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design of pressure vessel (thick and thin cylinders) may 2020


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diploma mechanical engineering
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mechanical engineering
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design of machine elements
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machine design
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design of pressure vessel
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thick cylinders
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thin cylinders
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thin spherical shell
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hoop stress
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longitudinal stress
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tangential stress
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radial stress
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lame's theory
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lame's equation for thick cylinders

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design of pressure vessel (thick and thin cylinders) may 2020

  1. 1. Design of Machine Elements Pressure Vessel Design. (Thin and Thick cylinders) Gaurav Mistry Assistant Professor Diwaliba Polytechnic, UTU.
  2. 2. ❑ Pressure Vessel Design Design of Machine Elements 2Gaurav Mistry The pressure vessels are designed with great care because rupture of a pressure vessel means an explosion which may cause loss of life and property. The material of pressure vessels may be brittle such as cast iron, or ductile such as mild steel. It caused 58 deaths and 150 reported injuries.
  3. 3. Design of Machine Elements 3Gaurav Mistry Based on different considerations, the pressure vessels are classified as follows: 1. According to ratio of Internal Diameter (d) to the Thickness of the shell (t) i.e. 𝒅 𝒕 ratio. (i) When 𝑑 𝑡 ratio is equal to or more than 15. 𝑑 𝑡 ≥ 15, then such vessel is known as thin cylinders. (ii) When 𝑑 𝑡 ratio is less than 15. 𝑑 𝑡 < 15, then such vessel is known as thick cylinders (It is also known as Based on Wall thickness) Note: 1. In Some reference books the ratio may be considered between 10 to 20. 2. Both Thin and Thick cylinders are designed with complete separate procedure. 2. According to geometric shape: (i) Cylindrical, (ii) Spherical, (iii) Conical or (iv) Elliptical. ❑ Pressure Vessel Design
  4. 4. Design of Machine Elements 4Gaurav Mistry 3. According to end conditions of pressure vessels. (i) Open ended pressure vessel. e.g. cylinder with piston (usually engine cylinders, hydraulic cylinders used in automobiles and machineries, etc.) (ii) Closed ended pressure vessels. E.g. receiver tank of an air compressor, boiler, containers, etc. 4. According to Shape of the end heads/ end covers: (i) Flat headed, (ii) Convex headed or (iii) Dished headed. 5. According to direction of fluid pressure acting on the wall of vessel: (i) Internal Pressure (ii) External Pressure (Internal Vacuum). 6. ❑ Pressure Vessel Design
  5. 5. Design of Machine Elements 5Gaurav Mistry 7. 8. 9. ❑ Pressure Vessel Design
  6. 6. Design of Machine Elements 6Gaurav Mistry 10. 11. According to type of joint used for the fabrication of vessel: (i) Riveted (ii) Welded (iii) Forged ❑ Pressure Vessel Design
  7. 7. Design of Machine Elements 7Gaurav Mistry Main Consideration for the design of Pressure Vessel ❑ Pressure Vessel Design Following considerations are made for the design of pressure vessel: 1. Material of the shell must be capable to resist the fluid pressure, thermal load and corrosion. 2. Different types of stresses induced in the pressure vessel are determined and suitable safety factor is to be adopted. 3. Based on the geometry of the vessels and various connections, stress concentration take place which is required to be properly taken care. 4. To prevent leakage of the toxic acids, poisonous fluids, etc., efficient leak proof joints are to be used. 5. Where ever necessary, to prevent the corrosion, the contact surface of the shell is lined with rubber, glass fibre or appropriate anticorrosive materials. 6. To prevent the possible failure of the vessel, suitable safety devices like valves, leakage indicating devices, etc., should be included in the design.
  8. 8. Design of Machine Elements 8Gaurav Mistry ❑ Pressure Vessel Design
  9. 9. Design of Machine Elements 9Gaurav Mistry Design of Thin Cylinders Stresses in a Thin Cylindrical Shell due to an Internal Pressure The analysis of stresses induced in a thin cylindrical shell are made on the following assumptions: 1. The effect of curvature of the cylinder wall is neglected. 2. The tensile stresses are uniformly distributed over the section of the walls. 3. The effect of the restraining action of the heads at the end of the pressure vessel is neglected. When a thin cylindrical shell is subjected to an internal pressure, it is likely to fail in the following two ways: 1. It may fail along the longitudinal section (i.e. circumferentially) splitting the cylinder into two troughs, as shown in Fig. (a). 2. It may fail across the transverse section (i.e. longitudinally) splitting the cylinder into two cylindrical shells, as shown in Fig. (b). ❑ Pressure Vessel Design
  10. 10. Design of Machine Elements 10Gaurav Mistry Design of Thin Cylinders Thus the wall of a cylindrical shell subjected to an internal pressure has to withstand tensile stresses of the following two types: (a) Circumferential or hoop stress (𝜎𝑡), and (b) Longitudinal stress. (𝜎𝑙) Circumferential or Hoop Stress (𝜎𝑡): Consider a thin cylindrical shell subjected to an internal pressure as shown in Fig. (a), (b) and (c) (c) hoop stress Longitudinal stress ❑ Pressure Vessel Design
  11. 11. Design of Machine Elements 11Gaurav Mistry Design of Thin Cylinders Circumferential or Hoop Stress (𝜎𝑡): A tensile stress acting in a direction tangential to the circumference is called circumferential or hoop stress. In other words, it is a tensile stress on longitudinal section (or on the cylindrical walls). Let p = Intensity of internal pressure, d = Internal diameter of the cylindrical shell, l = Length of the cylindrical shell, t = Thickness of the cylindrical shell, and 𝜎𝑡 = Circumferential or hoop stress for the material of the cylindrical shell. We know that the total force acting on a longitudinal section (i.e. along the diameter) of the Shell as shown in Fig (a) = Intensity of pressure × Projected rectangular area = p × d × l ...(i) and the total resisting force acting on the cylinder walls as shown in Fig (b) = 𝜎𝑡 × 2t × l ...(∵ of two sections) ...(ii) (b)(a) ❑ Pressure Vessel Design
  12. 12. Design of Machine Elements 12Gaurav Mistry Design of Thin Cylinders Circumferential or Hoop Stress (𝜎𝑡): From equations (i) and (ii), we have 𝜎𝑡 × 2t × l = p × d × l or The following points may be noted: 1. In the design of engine cylinders, a value of 6 mm to 12 mm is added in equation To permit reboring after wear has taken place. 2. In constructing large pressure vessels like steam boilers, riveted joints or welded joints are used in joining together the ends of steel plates. In case of riveted joints, the wall thickness of the cylinder, t = p × d 2 𝝈 𝒕 t = p × d 2 𝝈 𝒕 t = p × d 2 𝝈 𝒕 + (6 to 12) mm t = p × d 2 𝝈 𝒕 𝐱 𝜼 𝒍 Where, 𝜼𝒍 = 𝑬𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 𝒐𝒇 𝑳𝒐𝒏𝒈𝒊𝒕𝒖𝒅𝒊𝒏𝒂𝒍 𝒋𝒐𝒊𝒏𝒕 𝝈 𝒕 = p × d 2 t Here using hoop stress, the thickness (t) of cylinder wall will be calculated ❑ Pressure Vessel Design
  13. 13. Design of Machine Elements 13Gaurav Mistry Design of Thin Cylinders Longitudinal Stress (𝜎𝑙): Consider a closed thin cylindrical shell subjected to an internal pressure as shown in Fig (a) and (b). A tensile stress acting in the direction of the axis is called longitudinal stress. In other words, it is a tensile stress acting on the transverse or circumferential section (or on the ends of the vessel). We know that the total force acting on the Transverse direction (i.e. at the end of the vessel) as shown in Fig (a). (b) (a)= Intensity of pressure × Cross-sectional area ...(i) and total resisting force acting on cylinder wall as shown in Fig (b). = 𝜎𝑙 × 𝜋 d× t ...(ii) ❑ Pressure Vessel Design
  14. 14. Design of Machine Elements 14Gaurav Mistry Design of Thin Cylinders Longitudinal Stress (𝜎𝑙): From equations (i) and (ii), we have 𝜎𝑙 × 𝜋 d× t 𝝈𝒍 = p × d 4 t t = p × d 4 𝝈 𝒍 t = p × d 4 𝝈𝒍 𝐱 𝜼 𝒍 If 𝜼𝒍 𝒊𝒔 𝒕𝒉𝒆 𝒆𝒇𝒇𝒊cie𝒏𝒄𝒚 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝒋𝒐𝒊𝒏𝒕,then From above we see that the longitudinal stress is half of the circumferential or hoop stress. Therefore, the design of a pressure vessel must be based on the maximum stress i.e. hoop stress. Further, the diameter d and length l of the cylindrical shell are determined based on the required volume of the fluid to be stored in the vessel (maximum storage capacity V in litres) Maximum shear stress We know that according to maximum shear stress theory, the maximum shear stress is one-half the algebraic difference of the maximum and minimum principal stress. Since the maximum principal stress is the hoop stress (𝝈 𝒕) and minimum principal stress is the longitudinal stress (𝝈𝒍 ), therefore maximum shear stress, or 𝝈𝒍 = p × d 4 t = p × d 2 . 2. t = 𝝈 𝒕 2 𝝉 𝒎𝒂𝒙 = 𝝈 𝒕 −𝝈 𝒍 2 Here using longitudinal stress, the thickness (t) of Cylinder cover plate will be calculated ❑ Pressure Vessel Design
  15. 15. Design of Machine Elements 15Gaurav Mistry Design of Thin Spherical Shell: Consider a thin spherical shell subjected to an internal pressure as shown in Fig. 7.5. Let V = Storage capacity of the shell, p = Intensity of internal pressure, d = Diameter of the shell, t = Thickness of the shell, 𝝈 𝒕 = Permissible tensile (hoop) stress for the shell material. In designing thin spherical shells, we have to determine 1. Diameter of the shell, and 2. Thickness of the shell. ❑ Pressure Vessel Design
  16. 16. Design of Machine Elements 16Gaurav Mistry Design of Thin Spherical Shell: 1. Diameter of the shell (d) Internal diameter of the shell is determined based on its maximum storage capacity. We know that the storage capacity of the shell, 2. Thickness of the shell wall (t) As a result of the internal pressure, the shell is likely to rupture along the centre of the sphere. Therefore force tending to rupture the shell along the centre of the sphere or bursting force, Fig(a) , ...(i) ...(ii) and resisting force of the shell wall, Fig (b), Design of Thin Spherical Shell: ❑ Pressure Vessel Design
  17. 17. Design of Machine Elements 17Gaurav Mistry From equations (i) and (ii), we have Design of Thin Spherical Shell: 2. Thickness of the shell wall (t) ❑ Pressure Vessel Design
  18. 18. Design of Machine Elements 18Gaurav Mistry 1. A thin cylinder with 150 mm inside diameter and 15 mm plate thickness is subjected to internal pressure of 5 N/ 𝒎𝒎 𝟐 . Determine the hoop stress and longitudinal stress and maximum shear stress in the cylinder. Solution: Given data, d = 150 mm, t = 15 mm and p = 5 N/ 𝑚𝑚2, Now, we know Hoop stress = = 5 × 150 2 x 15 = 25 N/𝑚𝑚2. Longitudinal stress = = 5 × 150 4 x 15 = 12.5 N/ 𝑚𝑚2. Maximum shear stress == 𝟐𝟓 − 𝟏𝟐.𝟓 2 = 6.25 N/ 𝑚𝑚2. Numerical: Design of Thin cylinder 𝝈 𝒕 = p × d 2 t 𝝈𝒍 = p × d 4 t 𝝉 𝒎𝒂𝒙 = 𝝈 𝒕 −𝝈 𝒍 2 ❑ Pressure Vessel Design
  19. 19. Design of Machine Elements 19Gaurav Mistry 2. A spherical shell having capacity of 4000 litres is subjected to internal pressure of 1.2 N/mm2. Determine the thickness of shell wall. Permissible stress for shell material is 60 N/ 𝒎𝒎 𝟐 and joint efficiency is 75%. Solution: Given data, p = 1.2 N/ 𝑚𝑚2, capacity V = 4000 litres = 4000 x 106 𝑚𝑚3, 𝜂 = 0.75 Now (i) Diameter of shell can be calculated from d = ( 6 x 4000 x 106 𝜋 ) 1 3 = 1969.49 mm = 1970 mm = 2000 mm Now, = = 13.33 mm = 16 mm. Therefore, the diameter of shell is 2000 mm and the thickness is t = 16 mm Numerical: Design of Thin Spherical Shell. 1.2 × 2000 4 𝐱 𝟔𝟎 𝐱 𝟎.𝟕𝟓 ❑ Pressure Vessel Design
  20. 20. Design of Machine Elements 20Gaurav Mistry Design of Thick Cylinders When a cylindrical shell of a pressure vessel, hydraulic cylinder, Gunbarrel and a pipe is subjected to a very high internal fluid pressure, then the walls of the cylinder must be made extremely heavy or thick. In thin cylindrical shells, we have assumed that the tensile stresses are uniformly distributed over the section of the walls. But in the case of thick wall cylinders as shown in Fig. (a), the stress over the section of the walls cannot be assumed to be uniformly distributed. They develop both tangential and radial stresses with values which are dependent upon the radius of the element under consideration. The distribution of stress in a thick cylindrical shell is shown in Fig. (b) and (c) ❑ Pressure Vessel Design
  21. 21. Design of Machine Elements 21Gaurav Mistry Design of Thick Cylinders In the design of thick cylindrical shells, the following equations are mostly used: 1. Lame’s equation; 2. Birnie’s equation; 3. Clavarino’s equation; and 4. Barlow’s equation. We will particularly study about Lame’s theory and equation only. LAME’s THEORY: Lame’s theory is commonly used for the stress analysis across the walls of the thick cylinders which is based on MAXIMUM NORMAL STRESSES THEORY OR MAXIMUM PRINCIPLE STRESS THEORY. According to Lame’s theory, 1. Hoop stress or Tangential stresses are tensile in nature and it is maximum at the inner surface of the cylinder wall and minimum at the outer surface of the cylinder wall. (The equation for distribution of Maximum tangential stress and Minimum tangential stress across the wall of thick cylinder will be obtained later.) Tangential stress distribution ❑ Pressure Vessel Design
  22. 22. Design of Machine Elements 22Gaurav Mistry Design of Thick Cylinders 2. Radial stress is compressive in nature and it is maximum at inner surface and zero at the outer surface. (The equation for distribution of maximum radial stress and minimum radial stress across the wall of thick cylinder will be obtained later.) ______________________________________ Assumptions made in design of thick cylinders: 1. The material of the cylinder is homogeneous and isotropic and it follows the Hook’s Law. 2. The cross section of the cylinder is symmetrical to its longitudinal axis and load acting in the cylinder is also symmetrical to its longitudinal axis. 3. Cylinder is open at both the ends and the longitudinal stresses induced in the cylinder is negligible. Radial stress distribution ❑ Pressure Vessel Design
  23. 23. Design of Machine Elements 23Gaurav Mistry Design of Thick Cylinders LAME’s Expressions for Hoop Stress and Radial Stress on Thick Cylinder: As shown in Figure, for thick cylinder subjected to internal and external pressures the basic equations for tangential (hoop) and radial stresses at any radius R given by Lame are a shown below: Tangential or hoop stress 𝝈 𝒕: 𝝈 𝒕 = 𝐴 + 𝐵 𝑅2 …(1) Radial stress 𝝈 𝒓: 𝝈 𝒓 = 𝐴 − 𝐵 𝑅2 …(2) Where, A and B are Lame’s constants and R = Radius of cylinder in consideration (shown by Orange color at radius R from centre of Cylinder. R1 R2R2 R1 R2 R1 R2 ❑ Pressure Vessel Design
  24. 24. Design of Machine Elements 24Gaurav Mistry Design of Thick Cylinders Both equations (1) and (2) shown above are known as Lame’s General Equations. With the appropriate boundary conditions, constants A and B for both the equations can be determined to get the values of the hoop and radial stresses. Let, R1 = Inner radius of the cylinder R2 = Outer radius of the cylinder p1 = Internal pressure. (As it produces a compression on inner cylinder walls, it is considered as negative, maximum radial stress “- p1” .) p2 = External pressure. (Neglecting the external pressure, it will be considered “0”) t = Thickness of cylinder wall = R2 – R1. Now, applying the boundary conditions of radial stress to Lame’s general equation (2) of radial stress, we can determine the constant A and B. And further substituting back these constants in equation (1) and (2) ❑ Pressure Vessel Design
  25. 25. Design of Machine Elements 25Gaurav Mistry Design of Thick CylindersDesign of Thick Cylinders Therefore substituting the condition At R = R1, 𝝈 𝒓 = - p1 and At R = R2, 𝝈 𝒓 = 0. in equation (2) i.e. 𝝈 𝒓 = 𝐴 − 𝐵 𝑅2 , we get the value of A and B as 𝐴 = p1 x R1 2 (R2 2 − R1 2) …(3) and B= p1 x R1 2. R2 2 (R2 2 − R1 2) …(4) Now again substituting the value of A and B from equation (3) and (4) in equation (1) and equation (2) we get 𝑻𝒂𝒏𝒈𝒆𝒏𝒕𝒊𝒂𝒍 𝒉𝒐𝒐𝒑 𝒔𝒕𝒓𝒆𝒔𝒔 = 𝝈 𝒕 = p1 x R1 2 (R2 2 − R1 2) [ 1 + R2 2 R2 ] …(5) 𝑹𝒂𝒅𝒊𝒂𝒍 𝒔𝒕𝒓𝒆𝒔𝒔 = 𝝈 𝒓 = p1 x R1 2 (R2 2 − R1 2) [ 1 − R2 2 R2 ] …(6) Here the equation of tangential and radial stress at any radius R, for thick cylinder subjected to internal pressure only. External pressure is neglected. Here we can see that, 𝝈 𝒕 > 𝝈 𝒓 ❑ Pressure Vessel Design
  26. 26. Design of Machine Elements 26Gaurav Mistry Design of Thick Cylinders Tangential stress distribution The equation for distribution of Maximum tangential stress and Minimum tangential stress across the wall of thick cylinder subjected to internal pressure only: (a) Hoop stress will be maximum when, R = R1. Therefore, Substituting R with R1 in equation (5) 𝝈 𝒕 𝒎𝒂𝒙 = p1 x R1 2 (R2 2 − R1 2) [ 1 + R2 2 R1 2 ] 𝝈 𝒕 𝒎𝒂𝒙 = p1 x R2 2 + R1 2 (R2 2 − R1 2) …(7) (at internal surface) (b) Hoop stress will be minimum when, R = R2. Therefore, Substituting R with R2 in equation (5) 𝝈 𝒕 𝒎𝒊𝒏 = p1 x R1 2 (R2 2 − R1 2) [ 1 + R2 2 R2 2 ] 𝝈 𝒕 𝒎𝒊𝒏 = p1 x 2 R1 2 (R2 2 − R1 2) …(8) (at outer surface) ❑ Pressure Vessel Design
  27. 27. Design of Machine Elements 27Gaurav Mistry Design of Thick Cylinders Radial stress distribution The equation for distribution of maximum radial stress and minimum radial stress across the wall of thick cylinder subjected to internal pressure only: (a) Radial stress will be maximum when, R = R1. Therefore, Substituting R with R1 in equation (6) 𝝈 𝒓 𝒎𝒂𝒙 = p1 x R1 2 (R2 2 − R1 2) [ 1 − R2 2 R1 2 ] 𝝈 𝒓 𝒎𝒂𝒙 = p1 x (R1 2−R2 2) (R2 2 − R1 2) 𝝈 𝒓 𝒎𝒂𝒙 = − p1 …(9) (at internal surface) (b) Radial stress will be minimum when, R = R2. Therefore, Substituting R with R2 in equation (6) 𝝈 𝒓 𝒎𝒊𝒏 = p1 x R1 2 (R2 2 − R1 2) [ 1 − R2 2 R2 2 ] 𝝈 𝒓 𝒎𝒊𝒏 = p1 x R1 2 (R2 2 − R1 2) [0] 𝝈 𝒓 𝒎𝒊𝒏 = 0 …(10) (at outer surface) ❑ Pressure Vessel Design
  28. 28. Design of Machine Elements 28Gaurav Mistry Design of Thick Cylinders For the design of thick cylinder subjected to internal pressure only, the maximum shear condition is considered. Taking t = R2 – 𝑅1, from equation (7) , for 𝝈 𝒕 𝒎𝒂𝒙 > p1 , we get 𝑡 = 𝑅1 [ 𝝈 𝒕 𝒎𝒂𝒙 + p1 𝝈 𝒕 𝒎𝒂𝒙 − p1 - 1] or …(11) Considering permissible value of 𝝈 𝒕 𝒎𝒂𝒙 for the cylinder material, the thickness of the cylinder can be obtained from above equation. ❑ Pressure Vessel Design
  29. 29. Design of Machine Elements 29Gaurav Mistry Numerical: Design of Thick Cylinders ❑ Pressure Vessel Design A cast steel hydraulic cylinder having internal diameter of 200 mm is subjected to oil pressure of 31.5 N/ 𝒎𝒎 𝟐. Determine the thickness of the shell for the maximum permissible hoop stress of 70 N/ 𝒎𝒎 𝟐 . Solution: Given data, D = 200 mm therefore R = 100 mm , p1 = 31.5 N/ 𝑚𝑚2 and 𝝈 𝒕 𝒎𝒂𝒙 = 70 N/ 𝑚𝑚2 Based on Lame’s Equation: 𝑡 = 𝑅1 [ 𝝈 𝒕 𝒎𝒂𝒙 + p1 𝝈 𝒕 𝒎𝒂𝒙 − p1 - 1] 𝑡 = 100 [ 70 + 31.5 70 − 31.5 - 1] t = 100 [ 2.637 - 1] = 100 [1.6237 – 1] = 100 [0.6237] = 62.37 mm = 62.5 mm Therefore, the thickness of cylinder is t = 62.5 mm.
  30. 30. Design of Machine Elements 30Gaurav Mistry Main Applications of Pressure Vessel is as listed below: ❑ Pressure Vessel Design 1. To store and process water, gas, acid and various chemicals in the liquid or gaseous form. 2. In boilers, heat exchanger and condenser. 3. As a receiver tank for compressed air in air compressor. 4. As a cylinder in case of steam engine, automobile engine, air compressor, etc. 5. As a cylinder and accumulator in case of hydraulic and pneumatic systems. 6. As a pipe to carry steam, fluid, gas or air in the piping system. 7. As a cylinder to carry pressurized, oxygen, acetylene, hydrogen, argon, LPG, CNG, etc. automobile engine air compressor heat exchanger condenser
  31. 31. Design of Machine Elements 31Gaurav Mistry (Here, S stands for allowable stress and E stands for efficiency) Just for Information Slide 1
  32. 32. Design of Machine Elements 32Gaurav Mistry Just for Information Slide 2
  33. 33. Design of Machine Elements 33Gaurav Mistry REFERENCES: 1. A textbook of Machine design, R. S. Khurmi, S. Chand. 2. Design of Machine Elements, S. B. Soni, Atul prakashan. 3. www.google.com 4. Fundamental concepts of PVD, by Sandip G. Patel, former Assistant Professor of Mech/Auto Dept, CGPIT, UTU. ❑ Pressure Vessel Design
  • MagedAdel5

    Jun. 1, 2021
  • AruneshChandra1

    Nov. 22, 2020
  • vishaldeokate1

    Aug. 3, 2020
  • PihuPatel4

    Jul. 2, 2020
  • GauravMistry14

    Jun. 20, 2020

, diploma mechanical engineering , mechanical engineering , design of machine elements , machine design , design of pressure vessel , thick cylinders , thin cylinders , thin spherical shell , hoop stress , longitudinal stress , tangential stress , radial stress , lame's theory , lame's equation for thick cylinders

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