Stresses acting on an infinitesimal soil element are shown in the figure (with $$\sigma$$_z > $$\sigma$$_x). The major and minor principal stresses are $$\sigma$$₁ and $$\sigma$$₃, respectively. Considering the compressive stresses as positive, which one of the following expressions correctly represents the angle between the major principal stress plane and the horizontal plane?

For a linear elastic and isotropic material, the correct relationship among Young's modulus of elasticity (E), Poisson's ratio (v), and shear modulus (G) is

The linearly elastic planar structure shown in the figure is acted upon by two vertical concentrated forces. The horizontal beams UV and WX are connected with the help of the vertical linear spring with spring constant k = 20 kN/m. The fixed supports are provided at U and X. It is given that flexural rigidity EI = 10⁵ kN-m², P = 100 kN, and a = 5 m. Force Q is applied at the center of beam WX such that the force in the spring VW becomes zero.

The magnitude of force Q (in kN) is __________. (round off to the nearest integer)

For the square steel beam cross-section shown in the figure, the shape factor about Z - Z axis is S and the plastic moment capacity is M_p. Consider yield stress f_y = 250 MPa and a = 100 mm.

The values of S and M_p (rounded off to one decimal place) are