Consider two linearly elastic rods HI and IJ. Each of length b, as shown in the figure. The rods are co-linear, and confined between two fixed supports at if and J. Both the rods are initially stress free. The coefficient of linear thermal expansion is a for both the rods. The temperature of the rod IJ is raised by $$\Delta$$T whereas the temperature of rod HI remains unchanged. An external horizontal force P is now applied at node I. It is given that a = 10^$$-$$6 $$^\circ$$C^$$-$$1, $$\Delta$$T = 50$$^\circ$$C, b = 2m, AE = 10⁶N. The axial rigidities of the rods HI and U are 2 AE and AE, respectively.

To make the axial force in rod HI equal to zero, the value of the external force P (in N) is _________. (rounded off to the nearest integer).

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)