a$.$ Distinguish and explain the difference between the $($i$)$ secant, $($ii$)$ tangent and $($iii$)$ small strain

shear stiffness $(2$ marks$).$

b$.$ Select and calculate the most appropriate measure of shear stiffnessfor the stress$-$strain loop

shown in Fig$\mathrm{.}1.(2$ marks$).$

c$.$ Identify the main soil properties that influence the value of small strain stiffness, Gmax, and

how each one of them influences that value. $(2$ marks$).$

d$.$ Determine the small strain shear stiffness $($Gmax$)$ by means of using an empirical correlation

of your choice and explain the variables used for its calculation $(2$ marks$).$

e$.$ Calculate the shear modulus reduction curve $($G$/$Gmax$)$ using the Menq $(2003)$ and Darendelli

$(2001)$ equations, assuming plasticity of PI $=10\%,$ OCR $=1$ and a uniformity coefficient of Cu $=$

$1\mathrm{.}75(5$ marks$).$

f$.$ What is the damping ratio and why is such an important dynamic soil property? Would you

consider that there is energy dissipation in Figure $1?$ How many forms of damping are there?

$(3$ marks$).$

g$.$ Using the results obtained in section e$,$ calculate the damping ratio vs shear deformation

according to the equation provided by Darendeli $(2001).$ Assume the following: Number of

cycles N $=10,$ frequency of loading f $=1$Hz$,$ plasticity PI $=20\%$ and OCR $=1.$ Plot the damping

ratio calculation according to Masing and the adjusted expression provided by Darendeli

$(2001)(6$ marks$).$

h$.$ Plot the shear modulus reduction curve and damping ratio with respect to the shear

deformation. Using previous reference, explain the change in material behaviour with the

increase in shear deformation from very small to large deformations $(3$ marks$).$

Total