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A Ibrahim et al, more catastrophic and can cause considerable damage to life and property The safe design installation opera. tion and maintenance of pressure vessels are in accordance with codes such as American Society of Mechanical. Engineers ASME boiler and pressure vessel code 1 Therefore great emphasis should be placed on analytical. and experimental methods for determining their operating stresses. Spherical Pressure Vessel like the one shown in Figure 1 is preferred for storage of high pressure fluids A. spherical pressure vessel has approximately twice the strength of a cylindrical pressure vessel with the same. wall thickness A sphere is a very strong structure The distribution of stresses on the sphere s surfaces both in. ternally and externally are equal Spheres however are much more costly to manufacture than cylindrical ves. sels A spherical storage has a smaller surface area per unit volume than any other shape of vessel This means. that the quantity of heat transferred from warmer surroundings to the liquid in the sphere will be less than that. for cylindrical storage vessels, Pressure vessels are subjected to tensile forces within the walls of the container The normal stress in the walls. of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thick. ness of the walls 2 3 As a general rule pressure vessels are considered to be thin walled when the ratio of. radius r to wall thickness t is greater than 10 4 Pressure vessels fail when the stress state in the wall exceeds. some failure criterion 5 6 Therefore pressure vessels are designed to have a thickness proportional to the ra. dius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the. particular material used in the walls of the container Thus it is important to understand and quantify analyze. stresses in pressure vessels In this paper we will analyze the stresses in thin walled pressure vessels cylindrical. spherical shapes like the one shown in Figure 1 Figure 2 In addition a case study of internal stresses. developed in a soda can will be presented and discussed. Figure 1 Japanese gas companies added a touch of cha. racter to giant spherical gas tanks, Figure 2 Cylindrical pressure vessel in a chemical plant. A Ibrahim et al,2 Thin Walled Cylindrical Pressure Vessel. A thin walled circular tank AB subjected to internal pressure shown in Figure 3 A stress element with its faces. parallel and perpendicular to the axis of the tank is shown on the wall of the tank The normal stresses 1 and. 2 acting on the side faces of this element No shear stresses act on these faces because of the symmetry of the. vessel and its loading Therefore the stresses 1 and 2 are principal stresses Because of their directions the. stress 1 is called the circumferential stress or the hoop stress and the stress 2 is called the longitudinal. stress or the axial stress Each of these stresses can be calculated from static equilibrium equations. Several assumptions have been made to derive the following equations for circumferential and longitudinal. 1 Plane sections remain plane,2 r t 10 with t being uniform and constant.

3 Material is linear elastic isotropic and homogeneous. 4 Stress distributions throughout the wall thickness will not vary. 5 Weight of the fluid is considered negligible,3 Circumferential Stress. To determine the circumferential stress 1 make three sections cd and ef perpendicular to the longitudinal. axis and distance b apart Figure 3 a and a third cut in a vertical plane through the longitudinal axis of the. tank The resulting free body diagram is shown in Figure 3 b Acting on the longitudinal cut plane cefd are the. circumferential stresses 1 and the internal pressure p. The circumferential stresses 1 acting in the wall of the vessel have a resultant equal to 1 2bt where t. is the thickness of the wall Also the resultant force P1 of the internal pressure is equal to 2 pbr where r is. the inner radius of the cylinder Hence we have the following equation of equilibrium. 1 2bt 2pbr, From the above equation the circumferential stress in a pressurized cylinder can be found. Figure 3 Stresses in a circular cylindrical presure vessel. A Ibrahim et al, If there exist an external pressure po and an internal pressure pi the formula may be expressed as. 4 Longitudinal Stress, The longitudinal stress 2 is obtained from the equilibrium of a free body diagram shown in Figure 3 c The. stresses 2 acts longitudinally and have a resultant force equal to 2 2 rt The resultant force P2 of the. internal pressure is a force equal to p r 2 The equation of equilibrium for the free body diagram is. 2 2 rt p r 2, Solving the above equation for 2 lead to the following formula for the longitudinal stress in a cylindrical.

pressure vessel, If there exist an external pressure po and an internal pressure pi the formula may be expressed as. Comparing Equations 1 and 3 we find that the circumferential stress in a cylindrical vessel is equal to twice. the longitudinal stress, Due to this cylindrical pressure vessels will split on the wall instead of being pulled apart like it would under an. axial load,5 Stresses at the Outer Surface, The principal stresses 1 and 2 at the outer surface of a cylindrical vessel are shown on the stress element of. Figure 4 a The element is in biaxial stress stress in z direction is zero. The maximum in plane shear stresses occur on planes that are rotated 45 about the z axis. The maximum out of plane shear stresses occur on planes that are rotated 45 about x and y axes respec. Therefore the maximum absolute shear stress is,Occurs on a plane rotated by 45 about the x axis. 6 Stresses at the Inner Surface, The stress conditions at the inner surface of the wall of the vessel are shown in Figure 4 b The principal stresses.

A Ibrahim et al, Figure 4 Stresses in a circular cylindrical pressure vessel at a the outer surface b the inner surface. The three maximum shear stresses obtained by 45 rotations about the x y and z axes are. When r t is very large thin walled the term p 2 can be disregarded and the equations are the same as the. stresses at the outer,7 Spherical Pressure Vessel, A similar approach can be used to derive an expression for an internally pressurized thin wall spherical vessel. A spherical pressure vessel is just a special case of a cylindrical vessel. To find we cut the sphere into two hemispheres as shown in Figure 5 The free body diagram gives the. equilibrium condition 2 rt p r 2 hence, Any section that passes through the center of the sphere yields the same result Comparing Equations 1 3. and 10 yields that for the same p r and t the spherical geometry is twice as efficient in terms of wall. As shown in Figure 6 the internal pressure of the cylindrical vessel is resisted by the hoop stress in arch ac. tion whereas the axial stress does not contribute In the spherical vessel the double curvature means that all. stress directions around the pressure point contribute to resisting the pressure. A Ibrahim et al,Figure 5 Stresses in a spherical pressure vessel. Figure 6 a Spherical pressure vessel b Cylindrical pressure vessel. 8 Case Study Measuring Internal Pressure in a Soda Can Using Strain Gauges. The soda can is analyzed as a thin wall pressure vessel In a thin wall pressure vessel two stresses exist the lon. gitudinal stress L and the hoop stress H Figure 7 The longitudinal stress is a result of the internal. pressure acting on the ends of the cylinder and stretching the length of the cylinder as shown in Figure 8 The. hoop stress is the result of the radial action of the internal pressure that tends to increase the circumference of. The pressure developed in a soda can be determined by measuring the elastic strains of the surface of the soda. can Internal pressure for a pressurized soda can be derived using basic Hooke s law stress and strain relations. that relate change in hoop and axial strains to internal pressure Two strain gauges Measurements Group CEA. series gages was attached to the soda can Figure 9 to measure the change in strains as measured through the. voltage across a calibrated Wheatstone bridge M bond 200 adhesive Measurements Group Inc was used to. glue the strain gages to the surface of the soda can. The hoop stress for the thin walled cylinder can be calculated from Equation 1. p internal pressure psi,D mean diameter of cylinder in.

t wall thickness in, Similarly the longitudinal stress cylinder wall can be calculated from Equation 3. Equation 5 yields,A Ibrahim et al,Figure 7 Coca cola soda can. Figure 8 Longitudinal stress distribution, Figure 9 Strain gages attached to a soda can and strain indicator. vishay model 3800,A Ibrahim et al,Assuming that,The material is homogeneous and isotropic. The can is loaded only within its elastic range,A biaxial state of stress exists in the can.

The internal stresses developed in the soda can are proportional to the elastic strains of the outside surface of. the soda can as follow,E modulus of elasticity or Young s modulus psi. Poisson s ratio,H hoop strain in in,L longitudinal strain in in. Using Equations 11 and 12 with Equation 5 and simplifying results in. Thus the pressure can be calculated directly from the measured strains by substituting Equations 13 and 14. back into Equation 1 and 2 to get, Once we have Equations 15 and 16 then the internal pressure in the can may be directly calculated from. the measured longitudinal and hoop strains,9 Internal Pressure Results. Measured Values,Can thickness t 0 004 in,Can diameter D 2 59 in.

Young s Modulus E 10 106 psi assumed,Poisson s Ratio 0 3. The change in longitudinal and hoop strains were measured after the pressure was released from the cans The. results of the strains and corresponding pressures are shown in Table 1. 10 Conclusion, This paper presented a detailed stress analysis of the stresses developed in thin walled pressure vessels cylin. drical spherical Then a case study of a soda can that was analyzed as a thin wall pressure vessel was dis. cussed The elastic strains H L of the external surface of the soda can was determined through strain. gages attached to the can surface and connected to a strain indicator The longitudinal stress hoop stress and the. internal pressure were determined from equations of generalized Hooke s law for stress and strain Small varia. A Ibrahim et al, Table 1 Elastic strains and corresponding internal pressures in a Soda can. Test Strain 10 6 in in Internal Pressure psi,Longitudinal 275 42 5. Hoop 1248 45 4,Longitudinal 295 45 6,Hoop 1172 42 2.

Longitudinal 224 41 3,Hoop 1156 42, tions recorded in internal pressures calculated from the longitudinal strain L and the hoop strain H. References, 1 Rao K 2012 Companion Guide to the ASME Boiler and Pressure Vessel Code Volume 1 Fourth Edition ASME. http dx doi org 10 1115 1 859865, 2 Gere J and Timoshenko S 1997 Mechanics of Materials 4th Edition PWS 549. http dx doi org 10 1007 978 1 4899 3124 5, 3 Moss D 2013 Pressure Vessel Design Manual Elsevier http dx doi org 10 1016 b978 0 12 387000 1 10032 4. 4 Freyer D and Harvey J 1998 High Pressure Vessels Springer Berlin 11. http dx doi org 10 1007 978 1 4615 5989 4, 5 Annaratone D 2007 Pressure Vessel Design Springer Berlin 47 125 http dx doi org 10 1007 978 3 540 49144 6.

6 Shigley J 1983 Mechanical Engineering Design 4th Edition McGraw Hill New York 70. Stress Analysis of Thin Walled Pressure Vessels Ahmed Ibrahim The safe design installation opera tion and maintenance of pressure vessels are in accor dance with codes such as American Society of Mechanical Engineers ASME boiler and pressure vessel code 1 Therefore great emphasis should be pl aced on analytical and experimental methods for determining their operating stresses

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