When a simply-supported elastic beam of span L and flexural rigidity EI (E is the modulus of elasticity and I is the moment of inertia of the section) is loaded with a uniformly distributed load w per unit length, the deflection at the mid-span is

$\rm \Delta_0=\frac{5}{384}\frac{wL^4}{El}$

If the load on one half of the span is now removed, the mid-span deflection _______.

In the frame shown in the figure (not to scale), all four members (AB, BC, CD, and AD) have the same length and same constant flexural rigidity. All the joints A, B, C, and D are rigid joints. The midpoints of AB, BC, CD, and AD, are denoted by E, F, G, and H, respectively. The frame is in unstable equilibrium under the shown forces of magnitude 𝑃 acting at E and G. Which of the following statements is/are TRUE?

In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length of PQ, QR, and RP are 4, 3, and 5 units, respectively. The principal stresses are ________. (round off to one decimal place)

A 5 cm long metal rod AB was initially at a uniform temperature of T0°C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\rm\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$, where D is the thermal diffusivity of the metal, given as 1.0 cm2/s.

The temperature distribution in the rod is obtained as

$\rm T(x,t)=\Sigma_{n=1,3,5...}^{\infty}C_n\sin\frac{n\pi x}{5}e^{-\beta n^2t}$,

where x is in cm measured from A to B with 𝑥 = 0 at A, t is in s, 𝐶𝑛 are constants in °C, T is in °C, and β is in s−1 .

The value of β (in 𝑠−1 , rounded off to three decimal places) is ________.