Internal Erosion Suite

Appendix B - Summary of Hydraulic Shear Stress Equations

Table: Hydraulic shear stress in pipe or crack.

Transverse Flaw	Tailwater above Base of Pipe/Crack $(H_{2}>0)$	Tailwater below Base of Pipe/Crack $(H_{2}=0)$	Approximation for No Tailwater and Small Flaw
Cylindrical Pipe	$\tau = \gamma_{w}\left(\frac{H_{1}-H_{2}}{L}\right)\frac{D}{4} = \frac{\gamma_{w}iD}{4}$	$\tau = \gamma_{w}\left(\frac{H_{1}-H_{2}}{L}\right)\frac{D}{4} = \frac{\gamma_{w}iD}{4}$	$\tau = \gamma_{w}\left(\frac{H_{1}-H_{2}}{L}\right)\frac{D}{4} = \frac{\gamma_{w}iD}{4}$
Horizontal Crack	$\tau = \gamma_{w}\left(\frac{H_{1}-H_{2}}{L}\right) \frac{WX}{2W+2X} = \frac{\gamma_{w}iWX}{(2W+2X)}$	$\tau = \gamma_{w}\left(\frac{H_{1}-H_{2}}{L}\right) \frac{WX}{2W+2X} = \frac{\gamma_{w}iWX}{(2W+2X)}$	$\tau \approx \frac{\gamma_{w}iW}{2}, W<<X$
Vertical Rectangular Crack	$\tau = \frac{\gamma_{w}W(H_{1}^{2}-H_{2}^2)}{2L(H_{1}+H_{2}+W)}$	$\tau = \frac{\gamma_{w}H_{1}^{2}W}{2(H_{1}+W)L}$	$\tau \approx \frac{\gamma_{w}WH_{1}}{2L} = \frac{\gamma_{w}iW}{2}, W<<H_{1}$
Vertical Triangular Crack	$\tau = \frac{\gamma_{w}W}{6DL}\frac{(H_{1}^{3}-H_{2}^3)}{(H_{1}+H_2)\sqrt{1+\frac{W^2}{4D^2}}}$	$\tau = \frac{\gamma_{w}W}{6DL}\frac{(H_{1}^{2})}{\sqrt{1+\frac{W^2}{4D^2}}}$	$\tau \approx \frac{\gamma_{w}WH_{1}^{2}}{6DL} = \frac{\gamma_{w}iWH_{1}}{6D}, W<<D$

Table: Critical pipe diameter or crack width.

Transverse Flaw	Tailwater above Base of Pipe/Crack $(H_{2}>0)$	Tailwater below Base of Pipe/Crack $(H_{2}=0)$	Approximation for No Tailwater and Small W
Cylindrical Pipe	$D_{cr} = \frac{4\tau_{c}}{\gamma_{w}i}$	$D_{cr} = \frac{4\tau_c}{\gamma_{w}i}$	$D_{cr} = \frac{4\tau_c}{\gamma_{w}i}$
Horizontal Crack	$W_{cr} = \frac{-2L\tau_{c}X}{2L\tau_{c}-\gamma_{w}(H_{1}-H_{2})X}$	$W_{cr} = \frac{-2L\tau_{c}X}{2L\tau_{c}-\gamma_{w}H_{1}X}$	$W_{cr} \approx \frac{2\tau_c}{\gamma_{w}i}, W<<X$
Vertical Rectangular Crack	$W_{cr} = \frac{2L\tau_{c}(H_{1}+H_2)}{\gamma_{w}(H_{1}^{2}-H_{2}^{2})-2L\tau_c}$	$W_{cr} = \frac{2L\tau_{c}H_{1}}{\gamma_{w}H_{1}^{2}-2L\tau_c}$	$W_{cr} \approx \frac{2\tau_c}{\gamma_{w}i}, W<<H_{1}$
Vertical Triangular Crack	$W_{cr} = \frac{\tau_{c}(H_{1}+H_{2})}{\sqrt{\frac{\gamma_{w}^{2}(H_{1}^{3} - H_{2}^3)^2}{36D^{2}L^2} - \frac{\tau_{c}^{2}(H_{1}+H_{2})^2}{4D^2}}}$	$W_{cr} = \frac{\tau_{c}H_1}{\sqrt{\frac{\gamma_{w}^{2}H_{1}^6}{36D^{2}L^2} - \frac{\tau_{c}^{2}H_{1}^2}{4D^2}}}$	$W_{cr} \approx \frac{6D\tau_c}{\gamma_{w}iH_1}, W<<D$