Pressure vessel design using boundary element method with optimization

Pressure vessel design using boundary element
method with optimization
R.E.
Jr. D. Srivastava
Kettering
Michigan, USA
Abstract
Design of pressure vessels is covered by references such as the
Pressure
Vessel Code and textbooks devoted to pressure vessel design.
stress
analysis, particularly in the area of discontinuities, is generally left to the design
engineer. The type discontinuity addressed in this paper is the design of bolted

flanges for a pressure vessel. Work for this project involves
of the hub

contour using the
Pressure Vessel Code requirements as constraints. This

paper summa&es the initial work,

models and
of the model to classical techniques such as a
123. This

dimensional model will later be expanded to a
three-dimension model;
will
provide data for establishing allowable
sizes, based upon inspection
techniques and design life.
Pressure vessel design
Design of pressure vessels is governed by the
pressure vessel code
Other
such as

Moss


Harvey


and
Gill
provide
insight and guidelines to pressure vessel
Bolted
flange analysis is discussed
machine design textbooks such- as Norton
or
speciality books such as
or


or Company design practices or
criteria. Detailed stress analysis, particularly in the area of discontinuities, is
generally left to the design engineer.
type stresses are
by
4 of the
code
Stress limits, (allowable stress magnitudes),
based upon the type stress, are addressed by Appendix 4
Design Based


of the
code [
These are discussed later in this paper.

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268
The
discontinuity
in this paper is one associated with
design of bolted flanges for a pressure vessel. Figure la
one type design
which consists of the shell welded to the flange. Figure lb
another type
design, a
flange, with the weld located away from the shell/flange
discontinuity.
This is done to locate the weld in
area of lower bending stress,
improving the strength of the joint; and, also to locate it in an area where it may
require less weld material (cost) and can be more easily
Figure la Basic Configuration
Figure lb

Flange Outer Radius
Bolt Circle Radius
Shell (Vessel) Inner Radius
t or h
Shell (Wall) Thickness
H
Flange
Distance From Joint (Discontinuity) to Weld
Bending moments at a
such as a flange, will generally be


when the term
= 3.0, the moment effect is almost
zero.

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XIV



V
Poisson’s Ratio
a
Shell Radius
h
Vessel (Shell) Thickness
X
Axial Distance from Discontinuity
Using the
Code, previously mentioned references, and handbooks
such

vessel design could be a very complex task. With the
advent of the computer age, techniques such as the finite element method
and boundary element method (BEM) became very valuable design and analysis
aids.
Section VIII, Division 2 of the
Code defines several category of
stresses: Primary, Secondary, and Peak. A primary
is a normal or shear
stress developed by the imposed loading and necessary to
the laws of
equilibrium, such as the hoop (primary membrane) stress resulting
internal
pressure in a shell. Secondary stresses are normal or shear stress developed by the
constraint of adjacent parts or the self-constrain of a structure, a bending stress at
a gross structural discontinuity. The basic characteristic of a secondary stress is it
is self-limiting, local yielding and minor distortions can satisfy the conditions
which cause to stress to occur.

is a stress which does not cause any
noticeable distortion and is undesirable because it may be a possible source of a
fatigue crack or brittle fracture, for example, the stress at a local structural
discontinuity.
Definitions of all terms and tables of combinations and allowable
stresses are provided in Appendix 4 of Section VIII, Division 2.
Provisions for
plastic, limit, experimental, shakedown, and fatigue analysis are also available in
this same section of the Code.
Coupled with the stress analysis are design considerations for failure
analysis. Allowable
limits must be determined by the designer, even with
the Code allowable limits. Teak before failure”, design life, damage tolerance are
all factors which must be considered in the design process.
Once again textbooks
and handbooks, such as Collins
Dowling [
Ta& and Paris
Maddox
[
Brooks and Choudhury [
and
and

are available for
design reference.
Boundary element analysis with
The use of BEM is not new to pressure vessel analysis as evidenced by

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Element Technology XIV
Trevelyan
and Floyd
Fracture and crack
using BEM is also
evidenced by textbooks such as
Aliabadi


and

A test model, as shown by Figure 2, has been run to
BEM
results.
The model was based upon a flanged and bolted pipe. The model was
using equations from Table XIII, case 32 of Roark [
Results of this
model were comparable to the “hand calculations” of Roark
Like the boundary element method, optimization techniques have been
enhanced by the continued growth of computers.

is one of those
who has been part of this growth as well as others such
as Chanrupatala and Belegundu
The work being
performed in conjunction with this paper is based upon
design of a
flange, subject to the requirements
of the
Code, using the boundary element
method.

and

are the
software codes selected for this work In other words,
what is being accomplished is to optimize a design
based upon BEM analysis with constraints imposed by
codes such as the
Pressure Vessel Code.
A criterion or objective
must be
determined which will
inequality; and, possibly,
equality constraints. In turn, the objective must be
minimized (or maximized, depending upon the
problem.)
For a simple function, this means
determining where the first derivative is zero, and if
BEASY Model
the second derivative is positive or negative at those
points.
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, with
sufficient time and thought, this can be translated into cost, considering
costs, inspection costs, and material costs. Weight should provide a good working
model. Constraints will be to satisfy the stress limits of the
Code and the
weld to be in a low bending stress area
3.0). As the work progresses, cost
and multi-objective function problems will be developed. The
step will be to
crack propagation constraints into the models.
Conclusions
Initial results show theboundary element method will provide accurate predictions
of the stresses in a pressure vessel flange. With further development, an optimum
hub contour and weld location, subject to pressure vessel code and other

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27
constraints will be obtained. Once this methodology has been established, the
work will be expanded to three-dimensional models. Flange-opening under loads
can then be considered, with local
as a
of angular location, a
consideration in the methodology to be
References
The American Society of Mechanical Engineers, 1998
Boiler and
Pressure Vessel Code, Division 2 -Alternative



and


the Design
Section
Pressure Vessels,

1998
DR Moss, Pressure Vessel Design
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1987
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JF Harvey,
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Edition, Van
Nostrand Reinhold, 1991

Pressure Vessel Design Handbook,


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Pressure Vessel
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Machine Design An Integrated Approach, 2ne
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Stress




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Edition, McGraw-Hill, 1993.
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and ST
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Prentice-Hall, 1987.

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Boundary Element Technology XIV


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