} \geq \htmlClass{sym-A_s_min}{A_{s,\mathrm{min}}} \quad \Rightarrow \quad \htmlClass{sym-A_s}{1000\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_s_min}{900\ \mathrm{mm}^{2}}\) 0.90 PASS
Steel yields\(\frac{\htmlClass{sym-c}{c}}{\htmlClass{sym-d}{d}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{f_{y}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c}{28.38\ \mathrm{mm}}}{\htmlClass{sym-d}{367\ \mathrm{mm}}} \leq \frac{700\ \mathrm{MPa}}{700\ \mathrm{MPa} + \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}\) 0.12 PASS
Flexure adequate\(\htmlClass{sym-M_u}{M_{u}} \leq \htmlClass{sym-M_r}{M_{r}} \quad \Rightarrow \quad \htmlClass{sym-M_u}{47.5\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r}{120.4\ \mathrm{kN} \cdot \mathrm{m}}\) 0.39 PASS

Results

Quantity Description Value Unit
\(\htmlClass{sym-q_s}{q_{s}}\) Service bearing pressure 194.4 \(\mathrm{kPa}\)
\(\htmlClass{sym-q_u}{q_{u}}\) Factored bearing pressure 263.9 \(\mathrm{kPa}\)
\(\htmlClass{sym-V_c}{V_{c}}\) Factored shear resistance 225.4 \(\mathrm{kN}\)
\(\htmlClass{sym-M_r}{M_{r}}\) Factored flexural resistance 120.4 \(\mathrm{kN} \cdot \mathrm{m}\)

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 \[\begin{aligned} \htmlClass{sym-P_s}{P_{s}} &= \frac{\htmlClass{sym-D}{D} + \htmlClass{sym-L}{L}}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\htmlClass{sym-D}{100\ \mathrm{kN}} + \htmlClass{sym-L}{75\ \mathrm{kN}}}{\htmlClass{sym-L_f}{1\ \mathrm{m}}} \\ &= 175\ \mathrm{kN/m} \end{aligned}\]
S.4

NBCC Table 4.1.3.2.A

\[\begin{aligned} \htmlClass{sym-P_u}{P_{u}} &= \frac{\max\left(1.4 \cdot \htmlClass{sym-D}{D}, 1.25 \cdot \htmlClass{sym-D}{D} + 1.5 \cdot \htmlClass{sym-L}{L}\right)}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\max\left(1.4 \cdot \htmlClass{sym-D}{100\ \mathrm{kN}}, 1.25 \cdot \htmlClass{sym-D}{100\ \mathrm{kN}} + 1.5 \cdot \htmlClass{sym-L}{75\ \mathrm{kN}}\right)}{\htmlClass{sym-L_f}{1\ \mathrm{m}}} \\ &= 237.5\ \mathrm{kN/m} \end{aligned}\]
S.5 \[\begin{aligned} \htmlClass{sym-M_ws}{M_{\mathrm{ws}}} &= \frac{\htmlClass{sym-M_D}{M_{D}} + \htmlClass{sym-M_L}{M_{L}}}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\htmlClass{sym-M_D}{0\ \mathrm{kN} \cdot \mathrm{m}} + \htmlClass{sym-M_L}{0\ \mathrm{kN} \cdot \mathrm{m}}}{\htmlClass{sym-L_f}{1\ \mathrm{m}}} \\ &= 0\ \mathrm{N} \end{aligned}\]
S.6 \[\begin{aligned} \htmlClass{sym-M_wu}{M_{\mathrm{wu}}} &= \frac{\max\left(1.4 \cdot \htmlClass{sym-M_D}{M_{D}}, 1.25 \cdot \htmlClass{sym-M_D}{M_{D}} + 1.5 \cdot \htmlClass{sym-M_L}{M_{L}}\right)}{\htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\max\left(1.4 \cdot \htmlClass{sym-M_D}{0\ \mathrm{kN} \cdot \mathrm{m}}, 1.25 \cdot \htmlClass{sym-M_D}{0\ \mathrm{kN} \cdot \mathrm{m}} + 1.5 \cdot \htmlClass{sym-M_L}{0\ \mathrm{kN} \cdot \mathrm{m}}\right)}{\htmlClass{sym-L_f}{1\ \mathrm{m}}} \\ &= 0\ \mathrm{N} \end{aligned}\]

Assumed: each at its own worst combination

S.7 \[\begin{aligned} \htmlClass{sym-q_s}{q_{s}} &= \frac{\htmlClass{sym-P_s}{P_{s}}}{\htmlClass{sym-B}{B}} + \frac{6 \cdot \htmlClass{sym-M_ws}{M_{\mathrm{ws}}}}{\htmlClass{sym-B}{B}^{2}} \\ &= \frac{\htmlClass{sym-P_s}{175\ \mathrm{kN/m}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} + \frac{6 \cdot \htmlClass{sym-M_ws}{0\ \mathrm{N}}}{\left(\htmlClass{sym-B}{900\ \mathrm{mm}}\right)^{2}} \\ &= 194.4\ \mathrm{kPa} \end{aligned}\]

Assumed: linear bearing, no self-weight

S.8 \[\begin{aligned} \htmlClass{sym-q_s_min}{q_{s,\mathrm{min}}} &= \frac{\htmlClass{sym-P_s}{P_{s}}}{\htmlClass{sym-B}{B}} - \frac{6 \cdot \htmlClass{sym-M_ws}{M_{\mathrm{ws}}}}{\htmlClass{sym-B}{B}^{2}} \\ &= \frac{\htmlClass{sym-P_s}{175\ \mathrm{kN/m}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} - \frac{6 \cdot \htmlClass{sym-M_ws}{0\ \mathrm{N}}}{\left(\htmlClass{sym-B}{900\ \mathrm{mm}}\right)^{2}} \\ &= 194.4\ \mathrm{kPa} \end{aligned}\]
S.9 \[\begin{aligned} \htmlClass{sym-q_u}{q_{u}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-B}{B}} + \frac{6 \cdot \htmlClass{sym-M_wu}{M_{\mathrm{wu}}}}{\htmlClass{sym-B}{B}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{237.5\ \mathrm{kN/m}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} + \frac{6 \cdot \htmlClass{sym-M_wu}{0\ \mathrm{N}}}{\left(\htmlClass{sym-B}{900\ \mathrm{mm}}\right)^{2}} \\ &= 263.9\ \mathrm{kPa} \end{aligned}\]
S.10 \[\begin{aligned} \htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}} &= \frac{\htmlClass{sym-P_u}{P_{u}}}{\htmlClass{sym-B}{B}} - \frac{6 \cdot \htmlClass{sym-M_wu}{M_{\mathrm{wu}}}}{\htmlClass{sym-B}{B}^{2}} \\ &= \frac{\htmlClass{sym-P_u}{237.5\ \mathrm{kN/m}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} - \frac{6 \cdot \htmlClass{sym-M_wu}{0\ \mathrm{N}}}{\left(\htmlClass{sym-B}{900\ \mathrm{mm}}\right)^{2}} \\ &= 263.9\ \mathrm{kPa} \end{aligned}\]
S.11 \[\begin{aligned} \htmlClass{sym-l_c}{l_{c}} &= \htmlClass{sym-B}{B} - \htmlClass{sym-t_w}{t_{w}} \\ &= \htmlClass{sym-B}{900\ \mathrm{mm}} - \htmlClass{sym-t_w}{300\ \mathrm{mm}} \\ &= 600\ \mathrm{mm} \end{aligned}\]

Assumed: wall at one edge, one projection

S.12 \[\begin{aligned} \htmlClass{sym-d_b}{d_{b}} &= 16\ \mathrm{mm} \quad \left(\text{Bar diameter | 15M}\right) \end{aligned}\]
S.13 \[\begin{aligned} \htmlClass{sym-d}{d} &= \htmlClass{sym-h}{h} - \htmlClass{sym-c_c}{c_{c}} - \frac{\htmlClass{sym-d_b}{d_{b}}}{2} \\ &= \htmlClass{sym-h}{450\ \mathrm{mm}} - \htmlClass{sym-c_c}{75\ \mathrm{mm}} - \frac{\htmlClass{sym-d_b}{16\ \mathrm{mm}}}{2} \\ &= 367\ \mathrm{mm} \end{aligned}\]
S.14

CSA A23.3 Cl. 3.2

\[\begin{aligned} \htmlClass{sym-d_v}{d_{v}} &= \max\left(0.72 \cdot \htmlClass{sym-h}{h}, 0.9 \cdot \htmlClass{sym-d}{d}\right) \\ &= \max\left(0.72 \cdot \htmlClass{sym-h}{450\ \mathrm{mm}}, 0.9 \cdot \htmlClass{sym-d}{367\ \mathrm{mm}}\right) \\ &= 330.3\ \mathrm{mm} \end{aligned}\]

Branch: \(h \leq 350\ \mathrm{mm}\ \text{or}\ l_{c} < 2 \cdot d_{v}\) held

S.15 \[\begin{aligned} \htmlClass{sym-beta}{\beta} &= 0.21 \quad \left(\text{Shallow section or short projection | CSA A23.3 Cl. 11.3.6.2}\right) \end{aligned}\]
S.16 \[\begin{aligned} \htmlClass{sym-x_v}{x_{v}} &= \max\left(\htmlClass{sym-l_c}{l_{c}} - \htmlClass{sym-d_v}{d_{v}}, 0\ \mathrm{m}\right) \\ &= \max\left(\htmlClass{sym-l_c}{600\ \mathrm{mm}} - \htmlClass{sym-d_v}{330.3\ \mathrm{mm}}, 0\ \mathrm{m}\right) \\ &= 269.7\ \mathrm{mm} \end{aligned}\]
S.17 \[\begin{aligned} \htmlClass{sym-q_v}{q_{v}} &= \htmlClass{sym-q_u}{q_{u}} - \frac{\left(\htmlClass{sym-q_u}{q_{u}} - \htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}}\right) \cdot \htmlClass{sym-x_v}{x_{v}}}{\htmlClass{sym-B}{B}} \\ &= \htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} - \htmlClass{sym-q_u_min}{263.9\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-x_v}{269.7\ \mathrm{mm}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} \\ &= 263.9\ \mathrm{kPa} \end{aligned}\]

Assumed: q_u at the toe, the heavier side

S.18 \[\begin{aligned} \htmlClass{sym-V_u}{V_{u}} &= \frac{\htmlClass{sym-q_u}{q_{u}} + \htmlClass{sym-q_v}{q_{v}}}{2} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-x_v}{x_{v}} \\ &= \frac{\htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} + \htmlClass{sym-q_v}{263.9\ \mathrm{kPa}}}{2} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}} \cdot \htmlClass{sym-x_v}{269.7\ \mathrm{mm}} \\ &= 71.17\ \mathrm{kN} \end{aligned}\]
S.19

CSA A23.3 Cl. 11.3.4

\[\begin{aligned} \htmlClass{sym-f_v}{f_{v}} &= \min\left(\sqrt{\htmlClass{sym-f_c}{f_{c}} \cdot 1\ \mathrm{MPa}}, 8\ \mathrm{MPa}\right) \\ &= \min\left(\sqrt{\htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot 1\ \mathrm{MPa}}, 8\ \mathrm{MPa}\right) \\ &= 5\ \mathrm{MPa} \end{aligned}\]
S.20

CSA A23.3 Cl. 11.3.4

\[\begin{aligned} \htmlClass{sym-V_c}{V_{c}} &= \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-lambda_c}{\lambda_{c}} \cdot \htmlClass{sym-beta}{\beta} \cdot \htmlClass{sym-f_v}{f_{v}} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-d_v}{d_{v}} \\ &= \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-lambda_c}{1} \cdot \htmlClass{sym-beta}{0.21} \cdot \htmlClass{sym-f_v}{5\ \mathrm{MPa}} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}} \cdot \htmlClass{sym-d_v}{330.3\ \mathrm{mm}} \\ &= 225.4\ \mathrm{kN} \end{aligned}\]
S.21 \[\begin{aligned} \htmlClass{sym-q_f}{q_{f}} &= \htmlClass{sym-q_u}{q_{u}} - \frac{\left(\htmlClass{sym-q_u}{q_{u}} - \htmlClass{sym-q_u_min}{q_{u,\mathrm{min}}}\right) \cdot \htmlClass{sym-l_c}{l_{c}}}{\htmlClass{sym-B}{B}} \\ &= \htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} - \frac{\left(\htmlClass{sym-q_u}{263.9\ \mathrm{kPa}} - \htmlClass{sym-q_u_min}{263.9\ \mathrm{kPa}}\right) \cdot \htmlClass{sym-l_c}{600\ \mathrm{mm}}}{\htmlClass{sym-B}{900\ \mathrm{mm}}} \\ &= 263.9\ \mathrm{kPa} \end{aligned}\]
S.22 \[\begin{aligned} \htmlClass{sym-M_u}{M_{u}} &= \frac{\htmlClass{sym-q_f}{q_{f}} + 2 \cdot \htmlClass{sym-q_u}{q_{u}}}{6} \cdot \htmlClass{sym-L_f}{L_{f}} \cdot \htmlClass{sym-l_c}{l_{c}}^{2} \\ &= \frac{\htmlClass{sym-q_f}{263.9\ \mathrm{kPa}} + 2 \cdot \htmlClass{sym-q_u}{263.9\ \mathrm{kPa}}}{6} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}} \cdot \left(\htmlClass{sym-l_c}{600\ \mathrm{mm}}\right)^{2} \\ &= 47.5\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]
S.23 \[\begin{aligned} \htmlClass{sym-A_b}{A_{b}} &= 200\ \mathrm{mm}^{2} \quad \left(\text{Bar area | 15M}\right) \end{aligned}\]
S.24 \[\begin{aligned} \htmlClass{sym-A_s}{A_{s}} &= \frac{\htmlClass{sym-A_b}{A_{b}} \cdot \htmlClass{sym-L_f}{L_{f}}}{\htmlClass{sym-s}{s}} \\ &= \frac{\htmlClass{sym-A_b}{200\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}}}{\htmlClass{sym-s}{200\ \mathrm{mm}}} \\ &= 1000\ \mathrm{mm}^{2} \end{aligned}\]
S.25

CSA A23.3 Cl. 7.8.1

\[\begin{aligned} \htmlClass{sym-A_s_min}{A_{s,\mathrm{min}}} &= 0.002 \cdot \htmlClass{sym-h}{h} \cdot \htmlClass{sym-L_f}{L_{f}} \\ &= 0.002 \cdot \htmlClass{sym-h}{450\ \mathrm{mm}} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}} \\ &= 900\ \mathrm{mm}^{2} \end{aligned}\]
S.26

CSA A23.3 Cl. 10.1.7

\[\begin{aligned} \htmlClass{sym-alpha_1}{\alpha_{1}} &= 0.85 - \frac{0.0015 \cdot \htmlClass{sym-f_c}{f_{c}}}{1\ \mathrm{MPa}} \\ &= 0.85 - \frac{0.0015 \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}}}{1\ \mathrm{MPa}} \\ &= 0.8125 \end{aligned}\]
S.27

CSA A23.3 Cl. 10.1.7

\[\begin{aligned} \htmlClass{sym-beta_1}{\beta_{1}} &= 0.97 - \frac{0.0025 \cdot \htmlClass{sym-f_c}{f_{c}}}{1\ \mathrm{MPa}} \\ &= 0.97 - \frac{0.0025 \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}}}{1\ \mathrm{MPa}} \\ &= 0.9075 \end{aligned}\]
S.28 \[\begin{aligned} \htmlClass{sym-a}{a} &= \frac{\htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_s}{A_{s}} \cdot \htmlClass{sym-f_y}{f_{y}}}{\htmlClass{sym-alpha_1}{\alpha_{1}} \cdot \htmlClass{sym-phi_c}{\phi_{c}} \cdot \htmlClass{sym-f_c}{f_{c}} \cdot \htmlClass{sym-L_f}{L_{f}}} \\ &= \frac{\htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_s}{1000\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}}}{\htmlClass{sym-alpha_1}{0.8125} \cdot \htmlClass{sym-phi_c}{0.65} \cdot \htmlClass{sym-f_c}{25\ \mathrm{MPa}} \cdot \htmlClass{sym-L_f}{1\ \mathrm{m}}} \\ &= 25.75\ \mathrm{mm} \end{aligned}\]
S.29 \[\begin{aligned} \htmlClass{sym-c}{c} &= \frac{\htmlClass{sym-a}{a}}{\htmlClass{sym-beta_1}{\beta_{1}}} \\ &= \frac{\htmlClass{sym-a}{25.75\ \mathrm{mm}}}{\htmlClass{sym-beta_1}{0.9075}} \\ &= 28.38\ \mathrm{mm} \end{aligned}\]
S.30

CSA A23.3 Cl. 10.1

\[\begin{aligned} \htmlClass{sym-M_r}{M_{r}} &= \htmlClass{sym-phi_s}{\phi_{s}} \cdot \htmlClass{sym-A_s}{A_{s}} \cdot \htmlClass{sym-f_y}{f_{y}} \cdot \left(\htmlClass{sym-d}{d} - \frac{\htmlClass{sym-a}{a}}{2}\right) \\ &= \htmlClass{sym-phi_s}{0.85} \cdot \htmlClass{sym-A_s}{1000\ \mathrm{mm}^{2}} \cdot \htmlClass{sym-f_y}{400\ \mathrm{MPa}} \cdot \left(\htmlClass{sym-d}{367\ \mathrm{mm}} - \frac{\htmlClass{sym-a}{25.75\ \mathrm{mm}}}{2}\right) \\ &= 120.4\ \mathrm{kN} \cdot \mathrm{m} \end{aligned}\]

Questions

Where is the wall on the footing, and how does it change l_c?

Either, and the sheet asks. At one edge the footing projects past the wall on one side, l_c = B - t_w; centred, it projects both ways by half that. For the same bearing pressure, halving the projection roughly halves the shear at the wall face and quarters the moment.

Why is beta 0.21 for most footings under Cl. 11.3.6.2?

Cl. 11.3.6.2 allows 0.21 for a section no deeper than 350 mm, and for a footing whose projection l_c past the wall is shorter than twice d_v. Otherwise the size-effect value 230 / (1000 + d_v) of Cl. 11.3.6.3 applies, which is below 0.21 for any d_v over 95 mm.

Does q_s include the footing's own weight?

No. q_s is the pressure from the wall loads D and L alone, compared with q_a as entered. Add the footing and any soil over it to D, or compare with the net allowable pressure your geotechnical report gives.

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