$1.$ The equation for effective stress is shown below. Can effective stress become negative $($i$.$e$.$ can pore pressure exceed total stress$)?$ If yes, explain what may happen in a soil if effective stress becomes negative and provide an example of how this can happen. If no$,$ explain why this cannot happen. Hint: consult your textbook.

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$\backslash$sigma$^\{\backslash$prime$\}=\backslash$sigma$\_\{$t$\}-$u

$\backslash ]2.$ A series of soil strata is shown below.

a$.$ Plot the total stress, effective stress, and pore pressure for the geostatic case $($no fill added$).$

b$.$ Plot the total stress, effective stress, and pore pressure if a fill with a unit weight of $\backslash (19\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{\backslash$wedge$\}3\backslash )$ and a height of $3$ meters is uniformly placed on top of the site. Assume excess pore pressure dissipates instantly.

c$.$ If the coefficient of lateral earth pressure at point A is $0\mathrm{.}65,$ determine the horizontal effective stress for the conditions in part $($a$)$ and $($b$).$

d$.$ Draw Mohr's circle for point A for the conditions in part $($a$)$ and $($b$).7.$ A square foundation has a side length of $6$ feet and bears a load of $\backslash (40,000\backslash$mathrm$\{$lbs$\}\backslash ).$ It is placed directly on the ground surface $($see the figure below$).$ The groundwater table is coincident with the ground surface as shown.

a$.$ Determine the induced stress from the foundation using the $\backslash (2$: $1\backslash )$ approximation at depths of $3$ feet, $6$ feet, $9$ feet, and $12$ feet.

b$.$ For the depths in part $($a$),$ determine the total stress, effective stress, and pore pressure before and a long time after the foundation is placed. $($This process is the basis of settlement calculations$).6.$ A rectangular foundation is $8$ feet wide by $12$ feet long. It bears $\backslash (115,000\backslash$mathrm$\{$lbs$\}\backslash ).$ Determine the stress increase at the points shown below. All points are located $4$ feet below the foundation. $\backslash ((\backslash$mathrm$\{$x$\},\backslash$mathrm$\{$y$\})\backslash )$ coordinate pairs are shown on the figure.

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