Pressure vessel design using boundary elementmethod with optimization

R.E. Dippery, Jr. & D. SrivastavaKettering University, Michigan, USA

Abstract

Design of pressure vessels is covered by references such as the ASME PressureVessel Code and textbooks devoted to pressure vessel design. Detailed stressanalysis, particularly in the area of discontinuities, is generally left to the designengineer. The type discontinuity addressed in this paper is the design of boltedflanges for a pressure vessel. Work for this project involves optimization of the hubcontour using the ASMEZ Pressure Vessel Code requirements as constraints. Thispaper summa&es the initial work, developmentoftheBEM (BEASY) models andvefificafion of the model to classical techniques such as a “Roarkl’ [ 123. This two-dimensional model will later be expanded to a U l , three-dimension model; and,will will provide data for establishing allowable defti sizes, based upon inspectiontechniques and design life.

Pressure vessel design

Design of pressure vessels is governed by the ASME pressure vessel code [l].Other ttxtbooh such as Farr [2], Moss [3], Chuse [4], Harvey [5], Bednar [6], andGill [7], provide valuable insight and guidelines to pressure vessel desigrt Boltedflange analysis is discussed in machine design textbooks such- as Norton [8] orspeciality books such as Bickford or Blake [9, lo] or Company design practices orcriteria. Detailed stress analysis, particularly in the area of discontinuities, isgenerally left to the design engineer. Different type stresses are defined byAppendix 4 of the ASME code [l]. Stress limits, (allowable stress magnitudes),based upon the type stress, are addressed by Appendix 4 (Marulatmy Design Basedon St&zw AncJysis) of the ASME code [ 1). These are discussed later in this paper.

268

The type discontinuity addread in this paper is one associated withdesign of bolted flanges for a pressure vessel. Figure la illutrates one type designwhich consists of the shell welded to the flange. Figure lb ill- another typedesign, a hubbed flange, with the weld located away from the shell/flangediscontinuity. This is done to locate the weld in an area of lower bending stress,improving the strength of the joint; and, also to locate it in an area where it mayrequire less weld material (cost) and can be more easily inspected

Figure la - Basic Configuration

Figure lb - wubbed Cunfiguration

6 Flange Outer RadiusGC Bolt Circle Radiusr Shell (Vessel) Inner Radiust or h Shell (Wall) ThicknessH Flange Thic~sx Distance From Joint (Discontinuity) to Weld

Bending moments at a ciicontinuity, such as a flange, will generally belocalanddiminish~~litudeasthedistancefrornthediscontiouityisincreasedRe5erring toTitnoshenko[ 111, when the term px = 3.0, the moment effect is almostzero.

Technologv XIV 269d ”

B4 3(1-v*)Z-( h) 2

px= IOn

V Poisson’s Ratioa Shell Radiush Vessel (Shell) ThicknessX Axial Distance from Discontinuity

Using the ASME Code, previously mentioned references, and handbookssuch asRoark[ 12lpressure vessel design could be a very complex task. With theadvent of the computer age, techniques such as the finite element method (FEM)and boundary element method (BEM) became very valuable design and analysisaids.

Section VIII, Division 2 of the ASME Code defines several category ofstresses: Primary, Secondary, and Peak. A primary stress is a normal or shearstress developed by the imposed loading and necessary to satisfl the laws ofequilibrium, such as the hoop (primary membrane) stress resulting from internalpressure in a shell. Secondary stresses are normal or shear stress developed by theconstraint of adjacent parts or the self-constrain of a structure, a bending stress ata gross structural discontinuity. The basic characteristic of a secondary stress is itis self-limiting, local yielding and minor distortions can satisfy the conditionswhich cause to stress to occur. Apeak stress is a stress which does not cause anynoticeable distortion and is undesirable because it may be a possible source of afatigue crack or brittle fracture, for example, the stress at a local structuraldiscontinuity. Definitions of all terms and tables of combinations and allowablestresses are provided in Appendix 4 of Section VIII, Division 2. Provisions forplastic, limit, experimental, shakedown, and fatigue analysis are also available inthis same section of the Code.

Coupled with the stress analysis are design considerations for failureanalysis. Allowable defti limits must be determined by the designer, even withthe Code allowable limits. Teak before failure”, design life, damage tolerance areall factors which must be considered in the design process. Once again textbooksand handbooks, such as Collins [ 131, Dowling [ 141, Ta& and Paris [ 151, Maddox[ 161, Brooks and Choudhury [ 171, and Barson and Rolfe [18] are available fordesign reference.

Boundary element analysis with optimization

The use of BEM is not new to pressure vessel analysis as evidenced by

270 Element Technology XIVI

Trevelyan [19] and Floyd [20]. Fracture and crack growth using BEM is alsoevidenced by textbooks such as Prasad [2 I], Aliabadi [22][23], Monahan [24], andL&a0 [25]. A test model, as shown by Figure 2, has been run to veri@ BEMresults. The model was based upon a flanged and bolted pipe. The model wasveri&d using equations from Table XIII, case 32 of Roark [ 121. Results of thismodel were comparable to the “hand calculations” of Roark

Like the boundary element method, optimization techniques have beenenhanced by the continued growth of computers. Vande@Us [26] is one of those

Figure2-BEASY Model

who has been part of this growth as well as others suchas Chanrupatala and Belegundu [27J. The work beingperformed in conjunction with this paper is based upondesign of a hubbed flange, subject to the requirementsof the ASME Code, using the boundary elementmethod. BEASY@ [28] and VisualDOC@ [29] are thesoftware codes selected for this work In other words,what is being accomplished is to optimize a designbased upon BEM analysis with constraints imposed bycodes such as the ASME Pressure Vessel Code.

A criterion or objective Xmction” must bedetermined which will sati@ inequality; and, possibly,equality constraints. In turn, the objective must beminimized (or maximized, depending upon theproblem.) For a simple function, this meansdetermining where the first derivative is zero, and ifthe second derivative is positive or negative at thosepoints. Furthermoix, at those points, the design or

objective must satisfy all constraints. That is, the solution must be feasible.The actual problem is somewhat more complex. The first question

becomes what is to be minimized? In this case, it will be weight. Although, withsufficient time and thought, this can be translated into cost, considering f&ricationcosts, inspection costs, and material costs. Weight should provide a good workingmodel. Constraints will be to satisfy the stress limits of the ASME Code and theweld to be in a low bending stress area (px r 3.0). As the work progresses, costand multi-objective function problems will be developed. The Gnal step will be tointroduw crack propagation constraints into the models.

Conclusions

Initial results show theboundary element method will provide accurate predictionsof the stresses in a pressure vessel flange. With further development, an optimumhub contour and weld location, subject to pressure vessel code and other

Technologv XIVc 27 I”

constraints will be obtained. Once this methodology has been established, thework will be expanded to three-dimensional models. Flange-opening under loadscan then be considered, with local stiffening, as a f&tion of angular location, aconsideration in the methodology to be develope&

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