$3.(40$ points$)$ A four$-$story building is to be constructed on a square $(\backslash (40\backslash$times $40\backslash$mathrm$\{$ft$\}\backslash ))$ concrete mat foundation underlying the entire building, as shown in Figure $1.$ The design weight of each story is $\backslash (1,300\backslash$mathrm$\{$kips$\}\backslash ),$ and the thickness of the mat foundation is $4$ ft $.$ The project requires excavation of $15$ ft of the native sandy soil to reach the foundation grade $($Elev$.\backslash (=900\backslash$mathrm$\{$ft$\}\backslash ),$ followed by construction of the mat foundation and building. During the design phase of the project, an undisturbed sample of clay $\backslash (\backslash$left$($e$\_\{$o$\}=0\mathrm{.}91\backslash$right$)\backslash )$ was collected from the midpoint of the clay layer $($Point A$)$ and subjected to a consolidation test, yielding the $\backslash ($ e $\backslash )-$log $\backslash (\backslash$sigma$^\{\backslash$prime$\}\backslash )$ curve shown in Figure $2.$

nicitide $1.$ Pactannetruntion profile view.

FIGURE $2$: Consolidation test results for clay sample.

Estimate the ultimate settlement $($in$)$ directly under the center of the building due to primary consolidation of the clay layer, based on the net increase in vertical stress at Point A $($accounting for both the excavation and the construction$).$ Assume a concrete unit weight of $150$ pcf for the mat foundation.

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