A structural member under loading has a uniform state of plane stress which in usual notations is given by σ_x = 3P, σ_y = -2P and τ_xy = √2 P, where P > 0. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of P at which yielding starts (according to the maximum distortion energy theory) is

A linear elastic structure under plane stress conditions is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are as shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress σ_xx is ________

A shaft of length L is made of two materials, one in the inner core and the other in the outer rim, and the two are perfectly joined together (no slip at the interface) along the entire length of the shaft. The diameter of the inner core is d; and the external diameter of the rim is d₀, as shown in the figure. The modulus of rigidity of the core and rim materials are G_i and G₀, respectively. It is given that d₀ = 2d_i and G_i = 3G₀. When the shaft is twisted by application of a torque along the shaft axis, the maximum shear stress developed in the outer rim and the inner core turn out to be τ₀ and τ_i, respectively. All the deformations are in the elastic range and stress-strain relations are linear. Then the ratio τ_i/τ₀ is (round off to 2 decimal places).

For a dynamical system governed by the equation,

$\ddot{x}(t)+2 ζ \omega_n \dot{x}(t)+\omega_n^2x(t)=0$

the damping ratio ζ is equal to $\frac{1}{2\pi}\log_e2$. The displacement x of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be 4 mm. Neglecting higher powers (> 1) of the damping ratio, the displacement at the next peak in the positive direction will be _______ mm (in integer).