An initially stress-free massless elastic beam of length $ L $ and circular cross-section with diameter $ d(d\ll L) $ is held fixed between two walls as shown. The beam material has Young's moduls $ E $ and coefficeint of thermal expansion $\style{font-family:'Times New Roman'}{\mathrm\alpha}$.
If the beam is slowly and uniformly heated, the tempreature rise required to cause the beam to buckle is prportional to
A steel bar is held by two fixed supports as shown in the figure and is subjected to an increase of temperature $\triangle T=100\;^\circ\mathrm C$. If the coefficient of thermal expansion and Young’s modulus of elasticity of steel are $11\times {10}^{-6}/\xb0\mathrm{C}$ and 200 GPa, respectively, the magnitude of thermal stress (in MPa) induced in the bar is__________
A circular metallic rod of length 250 mm is placed between two rigid immovable walls as shown in the figure. The rod is in perfect contact with the wall on the left side and there is a gap of 0.2 mm between the rod and the wall on the right side. If the temperature of the rod is increased by $200^o\mathrm C$, the axial stress developed in the rod is __________ MPa.
Young’s modulus of the material of the rod is 200 GPa and the coefficient of thermal expansion is $10^{−5} $ per ${}^\circ\mathrm C$.
A circular rod of length ‘L’ and area of cross-section ‘A’ has a modulus of elasticity ‘E’ and coefficient of thermal expansion ‘α’. One end of the rod is fixed and other end is free. If the temperature of the rod is increased by ΔT, then
A 200 mm long, stress free rod at room temperature is held between two immovable rigid walls. The temperature of the rod is uniformly raised by 250°C. If the Young’s modulus and coefficient of thermal expansion are 200 GPa and 1×10^{−5} /°C, respectively, the magnitude of the longitudinal stress (in MPa) developed in the rod is _______
A steel cube, with all faces free to deform, has Young’s modulus, E, Poisson’s ratio, ν, and coefficient of thermal expansion, α. The pressure (hydrostatic stress) developed within the cube, when it is subjected to a uniform increase in temperature, ΔT, is given by
A solid steel cube constrained on all six faces is heated so that the temperature rises uniformly by $\Delta $T. If the thermal coefficient of the material is $\alpha $, Young’s modulus is E and the Poisson’s ratio is $\nu $ , the thermal stress developed in the cube due to heating is
A steel rod of length L and diameter D, fixed at both ends, is uniformly heated to a temperature rise of $\mathrm{\Delta}$T. The Young’s modulus is E and the co efficient of linear expansion is $\alpha $. The thermal stress in the rod is