Question 2.29: A sloping plane with a width of 8ft and a height of 10 ft is...

(a) The magnitude of the hydrostatic resultant force acting on the sloping plane is determined by applying Equation 2.171 F= \int{pdA} =\int_{0}^{L}{(\gamma h(y)+ p_{g})(wdy)} =\int_{0}^{L}{(\gamma (y_{o}+ y)\sin \alpha +p_{g} )(wdy)} , integrating over the length of the sloping planar gate using the top of the plane as a reference point for integration as follows:

W:= 8 ft                   L: 10 ft                 A:=w.L = 80 ft^{2}                 h_{o} := 6 ft

\alpha := 40 deg             y_{o}: = \frac{ h_{o}}{\sin (\alpha )} =9.334 ft           h(y):= (y_{o}+y) \sin (\alpha )

p_{g} :=14 \frac{Ib}{in^{2} } \frac{(12in)^{2} }{(1in)^{2}} =2.016 \times 10^{3} \frac{Ib}{ft^{2} }                 \gamma :=62.417 \frac{Ib}{ft^{3} }
F:= \int_{0}^{L}{(\gamma .h(y)+ p_{g}).wdy} =207288 Ib

Alternatively, the magnitude of the hydrostatic force acting on the sloping plane may be determined by applying Equation 2.172 F= (p_{ca}+p_{g})A=(\gamma h_{ca}+p_{g})A=(\gamma ( y_{o} + y_{ca})\sin \alpha + p_{g})A= (\gamma ( y_{o} + L/2)\sin \alpha + p_{g})(wL) as follows:

y_{ca}:= \frac{L}{2} =5 ft                                     h(ca):= y_{o}+ y _{ca} \sin (\alpha )= 9.214 ft
F:= (\gamma .h(ca)+ p_{g} ).A=2.073 \times 10^{5} Ib

(b) The location of the resultant hydrostatic force acting on the sloping plane is determined by applying Equation 2.174 y_{f} = \frac{M}{F} as follows:

y_{F}:= \frac{\int_{0}^{L}{y.(\gamma .h(y)+ p_{g}).wdy} }{\int_{0}^{L}{(\gamma .h(y) + p_{g} ).wdy} } =5.129 ft               h(F):= y_{o}+ y_{F} \sin (\alpha ) =9.297 ft

Alternatively, the location of the resultant force acting determined by applying Equation 2.177 z_{F} = \frac{\gamma (I_{x-ca}+ A z^{2}_{ca} )\sin \alpha + p_{g} z_{ca}A}{(\gamma ( z_{ca} \sin \alpha)+ p_{g})A}= \frac{\gamma (wL^{3}/12 +wl (y_{o}+L/2 )^{2})\sin \alpha + p_{g}(y_{o}+L/2 )(wL)}{(\gamma (y_{o}+L/2 )\sin \alpha+ p_{g} )(wL)} as follows:

I_{xca} := \frac{w. L^{3} }{12} =666.667 ft^{4}                           z(ca):= ( y_{o}+ y_{ca} )=14.334 ft
z(F):= \frac{\gamma .(I_{xca} +A. z^{2}_{ca} ) \sin (\alpha ) + p_{g}. z(ca).A}{[\gamma (z(ca) \sin (\alpha )) + p_{g}].A} = 14.463 ft
y(F):= z(F)- y_{o}=5.129 ft                       h(F):= z(F) \sin (\alpha )= 9.297 ft