The radial stress for a thick-walled cylinder is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface. The circumferential stress and longitudinal stresses are usually much larger for pressure vessels, and so for thin-walled instances, radial stress is usually neglected. 2.1 Pressure Vessel Design Model for Cylinders 2.1.1 Thick Wall Theory Thick-wall theory is developed from the Theory of Elasticity which yields the state of stress as a continuous function of radius over the pressure vessel wall . The state of stress is defined relative 2.1 Pressure Vessel Design Model for Cylinders 2.1.1 Thick Wall Theory Thick-wall theory is developed from the Theory of Elasticity which yields the state of stress as a continuous function of radius over the pressure vessel wall . The state of stress is defined relative Thick-walled cylinders are often used to contain very high pressures.^ The elastic stress solution for such cylinders was developed by Lame' and is well known (see for example, Reference 2). This solution shows that for a smooth cylinder the maximum stress occurs in the tangential direction at the inside diameter (ID). TecQuipment’s Thick Cylinder apparatus allows students to examine radial and hoop stresses and strains in the wall of a thick cylinder. They can then compare experiment results with the theoretical Lamé predictions. It clearly shows the principles, theories and analytical techniques, and provides effective, practical support to studies. A cylinder is said to be thick walled if the wall thickness is greater than 1/10th of its internal radius. Another criterion to distinguish between thin and thick shells is the internal pressure and the allowable stress, if the internal pressure exerted by the fluid is greater than 1/6th of the allowable Note L.4 Page 4 Definition of Stress (mathematically), (Fig. 4B) [see Ref. 1, p. 203] n r T = stress at point 0 on plane aa whose normal is n passing through point 0 = lim dF where dF is a force acting on area dA. dA dA → 0 n T to introduce the concept that n [Reference 1 uses the notation T is a stress vector] process of producing residual stresses in the wall of a thick-walled cylinder prior to use. An appropriate pressure, large enough to cause yielding within the wall, is applied to the inner surface of the cylinder and then removed. Upon release of this pressure, compressive residual stresses are Manuscript received November 11, 2008. process of producing residual stresses in the wall of a thick-walled cylinder prior to use. An appropriate pressure, large enough to cause yielding within the wall, is applied to the inner surface of the cylinder and then removed. Upon release of this pressure, compressive residual stresses are Manuscript received November 11, 2008. • The cross-sectional area of the cylinder wall is characterized by the product of its wall thickness and the mean circumference. i.e., • For the thin-wall pressure vessels where D >> t, the cylindrical cross-section area may be approximated by πDt. • Therefore, the longitudinal stress in the cylinder is given by: t pD Dt D p A P l 4 4 2 ... The FL 140 experimental unit is used to investigate direct stresses and strains occurring on a thick-walled cylinder subjected to internal pressure. The oil-filled cylinder is made up of two halves, and is sealed on both sides. Internal pressure is generated inside the vessel with a hydraulic pump. A pressure gauge indicates the internal pressure. Collapse of Thick Cylinders Under Radial Pressure and Axial Load Article (PDF Available) in Journal of Applied Mechanics 72(4) · July 2005 with 2,524 Reads How we measure 'reads' Collapse of Thick Cylinders Under Radial Pressure and Axial Load Article (PDF Available) in Journal of Applied Mechanics 72(4) · July 2005 with 2,524 Reads How we measure 'reads' Longitudinal Stress Thin Walled Pressure Vessel: When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial or longitudinal stress and is usually less than the hoop stress. Though this may be approximated to. Thin Wall Pressure Vessel Longitudinal Stress ... Collapse of Thick Cylinders Under Radial Pressure and Axial Load Article (PDF Available) in Journal of Applied Mechanics 72(4) · July 2005 with 2,524 Reads How we measure 'reads' Thick-Walled Cylinder with external pressure of 5330 psi. RADIAL STRESS HOOP STRESS-16-14-12-10-8-6-4-2 0 0 0.5 1 1.5 Radius (in.) Stress (KSI) Stresses for External Pressurization ± − − = 2 2 1 R r r r r p i o i s o o ( + is hoop, - is radial ) 8 The above cylinder has an internal diameter and a wall thickness of .If the applied internal pressure is , then the Hoop stress is and the Longitudinal stress is .. Section the above cylinder through a diametral plane and consider the equilibrium of the resulting half cylinder, where acts upon an area of . Collapse of Thick Cylinders Under Radial Pressure and Axial Load Article (PDF Available) in Journal of Applied Mechanics 72(4) · July 2005 with 2,524 Reads How we measure 'reads' A cylinder is considered to be ‘thick’ if the ratio of the inner diameter to the thickness of the walls is > 20: When we considered thin cylinders, we assumed that the hoop stress was constant across the thickness of the cylinder wall and we ignored any pressure gradient across the wall. Longitudinal stress in a thin-walled cylindrical pressure vessel (7.3.10) Note that this analysis is only valid at positions sufficiently far away from the cylinder ends, where it might be closed in by caps – a more complex stress field would arise there. and elastic-plastic stresses and their distribution in thick walled cylinders under internal pressure (Kihiu 2002). The analysis was done by computer simulation using 3-dimensional FEM procedures. Pressure vessel material was high strength SA-372 steel. Model cylinders had varying thickness ratio, varying cross-bore The theoretical treatment of thin cylinders assumes that the hoop stress is constant across the thickness of the cylinder wall (Fig. 6.1), and also that there is no pressure gradient across the wall. Neither of these assumptions can be used for thick Problems on Thick Cylinders. 1. A steel cylinder is 160 mm ID and 320 mm OD. If it is subject to an internal pressure of 150 MPa, determine the radial and tangential stress distributions and show the results on a plot (using a spreadsheet). Determine the maximum shear stress in the cylinder. Assume it has closed ends. ( σ t = 250 to 100 MPa, σ • A thick-wall cylinder is made of steel (E = 200 GPa and v = 0.29), has an inside diameter of 20mm, and has an outside diameter of 100mm. The cylinder is subjected to an internal pressure of 300 MPa. Determine the stress components and at r = a = 10mm, r = 25mm, and r = b = 50mm. • The external pressure = 0. The radial stress for a thick-walled cylinder is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface. The circumferential stress and longitudinal stresses are usually much larger for pressure vessels, and so for thin-walled instances, radial stress is usually neglected. The radial stress for a thick-walled cylinder is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface. The circumferential stress and longitudinal stresses are usually much larger for pressure vessels, and so for thin-walled instances, radial stress is usually neglected. 2.1 Pressure Vessel Design Model for Cylinders 2.1.1 Thick Wall Theory Thick-wall theory is developed from the Theory of Elasticity which yields the state of stress as a continuous function of radius over the pressure vessel wall . The state of stress is defined relative The Shaft will be subjected to an external pressure , and if and are the Hoop and Radial Stresses at a radius , the equilibrium equation will be obtained as for a "Thick Cylinder". i.e., The longitudinal Stress is zero and assuming that the longitudinal strain is constant, it follows that: = Constant The general equations to calculate the stresses are: Hoop Stress, (1) Radial Stress, (2) From a thick-walled cylinder, we get the boundary conditions: at and at . Applying these boundary conditions to the above simultaneous equations gives us the following equations for the constants A & B: (3) (4) Finally, solving the general equations with A & B gives Lamé’s equations: Hoop Stress, (5) Radial Stress, (6)

Stress in Thick-Walled Cylinders - or Tubes - Radial and tangential stress in thick-walled cylinders or tubes with closed ends - with internal and external pressure Stress, Strain and Young's Modulus - Stress is force per unit area - strain is the deformation of a solid due to stress.