(Solved): 2. Consider the dam and the sheet pile represented below. The structure is built on top of a soil l ...

2. Consider the dam and the sheet pile represented below. The structure is built on top of a soil layer. The permeability of the soil is $\mathrm{k}=2.5\times 10^{?4}\mathrm{cm}/\mathrm{s}$, and the saturated unit weight is ${?}_{ut}=18\mathrm{kN}/{\mathrm{m}}^3$. The flow net of this cross-section is presented in the figure. Assuming that the dam, the sheet pile, and the rock contact below the soil are impermeable, answer the following questions: a) Calculate the flow rate of water per unit width through the permeable layer, $\mathrm{q}$ (in units ${\mathrm{of}}_{\mathrm{cm}}3/\mathrm{s}$ per unit width). b) Plot the uplift pressure distribution below the dam. What is the resultant force due to uplift pressure (in units of $\mathrm{kN}$ per unit width)? Suggestion: consider 6 points distributed as indicated in the figure (P1 at $x=0\mathrm{m},\mathrm{P}2$ at $x=10\mathrm{m},\mathrm{P}3$ at $x=19.9\mathrm{m}$ (right before the sheet pile), $P4x=20.1\mathrm{m}$ (right after the sheet pile), P5 at $x=30\mathrm{m}$, and $\mathrm{P}6\mathrm{x}=40\mathrm{m}$ ). (Assume; ${?}_w=9.81\mathrm{kN}/{\mathrm{m}}^3$ ). Note: You do not need to identify the location of the resultant force. Make sure to draw your pressure diagram to scale. Use trapezoidal rule to get the area of the pressure distribution. c) Calculate the factor of safety against piping. Note: the element closest to the structure on the downstream side of the dam is $5\mathrm{m}$ in length vertically. d) Is the factor of safety high enough to be considered safe from piping? e) Assuming that the downstream water level is constant, calculate the maximum upstream water level if we want to avoid erosion at the downstream toe of the dam.