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

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Unformatted text preview: 3.1.E Cylindrical Thin-Walled pressure vessels 125 126 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 δR=1mm. What is the value of the pressure p? 127 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. R R t L δ p L 128 Problem 3 Part 1 A steel cylindrical thin-wall vessel contains a Circumferential volatile fuel under pressure. For safety reasons, gage (measures εθθ) the pressure inside the tank should not exceed a maximum value Pmax. 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, (εθθ)max should the operators take action and open the relief valve on the tank (to reduce the pressure)? Express (εθθ)max in terms of Pmax, 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). x If you have to account for the effects of these supports does your relation for (εθθ)max change? If so, what is the new expression? 129 130 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 T0, with no pressure in the tank, the tank fits without stress between a rigid wall WB and a rigid plate C. A spring of stiffness K fits exactly between the rigid plate C and a second rigid wall WA. The plate can only move in the horizontal direction, as indicated in the figure. The tank is free to expand radially. L C spring WA tank K 2R WB C WA uC WB 1) The internal pressure in the tank is brought to a value p. Obtain expressions for the resulting displacement of the plate uC, and for the change in tank wall thickness δt. What is the radial strain in the tank wall? What is the circumferential strain in the tank wall? 2) The temperature of the wall of the pressurized tank is now increased by ∆T. What are the new values for uP and δt? 131 132 133 134 135 136 137 Problem 4 (20 Pts) Consider two thin-walled pressure vessels between two frictionless fixed walls at A and C, separated by a movable frictionless rigid plate at B, as shown in the figure. Both vessels have an inner radius, R, and wall thickness, t. Both are made out of the same material with a Young’s modulus, E, and Poisson’s ratio, ν. The left vessel has length 2L, and the right vessel has length L as indicated. Initially, the pressure in both vessels is zero and there is no stress in the walls of the pressure vessels. If the left vessel is pressurized to a pressure p, while the pressure in the right vessel is kept to zero, the rigid plate at B will displace by uBx. Find an expression for uBx, the distance the plate moves, in terms of the pressure p, and the known quantities, R, L, t, E, and ν. 138 139 3.3.E Cylindrical Thin-Walled pressure vessels: Conceptual Questions For each of the following, cross the false (wrong) statements, circle the true (right) ones: E-1) The cylindrical walls of a vessel of outer radius R and wall thickness t (t<<R) are made of an isotropic linear elastic material with modulus E and Poisson’s ratio ν. The end caps are rigid. If the vessel is empty (zero gage pressure), the vessel fits precisely (without any stress) inside a rigid frictionless well of inner diameter 2R. With the vessel inside the rigid, frictionless well, a pressurized fluid is introduced in the vessel, so that the internal gage pressure in the vessel is now p. For these conditions, in the cylindrical walls of the vessel we will have: a) No change in outer radius and therefore zero radial strain. pR . t pR c) σ xx = . 2t b) σ θθ = d) εθθ = –ν εxx. e) σθθ = ν σxx. x ρ θ 2R 2R 2R p 140 E-2) A rigid block of weight W rests over a soda can. The material of the can is isotropic linear elastic with properties E, ν (0.5>ν > 0), α. The gage pressure inside the can is p. Under these conditions, the height of the can is H, its radius is R, and the wall thickness is t<<R. (1) The weight is removed but the can is still closed, with internal pressure p: (a) the axial stress changes but the circumferential stress stays the same (b) the height of the can changes but its radius stays identical (2) Then the can is opened, (so now the inside gage pressure is p=0), the height of the can becomes H2 and its radius becomes R2: (c) It is possible, for a special combination of W and p, to have H =H2 (d) It is possible, for a special combination of W and p, to have R =R2 W (0) p p p=0 H2 H1 H 2R (2) (1) 2R1 2R2 141 SOLUTIONS: E-1) The cylindrical walls of a vessel of outer radius R and wall thickness t (t<<R) are made of an isotropic linear elastic material with modulus E and Poisson’s ratio ν. The end caps are rigid. If the vessel is empty (zero gage pressure), the vessel fits precisely (without any stress) inside a rigid frictionless well of inner diameter 2R. With the vessel inside the rigid, frictionless well, a pressurized fluid is introduced in the vessel, so that the internal gage pressure in the vessel is now p. For these conditions, in the cylindrical walls of the vessel we will have: a) No change in outer radius and therefore zero radial strain. pR . t pR c) σ xx = . 2t b) σ θθ = d) εθθ = –ν εxx. e) σθθ = ν σxx. x ρ θ 2R 2R 2R p 142 E-2) A rigid block of weight W rests over a soda can. The material of the can is isotropic linear elastic with properties E, ν (0.5>ν > 0), α. The gage pressure inside the can is p. Under these conditions, the height of the can is H, its radius is R, and the wall thickness is t<<R. (1) The weight is removed but the can is still closed, with internal pressure p: (e) the axial stress changes but the circumferential stress stays the same (f) the height of the can changes but its radius stays identical (2) Then the can is opened, (so now the inside gage pressure is p=0), the height of the can becomes H2 and its radius becomes R2: (g) It is possible, for a special combination of W and p, to have H =H2 (h) It is possible, for a special combination of W and p, to have R =R2 W (0) p p p=0 H2 H1 H 2R (2) (1) 2R1 2R2 143 3.1.F Circular shafts in torsion Axisymmetric loading and deformation: Stresses and Strains in cylindrical coordinates 144 145 Comparison and correspondences between rods in torsion and bars in tension 146 147 148 149 150 151 3.2.F Circular shafts in torsion. Example problems Problem 1 The tapered AB shaft in the figure has a variable cross section with a radius defined by the equation R(x)=R0 eax, whereR0 (units:[m]) and a (units:[1/m]) are constants, and the origin of the x-axis is at A. The shaft is made of isotropic linear elastic material with modulus G. The shaft is fixed at B and subjected to a torque T at the free end, A, as indicated in the figure. a) Determine the angle of twist at the free end: ϕx(A). b) Determine the maximum shear stress in the shaft. [Friendly hint: ∫ecx dx= (1/c) ecx for any constant c] 152 153 Problem 2 The composite shaft of length L is constructed from an inner core of radius R and modulus Gc =3G*, and a sleeve of outer radius 2 R and modulus Gs =G*, bonded together. One end of the shaft is fixed and the other is free to rotate (see figure). A uniform distributed torque, t (t=constant), is applied to the shaft. B t L A x (a) Graph the internal moment (torque T ) as a function of x measured from the free end of the shaft. (b) Determine the rotation angle of the shaft, ϕx , as a function of x. Determine the angle of twist δϕ. (c) Determine the maximum shear stress, σ θ x max , in both materials . 154 155 Problem 3 x The shaft ABCD in the figure is 2a made from two segments: AB of a diameter a and length L/4, and BD of diameter 2a and length 2L. A B The shaft is confined by fixed supports at the walls A and D. A torque Q is applied to the shaft at L point C (at a distance L from the 4 fixed support D as indicated in the figure). Please answer the following questions [careful with the signs and conventions!] L D C Q 2L (a) Obtain the reaction moments at A and D (MAX, MDX). (b) For each of the three segments: AB, BC, CD : determine the internal torque [T AB, T BC T CD] determine the stress state along the surface of the shaft and draw the stress components acting on a surface element aligned with the x and θ axes as indicated to the right. x θ 156 157 158 159 160 161 3.3.F Circular shafts in torsion. Conceptual Questions For each of the following, cross the false (wrong) statements, circle the true (right) ones: F-1) The tapered shaft in the figure has a radius linearly varying between rA and rB, and it is subjected to a constant torque T as shown. (a) The maximum shear stress for the shaft will be at section A because it has the largest radius. (b) The shear strain at the outer radius will be linearly varying (along x) between section A and section B (c) At each section x, the shear strain will be linearly varying from the axis to the outer radius (d) Changes in temperature will only affect the shear stresses σxθ if an axial force is superimposed to the torque. F-2) The cylindrical shaft in the figure is confined by fixed supports at the walls A and B. A torque T is applied to the shaft at point C as indicated in the figure, with LAC > LBC. (a) The peak magnitude of shear stress in the shaft will be in section BC. (b) δϕAC = −δϕBC AC (c) T = −T x CB (d) At any point of AC σxθ ≥0 ; At any point of BC σxθ ≤ 0. 162 F-3) The homogeneous shaft of radius R (known) and length L (known) in the figure is composed of a linear isotropic elastic material of known shear modulus G. Under the effects of an unknown distribution of twisting moments, the bar is observed to deform. The value of the x-rotation along the bar is known, and it is given as a function of the distance from the fixed support, x, by the relation: ϕx(x) = 2ax2–2aLx, where a is a dimensional constant with units [1/m2], and L is the length of the bar. x a) End A is fixed, therefore the shear strain at any point of section A must be zero: εxθ (r, x=0) = 0. b) The angle of twist of shaft AB is zero: δϕ AB=0 c) At any distance (x) from the fixed support, the shear strain at the center (axis of symmetry) of the bar is zero: εxθ (r=0, x) = 0. d) The shear strain at any point of section B is zero: εxθ (r, x=L) = 0, therefore we can conclude that there is no concentrated twisting moment acting at B. e) Values of the shear stresses in the shaft cannot be determined because the distribution of applied twisting moments is unknown. F-4) Shaft AB has outer radius R and is fixed between rigid walls at the two ends. A twisting moment Q is applied at the mid-span (section C) as indicated. The left half of the shaft (AC) is solid, while the right half (CB) is hollow, (with an inner radius R/2). a) The magnitude of the maximum shear strain is the same in AC and CB. L/2 R b) The magnitude of the internal twisting moment, T, is the same in AC and CB. L/2 Q x C c) The magnitude of the maximum shear stress is the same in AC and CB. d) Equilibrium conditions yield Q = –TAC–TCB. e) Equilibrium conditions yield Q = –TxA –TxB . [TxA and TxB are the reaction moments at A and B] 163 SOLUTIONS F-1) The tapered shaft in the figure has a radius linearly varying between rA and rB, and it is subjected to a constant torque T as shown. (a) The maximum shear stress for the shaft will be at section A because it has the largest radius. (b) The shear strain at the outer radius will be linearly varying (along x) between section A and section B (c) At each section x, the shear strain will be linearly varying from the axis to the outer radius (d) Changes in temperature will only affect the shear stresses σxθ if an axial force is superimposed to the torque. F-2) The cylindrical shaft in the figure is confined by fixed supports at the walls A and B. A torque T is applied to the shaft at point C as indicated in the figure, with LAC > LBC. (a) The peak magnitude of shear stress in the shaft will be in section BC. (b) δϕAC = −δϕBC AC (c) T = −T x CB (d) At any point of AC σxθ ≥0 ; At any point of BC σxθ ≤ 0. 164 F-3) The homogeneous shaft of radius R (known) and length L (known) in the figure is composed of a linear isotropic elastic material of known shear modulus G. Under the effects of an unknown distribution of twisting moments, the bar is observed to deform. The value of the x-rotation along the bar is known, and it is given as a function of the distance from the fixed support, x, by the relation: ϕx(x) = 2ax2–2aLx, where a is a dimensional constant with units [1/m2], and L is the length of the bar. x a) End A is fixed, therefore the shear strain at any point of section A must be zero: εxθ (r, x=0) = 0. b) The angle of twist of shaft AB is zero: δϕ AB=0 c) At any distance (x) from the fixed support, the shear strain at the center (axis of symmetry) of the bar is zero: εxθ (r=0, x) = 0. d) The shear strain at any point of section B is zero: εxθ (r, x=L) = 0, therefore we can conclude that there is no concentrated twisting moment acting at B. e) Values of the shear stresses in the shaft cannot be determined because the distribution of applied twisting moments is unknown. F-4) Shaft AB has outer radius R and is fixed between rigid walls at the two ends. A twisting moment Q is applied at the mid-span (section C) as indicated. The left half of the shaft (AC) is solid, while the right half (CB) is hollow, (with an inner radius R/2). a) The magnitude of the maximum shear strain is the same in AC and CB. L/2 R b) The magnitude of the internal twisting moment, T, is the same in AC and CB. L/2 Q x C c) The magnitude of the maximum shear stress is the same in AC and CB. d) Equilibrium conditions yield Q = –TAC–TCB. e) Equilibrium conditions yield Q = –TxA –TxB . [TxA and TxB are the reaction moments at A and B] 165 ...
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This document was uploaded on 11/04/2011 for the course MME 512 at Miami University.

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