Chap 08

The Babcock & Wilcox Company

Steam 41 / Structural Analysis and Design 8-1

Chapter 8Structural Analysis and Design

Equipment used in the power, chemical, petroleumand cryogenic fields often includes large steel vessels.These vessels may require tons of structural steel fortheir support. Steam generating and emissions con-trol equipment, for example, may be comprised of pres-sure parts ranging from small diameter tubing to ves-sels weighing more than 1000 t (907 tm). A large fossilfuel boiler may extend 300 ft (91.4 m) above theground, requiring a steel support structure compa-rable to a 30 story building. To assure reliability, athorough design analysis of pressure parts and theirsupporting structural components is required.

Pressure vessel design and analysisSteam generating units require pressure vessel

components that operate at internal pressures of upto 4000 psi (27.6 MPa) and at steam temperatures upto 1100F (566C). Even higher temperature and pres-sure conditions are possible in advanced system de-signs. Maximum reliability can be assured only witha thorough stress analysis of the components. There-fore, considerable attention is given to the design andstress analysis of steam drums, superheater headers,heat exchangers, pressurizers and nuclear reactors.In designing these vessels, the basic approach is toaccount for all unknown factors such as local yield-ing and stress redistribution, variability in materialproperties, inexact knowledge of loadings, and inex-act stress evaluations by using allowable workingstresses that include appropriate factors of safety.

The analysis and design of complex pressure ves-sels and components such as the reactor closure head,shown in Fig. 1, and the fossil boiler steam drum,shown in Fig. 2, requires sophisticated principles andmethods. Mathematical equations based on the theoryof elasticity are applied to regions of discontinuities,nozzle openings and supports. Advanced computerizedstructural mechanics methods, such as the finite elementmethod, are used to determine complex vessel stresses.

In the United States (U.S.), pressure vessel con-struction codes adopted by state, federal and munici-pal authorities establish safety requirements for ves-sel construction. The most widely used code is theAmerican Society of Mechanical Engineers (ASME)

Boiler and Pressure Vessel Code. Key sections includeSections I, Rules for Construction of Power Boilers; III,Rules for Construction of Nuclear Power Plant Com-ponents; and VIII, Rules for Construction of PressureVessels. A further introduction to the ASME Code ispresented in Appendix 2.

Stress significanceStress is defined as the internal force between two

adjacent elements of a body, divided by the area overwhich it is applied. The main significance of a stressis its magnitude; however, the nature of the applied

Fig. 1 Head of nuclear reactor vessel.


8-2 Steam 41 / Structural Analysis and Design

load and the resulting stress distribution are also im-portant. The designer must consider whether the load-ing is mechanical or thermal, whether it is steady-stateor transient, and whether the stress pattern is uniform.

Stress distribution depends on the material prop-erties. For example, yielding or strain readjustmentcan cause redistribution of stresses.

Steady-state conditions An excessive steady-statestress due to applied pressure results in vessel mate-rial distortion, progresses to leakage at fittings andultimately causes failure in a ductile vessel. To pre-vent this type of failure a safety factor is applied tothe material properties. The two predominant prop-erties considered are yield strength, which establishesthe pressure at which permanent distortion occurs,and tensile strength, which determines the vesselbursting pressure. ASME Codes establish pressurevessel design safety factors based on the sophistica-tion of quality assurance, manufacturing control, anddesign analysis techniques.

Transient conditions When the applied stresses arerepetitive, such as those occurring during testing andtransient operation, they may limit the fatigue life ofthe vessel. The designer must consider transient con-ditions causing fatigue stresses in addition to thosecaused by steady-state forces.

Although vessels must have nozzles, supports andflanges in order to be useful, these features often em-body abrupt changes in cross-section. These changescan introduce irregularities in the overall stress pat-tern called local or peak stresses. Other constructiondetails can also promote stress concentrations which,in turn, affect the vessel’s fatigue life.

Strength theoriesSeveral material strength theories are used to de-

termine when failure will occur under the action ofmulti-axial stresses on the basis of data obtained fromuni-axial tension or compression tests. The three mostcommonly applied theories which are used to estab-lish elastic design stress limits are the maximum (prin-cipal) stress theory, the maximum shear stress theory,and the distortion energy theory.

Maximum stress theory The maximum stress theoryconsiders failure to occur when one of the three prin-cipal stresses (σ) reaches the material yield point (σy.p.)in tension:

σ σ= y.p. (1)

This theory is the simplest to apply and, with anadequate safety factor, it results in safe, reliable pres-sure vessel designs. This is the theory of strength usedin the ASME Code, Section I, Section VIII Division 1,and Section III Division 1 (design by formula Subsec-tions NC-3300, ND-3300 and NE-3300).

Maximum shear stress theory The maximum shearstress theory, also known as the Tresca theory,1 con-siders failure to occur when the maximum shear stressreaches the maximum shear stress at the yieldstrength of the material in tension. Noting that themaximum shear stress (τ) is equal to half the differ-ence of the maximum and minimum principal stresses,and that the maximum shear stress in a tension testspecimen is half the axial principal stress, the condi-tion for yielding becomes:

τσ σ σ

τ σ σ σ

= − =

= − =

max min y.p.

max min y.p.

2 22

(2)

The value 2τ is called the shear stress intensity. Themaximum shear stress theory predicts ductile mate-rial yielding more accurately than the maximum stresstheory. This is the theory of strength used in theASME Code, Section VIII Division 2, and Section IIIDivision 1, Subsection NB, and design by analysis,Subsections NC-3200 and NE-3200.

Distortion energy theory The distortion energytheory (also known as the Mises criterion1) considersyielding to occur when the distortion energy at a pointin a stressed element is equal to the distortion energyin a uni-axial test specimen at the point it begins to yield.While the distortion energy theory is the most accuratefor ductile materials, it is cumbersome to use and is notroutinely applied in pressure vessel design codes.

Design criteria1

To determine the allowable stresses in a pressurevessel, one must consider the nature of the loading andthe vessel response to the loading. Stress interpreta-tion determines the required stress analyses and theallowable stress magnitudes. Current design codesestablish the criteria for safe design and operation ofpressure vessels.

Stress classifications Stresses in pressure vesselshave three major classifications: primary, secondaryand peak.

Primary stresses (P) are caused by loadings whichare necessary to satisfy the laws of equilibrium withapplied pressure and other loads. These stresses arefurther divided into general primary membrane (Pm),local primary membrane (PL) and primary bending (Pb)stresses. A primary stress is not self-limiting, i.e., if thematerial yields or is deformed, the stress is not reduced.A good example of this type of stress is that produced

Fig. 2 Fossil fuel boiler steam drum.



by internal pressure such as in a steam drum. Whenit exceeds the vessel material yield strength, perma-nent distortion appears and failure may occur.

Secondary stresses (Q), due to mechanical loads ordifferential thermal expansion, are developed by theconstraint of adjacent material or adjacent compo-nents. They are self-limiting and are usually confinedto local areas of the vessel. Local yielding or minordistortion can reduce secondary stresses. Althoughthey do not affect the static bursting strength of avessel, secondary stresses must be considered in es-tablishing its fatigue life.

Peak stresses (F ) are concentrated in highly local-ized areas at abrupt geometry changes. Although noappreciable vessel deformations are associated withthem, peak stresses are particularly important inevaluating the fatigue life of a vessel.

Code design/analysis requirements Allowable stresslimits and design analysis requirements vary withpressure vessel design codes.

According to ASME Code, Section I, the minimumvessel wall thickness is determined by evaluating thegeneral primary membrane stress. This stress, limitedto the allowable material tension stress S, is calculatedat the vessel design temperature. The Section I regu-lations have been established to ensure that second-ary and peak stresses are minimized; a detailed analy-sis of these stresses is normally not required.

The design criteria of ASME Code, Sections VIIIDivision 1, and Section III Division 1 (design by for-mula Subsections NC-3300, ND-3300 and NE-3300),are similar to those of Section I. However, they requirecylindrical shell thickness calculations in the circum-ferential and longitudinal directions. The minimumrequired pressure vessel wall thickness is set by themaximum stress in either direction. Section III Divi-sion 1 and Subsections NC-3300 and ND-3300 per-mit the combination of primary membrane and pri-mary bending stresses to be up to 1.5 S at design tem-perature. Section VIII Division 1 permits the combi-nation of primary membrane and primary bendingstresses to be 1.5 S at temperatures where tensile oryield strength sets the allowable stress S, and a valuesmaller than 1.5 S at temperatures where creep gov-erns the allowable stress.

ASME Code, Section VIII Division 2 provides for-mulas and rules for common configurations of shellsand formed heads. It also requires detailed stressanalysis of complex geometries with unusual or cyclicloading conditions. The calculated stress intensities areassigned to specific categories. The allowable stressintensity of each category is based on a multiplier ofthe Code allowable stress intensity value. The Codeallowable stress intensity, Sm, is based on the materialyield strength, Sy, or tensile strength, Su. (See Table 1.)

The factor k varies with the type of loading:

k Loading

1.0 sustained

1.2 sustained and transient

1.25 hydrostatic test

1.5 pneumatic test

The design criteria for ASME Code, Section III Divi-sion 1, Subsection NB and design by analysis Subsec-tions NC-3200 and NE-3200 are similar to those forSection VIII Division 2 except there is less use of designformulas, curves, and tables, and greater use of designby analysis in Section III. The categories of stresses andstress intensity limits are the same in both sections.

Stress analysis methodsStress analysis of pressure vessels can be performed

by analytical or experimental methods. An analyticalmethod, involving a rigorous mathematical solutionbased on the theory of elasticity and plasticity, is themost direct and inexpensive approach when the prob-lem is adaptable to such a solution. When the prob-lem is too complex for this method, approximate ana-lytical structural mechanics methods, such as finiteelement analysis, are applied. If the problem is beyondanalytical solutions, experimental methods must beused.

Mathematical formulas2 Pressure vessels are com-monly spheres, cylinders, ellipsoids, tori or compositesof these. When the wall thickness is small compared toother dimensions, vessels are referred to as membraneshells. Stresses acting over the thickness of the vesselwall and tangential to its surface can be represented bymathematical formulas for the common shell forms.

Pressure stresses are classified as primary membranestresses since they remain as long as the pressure isapplied to the vessel. The basic equation for the lon-gitudinal stress σ1 and hoop stress σ2 in a vessel ofthickness h, longitudinal radius r1, and circumferen-tial radius r2, which is subject to a pressure P, shownin Fig. 3 is:

σ σ1

1

2

2r rPh

+ = (3)

From this equation, and by equating the total pres-sure load with the longitudinal forces acting on a trans-verse section of the vessel, the stresses in the commonlyused shells of revolution can be found.

Table 1Code-Allowable Stress Intensity

Basis for Allowable Stress Intensity Allowable Value at k = 1.0 Category Value (Lesser Value)

General primary kSm 2/3 Sy or 1/3 Su

membrane (Pm )

Local primary 3/2 kSm Sy or 1/2 Su

membrane (PL )

Primary membrane 3/2 kSm Sy or 1/2 Su

plus primary bending (Pm + Pb)

Range of primary plus 3 Sm 2 Sy or Su

secondary (Pm + Pb + Q)



1. Cylindrical vessel – in this case, r1 = ∞, r2 = r, and

σ1 2= Pr

h(4)

σ2 = Prh

(5)

2. Spherical vessel – in this case, r1 = r2 = r, and

σ1 2= Pr

h(6)

σ2 2= Pr

h(7)

3. Conical vessel – in this case, r1 = ∞, r2 = r/cos αwhere α is half the cone apex angle, and

σα1 2

= Prcosh (8)

σα2 = Pr

cosh (9)

4. Ellipsoidal vessel – in this case (Fig. 4), the instan-taneous radius of curvature varies with each po-sition on the ellipsoid, whose major axis is a andminor axis is b, and the stresses are given by:

σ12

2= Pr

h(10)

σ2 222

12= −

Ph

rrr (11)

At the equator, the longitudinal stress is the sameas the longitudinal stress in a cylinder, namely:

σ1 2= Pa

h(12)

and the hoop stress is:

σ2

2

212

= −

Pah

ab (13)

When the ratio of major to minor axis is 2:1, thehoop stress is the same as that in a cylinder of thesame mating diameter, but the stress is compres-sive rather than tensile. The hoop stress rises rap-idly when the ratio of major to minor axis exceeds2:1 and, because this stress is compressive, buck-ling instability becomes a major concern. For thisreason, ratios greater than 2:1 are seldom used.

5. Torus – in this case (Fig. 5), Ro is the radius of thebend centerline, θ is the angular hoop locationfrom this centerline and:

σ1 2= Pr

h(14)

σθθ2 2

2= ++

Prh

R r sinR r sin

o

o(15)

The longitudinal stress remains uniform around thecircumference and is the same as that for a straightcylinder. The hoop stress, however, varies for differ-ent points in the torus cross-section. At the bendcenterline, it is the same as that in a straight cylin-der. At the outside of the bend, it is less than this andis at its minimum. At the inside of the bend, or crotch,the value is at its maximum. Hoop stresses are depen-

Fig. 3 Membrane stress in vessels (courtesy Van Nostrand Reinhold).2

Fig. 4 Stress in an ellipsoid.2



dent on the sharpness of the bend and are inverselyproportional to bend radii. In pipe bending operations,the material thins at the outside and becomes thickerat the crotch of the bend. This is an offsetting factorfor the higher hoop stresses that form with smallerbend radii.

Thermal stresses result when a member is re-strained as it attempts to expand or contract due to atemperature change, ∆T. They are classified as sec-

ondary stresses because they are self-limiting. If thematerial is restricted in only one direction, the stressdeveloped is:

σ α= ± E T∆ (16)

where E is the modulus of elasticity and α is the coef-ficient of thermal expansion. If the member is re-stricted from expanding or contracting in two direc-

ASME Code calculations

In most U.S. states and Canadian provinces lawshave been established requiring that boilers and pres-sure vessels comply with the rules for the design andconstruction of boilers and pressure vessels in the ASMECode. The complexity of these rules and the amount ofanalysis required are inversely related to the factors ofsafety which are applied to the material properties usedto establish the allowable stresses. That is, when thestress analysis is simplified, the factor of safety is larger.When the stress analysis is more complex, the factoryof safety is smaller. Thus, overall safety is maintainedeven though the factor of safety is smaller. For condi-tions when material tensile strength establishes theallowable stress, ASME Code, Section IV, Rules for Con-struction of Heating Boilers, requires only a simplethickness calculation with a safety factor on tensilestrength of 5. ASME Code, Section I, Rules for Construc-tion of Power Boilers and Section VIII, Division 1, Rulesfor Construction of Pressure Vessels, require a morecomplex analysis with additional items to be considered.However, the factor of safety on tensile strength is re-duced to 3.5. Section III, Rules for Construction ofNuclear Components and Section VIII, Division 2, Rulesfor Construction of Pressure Vessels require extensiveanalyses which are required to be certified by a regis-tered professional engineer. In return, the factor of safetyon tensile strength is reduced even further to 3.0.

When the wall thickness is small compared to the di-ameter, membrane formulas (Equations 4 and 5) may beused with adequate accuracy. However, when the wallthickness is large relative to the vessel diameter, usuallyto accommodate higher internal design pressure, the mem-brane formulas are modified for ASME Code applications.

Basically the minimum wall thickness of a cylindri-cal shell is initially set by solving the circumferentialor hoop stress equation assuming there are no additionalloadings other than internal pressure. Other loadingsmay then be considered to determine if the initial mini-mum required wall thickness has to be increased tokeep calculated stresses below allowable stress values.

As an example, consider a Section VIII, Division 1,pressure vessel with no unreinforced openings and noadditional loadings other than an internal design pres-sure of 1200 psi at 500F. The inside diameter is 10 in.and the material is SA-516, Grade 70 carbon steel. Thereis no corrosion allowance required by this applicationand the butt weld joints are 100% radiographed. Whatis the minimum required wall thickness needed? Theequation for setting the minimum required wall thick-ness in Section VIII, Division 1, of the Code (paragraphUG-27(c)(l), 2001 Edition) is:

tPR

SE P=

− 0 6.

where

t = minimum required wall thickness, in.P = internal design pressure, psiR = inside radius, in.S = allowable stress at design temperature, psi (Sec-

tion II, Part D) = 20,000 psiE = lower of weld joint efficiency or ligament efficiency

(fully radiographed with manual penetrations) =1.0

For the pressure vessel described above:

P = 1200 psiR = 5 in.S = 20,000 psiE = 1.0

t = ( ) − ( ) =( ) ( )( , ) . ( . )

.1200

in.5

20 000 1 0 0 6 12000 311

Using commercial sizes, this plate thickness probablywould be ordered at 0.375 in.

If Equation 5 for simple hoop stress (see Figure be-low) is used alone to calculate the plate thickness usingthe specified minimum tensile strength of SA-516, Grade70 of 70,000 psi, the thickness h would be evaluated to be:

h = =1200 570 000

0 0857( )

,.

Therefore, the factor of safety (FS) based on tensilestrength is:

FS0.311

0.0857.6= = 3



tions, as is the case in pressure vessels, the resultingstress is:

σα

µ= ±

−E T∆1 (17)

where µ is Poisson’s ratio.

These thermal stress equations consider full re-straint, and therefore are the maximum that can becreated. When the temperature varies within a mem-ber, the natural growth of one fiber is influenced bythe differential growth of adjacent fibers. As a result,fibers at high temperatures are compressed and thoseat lower temperatures are stretched. The generalequations for radial (σr), tangential (σt), and axial (σz)thermal stresses in a cylindrical vessel subject to aradial thermal gradient are:

σαµr a

b

a

rEr

r ab a

Trdr Trdr=−( )

−−

−

∫ ∫1 2

2 2

2 2 (18)

σαµt a

b

a

rEr

r ab a

Trdr Trdr Tr=−( )

+−

+ −

∫ ∫1 2

2 2

2 22

(19)

σα

µz a

bEb a

Trdr T=−( ) −

−

∫1

22 2 (20)

whereE = modulus of elasticityµ = Poisson’s ratior = radius at any locationa = inside radiusb = outside radiusT = temperatureFor a cylindrical vessel in which heat is flowing

radially through the walls under steady-state condi-tions, the maximum thermal stresses are:

σα

µta

ainsideET

nba

bb a

nba

( ) =−( )

−−

2 11

2 2

2 2

��

(21)

σα

µtb

aoutsideET

nba

ab a

nba

( ) =−( )

−−

2 11

2 2

2 2

��

(22)

For relatively thin tubes and Ta > Tb, this can be sim-plified to:

σα

µta

E T= −−( )∆

2 1 (23)

σα

µtb

E T=−( )∆

2 1 (24)

To summarize, the maximum thermal stress for athin cylinder with a logarithmic wall temperaturegradient is one half the thermal stress of an elementrestrained in two directions and subjected to a tem-perature change ∆T (Equation 17). For a radial ther-mal gradient of different shape, the thermal stress canbe represented by:

σα

µ=

−KE T∆1 (25)

where K ranges between 0.5 and 1.0.

Alternating stresses resulting from cyclic pressurevessel operation may lead to fatigue cracks at highstress concentrations. Fatigue life is evaluated by com-paring the alternating stress amplitude with designfatigue curves (allowable stress versus number ofcycles or σ-N curves) experimentally established forthe material at temperature. A typical σ-N designcurve for carbon steel is shown in Fig. 6 and can beexpressed by the equation:

σaa

a

E

Nn

dTS d=

−

+

4100

10001� . ( ) (26)

where

σa = allowable alternating stress amplitudeE = modulus of elasticity at temperatureN = number of cyclesda = percent reduction in areaTS = tensile strength at temperature

The two controlling parameters are tensile strengthand reduction in area. Tensile strength is controllingin the high cycle fatigue region, while reduction inarea is controlling in low cycle fatigue. The usual di-

Fig. 5 Hoop stress variation in a bend.2

Fig. 6 Design fatigue curve.N, Number of Allowed Cycles

(6,890)

(689)

(68.9)

(6.90)

700F(371C) 800F

(427C)

900F(482C) 1000F

(538C)Val

ues

ofA

ltern

atin

g S

tress

Inte

nsity

,

, psi

(MPa)

a



vision between low and high cycle fatigue is 105 cycles.Pressure vessels often fall into the low cycle fatiguecategory, thereby demonstrating the importance of thematerial’s ability to deform in the plastic range with-out fracturing. Lower strength materials, with theirgreater ductility, have better low cycle fatigue resis-tance than do higher strength materials.

Practical operating service conditions subject manyvessels to the random occurrence of a number of stresscycles at different magnitudes. One method of ap-praising the damage from repetitive stresses to a ves-sel is the criterion that the cumulative damage fromfatigue will occur when the summation of the incre-ments of damage at the various stress levels exceedsunity. That is:

nN

=∑ 1 (27)

where n = number of cycles at stress σ, and N = num-ber of cycles to failure at the same stress σ. The ration/N is called the cycle damage ratio since it representsthe fraction of the total life which is expended by thecycles that occur at a particular stress value. Thevalue N is determined from σ-N curves for the mate-rial. If the sum of these cycle ratios is less than unity,the vessel is considered safe. This is particularly im-portant in designing an economic and safe structurewhich experiences only a relatively few cycles at ahigh stress level and the major number at a relativelylow stress level.

Discontinuity analysis method At geometricaldiscontinuities in axisymmetric structures, such as theintersection of a hemispherical shell element and acylindrical shell element (Fig. 7a), the magnitude andcharacteristic of the stress are considerably differentthan those in elements remote from the discontinu-ity. A linear elastic analysis method is used to evalu-ate these local stresses.

Discontinuity stresses that occur in pressure ves-sels, particularly axisymmetric vessels, are determinedby a discontinuity analysis method. A discontinuitystress results from displacement and rotation incom-patibilities at the intersection of two elements. Theforces and moments at the intersection (Fig. 7c) areredundant and self-limiting. They develop solely toensure compatibility at the intersection. As a conse-quence, a discontinuity stress can not cause failure inductile materials in one load application even if themaximum stress exceeds the material yield strength.Such stresses must be considered in cyclic load appli-cations or in special cases where materials can notsafely redistribute stresses. The ASME Code refers todiscontinuity stresses as secondary stresses. The ap-plication to the shell of revolution shown in Fig. 7outlines the major steps involved in the method usedto determine discontinuity stresses.

Under internal pressure, a sphere radially expandsapproximately one half that of a cylindrical shell (Fig.7b). The difference in free body displacement resultsin redundant loadings at the intersection if Elements(1) and (2) are joined (Fig. 7c). The final displacementand rotation of the cylindrical shell are equal to the

free body displacement plus the displacements due tothe redundant shear force Vo and redundant bendingmoment Mo (Fig. 7d).

The direction of the redundant loading is unknownand must be assumed. A consistent sign conventionmust be followed. In addition, the direction of loadingon the two elements must be set up consistently be-cause Element (1) reacts Element (2) loading and viceversa. If Mo or Vo as calculated is negative, the correctdirection is opposite to that assumed.

In equation form then, for Element (1):

δ δ β βδ δFINAL 1 FREE1= − +V o M oV M1 1 (28)

γ γ β βγ γFINAL 1 FREE1= + −V o M oV M1 1 (29)

Similarly for Element (2):

δ δ β βδ δFINAL 2 FREE2= + +V o M oV M2 2 (30)

γ γ β βγ γFINAL 2 FREE2= + +V o M oV M2 2 (31)

where

δµ

FREE1

2PR= −

Et

12 (32)

δ µ

γ γ

FREE 2

FREE1 FREE 2

PR

in this case

= −( )= =

2

21

0Et (33)

The constants β are the deflections or rotations due toloading per unit of perimeter, and are referred to asinfluence coefficients. These constants can be deter-mined for a variety of geometries, including rings andthin shells of revolution, using standard handbooksolutions. For example:

βδV1 = radial displacement of Element (1) due to unitshear load

Fig. 7 Discontinuity analysis.



βδM1 = radial displacement of Element (1) due to unitmoment load

βγV1 = rotation of Element (1) due to unit shear load βγM1 = rotation of Element (1) due to unit moment

loadBecause δFINAL 1 = δFINAL 2 and γFINAL 1 = γFINAL 2 from com-patibility requirements, Equations 28 through 31 canbe reduced to two equations for two unknowns, Vo andMo, which are solved simultaneously. Note that thenumber of equations reduces to the number of redun-dant loadings and that the force F can be determinedby static equilibrium requirements.

Once Vo and Mo have been calculated, handbooksolutions can be applied to determine the resultingmembrane and bending stresses. The discontinuitystress must then be added to the free body stress toobtain the total stress at the intersection.

Although the example demonstrates internal pres-sure loading, the same method applies to determin-ing thermally induced discontinuity stress. For morecomplicated geometries involving four or more un-known redundant loadings, commercially availablecomputer programs should be considered for solution.

Finite element analysis When the geometry of a com-ponent or vessel is too complex for classical formulasor closed form solutions, finite element analysis (FEA)can often provide the required results. FEA is a pow-erful numerical technique that can evaluate structuraldeformations and stresses, heat flows and tempera-tures, and dynamic responses of a structure. BecauseFEA is usually more economical than experimentalstress analysis, scale modeling, or other numericalmethods, it has become the dominant sophisticatedstress analysis method.

During product development, FEA is used to pre-dict performance of a new product or concept beforebuilding an expensive prototype. For example, a de-sign idea to protect the inside of a burner could be ana-lyzed to find out if it will have adequate cooling and fa-tigue life. FEA is also used to investigate field problems.

To apply FEA, the structure is modeled as an as-sembly of discrete building blocks called elements. Theelements can be linear (one dimensional truss or beam),plane (representing two dimensional behavior), or solid(three dimensional bricks). Elements are connected attheir boundaries by nodes as illustrated in Fig. 8.

Except for analyses using truss or beam elements,the accuracy of FEA is dependent on the mesh den-sity. This refers to the number of nodes per modeledvolume. As mesh density increases, the result accu-racy also increases. Alternatively, in p-method analy-sis, the mesh density remains constant while increasedaccuracy is attained through mathematical changesto the solution process.

A computer solution is essential because of the numer-ous calculations involved. A medium sized FEA may re-quire the simultaneous solution of thousands of equa-tions, but taking merely seconds of computer time. FEAis one of the most demanding computer applications.

FEA theory is illustrated by considering a simplestructural analysis with applied loads and specifiednode displacements. The mathematical theory is es-sentially as follows.

For each element, a stiffness matrix satisfying thefollowing relationship is found:

k d r { } = { } (34)

where

[k] = an element stiffness matrix. It is square anddefines the element stiffness in each direction(degree of freedom)

{d} = a column of nodal displacements for one element{r} = a column of nodal loads for one element

The determination of [k] can be very complex and itstheory is not outlined here. Modeling the whole struc-ture requires that:

K D R { } = { } (35)

where

[K] = structure stiffness matrix; each member of [K]is an assembly of the individual stiffnesscontributions surrounding a given node

{D} = column of nodal displacements for the struc-ture

{R} = column of nodal loads on the structure

In general, neither {D} nor {R} is completely known.Therefore, Equation 35 must be partitioned (rear-ranged) to separate known and unknown quantities.Equation 35 then becomes:

K K

K KDD

RR

s

o

o

s

11 12

21 22

=

(36)

where

Ds = unknown displacementsDo = known displacementsRs = unknown loadsRo = known loads

Equation 36 represents the two following equations:

K D K D Rs o o11 12 { } + { } = { } (37)

K D K D Rs o s21 22 { } + { } = { } (38)

Equation 37 can be solved for Ds and Equation 38 canthen be solved for Rs.

Using the calculated displacements {D}, {d} can be

Fig. 8 Finite element model composed of brick elements.



found for each element and the stress can be calcu-lated by:

σ{ } = { }E B d (39)

where

{σ} are element stresses[E] and [B] relate stresses to strains and strains todisplacements respectively

FEA theory may also be used to determine tempera-tures throughout complex geometric components. (Seealso Chapter 4.) Considering conduction alone, thegoverning relationship for thermal analysis is:

C T K T Q { } + { } = { }� (40)

where

[C ] = system heat capacity matrix{ �T } = column of rate of change of nodal

temperatures[K ] = system thermal conductivity matrix{T } = column of nodal temperatures{Q } = column of nodal rates of heat transfer

In many respects, the solution for thermal analy-sis is similar to that of the structural analysis. One im-portant difference, however, is that the thermal solu-tion is iterative and nonlinear. Three aspects of a ther-mal analysis require an iterative solution.

First, thermal material properties are temperaturedependent. Because they are primary unknowns, tem-perature assumptions must be made to establish theinitial material properties. Each node is first given anassumed temperature. The first thermal distributionis then obtained, and the calculated temperatures areused in a second iteration. Convergence is attainedwhen the calculated temperature distributions fromtwo successive iterations are nearly the same.

Second, when convective heat transfer is accountedfor, heat transfer at a fluid boundary is dependent onthe material surface temperature. Again, because tem-peratures are the primary unknowns, the solution mustbe iterative.

Third, in a transient analysis, the input parameters,including boundary conditions, may change with time,and the analysis must be broken into discrete steps.Within each time step, the input parameters are heldconstant. For this reason, transient thermal analysisis sometimes termed quasi-static.

FEA applied to dynamic problems is based upon thedifferential equation of motion:

M D C D K D R { } + { } + { } = { }�� (41)

where[M ] = structure lumped mass matrix[C ] = structure damping matrix[K ] = structure stiffness matrix{R} = column of nodal forcing functions

{ }, { } { }D D D� ��and are columns of nodal displacements,velocities, and accelerations, respectively.

Variations on Equation 41 can be used to solve forthe natural frequencies, mode shapes, and responsesdue to a forcing function (periodic or nonperiodic), orto do a dynamic seismic analysis.

Limitations of FEA involve computer and humanresources. The user must have substantial experienceand, among other abilities, he must be skilled in se-lecting element types and in geometry modeling.

In FEA, result accuracy increases with the num-ber of nodes and elements. However, computationtime also increases and handling the mass of data canbe cumbersome.

In most finite element analyses, large scale yield-ing (plastic strain) and deformations (including buck-ling instability), and creep are not accounted for; thematerial is considered to be linear elastic. In a linearstructural analysis, the response (stress, strain, etc.)is proportional to the load. For example, if the appliedload is doubled, the stress response would also double.For nonlinear analysis, FEA can also be beneficial.Recent advancements in computer hardware and soft-ware have enabled increased use of nonlinear analy-sis techniques.

Although most FEA software has well developed threedimensional capabilities, some pressure vessel analysesare imprecise due to a lack of acceptance criteria.

Computer software consists of commercially avail-able and proprietary FEA programs. This software canbe categorized into three groups: 1) preprocessors, 2)finite element solvers, and 3) postprocessors.

A preprocessor builds a model geometry and appliesboundary conditions, then verifies and optimizes themodel. The output of a finite element solver consistsof displacements, stresses, temperatures, or dynamicresponse data.

Postprocessors manipulate the output from the fi-nite element solver for comparison to acceptance cri-teria or to make contour map plots.

Application of FEA Because classical formulas andshell analysis solutions are limited to simple shapes,FEA fills a technical void and is applied in responseto ASME Code requirements. A large portion of TheBabcock & Wilcox Company’s (B&W) FEA supportspressure vessel design. Stresses can be calculated nearnozzles and other abrupt geometry changes. In addi-tion, temperature changes and the resulting thermalstresses can be predicted using FEA.

The raw output from a finite element solver can notbe directly applied to the ASME Code criteria. Thestresses or strains must first be classified as membrane,bending, or peak (Fig. 9). B&W pioneered the classi-fication of finite element stresses and these proceduresare now used throughout the industry.

Piping flexibility, for example, is an ideal FEA ap-plication. In addition, structural steel designers relyon FEA to analyze complex frame systems that sup-port steam generation and emissions control equipment.

Finite element analysis is often used for preliminaryreview of new product designs. For example, Fig. 10shows the deflected shape of two economizer fin con-figurations modeled using FEA.



Fracture mechanics methodsFracture mechanics provides analysis methods to

account for the presence of flaws such as voids orcracks. This is in contrast to the stress analysis meth-ods discussed above in which the structure was con-sidered to be free of those kinds of defects. Flaws maybe found by nondestructive examination (NDE) orthey may be hypothesized prior to fabrication. Frac-ture mechanics is particularly useful to design or evalu-ate components fabricated using materials that are moresensitive to flaws. Additionally, it is well suited to theprediction of the remaining life of components undercyclic fatigue and high temperature creep conditions.

During component design, the flaw size is hypoth-esized. Allowable design stresses can be determinedknowing the lower bound material toughness fromaccepted design procedures in conjunction with a fac-tor of safety.

Fracture mechanics can be used to evaluate theintegrity of a flawed existing structure. The defect,usually found by NDE, is idealized according to ac-cepted ASME practices. An analysis uses design orcalculated stresses based on real or hypothesized loads,and material properties are found from testing a speci-men of similar material. Determining allowable flawsizes strongly relies on accurate material properties andthe best estimates of structural stresses. Appropriatesafety factors are then added to the calculations.

During inspection of power plant components, mi-nor cracks or flaws may be discovered. However, theflaws may propagate by creep or fatigue and becomesignificant. The remaining life of components can not beaccurately predicted from stress/cycles to failure (σ/N)curves alone. These predictions become possible usingfracture mechanics.

Linear elastic fracture mechanics The basic conceptof linear elastic fracture mechanics (LEFM) was origi-nally developed to quantitatively evaluate suddenstructural failure. LEFM, based on an analysis of thestresses near a sharp crack, assumes elastic behaviorthroughout the structure. The stress distribution nearthe crack tip depends on a single quantity, termed the

stress intensity factor, KI. LEFM assumes that un-stable propagation of existing flaws occurs when thestress intensity factor becomes critical; this criticalvalue is the fracture toughness of the material KIC.

The theory of linear elastic fracture mechanics,LEFM, is based on the assumption that, at fracture,stress σ and defect size a are related to the fracturetoughness KIC, as follows:

K C aI = σ π (42a)

and

K KI IC≥ at failure (42b)

The critical material property, KIC, is compared tothe stress intensity factor of the cracked structure, KI,to identify failure potential. KI should not be confusedwith the stress intensity used elsewhere in ASMEdesign codes for analysis of unflawed structures. Theterm C, accounting for the geometry of the crack andstructure, is a function of the crack size and relevantstructure dimensions such as width or thickness.

C is exactly 1.0 for an infinitely wide center crackedpanel with a through-wall crack length 2a, loaded intension by a uniform remote stress σ. The factor Cvaries for other crack geometries illustrated in Fig. 11.Defects in a structure due to manufacture, in-serviceenvironment, or in-service cyclic fatigue are usuallyassumed to be flat, sharp, planar discontinuities wherethe planar area is normal to the applied stress.

ASME Code procedures for fracture mechanics de-sign/analysis are presently given in Sections III andXI which are used for component thicknesses of atleast 4 in. (102 mm) for ferritic materials with yieldstrengths less than 50,000 psi (344.7 MPa) and forsimple geometries and stress distributions. The basicconcepts of the Code may be extended to other ferriticmaterials (including clad ferritic materials) and morecomplex geometries; however, it does not apply toaustenitic or high nickel alloys. These procedures pro-vide methods for designing against brittle fracturesin structures and for evaluating the significance offlaws found during in-service inspections.

Fig. 9 Classification of finite element stress results on a vessel cross-section for comparison to code criteria.3

Fig. 10 Economizer tube and fin (quarter symmetry model) deformedshape plots before (upper) and after (lower) design modifications.



The ASME Code, Section III uses the principles oflinear elastic fracture mechanics to determine allow-able loadings in ferritic pressure vessels with an as-sumed defect. The stress intensity factors (KI) are cal-culated separately for membrane, bending, and ther-mal gradient stresses. They are further subdividedinto primary and secondary stresses before summingand comparison to the allowable toughness, KIR. KIR isthe reference critical stress intensity factor (tough-ness). It accounts for temperature and irradiationembrittlement effects on toughness. A safety factor of2 is applied to the primary stress components and afactor of 1 is applied to the secondary components.

To determine an operating pressure that is below thebrittle fracture point, the following approach is used:1. A maximum flaw size is assumed. This is a semi-

elliptical surface flaw one fourth the pressure ves-sel wall thickness in depth and 1.5 times the thick-ness in length.

2. Knowing the specific material’s nil ductility tem-perature, and the design temperature KIR can befound from the Code.

3. The stress intensity factor is determined based onthe membrane and bending stresses, and the ap-propriate correction factors. Additional determi-nants include the wall thickness and normal stressto yield strength ratio of the material.

4. The calculated stress intensity is compared to KIR.

The ASME Code, Section XI provides a procedureto evaluate flaw indications found during in-serviceinspection of nuclear reactor coolant systems. If anindication is smaller than certain limits set by SectionXI, it is considered acceptable without further analy-sis. If the indication is larger than these limits, Sec-tion XI provides information that enables the follow-ing procedure for further evaluation:

1. Determine the size, location and orientation of theflaw by NDE.

2. Determine the applied stresses at the flaw location(calculated without the flaw present) for all normal(including upset), emergency and faulted conditions.

3. Calculate the stress intensity factors for each ofthe loading conditions.

4. Determine the necessary material properties, in-cluding the effects of irradiation. A reference tem-perature shift procedure is used to normalize thelower bound toughness versus temperaturecurves. These curves are based on crack arrest andstatic initiation values from fracture toughness tests.The temperature shift procedure accounts for heatto heat variation in material toughness properties.

5. Using the procedures above, as well as a procedurefor calculating cumulative fatigue crack growth,three critical flaw parameters are determined:af = maximum size to which the detected flaw

can grow during the remaining service ofthe component

acrit = maximum critical size of the detected flawunder normal conditions

ainit = maximum critical size for nonarrestinggrowth initiation of the observed flaw un-der emergency and faulted conditions

6. Using these critical flaw parameters, determine ifthe detected flaw meets the following conditionsfor continued operation:

a a

a af crit

f init

<

<

0 1

0 5

.

. (43)

Elastic-plastic fracture mechanics (EPFM) LEFMprovides a one parameter failure criterion in terms ofthe crack tip stress intensity factor (KI), but is limitedto analyses where the plastic region surrounding thecrack tip is small compared to the overall component di-mensions. As the material becomes more ductile and thestructural response becomes nonlinear, the LEFM ap-proach loses its accuracy and eventually becomes invalid.

A direct extension of LEFM to EPFM is possible byusing a parameter to characterize the crack tip regionthat is not dependent on the crack tip stress. Thisparameter, the path independent J-integral, can char-acterize LEFM, EPFM, and fully plastic fracture me-chanics. It is capable of characterizing crack initiation,growth, and instability. The J-integral is a measureof the potential energy rate of change for nonlinearelastic structures containing defects.

The J-integral can be calculated from stressesaround a crack tip using nonlinear finite elementanalysis. An alternate approach is to use previouslycalculated deformation plasticity solutions in terms ofthe J-integral from the Electric Power Research In-stitute (EPRI) Elastic-Plastic Fracture AnalysisHandbook.4

The onset of crack growth is predicted when:

J JI IC≥ (44)

The material property JIC is obtained using AmericanSociety for Testing and Materials (ASTM) test E813-89, and JI is the calculated structural response.

Stable crack growth occurs when:

J a P J aI R,( ) = ( )∆

Fig. 11 Types of cracks.



anda a ao= + ∆ (45)

where

a = current crack sizeP = applied remote loadJR(∆a) = material crack growth resistance (ASTM

test standard E1152-87)∆a = change in crack sizeao = initial crack size

For crack instability, an additional criterion is:

∂ ∂ ≥ ∂ ∂J a J aR/ / (46)

Failure assessment diagrams Failure assessment dia-grams are tools for the determination of safety mar-gins, prediction of failure or plastic instability andleak-before-break analysis of flawed structures. Thesediagrams recognize both brittle fracture and net sec-tion collapse mechanisms. The failure diagram (seeFig. 12) is a safety/failure plane defined by the stressintensity factor/toughness ratio (Kr) as the ordinateand the applied stress/net section plastic collapse stressratio (Sr) as the abscissa. For a fixed applied stress anddefect size, the coordinates Kr, Sr are readily calcu-lable. If the assessment point denoted by these coor-dinates lies inside the failure assessment curve, nocrack growth can occur. If the assessment point liesoutside the curve, unstable crack growth is predicted.The distance of the assessment point from the failureassessment curve is a measure of failure potential ofthe flawed structure.

In a leak-before-break analysis, a through-wallcrack is postulated. If the resulting assessment pointlies inside the failure assessment curve, the crack willleak before an unstable crack growth occurs.

The deformation plasticity failure assessment dia-gram (DPFAD)5 is a specific variation of a failure as-sessment diagram. DPFAD follows the British PD 6493R-66 format, and incorporates EPFM deformation plas-ticity J-integral solutions. The DPFAD curve is deter-mined by normalizing the deformation plasticity J-

integral response of the flawed structure by its elasticresponse. The square root of this ratio is denoted byKr. The Sr coordinate is the ratio of the applied stressto the net section plastic collapse stress. Various com-puter programs are available which automate thisprocess for application purposes.

Subcritical crack growth Subcritical crack growthrefers to crack propagation due to cyclic fatigue, stresscorrosion cracking, creep crack growth or a combina-tion of the three. Stress corrosion cracking and creepcrack growth are time based while fatigue crackgrowth is based on the number of stress cycles.

Fatigue crack growth Metal fatigue, although stud-ied for more than 100 years, continues to plague struc-tures subjected to cyclic stresses. The traditional ap-proach to prevent fatigue failures is to base the allow-able fatigue stresses on test results of carefully madelaboratory specimens or representative structural com-ponents. These results are usually presented in cyclicstress versus cycles to failure, or σ/N, curves.

The significant events of metal fatigue are crackinitiation and subsequent growth until the net sectionyields or until the stress intensity factor of the struc-ture exceeds the material resistance to fracture. Tra-ditional analysis assumes that a structure is initiallycrack free. However, a structure can have cracks thatoriginate during fabrication or during operation.Therefore, fatigue crack growth calculations are re-quired to predict the service life of a structure.

Fatigue crack growth calculations can 1) determinethe service life of a flawed structure that (during itslifetime) undergoes significant in-service cyclic load-ing, or 2) determine the initial flaw size that can betolerated prior to or during a specified operating pe-riod of the structure.

The most useful way of presenting fatigue crackgrowth rates is to consider them as a function of thestress intensity difference, ∆K, which is the difference be-tween the maximum and minimum stress intensity factors.

To calculate fatigue crack growth, an experimen-tally determined curve such as Fig. 13 is used. Thevertical axis, da/dN, is the crack growth per cycle.ASME Code, Section XI contains similar growth ratecurves for pressure vessel steels.

Creep crack growth Predicting the remaining life offossil power plant components from creep rupture dataalone is not reliable. Cracks can develop at critical lo-cations and these cracks can then propagate by creepcrack growth.

At temperatures above 800F (427C), creep crackgrowth can cause structural components to fail. Op-erating temperatures for certain fossil power plantcomponents range from 900 to 1100F (482 to 593C).At these temperatures, creep deformation and crackgrowth become dependent on strain rate and timeexposure. Macroscopic crack growth in a creepingmaterial occurs by nucleation and joining ofmicrocavities in the highly strained region ahead ofthe crack tip. In time dependent fracture mechanics(TDFM), the energy release rate (power) parameter Ct

correlates7 creep crack growth through the relationship:

da dt bC tq/ = (47)

Fig. 12 Deformation plasticity failure assessment diagram in terms ofstable crack growth.



By using the energy rate definition, Ct can be de-termined experimentally from test specimens. The con-stants b and q are determined by a curve fit technique.Under steady-state creep where the crack tip stressesno longer change with time, the crack growth can becharacterized solely by the path independent energyrate line integral C*, analogous to the J-integral.

C* and Ct can both be interpreted as the differencein energy rates (power) between two bodies with in-crementally differing crack lengths. Furthermore, C*characterizes the strength of the crack tip stress sin-gularity in the same manner as the J-integral char-acterizes the elastic-plastic stress singularity.

The fully plastic deformation solutions from theEPRI Elastic-Plastic Fracture Handbook can then beused to estimate the creep crack tip steady-state pa-rameter, C*.

Significant data support Ct as a parameter for corre-lating creep crack growth behavior represented by Equa-tion 47. An approximate expression8 for Ct is as follows:

C C t tt T

nn= ( ) +

−−* /31 1 (48)

where tT is the transition time given by:

tK

n ECT

I=−( )+( )

1

1

2 2µ

* (49)

and µ is Poisson’s ratio, and n is the secondary creeprate exponent.

For continuous operation, Equation 48 is integratedover the time covering crack growth from the initialflaw size to the final flaw size. The limiting final flawsize is chosen based on fracture toughness or insta-bility considerations, possibly governed by cold startupconditions. For this calculation, fracture toughnessdata such as KIC, JIC or the JR curve would be used ina failure assessment diagram approach to determinethe limiting final flaw size.

Construction featuresAll pressure vessels require construction features

such as fluid inlets and outlets, access openings, andstructural attachments at support locations. Theseshell areas must have adequate reinforcement andgradual geometric transitions which limit local stressesto acceptable levels.

Openings Openings are the most prevalent con-struction features on a vessel. They can become ar-eas of weakness and may lead to unacceptable localdistortion, known as bell mouthing, when the vesselis pressurized. Such distortions are associated withhigh local membrane stresses around the opening.Analytical studies have shown that these high stressesare confined to a distance of approximately one holediameter, d, along the shell from the axis of the open-ing and are limited to a distance of 0.37 (dtnozzle)1/2 nor-mal to the shell.

Reinforcement to reduce the membrane stress nearan opening can be provided by increasing the vessel

wall thickness. An alternate, more economical stressreduction method is to thicken the vessel locallyaround the nozzle axis of symmetry. The reinforcingmaterial must be within the area of high local stressto be effective.

The ASME Code provides guidelines for reinforc-ing openings. The reinforcement must meet require-ments for the amount and distribution of the addedmaterial. A relatively small opening [approximatelyd<0.2 (Rts)1/2 where R is mean radius of shell and ts isthickness of shell] remote from other locally stressedareas does not require reinforcement.

Larger openings are normally reinforced as illus-trated in Figs. 14a and 14b. It is important to avoidexcessive reinforcement that may result in high sec-ondary stresses. Fig. 14c shows an opening with overreinforcement and Fig. 14a shows one with well pro-portioned reinforcement. Fig. 14b also shows a balanceddesign that minimizes secondary stresses at the nozzle/shell juncture. Designs a and b, combined with gener-ous radii r, are most suitable for cyclic load applications.

The ligament efficiency method is also used to com-pensate for metal removed at shell openings. Thismethod considers the load carrying ability of an areabetween two points in relation to the load carryingability of the remaining ligament when the two pointsbecome the centers of two openings. The ASME Codeguidelines used in this method only apply to cylindri-

Fig. 13 Relationship between da/dN and ∆K as plotted on logarithmiccoordinates.



cal pressure vessels where the circumferential stress istwice the longitudinal stress. In determining the thick-ness of such vessels, the allowable stress in the thick-ness calculation is multiplied by the ligament efficiency.

Nozzle and attachment loadings When external load-ings are applied to nozzles or attachment components,local stresses are generated in the shell. Several typesof loading may be applied, such as sustained, tran-sient and thermal expansion flexibility loadings. Thelocal membrane stresses produced by such loadingsmust be limited to avoid unacceptable distortion dueto a single load application. The combination of localmembrane and bending stresses must also be limitedto avoid incremental distortion under cyclic loading.

Finally, to prevent cyclic load fatigue failures, thenozzle or attachment should include gradual transi-tions which minimize stress concentrations.

Pressure vessels may require local thickening atnozzles and attachments to avoid yielding or incre-mental distortion due to the combined effects of ex-ternal loading, internal pressure, and thermal load-ing. Simple procedures to determine such reinforce-ment are not available, however FEA methods can beused. The Welding Research Council (WRC) BulletinNo. 107 also provides a procedure for determining lo-cal stresses adjacent to nozzles and rectangular at-tachments on cylindrical and spherical shells.

The external loadings considered by the WRC arelongitudinal moment, transverse moment, torsionalmoment, and axial force. Stresses at various insideand outside shell surfaces are obtained by combiningthe stresses from the various applied loads. These ex-ternal load stresses are then combined with internal pres-sure stresses and compared with allowable stress limits.

Use of the WRC procedure is restricted by limitationson shell and attachment parameters; however, experi-mental and theoretical work continues in this area.

Structural support componentsPressure vessels are normally supported by saddles,

cylindrical support skirts, hanger lugs and brackets,ring girders, or integral support legs. A vessel hasconcentrated loads imposed on its shell where thesesupports are located. Therefore, it is important thatthe support arrangements minimize local stresses inthe vessel. In addition, the components must providesupport for the specified loading conditions and with-stand corresponding temperature requirements.

Design criteriaStructural elements that provide support, stiffen-

ing, and/or stabilization of pressure vessels or compo-nents may be directly attached by welding or bolting.They can also be indirectly attached by clips, pins, orclamps, or may be completely unattached thereby trans-ferring load through surface bearing and friction.

Loading conditions In general, loads applied tostructural components are categorized as dead, live,or transient loads. Dead loads are due to the force ofgravity on the equipment and supports. Live loadsvary in magnitude and are applied to produce themaximum design conditions. Transient loads are timedependent and are expected to occur randomly for thelife of the structural components. Specific loadingsthat are considered in designing a pressure componentsupport include:

1. weight of the component and its contents duringoperating and test conditions, including loads dueto static and dynamic head and fluid flow,

2. weight of the support components,3. superimposed static and thermal loads induced by

the supported components,4. environmental loads such as wind and snow,5. dynamic loads including those caused by earth-

quake, vibration, or rapid pressure change,6. loads from piping thermal expansion,Fig. 14 Nozzle opening reinforcements.



7. loads from expansion or contraction due to pres-sure, and

8. loads due to anchor settlement.

Code design/analysis requirements Code require-ments for designing pressure part structural supportsvary. The ASME Code, Section I, only covers pressurepart attaching lugs, hangers or brackets. These mustbe properly fitted and must be made of weldable andcomparable quality material. Only the weld attach-ing the structural member to the pressure part is con-sidered within the scope of Section I. Prudent designof all other support hardware is the manufacturer’sresponsibility.

The ASME Code, Section VIII, Division 1, does notcontain design requirements for vessel supports; how-ever, suggested rules of good practice are presented.These rules primarily address support details whichprevent excessive local shell stresses at the attach-ments. For example, horizontal pressure vessel sup-port saddles are recommended to support at least onethird of the shell circumference. Rules for the saddledesign are not covered. However, the Code refers thedesigner to the Manual of Steel Construction, pub-lished by the American Institute of Steel Construction(AISC). This reference details the allowable stressdesign (ASD) method for structural steel building de-signs. When adjustments are made for elevated tem-peratures, this specification can be used for design-ing pressure vessel support components. Similarly,Section VIII, Division 2, does not contain design meth-ods for vessel support components. However, materi-als for structural attachments welded to pressure com-ponents and details of permissible attachment weldsare covered.

Section III of the ASME Code contains rules for thematerial, design, fabrication, examination, and instal-lation of certain pressure component and piping sup-ports. The supports are placed within three categories:

1. plate and shell type supports, such as vessel skirtsand saddles, which are fabricated from plate andshell elements,

2. linear supports which include axially loadedstruts, beams and columns, subjected to bending,and trusses, frames, rings, arches and cables, and

3. standard supports (catalog items) such as constantand variable type spring hangers, shock arrest-ers, sway braces, vibration dampers, clevises, etc.

The design procedures for each of these supporttypes are:

1. design by analysis including methods based on maxi-mum shear stress and maximum stress theories,

2. experimental stress analysis, and3. load rating by testing full size prototypes.

The analysis required for each type of support de-pends on the class of the pressure component beingsupported.

Typical support design considerationsDesign by analysis involves determining the stresses

in the structural components and their connections byaccepted analysis methods. Unless specified in an

applicable code, choosing the analysis method is thedesigner’s prerogative. Linear elastic analysis (coveredin depth here), using the maximum stress or maximumshear stress theory, is commonly applied to plate, shelltype and linear type supports. As an alternate, themethod of limit (plastic) analysis can be used forframed linear structures when appropriate load ad-justment factors are applied.

Plate and shell type supports Cylindrical shell skirtsare commonly used to support vertical pressure ves-sels. They are attached to the vessel with a minimumoffset in order to reduce local bending stresses at thevessel skirt junction. This construction also permitsradial pressure and thermal growth of the supportedvessel through bending of the skirt. The length of thesupport is chosen to permit this bending to occur safely.See Fig. 15 for typical shell type support skirt details.

In designing the skirt, the magnitudes of the loadsthat must be supported are determined. These nor-mally include the vessel weight, the contents of thevessel, the imposed loads of any equipment supportedfrom the vessel, and loads from piping or other attach-ments. Next a skirt height is set and the forces andmoments at the skirt base, due to the loads applied,are determined. Treating the cylindrical shell as abeam, the axial stress in the skirt is then determinedfrom:

σ = − ±PA

McI

v (50)

where

σ = axial stress in skirtPv = total vertical design loadA = cross-sectional areaM = moment at base due to design loadsc = radial distance from centerline of skirtI = moment of inertia

For thin shells (R/t > 10), the equation for the axialstress becomes:

σπ π

= − ±PRt

MR t

v

2 2 (51)

where

R = mean radius of skirtt = thickness of skirt

Because the compressive stress is larger than the ten-sile stress, it usually controls the skirt design. Usingthe maximum stress theory for this example, the skirtthickness is obtained by:

tPRF

MR F

v

A A

= +2 2π π (52)

where

FA = allowable axial compressive stress

The designer must also consider stresses caused bytransient loadings such as wind or earthquakes. Fi-nally, skirt connections at the vessel and support basemust be checked for local primary and secondary



bending stresses. The consideration of overall stresslevels provides the most accurate design.

Local thermal bending stresses often occur becauseof a temperature difference between the skirt and sup-port base. The magnitudes of these bending stressesare dependent upon the severity of this axial thermalgradient; steeper gradients promote higher stresses.To minimize these stresses, the thermal gradient atthe junction can be reduced by full penetration weldsat the skirt to shell junction, which permit maximumconduction heat flow through the metal at that point,and by selective use of insulation in the crotch regionto permit heat flow by convection and radiation. De-pending on the complexity of the attachment detail,the discontinuity stress analysis or the linear elasticfinite element method is used to solve for the thermalbending stresses.

Linear type supports Utility fossil fuel-fired steamgenerators contain many linear components that sup-port and reinforce the boiler pressure parts. For ex-ample, the furnace enclosure walls, which are con-structed of welded membraned tube panels, must bereinforced by external structural members (buckstays)to resist furnace gas pressure as well as wind and seis-mic forces. (See Chapter 23.) Similarly, chambers, suchas the burner equipment enclosure (windbox), requireinternal systems to support the enclosure and its con-tents as well as to reinforce the furnace walls. Thedesign of these structural systems is based on linearelastic methods using maximum stress theory allow-able limits.

The buckstay system is typically comprised of hori-zontally oriented beams or trusses which are attached

to the outside of the furnace membraned vertical tubewalls. As shown in Fig. 16, the buckstay ends are con-nected to tie bars that link them to opposing wallbuckstays thereby forming a self-equilibrating struc-tural system. The furnace enclosure walls are continu-ously welded at the corners creating a water-cooled,orthotropic plate, rectangular pressure vessel. Thestrength of the walls in the horizontal direction is con-siderably less than in the vertical direction, thereforethe buckstay system members are horizontally oriented.

The buckstay spacing is based on the ability of theenclosure walls to resist the following loads:

1. internal tube design pressure (P),2. axial dead loads (DL),3. sustained furnace gas pressure (PLs),4. transient furnace gas pressure (PLT),5. wind loads (WL), and6. seismic loads (EQ).

The buckstay elevations are initially establishedbased on wall stress checks and on the location of nec-essary equipment such as sootblowers, burners, accessdoors, and observation ports. These establishedbuckstay elevations are considered as horizontal sup-ports for the continuous vertical tube wall. The wallis then analyzed for the following load combinationsusing a linear elastic analysis method:

1. DL + PLs + P,2. DL + PLs + WL + P,3. DL + PLs + EQ + P, and4. DL + PLT + P.

Buckstay spacings are varied to assure that the wallstresses are within allowable design limits. Addition-ally, their locations are designed to make full use ofthe structural capability of the membraned walls.

The buckstay system members, their end connec-tions, and the wall attachments are designed for themaximum loads obtained from the wall analysis. Theyare designed as pinned end bending members accord-ing to the latest AISC ASD specification. This specifi-cation is modified for use at elevated temperatures anduses safety factors consistent with ASME Code, Sec-tions I and VIII. The most important design consider-ations for the buckstay system include:

1. stabilization of the outboard beam flanges or trusschords to prevent lateral buckling when subjectedto compression stress,

2. the development of buckstay to tie bar end con-nections and buckstay to wall attachments thatprovide load transfer but allow differential expan-sion between connected elements, and

3. providing adequate buckstay spacing and stiffnessto prevent resonance due to low frequency com-bustion gas pressure pulsations common in fossilfuel-fired boilers.

Fig. 15 Support skirt details.2



References1. Farr, J.R., and Jawad, M.H., Structural Analysis andDesign of Process Equipment, Second Ed., John Wileyand Sons, Inc., New York, New York, January, 1989.2. Harvey, J.F., Theory and Design of Pressure Vessels,Van Nostrand Reinhold Company, New York, New York,1985.3. Kroenke, W.C., “Classification of Finite ElementStresses According to ASME Section III Stress Categories,”Pressure Vessels and Piping, Analysis and Computers,American Society of Mechanical Engineers (ASME), June,1974.4. Kumar, V., et al., “An Engineering Approach for Elas-tic-Plastic Fracture Analysis,” Report EPRI NP-1931, Elec-tric Power Research Institute (EPRI), Palo Alto, Califor-nia, July, 1981.

5. Bloom, J.M., “Deformation Plasticity Failure Assess-ment Diagram,” Elastic Plastic Fracture Mechanics Tech-nology, ASTM STGP 896, American Society for Testingand Materials, Philadelphia, Pennsylvania, 1985.6. “Guidance on Methods for Assessing the Acceptabilityof Flaws in Fusion Welded Structure,” PD 6493:1991 Weld-ing Standards Committee, London, England, United King-dom, August 30, 1991.7. Saxena, A., “Creep Crack Growth Under Non-Steady-State Conditions,” Fracture Mechanics, Vol. 17, ASTM STP905, Philadelphia, Pennsylvania, 1986.8. Bassani, J.L., Hawk, D.E., and Saxena, A., “Evalua-tion of the Ct Parameter for Characterizing Creep CrackGrowth Rate in the Transient Region,” Third InternationalSymposium on Nonlinear Fracture Mechanics, ASTMSTP 995, Philadelphia, Pennsylvania, 1989.

BibliographyManual of Steel Construction (M016): Includes Code ofStandard Practice, Simple Shears, and Specification forStructural Joints Using ASTM A325 or A490 Bolts, NinthEd., American Institute of Steel Construction, July 1, 1989.Cook, R.D. et al., Concepts and Applications of FiniteElement Analysis, Fourth Ed., Wiley Publishers, NewYork, New York, October, 2001.Harvey, J.F., Theory and Design of Pressure Vessels,Second Ed., Chapman and Hall, London, England, UnitedKingdom, December, 1991.

Mershon, J.L., et al., “Local Stresses in Cylindrical ShellsDue to External Loadings on Nozzles,” Welding ResearchCouncil (WRC) Bulletin No. 297, Supplement to WRC Bulle-tin No. 107 (Revision 1), August, 1984, revised September, 1987.Thornton, W.A., Manual of Steel Construction: Load andResistance Factor Design (Manual of Steel Construction),Third Ed., American Institute of Steel Construction(AISC), November 1, 2001.Wichman, K.R., Hopper, A.G., and Mershon, J.L., “LocalStresses in Spherical and Cylindrical Shells Due to Ex-ternal Loadings,” Welding Research Council (WRC), Bul-letin No. 107, August, 1965, revised March, 1979, updatedOctober, 2002.

Fig. 16 Typical buckstay elevation, plan view.



A large steam drum is being lifted within power plant structural steel.

Chap 08

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