$[75$ pts$]$ You are a staff engineer for a local geotechnical engineering firm. As part of the geotechnical exploration for a project, several subsurface tests have been conducted. You are given the results of one traditional soil boring with Standard Penetration Testing $($SPT$).$ The results of this boring, labeled as B$-2,$ are presented on page $4$ of this assignment. Based on testing of collected soil samples, the encountered soils have the unit weights listed in Table $2.$ In addition, soil samples were taken from the first ML layer $($its depth ranging from $8$ to $\backslash (\backslash$mathbf$\{12\}\backslash$mathbf$\{$f t$\}\backslash )),$ on which CU triaxial tests were conducted under three different confining pressure levels. The CU triaxial testing results are summarized in Table $3.$

The figure below presents the general cross$-$section of two planned footings at your project site. The project structural engineer has given you design loads of $90$ kips for the columns and $7$ kips per linear foot for the wall loads. Refer to Boring B$-2$ on page $4$ for the soil profile at both footing locations.

Table $2.$ Summary of Soil Unit Weights from Boring B$-2.$

NOTE: Assume the asphalt and base course layers are removed and replaced with material identical to that underneath them.

Table $3.$ Summary of CU triaxial testing results of ML soil samples

$2$a$)[15$ pts$]$ Using the provided "Mohr circle for Triaxial test.xlsx$"$ to plot effective stress Mohr's circles of the three CU tests; $($Hint: you have to derive the effective Minor principal stress at failure and the effective Major principal stress at failure for each of the CU tests first; then you only need to change the values of the cells highlighted in Green in the excel spreadsheet: D$16\backslash$& D $17$ for CU test $1$; H $\backslash (16\backslash$& $\backslash )$ H$17$ for CU test $2$; and L $16\backslash$& L $17$ for CU test $\backslash (3)\backslash )$;

$2$b$)[14$ pts$]$ Manually draw a straight line that is the common tangent to all the three effective stress Mohr's circles; then you could either manually derive the mathematic equation of the common tangent line $($i$.$e$.,$ M$-$C shear strength line of the ML soil$)$; or identify two points on this line and input their coordinates in $($B$229,$ C $229)$ and $($B$230,$ C $230)($also highlighted in green$)$ of the Excel spreadsheet, followed by finding the equation by "Format Trendline". Based on your M$-$C shear strength line, determine the effective shear strength parameters of the ML soil $(\backslash ($ c$^\{\backslash$prime$\}\backslash )$ and $\backslash (\backslash$phi$^\{\backslash$prime$\}\backslash ))$; $2$c$)[14$ pts$]$ For the mid$-$point of the first ML layer, calculate its initial vertical effective stress and horizontal effective stress $($For the vertical effective stress, you can use your results from HW $4)($hint: using the derived effective frictional angle from step $\backslash (2$ b $\backslash )$ to calculate effective horizontal stress, with $\backslash (\backslash$sigma$\_\{$h$\}^\{\backslash$prime$\}=$K$\_\{0\}*\backslash$sigma$\_\{$v$\}^\{\backslash$prime$\}\backslash ),$ with $\backslash ($ K$\_\{0\}=1-\backslash$sin $\backslash$emptyset$^\{\backslash$prime$\}\backslash ))$ and draw the Mohr's circle corresponding to the initial effective stress state $($i$.$e$.,$ prior to the construction of any footing$)$;

$2$d$)[16$ pts$]$ Based on the induced vertical effective stresses from your HW$4,$ determine the induced vertical effective stress at the mid$-$point of the first ML layer under the center of each of the footings $($note that you need to determine the induced vertical effective stress for both of the footing types; assuming the footing load is applied so slow that no excess pore water pressure is induced$)$;

$2$e$)[4$ pts$]$ Determine the final vertical effective stress and final horizontal effective stress at the mid$-$point of the first ML layer, after the construction of either the column footing or the wall footing;

$2$f$)[8$ pts$]$ Draw the Mohr's circle for the state of the final effective stresses of the mid$-$point of the first ML layer after the construction of the footings;

$\backslash (2\backslash$mathrm$\{$~g$\})\backslash$quad$[4\backslash$mathrm$\{$pts$\}]\backslash )$ Based on its shear strength criterion determined from step $2$b$),$ please determine whether shear failure occurs or not at the mid$-$point of the first ML layer, after the construction of the column footing and the wall footing.