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Roadmap 3 - 3.1.E Cylindrical Thin-Walled pressure vessels...

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125 3.1.E Cylindrical Thin-Walled pressure vessels
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126
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127 3.2.E Cylindrical Thin-Walled pressure vessels: Example Problems Problem 1: The walls of a cylindrical pressure vessel of radius R = 1m, and wall thickness t =2cm are made of an isotropic linear elastic material with modulus E =10.0 GPa, Poisson’s ratio ν =0.2, and coefficient of thermal expansion α =1 x10 -6 /ºC. When a hot fluid at pressure p is introduced in the vessel, the temperature of the wall increases by T=100 ºC, and the radius of the vessel increases by δ R =1mm. What is the value of the pressure p ?
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128 Problem 2 : A thin walled cylindrical tank has outer radius R , wall thickness t and length L . The heads of the tank are flat and rigid, as shown, and the tank sits in a rigid well of inner radius R . No friction acts between the tank and the well. The walls of the tank are isotropic, linear elastic, with elastic constants E and ν . The tank is now subjected to an internal pressure p and the top end of the tank rises in the well by a distance δ . obtain an expression for δ in terms of the geometry of the problem, the material properties, and the applied internal pressure. p δ R R t L L
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129 Problem 3 Part 1 A steel cylindrical thin-wall vessel contains a volatile fuel under pressure. For safety reasons, the pressure inside the tank should not exceed a maximum value P max . A strain gage records the circumferential strain in the tank and transmits this information to the control room (the gage reads zero strain for zero pressure). At what value of strain, ( ε θθ ) max should the operators take action and open the relief valve on the tank (to reduce the pressure)? Express ( ε θθ ) max in terms of P max , the geometry of the vessel (radius R , thickness t ), and the material properties of the steel (Young’s modulus E , Poisson’s ratio ν ). Part 2 In part 1 we neglected the effects of the constraints on deformation imposed by the supports. The tank is attached to rigid supports that prevent any elongation along the axis x of the cylinder (but do not impose any displacement constraint in the circumferential direction at the location of the gage). If you have to account for the effects of these supports does your relation for ( ε θθ ) max change? If so, what is the new expression? Circumferential gage (measures ε θθ ) x
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130
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131 Problem 4: A cylindrical tank of radius R, length L, and wall thickness t is made of an isotropic linear elastic material with modulus E, Poisson’s ratio ν , and coefficient of thermal expansion α . At temperature T 0 , with no pressure in the tank, the tank fits without stress between a rigid wall W B and a rigid plate C. A spring of stiffness K fits exactly between the rigid plate C and a second rigid wall W A . The plate can only move in the horizontal direction, as indicated in the figure. The tank is free to expand radially.
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