Pressurized thick-walled cylinders

There are many important elastic problems that involve circular symmetry. Of interest here is the solution for a thick-walled cylinder under the action of internal and external pressures and respectively, as shown in Fig. 4.15. For this problem, the symmetry is such that the stresses will not depend on 6. Hence and all dx/d 0 terms vanish, which allows Eq. (4.28) to be reduced to the ordinary differential equation. 

For the case where there is only an internal pressure (F =G), Eqs. (4.32) and (4 33) reduce to 

The radial stress is found to be compressive and the tangential stress is tensile. The maximum value of the tensile stress is usually of concern, as failure processes usually initiate in such locations. For the above case, this occurs when r=a, i.e.. 

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Design of thick-walled cylinders, Design of removable closures, Manufacture of pressure vessels. Manufacture of tubing. 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Eig. 7. Pressure expansion curve of a thick-walled cylinder undergoing autofrettage (22). 

Creep of Thick-walled Cylinders. The design of relatively thick-walled pressure vessels for operation at elevated temperatures where creep caimot be ignored is of interest to the oil, chemical, and power industries. In steam power plants, pressures of 35 MPa (5000 psi) and 650°C are used. Quart2 crystals are grown hydrothermaHy, using a batch process, in vessels operating at a temperature of 340—400°C and a pressure of 170 MPa (25,000 psi). In general, in the chemical industry creep is not a problem provided the wall temperature of vessels made of Ni—Cr—Mo steel is below 350°C. 

The nylon ring may be considered as a thick wall cylinder subjected to this internal pressure (see Appendix D). At the inner surface of the ring there will be a hoop stress, <7, and a radial stress, Cr. Benham et al. shows these to be... 

Treating the bush as a thick wall cylinder subjected to this value of external pressure, then Benham et at. show that at the outer surface of the bush, the stresses are ... 

If the pressure in a thick-walled cylinder is raised beyond the yield pressure pei according to the equation (9), the yield will spread through the wall until it reaches the outer diameter [10]. For a perfectly elastic-plastic material the ultimate pressure for complete plastic deformation of the thick wall pCOmpi-pi, also called collapse pressure, can be calculated by equation (4.3-10). As the ductile materials used for high pressure equipment generally demonstrate strain... 

As a final result of the explanations about strengthening measures the admissible static internal pressure for thick-walled cylinders is compared in Fig. 4.3-7 for different design strategies according to the equations (4.3-9), (4.3-10), (4.3-12) and (4.3-13) and the explained assumptions and optimisations. In the case of the monobloc (A), the two-piece shrink fit and the autofrettaged cylinders the maximum stress at the inner diameter stays within the elastic limit (00.2). Comparatively much larger is the admissible pressure when complete plastic yielding occurs as shown for the collapse pressure (pCOmpi pi. = Pcoii D). 

A number of formulas have been developed for calculating the stress in thick walled cylinders due to internal pressure. From Barlow s formula ... 

Much information is available on the deformation and fatigue behavior of simple thick-walled cylinders [10-17], but it must be remembered that most process reactors will not be a simple hollow cylinder. Components such as connectors, threads and sleeves, windows, and removable closures make a complete analytical solution for a high-pressure system design problem quite involved. Useful design criteria for thick-walled vessels can be derived, however, under the assumption that the material of which the vessel is made is isotropic and that the cylinder is long (more than five diameters) and initially free from stress. The radial and tangential stresses in the walls are then only functions of the radius coordinate (r) and the internal pressure. Given the outer-to-inner wall radius ratio as o/i = w, and the yield point (To) of the material, the yield pressure (py) is... 

This approach is relevant to structures commonly used in pressure vessel construction. What this means is that the problem can be treated as a two-dimensional situation, wherein the stress or the strain components at every point in the body are fimctions only of the reference coordinates parallel to that plane. For example, a long, thick-walled cylinder subjected to internal pressure can be treated as a plane elastic problem. 

The reinforced thick wall cylinder problem with interior pressure has also been solved for the case when the outer shell is assumed to be elastic rather than rigid. (This case obtained by W.B. Woodward and J.R.M. Ra-dok is reported by E. H. Lee in Viscoelasticity Phenomenological Aspects (J.T. Bergen, Ed.), Wiley, (I960)). The solution for the circumferential stress is. 

Determine the stresses in a thick wall cylinder similar the one of Fig. 9.1 using the integral equation solution given by Eq. 9.58. Use the assumption of a step input in pressure as well as elastic response in bulk and Maxwellian in shear as in the earlier example. Compare your solution to that obtained using the correspondence principle. 

Calculations of open-circuit voltage sensitivity of such a sensor to pressure and sounds fluctuations are based on the determination of the circumferential strains in the external surface of a thick walled cylinder under uniform internal pressure. It is assumed that the strains at the outer surface of the cylinder arc the same as those existing inside the PVDF strip. Free-field voltage sensitivity was evaluated numerically fiir a PVDF strip wrapped around a cylinder, with its draw direction oriented circumferentially. In the case of is-pm-thick PVDF strip wrapped around a polytetrafluoroetbylene tube of tbe type used in extracorporeal circulalioo lines, the cakiilaled sensitivity was 11 mV/mmHg. 

Figure 11.1 Typical stress distribution of a thick-walled cylinder (OD/ID = 2) under internal pressure (300 MPa).

Li, H., Johnston, R., and Mackenzie, D. (2007) Effect of autofrettage in the thick-walled cylinder with a radial cross-bore. ASME Pressure Vessel and Piping Division Conference, July 22-26, 2007, San Antonio, TX, Paper No. PVP 2007-26319. 

WAKAI, T., HORIBQRI, M., POUSSARD, C., DRUBAY, B., A Comparison between Japanese and French A16 Defect Assessment Procedures for 2D Crack in the Thick-wall Cylinder Subjected to Thermal-Transient , Ninth International Conference on Pressure Vessel Technology (ICPVT-9), 9-14 April 2000, Sydney, Australia. 

Thin-wallcd and thick-walled cylinders arc forms of pipeline primarily used in the offshore industry to transport gas and hquid under pressure. Cylindrical pipes are also used as a stmctural load bearing member to restrain the motion of a platform, for example, in response to wind, waves and currents (to within a specified limit). When a pipe is exposed to a combination of internal and external pressure, the pipe material will be subjected to pressure loading and, as a result, will develop stresses in all directions. The resulting normal stresses induced by the pressure loading are functions of the pipe diameter, wall thickness and cylindrical shell boundary conditions (open-ended cylinder or closed-ended cylinder) (Kashani and Young, 2008). 

The second method is the Lame approach (Kashani and Young, 2008), which is based on displacement differential equations and is applicable to any cylindrical vessel with any diameter-to-wall-thickness ratio. The Lame method is often referred to as the solution for thick wall cylindrical pressure vessels. Equations for the hoop stress and radial stress in a thick-walled cylinder were developed by Lame in the early nineteenth century (Timoshenko and Goodier, 1969) ... 

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