CTBEST PVM 100 Engineering Principles

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100 Engineering Principles

AbstractThis section discusses the engineering principles and failure modes and stressdifferent shaped vessels, including cylindrical and spherical shells and hemisphical, ellipsoidal, torispherical, and conical heads. Equations for calculating stresare also given, including stresses from internal and external pressure, discontinand thermal stresses, and stresses at openings.

Contents Page

110 Introduction 100-3

111 Failure Modes

112 Loads

113 Summary of Stresses

114 Primary Stresses

115 Secondary Stresses

116 Peak Stresses

120 Stresses in Cylindrical Shells 100-9

121 General

122 Cylindrical Shells Under Internal Pressure

130 Stresses in Formed and Flat Components 100-1

131 Spherical Shells and Hemispherical Heads Under Internal Pressure

132 Ellipsoidal Heads Under Internal Pressure

133 Torispherical Heads Under Internal Pressure

134 Flat Plate Closures

135 Conical Sections Under Internal Pressure

140 Discontinuity Stresses 100-19

141 General

142 Calculation of Discontinuity Stresses

143 Discontinuities in Cylindrical Shells

Chevron Corporation 100-1 March 1990

100 Engineering Principles Pressure Vessel Manual

-28

144 Shell-to-Hemispherical Head Junction

145 Shell-to-Ellipsoidal Head Junction

146 Shell-to-Torispherical Head Junction

147 Shell-to-Cone Junction Without Knuckle

148 Concentric Toriconical Reducers

150 Stress Concentrations 100-22

151 General

152 Stresses at Openings

153 Reinforced Openings

154 Stresses from Locally Applied External Loads

155 Thermal Stresses

160 Stresses in Pressure Vessel Shells Due to External Pressure 100

161 General

162 Cylindrical Shells Subjected to External Pressure

163 Spherical Shells Subjected to External Pressure

164 Elliptical and Torispherical Heads Subjected to External Pressure

165 Conical Heads and Transitions Subjected to External Pressure

March 1990 100-2 Chevron Corporation

Pressure Vessel Manual 100 Engineering Principles

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110 IntroductionPressure vessels must be designed to resist all potential modes of failure undecombinations of internal and external loads that the vessel will be subjected to under normal operating conditions. This section gives a basic understanding ofdifferent failure modes that can occur, the various loadings that a pressure vesscan be subjected to that could cause it to fail, and how these loads develop thestresses in the pressure vessel shell.

111 Failure ModesThe major potential failure modes that must be considered for the design of a psure vessel are:

• Excessive plastic deformation (including creep or stress-rupture at high temperatures)

• Excessive elastic deformation or buckling

• Ductile bursting

• Brittle fracture

• Fatigue

112 LoadsThe forces applied to a vessel and its structural attachments are called loads, athe first requirement in vessel design is to determine the loads and the conditiowhich the vessel will be subjected in operation.

The major loads acting on a pressure vessel are caused by:

1. Internal pressure

2. External pressure

3. Weight of vessel and contents (including internal components that transmitloads to the pressure vessel)

4. Wind and seismic forces

5. Connecting piping and the weight of external appurtenances (platforms, etc

6. Differential thermal expansion (or temperature gradients)

7. Cyclic forces

These forces must be considered during design in order to prevent failure fromof the failure modes mentioned earlier.

The loads are usually static, or the amplitude and frequency of their fluctuationssuch that they can be considered to be so. However, cyclic loads of sufficient mtude can result in a fatigue failure, and it may be necessary to consider them in



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design of a pressure vessel. For example, pressure fluctuations that exceed 20the design pressure and cyclic temperature gradients greater than 50°F betweeadjacent locations can cause fatigue.

113 Summary of StressesA pressure vessel is subjected to various loads which develop stresses that aregorized by the ASME Code as:

• Primary• Secondary• Peak

The maximum allowable design stress for a pressure vessel is based on the strof the material. Stress limits depend on the category of the stress and its relatioship to the various potential failure modes. Figure 100-1 classifies typical stressin a vessel.

Under certain conditions, each category or combination of stresses can cause tvessel to fail in a different way. For this reason, the limits for each category of stress are related to the potential failure modes. Primary stress limits are set toprevent deformation and ductile burst. Primary plus secondary stress limits are to prevent plastic deformation leading to collapse. Peak stress limits are set to prevent fatigue failure due to cyclic loading.

Because of differences between the simplified equations used in the codes andcomplexity of the theoretical equations, a factor of safety is applied to various mrial properties used to determine allowable stress values. These safety factors daccording to the specific criteria of each section or division of the Code and according to the required levels for the actual stresses (see Section 200 of this manual).

Since each stress category requires a different safety factor to protect against failure, the designer has to evaluate each type of stress to achieve the most economical and safe design.

114 Primary StressesPrimary stresses are those developed in each component of a vessel due to sustained internal and external loads. The fundamental characteristic of primarstresses is that they are not self-limiting. In other words, no redistribution of loareduction of stress will occur despite yielding within the component; primary stresses are not reduced by the deformations they produce. Therefore, primarystresses that exceed the yield strength of the material will cause failure either bgross plastic deformation or by bursting.

Primary stresses are the most significant stresses that occur in pressure vessetheir limits for design are set both to prevent plastic deformation and to provide factor of safety against bursting.



Fig. 100-1 Pressure Vessels: Classification of Stresses for Some Typical Cases (ASME Code, Section VIII, Division 2, Table 4-120.1) (1 of 2) Courtesy of ASME

VesselComponent Location Origin of Stress Type of Stress Classification

Cylindrical or spherical shell

Shell plate remote from discontinuities

Internal pressure General membrane.

Gradient through plate thickness

Pm

Q

Axial thermal gradient Membrane

Bending

Q

Q

Junction with head or flange

Internal pressure Membrane

Bending

PL

-Q

Any shell or head

Any section across entire vessel

External load or moment, or internal pressure

General membrane aver-aged across full section. Stress component perpen-dicular to cross section

Pm

External load or moment

Bending across full section. Stress component perpen-dicular to cross section

Pm

Near nozzle or other opening

External load moment, or internal pressure

Local membrane.

Bending.

Peak (fillet or corner)

PL

Q

F

Any location Temperature differ-ence between shell and head

Membrane

Bending

Q

Q

Dished head or conical head

Crown Internal pressure Membrane

Bending

Pm

Pb

Knuckle or junction to shell

Internal pressure Membrane

Bending

PL

Q

Flat head Center region Internal pressure Membrane

Bending

Pm

Pb

Junction to shell Internal pressure Membrane

Bending

PL

Q



Primary stresses are further divided into primary membrane stresses and primary bending stresses because different stress limits are applied for design, depending on the type of primary stress.

Primary Membrane StressPrimary membrane stresses are tensile or compressive stresses that are essentially uniform through the entire cross-section of a pressure vessel component. Conse-quently, gross plastic deformation will occur when these stresses exceed the yield

Perforated head or shell

Typical ligament in a uniform pattern

Pressure Membrane (average through cross section).

Bending (average through width of ligament, but gradient through plate)

Peak

Pm

Pb

F

Isolated or a typical ligament

Pressure Membrane

Bending

Peak

Q

F

F

Nozzle Cross section perpendicular to nozzle axis

Internal pressure or external load or moment

General membrane (average across full section). Stress component perpendicular to section

Pm

External load or moment

Bending across nozzle section

Pm

Nozzle wall Internal pressure General membrane

Local membrane

Bending

Peak

Pm

PL

QF

Differential expansion Membrane

Bending

Peak

Q

Q

F

Cladding Any Differential expansion Membrane

Bending

F

F

Any Any Radial temperature distribution

Equivalent linear stress

Nonlinear portion of stress distribution

Q

F

Any Any Any Stress concentration (notch effect)

F

Fig. 100-1 Pressure Vessels: Classification of Stresses for Some Typical Cases (ASME Code, Section VIII, Division 2, Table 4-120.1) (2 of 2) Courtesy of ASME

VesselComponent Location Origin of Stress Type of Stress Classification



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strength of the material. Examples of primary membrane stresses in pressure vshells are:

• Circumferential and longitudinal stress attributable to internal pressure

• Longitudinal stress in horizontal vessels due to bending between saddle supports

• Axial compression due to the weight of a vertical vessel

• Stresses in a nozzle neck in the area of reinforcement due to internal pressand to external forces and moments attributable to piping connections. (Extions are those related to discontinuity effects.)

• Axial tensile and compressive stresses due to wind and earthquake loads

The design limit for primary membrane stresses is the maximum allowable desstress for the material of construction at the design temperature. Continuous primary membrane stresses cannot exceed two-thirds of the yield strength. However, stresses that act intermittently and for relatively short durations (for example, those attributable to wind and earthquake loads) can be increased to 1.2 times the maximum allowable design stress.

Primary Bending StressesPrimary bending stresses vary from tension to compression through the cross-section of a pressure vessel shell component. They are generally at a maximumthe surface. Higher average stresses are required to produce failure by plastic mation in bending than for uniform tensile or compressive loads. Bending stresare most likely to be the predominant primary stress in the following cases:

1. The bending stress in the center of a flat head.

2. The bending stress between the ligaments of closely spaced openings.

The stress limits for components, when primary bending stresses predominate,1.5 times the maximum allowable design stress for the material of construction the design temperature. This higher stress limit is usually incorporated into the design rules and equations for components that conform to the acceptable desdetails depicted in the ASME Code. The allowable design stress can be multiplby 1.5 only if a stress analysis is made of the component.

Local Primary Membrane StressLocal primary membrane stress is a subcategory of primary membrane stress tdeveloped by sustained internal and external loads similar to primary membranstresses. A local primary membrane stress exceeds the stress limit for a primarmembrane stress, but as the higher stress is localized it can be redistributed tosurrounding portions of the pressure vessel if yielding occurs. Although the redbution of stress upon localized yielding normally prevents failure of the pressurevessel, the plastic deformation associated with such yielding is unacceptable. Tfore, the stress limit for a localized stress is set at 1.5 times the maximum allow



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stress of the material of construction at the design temperature, which can be ahigh as the minimum yield strength.

In order to prevent excessive elastic distortion, a local primary membrane stresnot permitted to extend in a longitudinal direction more than R(t)1/2, where R is the radius of curvature of the vessel component and t is its thickness. Furthermore,vidual regions of localized stress must be separated by at least 2.5 × R(t)1/2.

Examples of local primary membrane stresses in pressure vessels are:

• Membrane stress at head-to-shell junctions.• Membrane stress at conical-transition-to-cylindrical-shell junctions.• Membrane stress in the shell at nozzles.• Membrane stress at vessel supports or external attachments.

115 Secondary StressesSecondary stresses differ from primary stresses because they are self-limiting. Secondary stresses develop at structural discontinuities.

Examples of secondary stresses are:

1. Bending stresses at head-to-shell junctions.

2. Bending stress at conical-transition-to-cylindrical-shell junction.

3. Bending stress in the shell at nozzles.

4. Bending stress at vessel supports and external attachments.

5. Thermal stresses produced by temperature gradients in the shell, or by diffences in temperature between the nozzle and shell.

Unlike primary stresses, secondary stresses are reduced in magnitude by the loyielding, before gross plastic deformation or bursting can occur. The first appliction of load during hydrotest will generally suffice to significantly reduce the secondary stresses in a pressure vessel, but subsequent load applications coufurther reduce the secondary stresses.

The stress limit for secondary stresses is 3.0 times the maximum allowable desstress for the material of construction at the design temperature. Therefore, thesecondary stress is permitted to be as high as twice the yield strength, but it is reduced in magnitude by local yielding. Unless a detailed stress analysis is madstructural discontinuities that develop secondary stresses should be separated distance of at least 2.5 × R(t)1/2 to avoid additive effects that could increase the total secondary stress above 3.0 times the maximum allowable design stress.

A distinction must be made between local primary membrane stresses and secondary stresses. Like secondary stresses, local primary membrane stressesdevelop at structural discontinuities, and are essentially self-limiting. However, tare categorized as primary stresses because the plastic deformation associatethe yielding (required to redistribute the local membrane stress) may be excess



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Therefore, in effect, the membrane component of the stress developed by the sconstraint at structural discontinuities is categorized as a primary stress, wherethe bending component of the stress is categorized as a secondary stress.

116 Peak StressesPeak stresses in pressure vessels are generally the highest stresses that exist various separate components of a vessel. They are distinguished from primary secondary stresses in that they do not produce significant distortion, but they nenot be localized nor necessarily self-limiting. They are developed at locations ohigh stress concentration (i.e., acute structural discontinuities) and by certain tyof thermal stress. Peak stresses are of consequence only with regard to the poinitiation of fatigue failure under cyclic loading conditions, and brittle fracture if the material lacks adequate toughness. The stress limit for peak stresses is thrtimes the allowable design stress for the material of construction.

Examples of peak stresses in pressure vessels are:

1. Stresses at corners and fillets of nozzles.

2. Thermal stresses in the shell related to cladding or weld overlay.

3. Thermal stresses in the shell due to rapid change in temperature of vesselcontents.

120 Stresses in Cylindrical Shells

121 GeneralPressure vessels basically consist of a cylindrical or spherical body, with hemisical, ellipsoidal, torispherical, conical, toriconical, or flat end-closures. The varioshell components are usually welded together, forming a shell with a common rtional axis. Occasionally, components can be bolted together by utilizing flange

All structures with shapes resembling curved plates are referred to as shells. Wshells are formed of plate where the thickness is small in comparison with othedimensions, they are called “membranes.” This condition is defined when the rabetween the radius of curvature “R” and the wall thickness “t” is greater than 10

Stresses in thin shells, called membrane stresses, are average tension or compsion stresses acting tangent to the surface of the shell and are assumed to be edistributed through the wall thickness. Membrane stresses are calculated by neglecting bending. Bending stresses due to concentrated external loads are ointensity only in close proximity to the area where the load is applied.

122 Cylindrical Shells Under Internal PressureThe most important case in vessel design is a thin shell surface of revolution subjected to internal pressure. The internal pressure can be a uniform gas pres



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or a liquid pressure varying along the axis of rotation due to the liquid head. In tlatter case usually two calculations are performed to determine the stresses duthe equivalent gas pressure plus the stress in the lowest part of the shell due toliquid weight.

Stresses in a closed-end-cylindrical shell under internal pressure Pi, computed from the conditions of static equilibrium (Figure 100-2) are the longitudinal (meridionstress:

(Eq. 100-1)

where:Pi = Internal pressure

R = Radius

t = Thickness

and the circumferential (hoop) stress:

(Eq. 100-2)

As can be seen from these two equations, the hoop stress is always greater andetermines the required thickness of the shell.

The equations above are accurate for thin wall cylinders (R/t > 10) under internpressure. However, for thick wall cylinders (R/t < 10), the variation in stress fromthe inner to the outer surface becomes appreciable, and the above equations asatisfactory.

σL

PiR

2t---------=

Fig. 100-2 Stresses in a Cylindrical (Closed End) Shell Under Internal Pressure From Pressure Vessel Design Handbook by Henry Bednar, 2nd ed., ©1985. Used with the permission of VNR.

σt

PiR

t---------=



l and 3.

e

The Lamé, or thick-cylinder equations, are used to calculate the stresses (radiacircumferential) at any radius, r, in a thick wall cylinder as shown in Figure 100-

The circumferential stress is given by

(Eq. 100-3)

and the radial stress is given by

(Eq. 100-4)


a = Inside radius

b = Outside radius

These equations show that both stresses are maximum at the inner surface. Thmaximum tensile stress (σr) at the inner surface is

(Eq. 100-5)

Fig. 100-3 Stresses in a Thick-Walled Cylinder From Pressure Component Construction by John F. Harvey, ©1980. Used with the permission of VNR.

σt

a2Pi

b2

a2

–----------------- 1

b2

r2

-----+

=

σr

a2Pi

b2

a2

–----------------- 1

b2

r2

-----–

=

σtmaxPi a

2b

2+( )

b2

a2

–---------------------------=



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The radial stress (σr) is always a compressive stress and smaller than the maximtensile stress (σt max). The maximum tensile stress is always greater than the internal pressure, but approaches this value as the wall thickness increases. Thdifference between the minimum tensile stress at the outside surface and the maximum tensile stress at the inside surface is the magnitude of the internal prsure. Therefore, for very high internal pressures it is necessary to use comparahigh-yield materials.

For thin walls, there is little difference between the maximum tensile stress giveby the thick-cylinder equation and that given by the thin-cylinder or average-streequation. For thick walls, however, the difference between the values of the twoequations is significant. For example, at a wall thickness of 10% of the radius (R/t = 10), the maximum stress is only 5% higher than the average stress. Howat a ratio R/t = 6, the maximum stress is 37% higher than the average stress. Fthis reason the ASME-Code equations approximate the more accurate thick-waequations, and are used for all thicknesses.

130 Stresses in Formed and Flat Components

131 Spherical Shells and Hemispherical Heads Under Internal PressureA sphere is the ideal shape for containing internal pressure because: (1) the loadeveloped in the shell are lower than for any other shape; (2) a sphere has the lowest surface area for the volume contained; and (3) a sphere will have the lowweight. However, because spheres are more difficult to fabricate, hemisphericaheads fitted to cylindrical shells are more often used for large diameter or high sure vessels.

The stress determined from the diagram in Figure 100-4 is,

(Eq. 100-6)


R = Radius

t = Thickness

Note that the longitudinal stress (σL) and hoop stress (σt) are the same because of the uniform geometry of the sphere. Therefore, the maximum stress in a spherehalf that for a cylinder of the same diameter. Equation 100-6 assumes a uniformstress distribution through the thickness, and is adequate for relatively thin headAs the wall thickness increases with respect to the radius, however, the assumpthat the stress is uniformly distributed through the wall is invalid. A “thick sphereequation must then be used, having the maximum stress at the inside surface, similar to the equations for a thick-walled cylinder. Like the stress equations for

σL σt

PiR

2t---------= =



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ad.

ator Ph/t rs at

ess ause due

ts

the -l

cylindrical shells, the ASME Code uses an approximation of the thick-sphere eqtion and applies it to all thicknesses.

132 Ellipsoidal Heads Under Internal PressureEllipsoidal heads, shown in Figure 100-5, are frequently used for the end closucylindrical shells. Since the radius of curvature varies from point to point, the eqtions for calculating the stress are more complicated than for cylinders or spherBoth the meridional radius of curvature (RM) and the longitudinal radius of curva-ture (RL) vary gradually from point to point on the ellipse and therefore so do themeridional stress (σx) and the latitudinal stress (σϕ).

Figure 100-6 gives the equations for calculating the stresses in an ellipsoidal heThe meridional stress (σx) remains tensile throughout the ellipsoid for all R/h ratios, being maximum at the crown (point 1) and diminishing in value to a minimum at the equator (point 2) (see Figure 100-5).

The latitudinal hoop stress (σϕ) is also tensile in the crown region, but decreases toward the equator. For ratios R/h greater than 1.42, the hoop stress in the equarea becomes compressive. If R/h = 2, a maximum tensile meridional stress of occurs at point 1, and a compressive latitudinal stress of equal magnitude occupoint 2. As R/h increases further, the greatest stress in the crown will still be tension at point 1, but will be far exceeded in magnitude by the compressive strin the knuckle area at the equator. This is a potentially dangerous situation becthis compressible stress can cause local buckling of thin heads and local failureto the high shear stress developed. For this reason, the ASME Code restricts the major-to-minor-axis ratio of elliptical heads to a maximum of 2.

Using membrane stress alone for the design, without including the various effecof the discontinuity stresses at the head-to-shell junction, would result in insuffi-cient head thickness. To simplify the design procedure, the ASME Code relatesequation for the thickness of ellipsoidal heads to the tangential stress of a cylindrical shell of radius R (the major radius of the ellipse), modified by an empiricacorrective stress-intensification factor, “K.”

Fig. 100-4 Stresses in a Hemispherical Head Under Internal Pressure From Structural Analysis and Design of Process Equipment by Jawad and Farr, ©1984 by John Wiley & Sons, Inc.



Fig. 100-5 Geometry of Ellipsoidal Head From Pressure Vessel Design Handbook by Henry Bednar, 2nd ed., ©1985. Used with the permission of VNR.

Fig. 100-6 Equations for Calculating Stresses in Ellipsoidal Heads

At Any Point X

At Center of Head

At Tangent Line

σx σϕ

σx

PiRL2t-----------= σϕ

PRLt--------- 1

RL2Rm----------–

=

σx

PiR2

2th-----------= σϕ σx=

σx

PiR

2t--------= σϕPR

t------- 1R

2

2h2--------–

=



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133 Torispherical Heads Under Internal PressureA torispherical head, sometimes referred as a “dished head,” simulates an elliphead with a compound curve composed of a crown radius that is a spherical segment and a knuckle, as shown in Figure 100-7. The knuckle radius should blarge enough to minimize the latitudinal stress in this region. For this reason, thASME Code specifies a minimum knuckle radius of 6% of the crown radius. Thmaximum inside crown radius for a torispherical head equals the outside diameof the cylindrical shell it caps. Under internal pressure the maximum membranestress in the crown region is the same as in the cylindrical shell it caps. The typheads using the radii values approved by the ASME Code are usually called “ASME 6% Flanged and Dished Heads.”

The abrupt change in radius from L to r in torispherical heads introduces large discontinuity stresses which are absent in elliptical heads. This factor makes diheads suitable for use only in low pressure applications (under 150 psi), whereshallower depth and lower fabrication costs make them a desirable shape.

Figure 100-8 gives the equations for calculating the meridional stresses (σx) and latitudinal stresses (σϕ) in torispherical heads. The maximum calculated compressive latitudinal stress in the knuckle occurs at point “a.” In the spherical cap themeridional and latitudinal stresses are both tensile stresses. The actual comprestress in the knuckle is reduced by the tensile stress in the spherical segment, resulting lower membrane stress at point “a.”

As with the design of semiellipsoidal heads, the ASME Code uses a simplified procedure for stress calculations, introducing an empirical correction factor “M”

Fig. 100-7 Geometry of a Torispherical Head From Pressure Vessel Design Handbook by Henry Bednar, 2nd ed., ©1985. Used with the permission of VNR.



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lt of

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.

by

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into the equation for membrane stress in the crown region. This correction factocompensates for the discontinuity stresses at the shell-to-head junction. In pracapplications it has been found that the ASME Code equation for the thickness otorispherical head under internal pressure gives conservative results for the maof head designs, but is not adequate for large ratios of R/t.

During hydrotests, large, thin-wall torispherical heads have collapsed as a resuelastic buckling, plastic yielding, or a combination of both. ASME Code, SectionVIII, Division 2, Appendix 4, provides a method of checking the plastic collapsepressure of torispherical heads against the hydrotest pressure of the vessel.

134 Flat Plate ClosuresThe unstayed flat head, or cover, is a common type of closure for vessels. It maintegrally formed with the shell, or welded to it as shown in Figure 100-9. It can also be attached by bolts or some quick opening device. Plates may be arbitrarclassified into three groups:

1. Thick plates in which the shear stress is the most important.

2. Medium thickness plates in which bending stresses are the most important

3. Thin plates whose strength depends mainly on the direct tension producedthe stretching of middle plane.

Most actual flat plate closures for pressure vessels are included in the second g

The stresses in circular flat plates of constant thickness are calculated based uthe assumption that the edges are either simply supported, or fully fixed. In actu

Fig. 100-8 Equations for Calculating Stresses in Torispherical Heads

At Junction of Crown and Knuckle

In Crown

In Knuckle

At Tangent Line

σx σϕ

σx

PiL

2t-------= σϕ

PiL

4t------- 3LR---–

=

σx

PiL

2t-------= σϕ σx=

σx

PiL

2t-------= σϕ

PiL

t------- 1L2r-----–

=

σx

PiR

2t--------= σx

PiR

t--------=



on

enter

designs, neither of these edge conditions is actually realized, the actual conditibeing somewhere between.

With edges assumed simply supported, the maximum stress is located at the cand equals:

(Eq. 100-7)


D = Diameter

t = Thickness

With fully fixed edges, the maximum stress is radial and located at the edge:

Fig. 100-9 Integral or Welded Flat Heads From Structural Analysis and Design of Process Equipment by Jawad and Farr, ©1984 by John Wiley and Sons, Inc.

σmax 0.309PiDt----

2=



plate.

mit.

ns

-

er, ng ll, .

ion.

(Eq. 100-8)

where D is the diameter of the plate, P the pressure, and t the thickness of the

The basic equations used by the ASME Code introduce a variable “C” factor depending on the details of the corner construction. The maximum deflection islimited to one half of the thickness, and all stresses are kept within the elastic li

135 Conical Sections Under Internal PressureConical shapes are used mainly as bottom end closures, or as transition sectiobetween cylinders with different diameters. The circumferential stress (σt) and longitudinal stress (σL) in a conical section, as shown in Figure 100-10, are calculated by essentially the same equations as for cylindrical shells, in which R hasbeen replaced by R/cosα:

(Eq. 100-9)

(Eq. 100-10)


R = Radius

t = Thickness

α = half apex angle

The largest stresses, however, will occur at the junction of the cone to the cylindwhich must always be considered as part of the cone design. The end supportiforce at section “a-a” (Figure 100-10) is actually sustained by the cylindrical sheas shown in Figure 100-11. This arrangement will produce an unbalanced force(PRtanα)/2 pointing inward, which develops a compressive stress at the junctionThis force increases with the angle α and therefore, the ASME Code limits this angle to 30° and sets special rules (UA-5b and c) for reinforcement of the junctOtherwise, the Code uses the membrane-thickness formula to determine the maximum stress and the minimum thickness of a conical shell.

σmax 0.188PDt----

2=

σt

PiR

t αcos--------------=

σL

PiR

2t αcos------------------=



s ents ch ld nts s in ther

140 Discontinuity Stresses

141 GeneralPressure vessels consist of axially symmetrical elements of different geometrieand thicknesses, and sometimes different materials. If these individual componwere allowed to expand freely as separate sections under internal pressure, eaelement would have an edge radial displacement and an edge rotation that woudiffer from those of the adjacent component. However, since all these componeform a continuous structure and must deflect and rotate together, the differencemovement at junctions result in local deformations and induce local stresses. Oitems, such as stiffening rings and internal bulkheads, also affect the cylinder deformation and introduce local stresses.

Fig. 100-10 Geometry of a Conical Head From Pressure Vessel Design Handbook by Henry Bednar, 2nd ed., ©1985. Used with the permission of VNR.

Fig. 100-11 Force Diagram at Cone-to-Cylinder Junction From Pressure Vessel Design Handbook by Henry Bednar, 2nd ed., ©1985. Used with the permission of VNR.



., an

sure, ign. here ve.

n n

lu-and

and nts,

Stresses created by the interaction of two shell components at their junction (i.eabrupt change in geometry of the vessel shell, or a structural discontinuity) are called discontinuity stresses. Under static loads, such as constant internal presand with ductile materials, discontinuity stresses can be kept low by proper desThey become important, however, under cyclic loads, or at low temperatures wthe ductility of the material is reduced. Discontinuity stresses must be added tomembrane stresses developed by other loads, as discussed in Section 113 abo

There are two categories of structural discontinuities: gross and local (Figure 100-12).

• Gross structural discontinuities affect a relatively large portion of a structureand have significant effect on the overall stress pattern. All of the junctions between shell components fall into this category.

• Local structural discontinuities are sources of stress or strain intensificationthat affect only a small volume of material and do not have a significant effect upon the overall stress pattern. They usually produce peak stresses (Section 116).

142 Calculation of Discontinuity StressesDiscontinuity stresses can be evaluated using the general bending theory of thicylindrical shells. Since this method uses edge forces and moments as unknowquantities, it is called the Force Method.

The Kalnin's computer program, “Stress Analysis of Thin Elastic Shells of Revotion,” can be used to calculate all the stresses due to pressure, discontinuities, thermal loads. This program uses the Force Method to determine discontinuity stresses. The vessel is divided into simple shell elements, then the edge forcesmoments, or the elastic deformations for the matching edges of adjoining elemeare balanced in order to solve the problem.

Fig. 100-12 Structural Discontinuities in a Pressure Vessel From Structural Analysis and Design of Process Equipment by Jawad and Farr, ©1984 by John Wiley & Sons, Inc.



ns , ,

e d ction an

r gn

other

an in ads

-apex qua-al-

0°, a be ompli-

143 Discontinuities in Cylindrical ShellsDiscontinuities in cylindrical shells occur when the shell is constructed of portioof different thicknesses and/or different materials. If the cylinder is long enoughthe effect of the edge forces will dissipate to a small value within short distanceand their overall effect on the shell can be neglected.

144 Shell-to-Hemispherical Head JunctionA hemispherical head is almost always thinner than a cylindrical shell (see Section 130), and, therefore, a structural discontinuity exists at the junction of thhead with the shell. This discontinuity stress is negligible, however, when taperetransitions are used. Discontinuity stresses at a hemispherical head-to-shell junare lower than at the junction of the shell with any other type of closure. This is important factor in selecting a hemispherical head for a large diameter vessel subjected to high operating temperatures and high internal pressures.

145 Shell-to-Ellipsoidal Head JunctionFor a 2:1 elliptical head, the ASME Code equation gives a thickness equal to overy close to that of the adjoining shell. This value produces a satisfactory desiwith low, acceptable discontinuity stresses. Consequently, this type of head is commonly used for all pressure levels.

146 Shell-to-Torispherical Head JunctionDiscontinuity stresses in shell-to-torispherical head junctions are due to sharp changes in the radius of curvature at points “a” and “2” of Figure 100-7. Since these points are close together, the edge loadings at both locations affect eachto a large degree.

Since the total combined stress in the knuckle region is several times greater tha standard 2:1 elliptical head under the same internal pressure, torispherical heare only suitable for low pressure (<150 psi) applications.

147 Shell-to-Cone Junction Without KnuckleThe thickness of a conical head or sections under internal pressure, with a halfangle smaller than 30°, is calculated by simple ASME Code membrane-stress etions and the ASME Code rules for reinforcement at the junction. No special anysis 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 3more detailed analysis of discontinuity stresses is necessary. This analysis canmade by using the Force Method, but as the equations for this case are more ccated, 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.



od

ng a nd

of ,

bri-ction.

nd the .

ption ss, s

ns, d in

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

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

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

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

150 Stress Concentrations

151 GeneralThe normal equations for stresses in pressure vessels are based on the assumthat there is continuous elastic action throughout the member, and that the strefor simple tension and compression, is uniformly distributed over the entire crossection.

Abrupt changes in section geometries, however, can invalidate these assumptioleading to great irregularities in stress distribution, with large stresses develope

Fig. 100-13 Toriconical Reducer Courtesy of the ASME



tra-ns

also o the ic or

ady as a the ones.

deter-ne t to

The

ten-

and ing

rated

o the

le omes ein-

ay e

e e ngi-

ater

o

a small portion of the member. These are called peak stresses or stress concentions. In pressure vessels they occur at transitions between thick and thin portioof the shell, and at openings, nozzles, or other attachments.

The importance of these stresses depends not only on their absolute value, buton material properties, such as ductility, the relative proportion of the stressed tunstressed part of the member, and on the type of loading on the member (statcyclic).

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

Stress concentrations create peak stresses (Section 116), and they are used tomine the design fatigue life of the vessel. Besides keeping the primary membrastresses within the limits set by allowable tensile stresses, it is equally importankeep 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.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 insity factors and stress concentration factors in determining peak stresses.

152 Stresses at OpeningsAll pressure vessels must be provided with openings to get the process fluid in out, and to provide entry for maintenance and inspection. When a circular openis 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 fromthe opening, the stress decreases until the nominal stress (stress in the unperfoplate) is reached. The ratio of the maximum stress at the edge of the opening tnominal stress is the stress intensity, or concentration factor.

Figure 100-14 illustrates the concentration of stress for an opening of radius r incylindrical and spherical shells. This figure shows that at a distance from the hoedge equal to the radius of the hole, the effect of the opening on the stress becnegligible. This distance is usually accepted as the boundary limit for effective rforcement.

Occasionally, an elliptical opening is used for special purposes, such as a manwor handhole. For elliptical openings, the maximum stress occurs at the end of thminor axis. Because the hoop stress is always greatest in a cylindrical shell (seSection 120), the most favorable alignment for an elliptical opening is to have thminor axis of the ellipse perpendicular to the hoop direction (or parallel to the lotudinal axis of the vessel). Otherwise the stress concentration factor will be grethan for a circular opening. The minimum stress concentration is obtained by making the elliptical opening with the lengths of the axis inversely proportional t



re s an

for the ss s

of l

ing n in

g,

by

l, as d ed

the applied stresses: for a cylindrical vessel subjected to internal pressure, whethe hoop stress is double that of the meridional (longitudinal stress), this requireellipse with an axis ratio of 1:2.

153 Reinforced OpeningsBecause stresses around openings are higher than the normal design stressesplate thickness, additional material must be provided to carry the additional strein the shell around the opening. The additional material provided is referred to areinforcement.

The basic concept of reinforcement of openings is that the cross-sectional areamaterial removed by an opening must be replaced by adding additional materiaadjacent 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 theopening.

The two basic requirements for reinforcement are:

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

2. The reinforcing material must be placed immediately adjacent to the openinbut 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, orextra thickness in the shell and nozzle wall.

It is most common for the reinforcement to be added to the outside of the vesseshown in Figure 100-15a. However, on some vessels, the reinforcement is addeon the inside, as shown in Figure 100-15b. The best configuration is the “balanc

Fig. 100-14 Variations in Stress in Region of a Circular Hole in (a) Cylinder and (b) Sphere Subjected to Internal Pressure From Pressure Component Construction by John F. Harvey. ©1980. Used with the permission of VNR.



of ein-s ly. er

ld

reinforcement,” shown in Figure 100-15c, which consists of about 35% to 40% the reinforcement on the inside and the remainder on the outside. A balanced rforcement introduces very little local bending moments and stresses. The stresconcentration factor in this case is 20% lower than for outside reinforcement onIt may, however, be difficult to place reinforcement on the inside of a vessel, eithbecause the vessel interior is not accessible or because the reinforcement wouinterfere with the flow or drainage.

Fig. 100-15 Methods of Reinforcing Nozzle Openings with Reinforcement Pad

a) Reinforcement added to outside of opening

b) Reinforcement added to inside of opening

c) Reinforcement added to both inside and outside



d

n

lace

eld.

lap-ir

diam-.

Stress concentration factors can be further reduced by using integrally reinforcenozzles (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 betweethe shell and the nozzle minimizes stress concentrations due to discontinuities.

A common practice in vessel design, in excess of Code requirements, is to repall 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 w

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

Fig. 100-16 Methods of Adding Reinforcement Material ASME Code, Section VIII, Division 1, Figure UW-16.1. Courtesy of the ASME



ents. , and t of

sses sses.

ed by r pads al

iffer-iron-

n-to

e ter

sel su-tresses

ce,

ssel e

the is a-er to d

the rete

154 Stresses from Locally Applied External LoadsMost vessels are also subject to loadings at the supports, nozzles, and attachmThese loadings produce deflections, edge rotations, shears, bending momentsmembrane forces. The effect rapidly decreases with the distance from the poinapplication, where the maximum stress occurs. In practical applications, the number of variables is considerable, and some judgement must be exercised tochoose the important ones and eliminate those of minor importance.

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

If the maximum stress at the attachment is too great, the shell must be reinforca pad, or by increasing the thickness of the reinforcing pad required for internalpressure. To avoid stress concentrations at the corners of square or rectangulaor structural clips, provide a radius of five to ten times the pad thickness, generCompany practice.

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

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

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

Another location where thermal stresses are likely to occur in a hot pressure veis the support skirt. At the shell-skirt junction the temperature of the shell and thskirt will be nearly the same. However, the skirt temperature will decrease fromjoint down. The temperature difference causes a rotation of the skirt end, whichrestrained by the welded joint. In addition to the thermal stresses, radial deformtion of the shell under internal pressure will cause discontinuity stresses. In ordminimize thermal stresses at this location, the shell insulation is usually extendebelow the skirt-to-shell weld. The skirt should also be long enough to minimize temperature difference between the bolted-down base of the skirt and the concfoundation, in order to prevent any distortion and local thermal stress at this location.



pres-f lastic is

e.

,

tion.

l ts, as atmo-ssure.

al pres-es-ause s

re am-ntial that

e the es.

Under certain conditions, application of a steady mechanical load (like internal sure) to a vessel subject to cyclic operating temperature may produce cycling ocombined thermal and mechanical stresses and a progressive increase in the p(permanent) strain in the entire vessel. The action of cyclic, progressive yieldingcalled thermal ratcheting. It may lead to large distortions and ultimately to failur

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

160 Stresses in Pressure Vessel Shells Due to External Pressure

161 GeneralExternal pressure on a vessel most commonly occurs when a vacuum or partiavacuum is created inside of the vessel by (1) design, (2) discharge of its conten(3) steam-out cleaning (condensation of steam), or (4) mechanical action, suchon a compressor suction, during off-design events. In these circumstances, thespheric pressure surrounding the vessel becomes greater than the internal pre

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

162 Cylindrical Shells Subjected to External PressureThe 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 abovyield strength of the material, and the ordinary membrane stress equation appliThis 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:

(Eq. 100-11)

AK2----

tDo-------

2=



i-

terial.

l

for for a

that ge

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

(Eq. 100-12)

where E is the modulus of elasticity of the material.

163 Spherical Shells Subjected to External PressureThe critical compressive stress (σcr) for buckling (collapse) of a spherical shell hasbeen experimentally determined to be approximated by:

(Eq. 100-13)

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

164 Elliptical and Torispherical Heads Subjected to External PressureThe knuckles of elliptical and torispherical heads under external pressure are intension. The critical region of these heads with regard to buckling under externapressure is the central crown, which is a spherical segment. The critical stress buckling of the crown is essentially the same as approximated by the equation sphere (Section 162) using the crown radius of the head.

165 Conical Heads and Transitions Subjected to External PressureExperimental research that compares conical and cylindrical shells has shown the buckling of a conical shell is similar to that of a cylindrical shell with a lengthequal to the slant length of the cone and an outside diameter equal to the averaoutside diameter of the cone.

σc AEKE2

--------t

Do-------

2⋅=

σcr0.125E

R/t-----------------=


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