Circumferential stresses

When the operation of the hoUow-ftber membrane is to be reversed, and permeation from the bore to outer 2one is required, circumferential stress and pressure drop along the fiber capiUary (bore) must be considered in the design of the fiber unit. The circumferential stress, S is expressed as... 

For straight metal pipe under internal pressure the formula for minimum reqiiired w thickness is applicable for D /t ratios greater than 6. Tme more conservative Barlow and Lame equations may also be used. Equation (10-92) includes a factor Y varying with material and temperature to account for the redistribution of circumferential stress which occurs under steady-state creep at high temperature and permits slightly lesser thickness at this range. 

Figure 9.2 Longitudinal stress-corrosion cracks in a heat exchanger tnbe the broad gap between the crack faces reveals that high-level residual hoop (circumferential) stresses from the tube-forming operation provided the stress component required for SCC.

The longitudinal orientation of these cracks reveals that hoop (circumferential) stresses caused by internal pressurization provided the necessary stresses. Ammonia was the specific corrodent involved. 

A cylindrical tube in a chemical plant is subjected to an excess internal pressure of 6 MN m , which leads to a circumferential stress in the tube wall. The tube wall is required to withstand this stress at a temperature of 510°C for 9 years. A designer has specified tubes of 40 mm bore and 2 mm wall thickness made from a stainless alloy of iron with 15% by weight of chromium. The manufacturer s specification for this alloy gives the following information ... 

Figure 6-6 Effect of Material Properties on Circumferential Stress Gq at the Edge of a Circular Hole in an Orthotropic Plate under a, (After Greszczuk [6-11])...

The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

The principal stresses (see Section 13.3.1) acting at a point in the wall of a vessel, due to a pressure load, are shown in Figure 13.1. If the wall is thin, the radial stress comparison with the other stresses, and the longitudinal and circumferential stresses o and <72 can be taken as constant over the wall thickness. In a thick wall, the magnitude of the radial stress will be significant, and the circumferential stress will vary across the wall. The majority of the vessels used in the chemical and allied industries are classified as thin-walled vessels. Thick-walled vessels are used for high pressures, and are discussed in Section 13.15. 

The longitudinal and circumferential stresses due to pressure (internal or external), given by ... 

The vessel is subjected to an internal pressure from the compressed air, which we shall designate as p. The internal pressure is uniformly distributed over the internal surfaces of the vessel, giving rise to both circumferential stress, also known as hoop stress, and longitudinal stress, (see Figure 8.7). We will examine each of these stresses independently before we begin the material selection process. In our development, we will make the following assumptions ... 

Fig. 8 Stress versus time to failure diagram of internally pressurized HD- and MD-polyethylene pipes three failure modes are indicated. The circumferential stress a is calculated according to the standard formula o =p(d -s)/2s (with p internal pressure, d pipe diameter, s pipe wall thickness)...

As a representative example, three of the failure modes of internally pressurized polyethylene pipes are indicated in Fig. 8 in the plot of circumferential stress <7f versus time to failure ff. (Not shown is the regime of brittle fracture by unstable crack propagation, since it only occurs at much lower temperatures and frequently requires a special crack initiation procedure). 

This expression shows that the critical circumferential stress falls with increasing pipe diameter D. This relation and the difficulty to extrude and keep in shape plastic pipes with large wall thicknesses s limit their upper size (at this time the largest external diameter of an extruded (HDPE) pipe is 1400 mm). 

Thick-walled cylindrical and spherical shells (internal pressure), minimum thickness based upon circumferential stress (longitudinal joints)... 

Formula as given above for circumferential stress, except that Z = [P/(FE)j + 1. 

This indicates that circumferential stresses are more important, and they determine the minimum shell thickness for this tank. The BHP steel plate available and closest to this specification (Ref. T2) is... 

Figure 8. Circumferential stresses in the sphere at different times.

Very informative plots are presented in figure 6. The envelope of AE occurrence rates and the theoretically estimated evolution of circumferential stresses based on the viscoelastic model are presented together in this figure. We can... 

Figure 6. Envelope of AE occurrence rates and the theoretical curve circumferential stresses.

The exact calculation of radial and circumferential stresses in each layer requires the solution of N+2 linear equations in N-r2 unknowns, namely the N-H radial displacements of the layer boundaries, and the longitudinal strain of the vessel. We simplify by assuming that the internal pressure is applied only to the steel shell, and that the other layers follow the expansion of the steel. We also assume a condition of plane stress that is, no stress in the axial direction of the cylindrical vessel. We also consider the layer as being flat when layer stresses are being computed. 

Tji Temperature at outside of layer n T Average temperature in a lining 5t Temperature drop across a layer A Change in a parameter when conditions change S Circumferential stress e Circumferential strain d Thickness, inches W Width of brick or mortar, inches... 

The ellipsoidal shape is usual for closures of vessels that are six feet in diameter or greater. The hemispherical shape is preferred for vessels of a lesser diameter. The basic relationships for thin cylindrical shells under internal pressure assume that circumferential stress is dependent on the pressure and vessel diameter, but independent of the shell thickness. 

It can be seen from TaUe 2 diat the stresses at the metal-ceramic interfuse are extremely high. For the radial stress and die circumferential stress 099, the metal side is in compressive state and the ceramic side in tensile state, nhich due to the diffidence thermal expansion coefiBcient The sfress state of die axial stress 022 in Z-a s direction is quite the contrary. Such large stresses at the interfrice can easily lead to nqiture failure of the two layers materiaL Also, the e q>eiimental results prove to be the same. 

Wind, seismic and vibrational stresses and accumulated dead weight compression loadings primarily affect the axial stress and produce only a small effect as a result of Poisson s relationship on the circumferential stress. Therefore, the shell thickness of the upper portion of a tall vertical vessel designed to operate under either internal pressure or vacuum is determined by the circumferential stress. 

We will now consider the special problems in tall tower design which are not described in the ASME Code for Unfired Pressure Vessels. As discussed previously, circumferential stresses control the design of cylindrical vessels if external loads are of small magnitude. In tall vertical vessels, four major factors (wind load, seismic loads, dead weight and vibration) may contribute to axial stresses — in addition to axial stress produced by the operating pressure or vacuum of the vessel. 

The bending stress induced by the cantilever beam action is zero at the top of the tower and a maximum at the base. The bending stress produces a compressive axial stress on the downwind side of the column and a corresponding tensile stress on the upwind side. Thus, regardless of whether a taU vertical tower is operated under vacuum or under internal pressure, there will be an increase of the axial stresses on one side and a subtraction on the opposite side. When this combination of axial stresses equals or exceeds the combined circumferential stress, the axial stresses, rather than the circumferential stresses, will control the thickness requirement of the shell. 

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