PROBLEM $1(30$ points$)$

Following are the results of four direct shear tests performed on a sample of a silty$-$clayey$-$sand collected from the middle of a $18$ ft layer extending from the ground surface. The soil unit weight is $19k\frac{N}{m^3}.$ The specimen diameter is $63\mathrm{.}5$ mm $.$ The raw data from the tests are as follows:

$\backslash$table$[[$Test No$.,$Normal Force $($N$),$Shear Force $($N$)],[1,379,420],[2,726,474],[3,1074,531],[4,1422,588]]$

a$.$ During the test, what quantitative measurement indicated that the specimen has failed?

b$.$ Find the shear strength parameters: cohesion and angle of internal friction

c$.$ What is the shear strength at the bottom of the layer?

d$.$ Draw the Mohr's circle for the third direct shear test

e$.$ What are the stress conditions at failure on the failure plane for the third test?

f$.$ Draw the above stress conditions on a soil element ABCD

g$.$ What are the principal stresses corresponding to the third test?

h$.$ What are the inclinations of the planes where the principal stresses act?

i$.$ Draw the direct$-$shear specimen and show the principal planes and principal stresses.

j$.$ For the third test, what is the inclination of the plane where the maximum shear stress acts? What is the normal stress on that plane?

k$.$ Which one is bigger: failure shear stress or maximum shear stress? WHY?

Assume that the results of Part $($b$)$ are valid, i$.$e$.$ the Mohr$-$Coulomb failure envelope is correct. If you now perform an unconfined compression test on the "same" soil, what will be the compressive strength at failure? Determine with a Mohr's Circle.