Problem $1$ GEOTECHNICAL ENGINEERING $-$ SHEAR STRENGTH

a$)$ Direct shear tests were performed on a dry, sandy soil. Each circular test specimen had a $3"$ diameter. Test results for test specimens $1-4$ are presented in the following table. Complete the following task:

$\backslash$table$[[$Test No$.,$Normal Force $($lb$),\backslash$table$[[$Shear Force $($lb$)],[($a$)$ Failure$]]],[1,23,13],[2,32,18\mathrm{.}3],[3,70,37],[4,95,55]]$

For each test specimen above, complete the normal and shear stress calculations necessary to generate the Mohr Coulomb failure criterion for this soil. Stresses should be calculated in $\text{'}$psf$\text{'}.$ Full supporting calculations are required for all four test specimens $$ You will lose points for any specimens that do not have supporting work.

Figure to be completed by pencil on engineering or graph paper: Generate the $x$ and $y$ axis for your failure curve. Both axes should use the same scale and extend from $0-2000$ psf$.$ Make it as large as possible on your paper, generate tick marks every $200psf,$ and include labels and units. Use straight edges and prepare your work professionally.

Display all $4$ data points and plot the failure curve.

Calculate the effective friction angle for the soil using geometry. Check your work with a protractor.

Label both shear strength parameters and their values directly on the failure curve in the appropriate locations.

b$)$ Now assume the sandy soil samples were acquired from a sampler retrieved at a $12ft$ depth in a homogeneous subsurface. The water table was located $4ft$ from the ground surface. $=115pcf$ and ${}_{\text{s}\text{a}\text{t}}=120$ pcf$.$ Draw a sketch of the subsurface, label the POI, and calculate the shear strength of the soil at this location based on the lab data generated from part $($a$).$

c$)$ If the shear stress at the same POI was measured and read $425psf$ in the field, calculate the FS against a shear failure at this POI.