calcsheet
Seismic loads, rectangular tank
Seismic design forces and overturning moments for a rectangular liquid-containing tank. Calculate seismic design forces for rectangular liquid-containing tanks per ACI 350.3-06. The calculator separates the liquid mass into impulsive and convective components, determines corresponding seismic coefficients, and computes wall pressures and base shear. Input tank geometry, liquid properties, and site seismic parameters for a complete load determination.
Given
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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/rect-tank-seismic.pdf.
Checks
| Check | D/C | Utilisation | Result |
|---|---|---|---|
| Freeboard adequate\(\htmlClass{sym-h_freeboard}{h_{\mathrm{freeboard}}} \geq 0\ \mathrm{m} \quad \Rightarrow \quad \htmlClass{sym-h_freeboard}{1.2\ \mathrm{m}} \geq 0\ \mathrm{m}\) | 0.00 | PASS |
Results
| Quantity | Description | Value | Unit |
|---|---|---|---|
| \(\htmlClass{sym-V}{V}\) | Total design base shear | 894.9 | \(\mathrm{kN}\) |
| \(\htmlClass{sym-V_t}{V_{t}}\) | Fluid design shear at the top of the foundation | 715.5 | \(\mathrm{kN}\) |
| \(\htmlClass{sym-M_s}{M_{s}}\) | Overturning moment at the base of the shell | 3.44 | \(\mathrm{MN} \cdot \mathrm{m}\) |
| \(\htmlClass{sym-M_s_prime}{{M'}_{s}}\) | Overturning moment including base pressure | 6.099 | \(\mathrm{MN} \cdot \mathrm{m}\) |
| \(\htmlClass{sym-T_c}{T_{c}}\) | Convective (sloshing) period | 4.992 | \(\mathrm{s}\) |
| \(\htmlClass{sym-h_freeboard}{h_{\mathrm{freeboard}}}\) | Freeboard above the fluid surface | 1.2 | \(\mathrm{m}\) |
Derivation
S.1
\[\begin{aligned}
\htmlClass{sym-rho_c}{\rho_{c}} &= \frac{\htmlClass{sym-gamma_conc}{\gamma_{\mathrm{conc}}}}{\htmlClass{sym-g}{g}} \\
&= \frac{\htmlClass{sym-gamma_conc}{23.56\ \mathrm{kN/m³}}}{\htmlClass{sym-g}{9.78\ \mathrm{m} \cdot \mathrm{s}^{-2}}} \\
&= 2409\ \mathrm{kg} \cdot \mathrm{m}^{-3}
\end{aligned}\]
S.2
\[\begin{aligned}
\htmlClass{sym-rho_L}{\rho_{L}} &= \frac{\htmlClass{sym-gamma_fluid}{\gamma_{\mathrm{fluid}}}}{\htmlClass{sym-g}{g}} \\
&= \frac{\htmlClass{sym-gamma_fluid}{9.81\ \mathrm{kN/m³}}}{\htmlClass{sym-g}{9.78\ \mathrm{m} \cdot \mathrm{s}^{-2}}} \\
&= 1003\ \mathrm{kg} \cdot \mathrm{m}^{-3}
\end{aligned}\]
S.3
\[\begin{aligned}
\htmlClass{sym-B}{B} &= 1\ \mathrm{m}
\end{aligned}\]
S.4
\[\begin{aligned}
\htmlClass{sym-W_fluid}{W_{\mathrm{fluid}}} &= \htmlClass{sym-L}{L} \cdot \htmlClass{sym-B}{B} \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \cdot \htmlClass{sym-gamma_fluid}{\gamma_{\mathrm{fluid}}} \\
&= \htmlClass{sym-L}{18\ \mathrm{m}} \cdot \htmlClass{sym-B}{1\ \mathrm{m}} \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \cdot \htmlClass{sym-gamma_fluid}{9.81\ \mathrm{kN/m³}} \\
&= 1.616\ \mathrm{MN}
\end{aligned}\]
S.5
\[\begin{aligned}
\htmlClass{sym-h_freeboard}{h_{\mathrm{freeboard}}} &= \htmlClass{sym-h_wall}{h_{\mathrm{wall}}} - \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= \htmlClass{sym-h_wall}{10.35\ \mathrm{m}} - \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 1.2\ \mathrm{m}
\end{aligned}\]
S.6
\[\begin{aligned}
\htmlClass{sym-W_i}{W_{i}} &= \frac{\tanh\left(0.866 \cdot \frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}\right)}{0.866 \cdot \frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}} \cdot \htmlClass{sym-W_fluid}{W_{\mathrm{fluid}}} \\
&= \frac{\tanh\left(0.866 \cdot \frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}\right)}{0.866 \cdot \frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}} \cdot \htmlClass{sym-W_fluid}{1.616\ \mathrm{MN}} \\
&= 887.6\ \mathrm{kN}
\end{aligned}\]
S.7
\[\begin{aligned}
\htmlClass{sym-W_c}{W_{c}} &= 0.264 \cdot \frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}} \cdot \tanh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right) \cdot \htmlClass{sym-W_fluid}{W_{\mathrm{fluid}}} \\
&= 0.264 \cdot \frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}} \cdot \tanh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right) \cdot \htmlClass{sym-W_fluid}{1.616\ \mathrm{MN}} \\
&= 774.2\ \mathrm{kN}
\end{aligned}\]
S.8
\[\begin{aligned}
\htmlClass{sym-W_walls}{W_{\mathrm{walls}}} &= \htmlClass{sym-gamma_conc}{\gamma_{\mathrm{conc}}} \cdot 2 \cdot \htmlClass{sym-t_wall}{t_{\mathrm{wall}}} \cdot \htmlClass{sym-h_wall}{h_{\mathrm{wall}}} \cdot \htmlClass{sym-B}{B} \\
&= \htmlClass{sym-gamma_conc}{23.56\ \mathrm{kN/m³}} \cdot 2 \cdot \htmlClass{sym-t_wall}{650\ \mathrm{mm}} \cdot \htmlClass{sym-h_wall}{10.35\ \mathrm{m}} \cdot \htmlClass{sym-B}{1\ \mathrm{m}} \\
&= 317\ \mathrm{kN}
\end{aligned}\]
S.9
\[\begin{aligned}
\htmlClass{sym-W_bottom}{W_{\mathrm{bottom}}} &= \htmlClass{sym-L}{L} \cdot \htmlClass{sym-B}{B} \cdot \htmlClass{sym-gamma_conc}{\gamma_{\mathrm{conc}}} \cdot \htmlClass{sym-t_bottom}{t_{\mathrm{bottom}}} \\
&= \htmlClass{sym-L}{18\ \mathrm{m}} \cdot \htmlClass{sym-B}{1\ \mathrm{m}} \cdot \htmlClass{sym-gamma_conc}{23.56\ \mathrm{kN/m³}} \cdot \htmlClass{sym-t_bottom}{0\ \mathrm{mm}} \\
&= 0\ \mathrm{N}
\end{aligned}\]
S.10
\[\begin{aligned}
\htmlClass{sym-S_ds}{S_{\mathrm{ds}}} &= \htmlClass{sym-S_s}{S_{s}} \cdot \htmlClass{sym-F_a}{F_{a}} \\
&= \htmlClass{sym-S_s}{1.06} \cdot \htmlClass{sym-F_a}{1} \\
&= 1.06
\end{aligned}\]
S.11
\[\begin{aligned}
\htmlClass{sym-S_d1}{S_{d1}} &= \htmlClass{sym-S_1}{S_{1}} \cdot \htmlClass{sym-F_v}{F_{v}} \\
&= \htmlClass{sym-S_1}{0.5} \cdot \htmlClass{sym-F_v}{1} \\
&= 0.5
\end{aligned}\]
S.12
\[\begin{aligned}
\htmlClass{sym-T_s}{T_{s}} &= \frac{\htmlClass{sym-S_d1}{S_{d1}}}{\htmlClass{sym-S_ds}{S_{\mathrm{ds}}}} \\
&= \frac{\htmlClass{sym-S_d1}{0.5}}{\htmlClass{sym-S_ds}{1.06}} \\
&= 0.4717
\end{aligned}\]
S.13
\[\begin{aligned}
\htmlClass{sym-C_t}{C_{t}} &= 0.4 \cdot \htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \\
&= 0.4 \cdot \htmlClass{sym-S_ds}{1.06} \\
&= 0.424
\end{aligned}\]
S.14
\[\begin{aligned}
\htmlClass{sym-m_w}{m_{w}} &= \htmlClass{sym-h_wall}{h_{\mathrm{wall}}} \cdot \htmlClass{sym-t_wall}{t_{\mathrm{wall}}} \cdot \htmlClass{sym-rho_c}{\rho_{c}} \\
&= \htmlClass{sym-h_wall}{10.35\ \mathrm{m}} \cdot \htmlClass{sym-t_wall}{650\ \mathrm{mm}} \cdot \htmlClass{sym-rho_c}{2409\ \mathrm{kg} \cdot \mathrm{m}^{-3}} \\
&= 16206\ \mathrm{kg} \cdot \mathrm{m}^{-1}
\end{aligned}\]
S.15
\[\begin{aligned}
\htmlClass{sym-m_i}{m_{i}} &= \frac{\frac{\htmlClass{sym-W_i}{W_{i}}}{\htmlClass{sym-W_fluid}{W_{\mathrm{fluid}}}} \cdot \htmlClass{sym-L}{L}}{2} \cdot \htmlClass{sym-h_wall}{h_{\mathrm{wall}}} \cdot \htmlClass{sym-rho_L}{\rho_{L}} \\
&= \frac{\frac{\htmlClass{sym-W_i}{887.6\ \mathrm{kN}}}{\htmlClass{sym-W_fluid}{1.616\ \mathrm{MN}}} \cdot \htmlClass{sym-L}{18\ \mathrm{m}}}{2} \cdot \htmlClass{sym-h_wall}{10.35\ \mathrm{m}} \cdot \htmlClass{sym-rho_L}{1003\ \mathrm{kg} \cdot \mathrm{m}^{-3}} \\
&= 51326\ \mathrm{kg} \cdot \mathrm{m}^{-1}
\end{aligned}\]
S.16
\[\begin{aligned}
\htmlClass{sym-T_c}{T_{c}} &= 2 \cdot \pi \cdot \sqrt{\frac{\htmlClass{sym-L}{L}}{3.16 \cdot \htmlClass{sym-g}{g} \cdot \tanh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right)}} \\
&= 2 \cdot \pi \cdot \sqrt{\frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{3.16 \cdot \htmlClass{sym-g}{9.78\ \mathrm{m} \cdot \mathrm{s}^{-2}} \cdot \tanh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right)}} \\
&= 4.992\ \mathrm{s}
\end{aligned}\]
Branch: \(T_{c} \leq \frac{1.6\ \mathrm{s}}{T_{s}}\) did not hold
S.17
\[\begin{aligned}
\htmlClass{sym-C_c}{C_{c}} &= \frac{2.4 \cdot \htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \cdot 1\ \mathrm{s}^{2}}{\htmlClass{sym-T_c}{T_{c}}^{2}} \\
&= \frac{2.4 \cdot \htmlClass{sym-S_ds}{1.06} \cdot 1\ \mathrm{s}^{2}}{\left(\htmlClass{sym-T_c}{4.992\ \mathrm{s}}\right)^{2}} \\
&= 0.1021
\end{aligned}\]
S.18
\[\begin{aligned}
\htmlClass{sym-epsilon}{\epsilon} &= \min\left(0.0151 \cdot \left(\frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}\right)^{2} - 0.1908 \cdot \frac{\htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}} + 1.021, 1\right) \\
&= \min\left(0.0151 \cdot \left(\frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}\right)^{2} - 0.1908 \cdot \frac{\htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}} + 1.021, 1\right) \\
&= 0.7041
\end{aligned}\]
Branch: \(\frac{L}{h_{\mathrm{fluid}}} < 0.75\) did not hold
S.19
\[\begin{aligned}
\htmlClass{sym-h_i_prime}{{h'}_{i}} &= \left(\frac{\frac{0.866 \cdot \htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}}{2 \cdot \tanh\left(\frac{0.866 \cdot \htmlClass{sym-L}{L}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}\right)} - \frac{1}{8}\right) \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= \left(\frac{\frac{0.866 \cdot \htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}}{2 \cdot \tanh\left(\frac{0.866 \cdot \htmlClass{sym-L}{18\ \mathrm{m}}}{\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}\right)} - \frac{1}{8}\right) \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 7.184\ \mathrm{m}
\end{aligned}\]
S.20
\[\begin{aligned}
\htmlClass{sym-h_c_prime}{{h'}_{c}} &= \left(1 - \frac{\cosh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right) - 2.01}{\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}} \cdot \sinh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right)}\right) \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= \left(1 - \frac{\cosh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right) - 2.01}{\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}} \cdot \sinh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right)}\right) \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 7.763\ \mathrm{m}
\end{aligned}\]
S.21
\[\begin{aligned}
\htmlClass{sym-H_w}{H_{w}} &= \frac{\htmlClass{sym-h_wall}{h_{\mathrm{wall}}}}{2} \\
&= \frac{\htmlClass{sym-h_wall}{10.35\ \mathrm{m}}}{2} \\
&= 5.175\ \mathrm{m}
\end{aligned}\]
S.22
\[\begin{aligned}
\htmlClass{sym-h_roof}{h_{\mathrm{roof}}} &= \htmlClass{sym-h_wall}{h_{\mathrm{wall}}} + \frac{\htmlClass{sym-t_roof}{t_{\mathrm{roof}}}}{2} \\
&= \htmlClass{sym-h_wall}{10.35\ \mathrm{m}} + \frac{\htmlClass{sym-t_roof}{0\ \mathrm{mm}}}{2} \\
&= 10.35\ \mathrm{m}
\end{aligned}\]
Branch: \(\frac{L}{h_{\mathrm{fluid}}} < 1.333\) did not hold
S.23
\[\begin{aligned}
\htmlClass{sym-h_i}{h_{i}} &= 0.375 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= 0.375 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 3.431\ \mathrm{m}
\end{aligned}\]
S.24
\[\begin{aligned}
\htmlClass{sym-h_c}{h_{c}} &= \left(1 - \frac{\cosh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right) - 1}{\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}} \cdot \sinh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}}{\htmlClass{sym-L}{L}}\right)}\right) \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= \left(1 - \frac{\cosh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right) - 1}{\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}} \cdot \sinh\left(\frac{3.16 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}}{\htmlClass{sym-L}{18\ \mathrm{m}}}\right)}\right) \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 5.357\ \mathrm{m}
\end{aligned}\]
S.25
\[\begin{aligned}
\htmlClass{sym-P_w}{P_{w}} &= \frac{\htmlClass{sym-I}{I} \cdot \htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \cdot \htmlClass{sym-epsilon}{\epsilon} \cdot \htmlClass{sym-W_walls}{W_{\mathrm{walls}}}}{\htmlClass{sym-R_wi}{R_{\mathrm{wi}}}} \\
&= \frac{\htmlClass{sym-I}{1.5} \cdot \htmlClass{sym-S_ds}{1.06} \cdot \htmlClass{sym-epsilon}{0.7041} \cdot \htmlClass{sym-W_walls}{317\ \mathrm{kN}}}{\htmlClass{sym-R_wi}{2}} \\
&= 177.4\ \mathrm{kN}
\end{aligned}\]
S.26
\[\begin{aligned}
\htmlClass{sym-P_r}{P_{r}} &= \frac{\htmlClass{sym-I}{I} \cdot \htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \cdot \htmlClass{sym-W_r}{W_{r}}}{\htmlClass{sym-R_wi}{R_{\mathrm{wi}}}} \\
&= \frac{\htmlClass{sym-I}{1.5} \cdot \htmlClass{sym-S_ds}{1.06} \cdot \htmlClass{sym-W_r}{5\ \mathrm{kN}}}{\htmlClass{sym-R_wi}{2}} \\
&= 3.975\ \mathrm{kN}
\end{aligned}\]
S.27
\[\begin{aligned}
\htmlClass{sym-P_i}{P_{i}} &= \frac{\htmlClass{sym-I}{I} \cdot \htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \cdot \htmlClass{sym-W_i}{W_{i}}}{\htmlClass{sym-R_wi}{R_{\mathrm{wi}}}} \\
&= \frac{\htmlClass{sym-I}{1.5} \cdot \htmlClass{sym-S_ds}{1.06} \cdot \htmlClass{sym-W_i}{887.6\ \mathrm{kN}}}{\htmlClass{sym-R_wi}{2}} \\
&= 705.6\ \mathrm{kN}
\end{aligned}\]
S.28
\[\begin{aligned}
\htmlClass{sym-P_c}{P_{c}} &= \htmlClass{sym-C_c}{C_{c}} \cdot \htmlClass{sym-I}{I} \cdot \htmlClass{sym-W_c}{W_{c}} \\
&= \htmlClass{sym-C_c}{0.1021} \cdot \htmlClass{sym-I}{1.5} \cdot \htmlClass{sym-W_c}{774.2\ \mathrm{kN}} \\
&= 118.5\ \mathrm{kN}
\end{aligned}\]
S.29
\[\begin{aligned}
\htmlClass{sym-V_t}{V_{t}} &= \sqrt{\htmlClass{sym-P_c}{P_{c}}^{2} + \htmlClass{sym-P_i}{P_{i}}^{2}} \\
&= \sqrt{\left(\htmlClass{sym-P_c}{118.5\ \mathrm{kN}}\right)^{2} + \left(\htmlClass{sym-P_i}{705.6\ \mathrm{kN}}\right)^{2}} \\
&= 715.5\ \mathrm{kN}
\end{aligned}\]
S.30
\[\begin{aligned}
\htmlClass{sym-V}{V} &= \sqrt{\left(\htmlClass{sym-P_i}{P_{i}} + \htmlClass{sym-P_w}{P_{w}} + \htmlClass{sym-P_r}{P_{r}}\right)^{2} + \htmlClass{sym-P_c}{P_{c}}^{2} + \htmlClass{sym-P_eg}{P_{\mathrm{eg}}}^{2}} \\
&= \sqrt{\left(\htmlClass{sym-P_i}{705.6\ \mathrm{kN}} + \htmlClass{sym-P_w}{177.4\ \mathrm{kN}} + \htmlClass{sym-P_r}{3.975\ \mathrm{kN}}\right)^{2} + \left(\htmlClass{sym-P_c}{118.5\ \mathrm{kN}}\right)^{2} + \left(\htmlClass{sym-P_eg}{0\ \mathrm{kN}}\right)^{2}} \\
&= 894.9\ \mathrm{kN}
\end{aligned}\]
S.31
\[\begin{aligned}
\htmlClass{sym-M_w}{M_{w}} &= \htmlClass{sym-P_w}{P_{w}} \cdot \htmlClass{sym-H_w}{H_{w}} \\
&= \htmlClass{sym-P_w}{177.4\ \mathrm{kN}} \cdot \htmlClass{sym-H_w}{5.175\ \mathrm{m}} \\
&= 918.3\ \mathrm{kN} \cdot \mathrm{m}
\end{aligned}\]
S.32
\[\begin{aligned}
\htmlClass{sym-M_r}{M_{r}} &= \htmlClass{sym-P_r}{P_{r}} \cdot \htmlClass{sym-h_roof}{h_{\mathrm{roof}}} \\
&= \htmlClass{sym-P_r}{3.975\ \mathrm{kN}} \cdot \htmlClass{sym-h_roof}{10.35\ \mathrm{m}} \\
&= 41.14\ \mathrm{kN} \cdot \mathrm{m}
\end{aligned}\]
S.33
\[\begin{aligned}
\htmlClass{sym-M_i}{M_{i}} &= \htmlClass{sym-P_i}{P_{i}} \cdot \htmlClass{sym-h_i}{h_{i}} \\
&= \htmlClass{sym-P_i}{705.6\ \mathrm{kN}} \cdot \htmlClass{sym-h_i}{3.431\ \mathrm{m}} \\
&= 2.421\ \mathrm{MN} \cdot \mathrm{m}
\end{aligned}\]
S.34
\[\begin{aligned}
\htmlClass{sym-M_c}{M_{c}} &= \htmlClass{sym-P_c}{P_{c}} \cdot \htmlClass{sym-h_c}{h_{c}} \\
&= \htmlClass{sym-P_c}{118.5\ \mathrm{kN}} \cdot \htmlClass{sym-h_c}{5.357\ \mathrm{m}} \\
&= 635.1\ \mathrm{kN} \cdot \mathrm{m}
\end{aligned}\]
S.35
\[\begin{aligned}
\htmlClass{sym-M_eg}{M_{\mathrm{eg}}} &= \htmlClass{sym-P_eg}{P_{\mathrm{eg}}} \cdot \htmlClass{sym-h_eg}{h_{\mathrm{eg}}} \\
&= \htmlClass{sym-P_eg}{0\ \mathrm{kN}} \cdot \htmlClass{sym-h_eg}{0\ \mathrm{m}} \\
&= 0\ \mathrm{N} \cdot \mathrm{m}
\end{aligned}\]
S.36
\[\begin{aligned}
\htmlClass{sym-M_s}{M_{s}} &= \sqrt{\left(\htmlClass{sym-M_i}{M_{i}} + \htmlClass{sym-M_w}{M_{w}} + \htmlClass{sym-M_r}{M_{r}}\right)^{2} + \htmlClass{sym-M_c}{M_{c}}^{2} + \htmlClass{sym-M_eg}{M_{\mathrm{eg}}}^{2}} \\
&= \sqrt{\left(\htmlClass{sym-M_i}{2.421\ \mathrm{MN} \cdot \mathrm{m}} + \htmlClass{sym-M_w}{918.3\ \mathrm{kN} \cdot \mathrm{m}} + \htmlClass{sym-M_r}{41.14\ \mathrm{kN} \cdot \mathrm{m}}\right)^{2} + \left(\htmlClass{sym-M_c}{635.1\ \mathrm{kN} \cdot \mathrm{m}}\right)^{2} + \left(\htmlClass{sym-M_eg}{0\ \mathrm{N} \cdot \mathrm{m}}\right)^{2}} \\
&= 3.44\ \mathrm{MN} \cdot \mathrm{m}
\end{aligned}\]
S.37
\[\begin{aligned}
\htmlClass{sym-M_i_prime}{{M'}_{i}} &= \htmlClass{sym-P_i}{P_{i}} \cdot \htmlClass{sym-h_i_prime}{{h'}_{i}} \\
&= \htmlClass{sym-P_i}{705.6\ \mathrm{kN}} \cdot \htmlClass{sym-h_i_prime}{7.184\ \mathrm{m}} \\
&= 5.069\ \mathrm{MN} \cdot \mathrm{m}
\end{aligned}\]
S.38
\[\begin{aligned}
\htmlClass{sym-M_c_prime}{{M'}_{c}} &= \htmlClass{sym-P_c}{P_{c}} \cdot \htmlClass{sym-h_c_prime}{{h'}_{c}} \\
&= \htmlClass{sym-P_c}{118.5\ \mathrm{kN}} \cdot \htmlClass{sym-h_c_prime}{7.763\ \mathrm{m}} \\
&= 920.2\ \mathrm{kN} \cdot \mathrm{m}
\end{aligned}\]
S.39
\[\begin{aligned}
\htmlClass{sym-M_s_prime}{{M'}_{s}} &= \sqrt{\left(\htmlClass{sym-M_i_prime}{{M'}_{i}} + \htmlClass{sym-M_w}{M_{w}} + \htmlClass{sym-M_r}{M_{r}}\right)^{2} + \htmlClass{sym-M_c_prime}{{M'}_{c}}^{2}} \\
&= \sqrt{\left(\htmlClass{sym-M_i_prime}{5.069\ \mathrm{MN} \cdot \mathrm{m}} + \htmlClass{sym-M_w}{918.3\ \mathrm{kN} \cdot \mathrm{m}} + \htmlClass{sym-M_r}{41.14\ \mathrm{kN} \cdot \mathrm{m}}\right)^{2} + \left(\htmlClass{sym-M_c_prime}{920.2\ \mathrm{kN} \cdot \mathrm{m}}\right)^{2}} \\
&= 6.099\ \mathrm{MN} \cdot \mathrm{m}
\end{aligned}\]
S.40
\[\begin{aligned}
\htmlClass{sym-q_hy_base}{q_{\mathrm{hy},\mathrm{base}}} &= \htmlClass{sym-gamma_fluid}{\gamma_{\mathrm{fluid}}} \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} \\
&= \htmlClass{sym-gamma_fluid}{9.81\ \mathrm{kN/m³}} \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} \\
&= 89.76\ \mathrm{kPa}
\end{aligned}\]
S.41
\[\begin{aligned}
\htmlClass{sym-p_wy}{p_{\mathrm{wy}}} &= \frac{\htmlClass{sym-S_ds}{S_{\mathrm{ds}}} \cdot \htmlClass{sym-I}{I} \cdot \htmlClass{sym-epsilon}{\epsilon} \cdot \htmlClass{sym-gamma_conc}{\gamma_{\mathrm{conc}}} \cdot \htmlClass{sym-t_wall}{t_{\mathrm{wall}}}}{\htmlClass{sym-R_wi}{R_{\mathrm{wi}}}} \\
&= \frac{\htmlClass{sym-S_ds}{1.06} \cdot \htmlClass{sym-I}{1.5} \cdot \htmlClass{sym-epsilon}{0.7041} \cdot \htmlClass{sym-gamma_conc}{23.56\ \mathrm{kN/m³}} \cdot \htmlClass{sym-t_wall}{650\ \mathrm{mm}}}{\htmlClass{sym-R_wi}{2}} \\
&= 8.572\ \mathrm{kPa}
\end{aligned}\]
S.42
\[\begin{aligned}
\htmlClass{sym-p_iy_base}{p_{\mathrm{iy},\mathrm{base}}} &= \frac{\left(4 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} - 6 \cdot \htmlClass{sym-h_i}{h_{i}}\right) \cdot \frac{\htmlClass{sym-P_i}{P_{i}}}{2}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}^{2} \cdot \htmlClass{sym-B}{B}} \\
&= \frac{\left(4 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} - 6 \cdot \htmlClass{sym-h_i}{3.431\ \mathrm{m}}\right) \cdot \frac{\htmlClass{sym-P_i}{705.6\ \mathrm{kN}}}{2}}{\left(\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right)^{2} \cdot \htmlClass{sym-B}{1\ \mathrm{m}}} \\
&= 67.48\ \mathrm{kPa}
\end{aligned}\]
S.43
\[\begin{aligned}
\htmlClass{sym-p_iy_top}{p_{\mathrm{iy},\mathrm{top}}} &= \frac{\left(6 \cdot \htmlClass{sym-h_i}{h_{i}} - 2 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}\right) \cdot \frac{\htmlClass{sym-P_i}{P_{i}}}{2}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}^{2} \cdot \htmlClass{sym-B}{B}} \\
&= \frac{\left(6 \cdot \htmlClass{sym-h_i}{3.431\ \mathrm{m}} - 2 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right) \cdot \frac{\htmlClass{sym-P_i}{705.6\ \mathrm{kN}}}{2}}{\left(\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right)^{2} \cdot \htmlClass{sym-B}{1\ \mathrm{m}}} \\
&= 9.64\ \mathrm{kPa}
\end{aligned}\]
S.44
\[\begin{aligned}
\htmlClass{sym-p_cy_base}{p_{\mathrm{cy},\mathrm{base}}} &= \frac{\left(4 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}} - 6 \cdot \htmlClass{sym-h_c}{h_{c}}\right) \cdot \frac{\htmlClass{sym-P_c}{P_{c}}}{2}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}^{2} \cdot \htmlClass{sym-B}{B}} \\
&= \frac{\left(4 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}} - 6 \cdot \htmlClass{sym-h_c}{5.357\ \mathrm{m}}\right) \cdot \frac{\htmlClass{sym-P_c}{118.5\ \mathrm{kN}}}{2}}{\left(\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right)^{2} \cdot \htmlClass{sym-B}{1\ \mathrm{m}}} \\
&= 3.154\ \mathrm{kPa}
\end{aligned}\]
S.45
\[\begin{aligned}
\htmlClass{sym-p_cy_top}{p_{\mathrm{cy},\mathrm{top}}} &= \frac{\left(6 \cdot \htmlClass{sym-h_c}{h_{c}} - 2 \cdot \htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}\right) \cdot \frac{\htmlClass{sym-P_c}{P_{c}}}{2}}{\htmlClass{sym-h_fluid}{h_{\mathrm{fluid}}}^{2} \cdot \htmlClass{sym-B}{B}} \\
&= \frac{\left(6 \cdot \htmlClass{sym-h_c}{5.357\ \mathrm{m}} - 2 \cdot \htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right) \cdot \frac{\htmlClass{sym-P_c}{118.5\ \mathrm{kN}}}{2}}{\left(\htmlClass{sym-h_fluid}{9.15\ \mathrm{m}}\right)^{2} \cdot \htmlClass{sym-B}{1\ \mathrm{m}}} \\
&= 9.801\ \mathrm{kPa}
\end{aligned}\]
Questions
Why does R_wi reduce P_i but not P_c?
Impulsive mass moves with the walls at a short period, and ACI 350.3 lets that response be reduced by R_wi for the ductility and damping of the structure. Convective mass sloshes near the free surface at a long period, T_c, and is taken as an elastic response with R_wc = 1.0, so P_c is not reduced at all.
Why are P_i and P_c combined as a square root of the sum of squares in V?
The two respond at very different periods, so their peaks are not expected at the same instant. V adds the impulsive force, the walls and the roof directly, P_i + P_w + P_r, because they move together, then combines that with P_c and the earth pressure P_eg by square root of the sum of squares. M_s does the same with the moments.
Does the check on h_freeboard allow for the sloshing wave?
No. freeboard_adequate only checks that the fluid surface is below the top of the wall. ACI 350.3 gives the maximum sloshing height of a rectangular tank as d_max = (L/2) C_c I; compare that with h_freeboard yourself where the wave must not reach the roof or the wall top.
Why is B a unit width of 1 m?
The sheet works a one-metre strip of the tank across the direction of shaking, so W_fluid, W_i, W_c and every force and moment are per metre of tank width. Multiply by the tank's width for the whole tank, and note that W_walls counts only the two walls perpendicular to the shaking.