Problem $1.$ Calculate the following using Figure $8\mathrm{.}7$ on pg$.311$ of your textbook: a$.$ Largest hydraulic gradient at the exit. Factor of safety against piping. Is piping an issue? b$.$ Total and effective vertical stress at Points A and B$.$ Assume that the total unit weight of the clean sand above the water table is $110$ pcf and the total unit weight of the clean sand below the water table is $114$ pcf$.$ c$.$ As you have found, the effective stress at B is negative which is impossible. In these cases and when the soil is a sand like in this particular problem, the effective stress is actually zero and the soil is under what is called a quick condition. This condition is caused by the fact that water is flowing upward with a large hydraulic gradient. This results in an upward heave of the soil within this zone and would ultimately collapse the excavation. This type of failure is usually avoided by keeping the effective stress within this zone well above zero either by extending the sheet pile walls or by covering the exit zone with a layer of gravel or other highly pervious fill $($often called a gravel blanket$).$ If the gravel you are sourcing has a unit weight of $120\mathrm{.}2$ pcf$,$ calculate the thickness of the layer of gravel you would need to place in the excavation to bring the effective stress at B up to the design value of $520$ psf$.$ Remember that the water level $($i$.$e$.$ phreatic surface$)$ will now be on the top of the gravel layer, but you do not need to extend the flow net into this layer or consider seepage forces because of the high permeability.