## Spherical Shells Subjected to External Pressure

The critical compressive stress (scr) for buckling (collapse) of a spherical shell has been experimentally determined to be approximated by:

where R is the radius of the sphere, and E is the modulus of elasticity of the material.

## Cylindrical Shells Subjected to External Pressure

The critical collapse pressure (Pc) of a cylindrical shell under external pressure depends on two characteristic geometric ratios: t/Do and L/Do, where t is the shell thickness, L is the unstiffened length, and Do is the outside diameter. If L is short enough, the cylinder can fail by plastic yielding in compression at a stress above the yield strength of the material, and the ordinary membrane stress equation applies. This type of failure, however, is likely only with heavy wall cylinders.

The critical strain, A, at which a thin wall cylinder under external pressure will collapse can be approximated by:

where K is a factor that depends on the length-to-radius, L/R, and Do/t ratios. The critical compressive stress (Qc) corresponding to the above critical strain is approximated by:

where E is the modulus of elasticity of the material.

## Stresses in Pressure Vessel Shells Due to External Pressure

External pressure on a vessel most commonly occurs when a vacuum or partial vacuum is created inside of the vessel by (1) design, (2) discharge of its contents, (3) steam-out cleaning (condensation of steam), or (4) mechanical action, such as on a compressor suction, during off-design events. In these circumstances, the atmospheric pressure surrounding the vessel becomes greater than the internal pressure.

Theoretically, the equations for internal pressure could be used to calculate the membrane compressive stresses in the shell of a pressure vessel under external pressure, if the pressure (P) is replaced by (-P). Thin wall vessels under external pressure, however, fail at stresses much lower than predicted by the equations, because of elastic or plastic instability, or buckling of the shell. In addition to the properties of the material and the operating temperature, the principal governing factors are geometrical: the unstiffened shell length, the shell thickness, and the outside diameter. Buckling or collapse is assumed to occur at a critical strain, when the potential energy of the external pressure exceeds the strain energy, caused by bending, that the cylinder can accommodate.

## Pressure Vessels Thermal Stresses

Thermal-expansion problems can occur whenever there is: (1) a considerable difference between the vessel operating temperature and the temperature of the environment surrounding the vessel; (2) restricted expansion or contraction; or (3) a temperature gradient within a vessel component that creates a differential expansion. Problems due to external constraint are solved differently than those due to internal constraint.

Thermal stresses are secondary stresses (see Section 115). They will not cause failure in ductile materials on their first application, but they can cause failure after repeated cycling, because of thermal fatigue.

Because the difference in temperature between the inside and outside of a vessel depends mainly on the thickness of the shell and insulation, thick-wall and uninsulated vessels are more susceptible to failure caused by thermal stresses. The stresses are compressive at the inner surface, where the temperature is the highest, and tensile at the outside. Failure from fatigue most likely initiates at the outer surface, where thermal stresses add to the tensile stresses from internal pressure.

Another location where thermal stresses are likely to occur in a hot pressure vessel is the support skirt. At the shell-skirt junction the temperature of the shell and the skirt will be nearly the same. However, the skirt temperature will decrease from the joint down. The temperature difference causes a rotation of the skirt end, which is restrained by the welded joint. In addition to the thermal stresses, radial deformation of the shell under internal pressure will cause discontinuity stresses. In order to minimize thermal stresses at this location, the shell insulation is usually extended below the skirt-to-shell weld. The skirt should also be long enough to minimize the temperature difference between the bolted-down base of the skirt and the concrete foundation, in order to prevent any distortion and local thermal stress at this location.

Under certain conditions, application of a steady mechanical load (like internal pressure) to a vessel subject to cyclic operating temperature may produce cycling of combined thermal and mechanical stresses and a progressive increase in the plastic (permanent) strain in the entire vessel. The action of cyclic, progressive yielding is called thermal ratcheting. It may lead to large distortions and ultimately to failure.

In practice, thermal stresses can be minimized by reducing external constraints, providing local flexibility capable of absorbing the expansion, selecting proper materials (or a combination of materials), and by selective use of thermal insulation.

## Stresses from Locally Applied External Loads

Most vessels are also subject to loadings at the supports, nozzles, and attachments. These loadings produce deflections, edge rotations, shears, bending moments, and membrane forces. The effect rapidly decreases with the distance from the point of application, where the maximum stress occurs. In practical applications, the number of variables is considerable, and some judgement must be exercised to choose the important ones and eliminate those of minor importance.

The procedure for determining local stresses is based on the concepts in Welding Research Council Bulletin No. 107. Calculation sheets are provided for local stresses at nozzles and attachments, which must be added to all the other calculated stresses.

If the maximum stress at the attachment is too great, the shell must be reinforced by a pad, or by increasing the thickness of the reinforcing pad required for internal pressure. To avoid stress concentrations at the corners of square or rectangular pads or structural clips, provide a radius of five to ten times the pad thickness, general Company practice.

## Pressure Vessels Reinforced Openings

Because stresses around openings are higher than the normal design stresses for the plate thickness, additional material must be provided to carry the additional stress in the shell around the opening. The additional material provided is referred to as reinforcement.

The basic concept of reinforcement of openings is that the cross-sectional area of material removed by an opening must be replaced by adding additional material adjacent to the opening. It is assumed that the material added adjacent to the opening has the same load carrying capabilities as the material removed for the opening.

The two basic requirements for reinforcement are:

1. Enough metal reinforcement must be added to compensate for the weakening effect caused by the opening, while still preserving the general strain pattern in the vessel. Adding an excessive amount of material for reinforcement will create a “hard spot” on the vessel that will not allow its natural deformation under pressure, creating local overstressing.

2. The reinforcing material must be placed immediately adjacent to the opening, but suitably disposed in profile and contour so as not to introduce a stress concentration itself.

The reinforcement is usually provided by a separate welded reinforcing pad, or by extra thickness in the shell and nozzle wall.

It is most common for the reinforcement to be added to the outside of the vessel, as shown in Figure 100-15a. However, on some vessels, the reinforcement is added on the inside, as shown in Figure 100-15b. The best configuration is the “balanced reinforcement,” shown in Figure 100-15c, which consists of about 35% to 40% of the reinforcement on the inside and the remainder on the outside. A balanced reinforcement introduces very little local bending moments and stresses. The stress concentration factor in this case is 20% lower than for outside reinforcement only. It may, however, be difficult to place reinforcement on the inside of a vessel, either because the vessel interior is not accessible or because the reinforcement would interfere with the flow or drainage.

Stress concentration factors can be further reduced by using integrally reinforced nozzles (Figure 100-16) which provide a more gradual transition in thickness between the shell, reinforcement, and nozzle. Equally important is the detailed shape of the integral reinforcement. The use of generous transition radii between the shell and the nozzle minimizes stress concentrations due to discontinuities.

A common practice in vessel design, in excess of Code requirements, is to replace all the metal area removed by an opening. This is a Company practice for the design of new vessels, in order to take full advantage of all the strength of the weld.

When several openings are closely spaced, their arrangement requires special consideration, because their individual effects and reinforcements become overlapping. Keeping the spacing between two openings at no less than the sum of their diameters—measured from their centerlines—will maintain the basic average membrane stress in the vessel wall. If the distance is less than the sum of their diameters, the ASME Code sets special rules for reinforcement of multiple openings.

## Pressure Vessels Stresses at Openings

All pressure vessels must be provided with openings to get the process fluid in and out, and to provide entry for maintenance and inspection. When a circular opening is made in a plate subjected to uniform tension, a high concentration of stress occurs near the hole, with its maximum value at the edge of the hole. Away from the opening, the stress decreases until the nominal stress (stress in the unperforated plate) is reached. The ratio of the maximum stress at the edge of the opening to the nominal stress is the stress intensity, or concentration factor.

Figure 100-14 illustrates the concentration of stress for an opening of radius r in cylindrical and spherical shells. This figure shows that at a distance from the hole edge equal to the radius of the hole, the effect of the opening on the stress becomes negligible. This distance is usually accepted as the boundary limit for effective reinforcement.

Occasionally, an elliptical opening is used for special purposes, such as a manway or handhole. For elliptical openings, the maximum stress occurs at the end of the minor axis. Because the hoop stress is always greatest in a cylindrical shell (see Section 120), the most favorable alignment for an elliptical opening is to have the minor axis of the ellipse perpendicular to the hoop direction (or parallel to the longitudinal axis of the vessel). Otherwise the stress concentration factor will be greater than for a circular opening. The minimum stress concentration is obtained by making the elliptical opening with the lengths of the axis inversely proportional to the applied stresses: for a cylindrical vessel subjected to internal pressure, where the hoop stress is double that of the meridional (longitudinal stress), this requires an ellipse with an axis ratio of 1:2.

## Pressure Vessels Stress Concentrations

The normal equations for stresses in pressure vessels are based on the assumption that there is continuous elastic action throughout the member, and that the stress, for simple tension and compression, is uniformly distributed over the entire cross section.

Abrupt changes in section geometries, however, can invalidate these assumptions, leading to great irregularities in stress distribution, with large stresses developed in a small portion of the member. These are called peak stresses or stress concentrations. In pressure vessels they occur at transitions between thick and thin portions of the shell, and at openings, nozzles, or other attachments.

The importance of these stresses depends not only on their absolute value, but also on material properties, such as ductility, the relative proportion of the stressed to the unstressed part of the member, and on the type of loading on the member (static or cyclic).

For example, stress concentrations in a pressure vessel subjected to only a steady pressure are of little importance if the vessel is made of a ductile material such as a mild steel. A ductile material yields at these highly stressed locations, allowing the stress to be transferred from the overstressed fibers to adjacent understressed ones. If the load is repetitive (cyclic), however, the stresses can become significant.

Stress concentrations create peak stresses (Section 116), and they are used to determine the design fatigue life of the vessel. Besides keeping the primary membrane stresses within the limits set by allowable tensile stresses, it is equally important to keep stress concentrations within acceptable limits when fatigue is a factor.

A rigorous mathematical analysis of peak stresses is frequently impossible or impracticable, and therefore experimental methods of stress analysis are used. The ASME Code, Section VIII, Division 1, does not require a consideration of peak stresses, but Division 2 gives some design rules to permit considering stress intensity factors and stress concentration factors in determining peak stresses.

## Concentric Toriconical Reducers Discontinuity Stress

At high pressures (over 150 psi), where discontinuity stresses at the cone-to-cylinder junction can reach values above allowable limits, conical reducers having a knuckle radius at the large cylinder and a flare (reintrant knuckle) at the small end are preferred, as shown in Figure 100-13.

Although more expensive to fabricate, toriconical reducers have the advantage of moving the circumferential weld joints away from the high discontinuity stresses, and allowing better fit-up with the cylindrical shells. The knuckles are usually fabricated in the form of toroidal rings of the same plate thickness as the conical section.

The ASME Code specifies only the minimum value for the knuckle radius at the large end (RL), but has no dimensional requirements for the radius at the small end (Rs). In most cases, the same plate thickness is used for the entire reducer and the required radii RL and Rs are determined using the maximum membrane stresses.

## Shell-to-Cone Junction Without Knuckle Discontinuity Stress

The thickness of a conical head or sections under internal pressure, with a half-apex angle smaller than 30°, is calculated by simple ASME Code membrane-stress equations and the ASME Code rules for reinforcement at the junction. No special analysis of discontinuity stresses is normally required. When, in addition to internal pressure, there are external loads, or when the half-apex angle is larger than 30°, a more detailed analysis of discontinuity stresses is necessary. This analysis can be made by using the Force Method, but as the equations for this case are more complicated, simplified approximate solutions are available in Reference 2. (See the reference section of this manual.) The computer program described above in Section 142 can also be used.

The discontinuity stress can be further increased by a poor fit-up of the joint. Good alignment of a cone with a cylinder is difficult to achieve in practice.