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Compromise: An effective approach for the design of pressure vessels using composite materials
Compurers & Srrucrures Vol. 33. No. 6. pp. Printed in Great Britain.
1465-1477,
COMPROMISE: DESIGN
1989 0
MM57949/89 $3.00 + 0.00 1989 Pergamon Press plc
AN EFFECTIVE APPROACH FOR THE OF PRESSURE VESSELS USING COMPOSITE MATERIALS
HARSHAVARDHAN KARANDIKAR,t
RAMESH SRINIVASAN,~ FARROKH MISTREE~$ and WILL] J. FUCHS$
tSystems Design Laboratory, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A. §Lemmerz-Werke KGaA, Postfach 1120, D-5330, Konigswinter 1, Federal Republic of Germany (Received
8
November 1988)
Abstract-The
factors involved in designing structures made of metals and composite materials differ significantly because of the inherent non-isotropy associated with the latter. In this paper, a compromise Decision Support Problem template for the design of pressure vessels made of composite materials is presented. Membrane theory and classical laminate analysis have been utilized for the structural analysis. A novel criterion for reducing the stress concentration at the “knuckle” region of the pressure vessel has been used. To support the design methodology, a mathematical template for the design of the pressure vessel is detailed and solved. The results are examined and insights noted and discussed. The emphasis, however, is on the method and its capabilities and not on numbers obtained as a result of solving the
NOTATION boss diameter of the pressure vessel chamber diameter of the pressure vessel boss radius of the pressure vessel chamber radius of the pressure vessel length of the cylindrical section of the pressure vessel fiber orientations in each layer of the pressure vessel thicknesses in each layer of the cylindrical section
thickness in the hemispherical section internal pressure in the pressure vessel temperature difference across the walls of the pressure vessel underachievement of ith system goal, i = 1,2,...,6 overachievement of ith system goal, i = 1,2, . . ,6 meridian angle in the hemispherical section circumferential angle in the hemispherical section local stress along the fiber direction local stress perpendicular to the fiber direction local shear stress local stress matrix global stress matrix bond break fiber break strain in the longitudinal direction strain in the hoop direction strain in the meridian direction strain in the circumferential direction local strain matrix deflection of the hemisuherical shell normal to midsurface in the “knuckle” region deflection of the cylindrical shell normal to midsurface in the “knuckle” region deflection along the meridian direction stress resultant along the axial direction of the cylindrical section due to pressure load stress resultant along the hoop direction of the cylindrical section due to pressure load stress resultant along the circumferential direction of the hemispherical section
stress resultant along the meridian direction of the hemispherical section stress resultant vector due to pressure load temperature stress resultant vector reduced stiffness matrix for each layer of the pressure vessel stiffness matrix for the whole structure density of the composite material acceleration due to gravity Young’s modulus along the fiber direction of the composite material Young’s modulus perpendicular to the fiber direction of the composite material Poisson’s ratio of the composite material shear modulus of the composite material thermal coefficient in the principal directions (global)
N
2 A P g 4 E2
VI2
G %
1. INTRODUCTION TO THE DESIGN OF COMPOSITE PRESSURE VESSELS Pressure of
vessels are mostly
standard
conical).
geometries
The use of these standard
the
stress
The
difficulty
between
and
should be addressed.
deformation
of calculating
cylindrical geometries
under
hence
bending
important.
similar
loading,
This is due
of the different shell are not the same
and shear at the connection
The
high
stresses
easier.
at connections
different shell types still remains.
types,
and makes
calculations stresses
to the fact that the deformations
and
become
at the connection
de-
crease rapidly away from it, so that for the most part of
the shell the membrane
Thus,
to design
the membrane The bending
$ To whom correspondence
built using shell sections
(spherical,
tures is much
1465
stresses
a shell structure theory theory
more
are important.
one must
and the bending for composite
complicated
use both
theory [l].
material
struc-
than for structures
1466
HARSHAVARDHAN
made of homogeneous and isotropic materials and therefore involves time-consuming analysis. In the earlier stages of design, when rapid prototyping is desirable, we believe that it is not cost-effective to use these in-depth and time-consuming methods of analysis. In a recent paper [2], we show that it is possible to obtain a design of a class of shell structures made of composite materials using membrane theory alone. We postulate that by requiring the strains (or deformations) at the connection between two shell geometries be the same and then “optimally” tailoring the material we should be able to use a simplified theory for structural analysis. In this paper, the efficacy of this postulate is established. We use as an example a cylindrical pressure vessel with hemispherical end closures. The pressure vessel is subjected to an internal pressure load and a constant temperature difference across its thickness. Classical laminate analysis coupled with membrane theory for shells is used here for the structural analysis. Use of classical laminate analysis ensures that temperature loads can be taken into consideration unlike the more popular netting analysis. The deformations and stresses of the shell are calculated with the basic equations of the membrane theory for composite material shells [3,4]. This approach is appropriate for use in the earlier stages of design. The design is obtained by formulating and solving a compromise Decision Support Problem template. The efficacy of using a compromise Decision Support Problem to model and solve different aspects of structural design problems has appeared in this journal. The efficacy of using a compromise Decision Support Problem to solve multiobjective structural design problems was illustrated by Kuppuraju and Mistree [S]. Subsequently, Shupe and Mistree [6] have shown how to include reserve strength, a feature of damage tolerance, in design of structures and Shupe et al. [7] have shown how to approach the hierarchical design of structures. These papers provide a basis for the current development. We have written this paper with two aims in mind. Our first aim is to demonstrate the effectiveness of using a novel criterion to reduce stress concentration at the “knuckle” region of the pressure vessel. Our second aim is to examine the efficacy of using the compromise DSP in the “designing for concept” phase of composite pressure vessels design. 1.1. Overview of a compromise Problem
Decision Support
We make a distinction between “designing for concept” and “designing for manufacture”. We use the term designing for concept in the early stages of project initiation and the term designing for manufacture in the later stages. The difference stems primarily from the observation that in the earlier stages of project initiation the design specifications and knowledge about the product are incomplete and
KARANDIKAR et al.
the work presented in this paper falls clearly into the designing for concept category. The compromise Decision Support Problem, hereafter referred to as the compromise DSP, is an important element in the Decision Support Problem Technique under development at the University of Houston [S]. A detailed description of these terms and the role of compromise in designing for concept is given in [9]. Compromise DSPs refer to a class of constrained, multiobjective optimization problems that have widespread engineering applications. To date, ship design has been the largest single application of the compromise DSP formulation [lo- 121. Applications involving the design of damage tolerant structural [6] and mechanical systems [13- 151, the design of aircraft [16, 171, mechanisms [13, 181, a solar-thermalpowered agricultural-water pumping system [19,20], design using composite materials [l, 211 and data compression [ 181have also been reported. DSPs have been developed for hierarchical design: selectioncompromise [7, 11,221, compromise-compromise [23] and selection-selection [24]. The compromise DSP is solved using a unique optimization scheme called Adaptive Linear Programming. This scheme is described in [25,26]. The terms compromise DSP and Goal Programming [27] are synonymous to the extent that they refer to multiobjective optimization models; they both share the concept of deviation variables to model and evaluate the “goodness” of the solution with respect to the target values of goals. What distinguishes the compromise DSP formulation is the fact that it is tailored to handle common engineering design situations in which physical limitations manifest themselves as system constraints (mostly inequalities) and bounds. These constraints and bounds are handled separately from the system goals, contrary to the goal programming formulation in which everything is converted into goals. The terms compromise DSP and Mathematical Programming (for example, Vanderplaats [28]) are synonymous to the extent that they refer to system constraints that must be satisfied for feasibility. They differ in the manner in which the “goodness” of the solution is modeled and evaluated. In the compromise DSP the goodness is modeled by system goals (that are a function of both the system and the deviation variables) and a measure of the goodness is provided by the deviation function. The deviation function is always modeled using deviation variables. This is in contrast to traditional mathematical programming where multiple objectives are not explicitly modeled and a measure of goodness is provided by an objective that is modeled as a weighted function of the system variables only. In the compromise DSP formulation (unlike in traditional mathematical programming and goal programming) special emphasis is placed on the bounds of the system variables. In effect the compromise DSP is a hybrid formulation. The traditional mathematical
Compromise: design of composite material pressure vessels programming formulation is a subset of the compromise DSP and the compromise DSP is a subset (with a twist) of generalized goal programming. A compromise DSP is defined using the following descriptors: system and deviation variables, system constraints and bounds, system goals and a deviation function. The structure of the compromise DSP is given below. Given Find
Satisfy
Min~ize
A candidate alternative. The values of the system variables (they describe the physical attributes of an artifact). The values of the deviation variables (they indicate the extent to which the goals are achieved). System constraints: These must be satisfied for the solution to be feasible. System goals: These need to achieve a specified target value as far as possible. Bounds: Lower and upper limits on the system variables. An objective that quantifies the deviation of the system performance from that implied by the set of goals and their associated priority levels or relative weights.
The system descriptors have been described in [6,8, 10, 16,291 and hence will not be repeated here. 1.2. The genesis of the work This study was initiated by a grant from the Shell Development Company, Houston, U.S.A. in 1986 to examine the efficacy of using Decision Support Problems in the design of composite material structures. We used a pressure vessel made of composite materials as an example. This was completed in 1987. We have used the same example, developed for the earlier study, to understand the interaction between design and manufacturing by posing and answering the following questions. What is the failure and deflection behavior of the pressure vessel? This question is answered by conducting a parametric study to gain an understanding of the failure and deflection behavior of constituent shells of the pressure vessel. Is the use of the compromise Decision Support Problem appropriate for designing composite pressure vessels? This question is answered by designing a composite pressure vessel using a compromise DSP template (defined in the next section). We believe that material selection and dimensional synthesis have to be integrated before attempting to integrate manufacturing and design. We pose the following question: Can the processes of material selection and dimensional synthesis be integrated using a compromise DSP template? This question is answered by developing an under-
1467
standing of and modeling the interactions between material selection and dimensional synthesis using the compromise DSP template. 4. Can the processes of design and manufacturing be integrated using a compromise DSP template? This question is answered by developing an understanding of and modeling the interaction between the processes of manufacturing and design via the formulation and solution of a compromise DSP template. The first question has been addressed in [21] and we are in the process of developing answers to questions three and four. In this paper we address the second question. 2. A
COMPROMISE DECISION SUPPORT PROBLEM TEMPLATE FOR THE DJXSIGN OF A COMPOSITE MATERIAL PRESSURE VESSEL
Central to the DSP Technique and the associated software system (called DSIDES) is a scheme to represent design information in a knowledge base. This requires the conceptual categorization of knowledge in terms of representation as well as the role it plays in capturing the DSP process and domainspecific information about the product. The knowledge base includes two types of knowledge: knowledge about the process of design and knowledge about the product being designed. The former, known as procedural knowledge, is defined by Rich [30] as a set of well-defined procedures representing information about doing things. The latter, namely declarative knowledge [30], is defined as a set of facts represented (usually) according to the protocol defined by procedural knowledge. In our case, the knowledge about the process (procedural knowledge), is embodied in the Decision Support Problem Technique. A template, therefore, is a collection of the mathematical forms of a set of DSPs relevant to a specific domain. It provides the means to customize group information collected for a specific domain. Further information is provided in [29]. 2.1. Problem stutement Design a cylindrical pressure vessel with hemispherical end closures (see Fig. 1). It is subjected to internal pressure loads and a constant temperature
p = const.
AT =
-1-I Fig. 1. A schematic of the pressure vessel.
1468
HARSHAVARDHAN KARANDIKARet
load across its thickness. The volume of the pressure vessel should be as close as possible to 10’ mm’. The performance factor of the pressure vessel is to be maximized. There is experimental evidence to suggest that it is advantageous to keep the ratios of the boss diameter to the chamber diameter between l/l0 and l/S. Needless to state that the pressure vessel should not fail under the given loading conditions. The design of the pressure vessel involves the determination of the following dimensions: boss diameter chamber diameter length of the cylind~cal section fiber orientation of each layer thickness of each layer
d D L %,@2
&I,f,.
The loading and the structural configuration pressure vessel follow:
of the
Loading symmetrical internal pressure p0 (1800 psi or 12.402 MPa) constant temperature difference AT (1OOC) (AT = internal operating temperature - ambient temperature). Structural configuration Thickness distribution hemispherical shell: t2(d) = &,/sin 4 cylindrical shell: t = to+ 2t, (constant along the lon~tudinal direction) Laminate configuration hemispherical shell: one symmetrical angle-ply laminate, fiber angle CQand thickness tz(4) cylindrical shell: two symmetrical angle-ply laminates, fiber angles o,, tlr and thicknesses to, 21,. 2.2. rde~ti~cut~o~ and yodeling of system descriptors A compromise DSP template is formulated using system descriptors, namely the system variables, the deviation variables (associated with the system goals), system constraints, system goals, bounds on the system variables and the objective. The first step in developing a compromise DSP template is to examine the problem statement and identify the system variables and also the factors that influence the design. In the compromise DSP these factors are modeled as system constraints, system goals and bounds. The objective in a compromise DSP is to minimize the difference between that which is sought and that which needs to be achieved. This difference is modeled as a deviation function. System variables. The system variables for this problem are as follows: Radius of the vessel Boss radius Length of the vessel
R (mm) r (mm) L (mm)
al.
Hemispherical sections thickness at 4 = 90” fiber orientation Cylindrical section thickness of the inner and outer layers fiber orientation in the inner and outer layers
to (mm) a2 (rad) t, (mm) tl, (rad).
System constraints. The most natural system constraints that arise in structural design of the pressure vessel are the failure criteria. Since no liner was specified for the pressure vessel, first-ply failure was assumed in the computations. Failure criteria have been in use for isotropic materials for centuries [31]. They can be divided into non-interactive criteria (such as maximum stress or maximum strain) and interactive criteria (for example, quadratic approximations). Non-interactive criteria are commonly used to model failure for brittle materials whereas interactive criteria are used for ductile materials that fail by yielding. For fiber-reinforced composite materials the failure criteria, as can be expected, involve extensions of those developed for isotropic materials. In this work we have used interactive failure criteria instead of non-interactive for two reasons, namely (i) these criteria are simple to use because they involve single-valued functions in contrast to non-interactive criteria, and (ii) their use results in a smaller number of constraints being specified in the template. A variation of the Halpin-Tsai failure criterion, as reported in [32], has been used in the work reported here. This criterion includes the checking of bond break and fiber break in each layer of the laminate. In general, the choice of the specific failure criteria to be used in a compromise DSP template is left to the template developer. New criteria can be very easily substituted in these templates. The stresses on the structure due to the loading are obtained by using classical laminate analysis and membrane theory (see Appendix). The hemispherical portion is checked for failure only at 4 = 90” since this is where the section is the thinnest and the pressure load is the highest. A parametric study of the stress behavior of the hemispherical shell was conducted to confirm this conjecture and is reported in [21]. Hence, we ensure against bonding break failure by specifying, [@,,i~,WB)” + (u~/~~~~)2 f
(~,2/~12BB)21 G 1
(11
and against fiber break failure with
K~,,bHFd216 1.
(2)
Since a 2D theory is used for computing stresses, the outer layer and the inner layer in the cylindrical section can be combined for checking for failure.
Compromise: design of composite material pressure vessels Thus the cylindrical section of the pressure vessel can be considered to consist of the following two layers: The combined outer and inner layers of the cylindrical section (k = 1). The middle layer of the cylindrical section (k = 2). Since the preceding constraints are imposed for each layer of the structure there are four constraints for the cylindrical portion and two constraints for the hemispherical portion. Additional constraints are imposed to model the desired geometrical relationships between certain variables. One of them is that the chamber diameter has to be greater than the boss diameter, R-r>&
(3)
Another constraint arises due to the fact that the winding angle in the hemisphe~cal section is limited by the ratios of the boss and chamber diameters, t12- sin-‘(r/R)
3 0.
(hemispherical and cylind~cal), at the junction (the “knuckle” area) of these shells. A new approach for reducing this stress concentration is proposed here, i.e., the stress concentration will be largely eliminated if the deformations of the hemispherical and cylindrical shells at the interface are identical. This, we believe, can be accomplished by tailoring the material thereby changing the stiffnesses of the different shells that form the connection. The deformations consist of the deflections along the meridian direction, circumferential direction and normal to the mid-surface and the rotations. In this study, the stress concentration is sought to be reduced by matching the deflection normal to the mid-surface for the two shells at their interface (see Appendix for the derivation of the deflections). This criterion is used as the primary system goal and is formulated as a system goal as shown in eqn (9). Since the absolute value of the difference in deflections is being used, d; is always zero and hence only the deviation variable, d;, is to be minimized. This is included in the deviation function [see eqn (13)]
(4) I(w,~,,- w&I + d; -d:
An additional constraint is imposed due to the fact that the winding angle in the cylindrical section is limited by the ratio of the chamber diameters and the length of the section, tll - tan-‘(2R/(L
- 2R)) 2 0.
(5)
It has been determined ex~rimentally that better pressure vessel performance can be expected when the boss-to-chamber diameter ratio is between l/l0 and l/S [33, 341. Further, it is seen that many of the vessels reported in [33] have length-to-chamber diameter ratios between 1 and 3. A higher L introduces a large variation in longitu~nal fiber tension on the cylindrical section of the mandrel when using an in-plane filament winding technique. Hence the following four constraints are formulated to keep the boss-tochamber diameter ratio between l/l0 and l/.5 and length-to-chamber diameter ratio between 1 and 3: lO.r-R30 5.r
-R
(6) (7)
L-2RaO
(8)
L -6RGO.
(9)
System goals. It has been seen and reported in [33,35] that a stress concentration is induced due to differing structural response of different shell types
t The rules for fo~~lating the goals and the inclusion of the deviation variables in the deviation function are presented in [5, 7, 10, 15, 161 and are therefore not repeated here.
1469
= 0.
(IO)
Pressure vessel designs are evaluated using a number of criteria, such as weight, strength-to-density ratio, burst pressure, performance factor and the factor of safety in different directions [34]. The performance factor, defined as the (burst pressure) * (volume)/ weight, is very popular for the following reasons: (i) a variety of different pressure posite and homogeneous, can (ii) it approximates the condition safety in all directions of the
vessels, both combe compared, and of equal factor of pressure vessel.
The performance factor appears as a system goal in the compromise DSP fo~ulation [see eqn (IO)]. Since a high performance is desirable, a large target value of 10’ is set for the performance factor goal and the deviation variable, d;. is included in the deviation function and minimized,t P * Voiume/(Weight
* 10”) + d; - d+ = 1
(11)
where Volume = n * R2. [(4/3) ( R + L] Weight = 2nRpg{2.
[R . to. (n/2 - &,,i,)]
&,i, = sin-‘(r/R).
If instead of performance factor, weight alone were to be used as a criterion for evaluation then it can be modeled as a system goal as shown in eqn (I 1). Since we desire to minimize the weight, the deviation
1470
HARSHAVARDHANKARANDIKAR
variable, d,+ , is included in the deviation function and minimized, Weight + d; - d: = 0.
(12)
The volume of the pressure vessel is also represented as a system goal. In this case we have used the customer’s specification to normalize the system goal. Since we wish to achieve the customer’s specified volume both the deviation variables have to be minimized and included in the deviation function, Volume/(lO’) +- d; - d: = 1.
(13)
Numerical bounds are placed on the variables. The specific values stem from a designer’s judgment and experience. The bounds limit the feasible design space (which is computationally advantageous). The minimum layer thickness that can be wound imposes the lower bound on the thicknesses. The bounds on the winding angle in the hemispherical section are due to the limitations of the manufacturing process. The other bounds reflect a designer’s judgment and experience. Deviation function. The deviation function consists of those deviation variables that are identified for minimization while formulating the system goals, i.e., d: , d; , d: , d; , and d: The volume (being specified by the customer) and the deflection match are the most important goals and hence the deviation variables, d; , dJ , and d: , are given the highest priority for solution. The system goals corresponding to performance factor and weight are the next important and hence the deviation variables, d; and d:, are given the next highest priority. All the other deviation variables are given the lowest priority. A pre-emptive formulation of deviation function [8] follows: Bounds.
et al.
Hemispherical section thickness at 4 = 90” fiber orientation Cylindrical section thickness of the inner and outer layers fiber orientation in the inner and outer layers
to (mm) a2 (rad) t, (mm) CI,(rad).
Satisfy System constraints
Failure criteria for each layer: bonding break failure [(all !c,188)2 + h/%2BB)2
+ (~,2/~,2Ed21 G 1 (1)
break failure of the fiber [(al I lo Id21
G 1
(2)
Geometry vessel radius is greater than the boss radius R-r>0
(3)
minimum fiber angle permissible in the hemispherical section t12- sin-‘(r/R)
> 0
(4)
minimum fiber angle permissible in the cylindrical section CI,- tan-‘(2R/(L
- 2R)) > 0
(5)
boss-to-chamber diameter ratios have to be between l/l0 and l/S
Z={(d:+d;+d:),(d;+d:), (d; +d:
2.3. The mathematical formulation DSP
+d;)}.
of the compromise
Given
Loading internal pressure, P temperature difference across the thickness, AT Pressure vessel construction (see Fig. 1) thickness distribution laminate configuration Find
Dimensions
lO.r-RaO
(6)
5.r-R
(7)
(14)
length-to-chamber tween 1 and 3
diameter ratios have to be be-
L-2R>O
(8)
L-6R
(9)
System goals
Deflection match at the interface of the hemispher ical and cylindrical sections I(%,, - w&l + d; -d:
= 0
(10)
of the pressure vessel Performance factor
Radius of the vessel Boss radius Length of the vessel
R (mm) r (mm) L (mm)
P * Volume/(Weight
* 10’) + d; - d: = 1 (11)
Compromise: design of composite material pressure Weight Weight + d; - d: = 0
(12)
Volume Volume/( 10’) + d; - d: = 1.
(13)
Bounds on each system variable:
0.115 < t,, t, G20.0 0.2094 < 01~< 0.4363 0.0 < tlr < 1.5708 50.0 < R < 300 5.0
variable:
d,,d:,d;,d:,d;,d:,d;,d:~O.O. Minimize
Deviation function Z = {(d: + d; + d: ), (d; + d:), (d, + d: + d;)}.
The template is solved using System [24,25,34].
3. IMPLEMENTATION
the UH
AND DISCUSSION
(14)
DSIDES
OF RESULTS
3.1. Implementation of the compromise DSP template Based on our experience in developing complex templates such as the one presented in Sec. 2.2, we developed the template in three steps. In this context, two cases and two parametric studies were implemented and a description of each follows. Analysis of the results is presented in Sec. 3.2. The deviation function [see eqn (14)] that is given in the mathematical form of the template can be implemented in three ways.
vessels
1471
in the literature as has been indicated in [l] and presented here to establish the congruency of results. The number of system variables is 4, the number of system constraints is 6 while the number of goals is 2. While the length and diameters are fixed, the thicknesses and the orientations of the various layers are retained as variables. The constraints that are included in the template pertain to checking for bond break and fiber break in each of the three layers. Only the bounds pertaining to the system variables that are included in the template are considered. The system goals considered pertain to deflection match [eqn (lo)] and minimization of weight [eqn (12)]. Comprehensive case. This is an expanded version of the first case. The number of system variables is 7 with 13 system contraints and 4 system goals. The diameters and the length are included as variables. Seven constraints concerned with the geometry of the pressure vessel [eqns (3)-(9)] and two more goals concerned with achieving the customer specification on volume and maximizing the performance factor [eqns (11) and (13)] appear in this template. Hence, this case is much more representative of real-world problems than the exploratory case. This study was carried out with only the carbon fiber-epoxy resin system. Parametric studies. Parametric studies are very important during designing for concept which typically occurs early in the design process. Two parametric studies are presented to demonstrate the ease with which such studies can be conducted using the compromise DSP template in the designing for concept phase of the design process. In Parametric Study A, the behavior of the performance factor for changes in d/D and L/D ratio are examined. This is done by fixing d/D and L/D ratios using equality constraints [see eqns (15) and (16)] and solving the template. The performance factor at the solution is plotted against d/D at different L/D ratios in Fig. 2. It is observed that the fiber angle of the middle layer always converges to the upper bound of 25” in both the exploratory and confirmatory cases (see Tables 3 and
1. By obtaining the lexicographic minimum of the deviation function using a multiplex algorithm [27]. 2. By obtaining the minimum of the deviation function using a pseudo-pre-emptive formulation [8]. 3. By obtaining the minimum of the deviation function using an Archimedean formulation [8]. The choice of the formulation depends on the maturity of the template. The Archimedean formulation is used in all the cases and parametric studies. Exploratory case. The major dimensions of the filament wound pressure vessel, the radii and the length, are fixed in this case. This is an elementary case which has been studied and reported extensively
Fig. 2. Parametric Study A: performance factor vs d/D ratio.
1472
HARSHAVARDHAN KARANDIKAR et al.
target values for the performance factor. The results are shown in Fig. 3. d/D-x=0
wherex=O.1,0.2
,...,
L/D -y=O
where y = 1,2,3.
0.7
(15) (16)
The system variables, system constraints and goals for the three cases are summarized in Table 1. The material properties used in this study are presented in Table 2.
o.oo*+o
2SOc+6
3.2. solution
1.001,+7
7.50*+6
s.oor+6 YoMlne (mu?,
Fig. 3. Parametric Study B: changes in target volume.
4). Hence, in the next parametric study the upper bound of fiber angle of the middle layer is increased from 25” to 45”. The results for this are also shown in Fig. 2. In Parametric Study B, the behavior of the system variables at different target volumes is examined. This is done by solving the template at different
analysis
Exploratory case. The solution for this case is presented in Table 3. After complete development the solution behavior of this template is seen to be very stable. This case is solved for different material combinations and since the diameters are fixed, d/D ratios are also varied.
(i) Carbon fiber-epoxy resin at d/D = l/S (ii) Carbon fiber-epoxy resin at d/D = l/l0 (iii) Glass fiber-epoxy resin at d/D = l/S.
Table 1. The compromise DSP templates: variables, constraints and goals
System variables
Exploratory case
case, Parametric Study B
fflrti~~zr~l
&,,~,,~2,~,,R,r,~
System constraints (nos.)
(1>2)
System goals (nos.)
(10, 12)
Fixed variables
R, r, L
R r
Parametric Studv A to.
5,
a,,
5.
R,
(1,2,3,4,5,6,7,8,9)
(t,2,3,4,5,15,16)
(10, 11, 12,13) -
(10, 11,13) -
r,
L
80.0
(l/S or l/lO)R 480.0
L
Table 2. Material constants used in the analysis Material? E-glass fiberepoxy resin Density (kg/mm3) Poisson’s ratio Shear modulus (N/mm2) Young’s modulus {N/mm~) fiber direc. transverse direc. Thermal coefficient (I /OK) fiber direc. transverse direc. Bond strength (N/mm*), fiber direc., X transverse direc., Y Bond strength, shear (N/mm2), V Fiber strength (N/mm2), 2
Carbon fiberepoxy resin
Kevlar 49epoxy resin
1.55E6
G
1.94E-6 0.25 4500
0.3 4600
1.32E-6 0.25 2800
E, EZ
37232 8274
140556 9646
82000 4000
arl 872
6.3E-6 20.5B6
0.23E-6 22SE-6
- 3.6E-6 57.6&6
(t=s) (camp) (tens) (camp)
820.5 517.0 48.3 137.9
1999.5 1559.2 54.0 150.
1158.0 276.0 Il.0 83.0
(tens) (camp)
68.9 1140 1045
90.9 1620 1480
69.0 1930 174s
VI2
t The composites have 60% fiber volume fraction.
and
Compromise: design of composite material pressure vessels
1473
Table 3. Exploratory case: solution Deflection (mm)
Final design (r-mm; a-degrees)
Initial design (f-mm; a-degrees) t0 fl a2 XI fo Carbon fiber-epoxy resin; d/D = l/S 6.0 6.0 12 zt 5.29 2.0 2.0 18 5.291 3.0 1.0 12 30 5.291
‘I
aI
g2
0.193 0.193 0.193
25 25 25
Wsph
89.91 89.73 89.78
0.7105 0.7105 0.7105
WCYI
0.7104 0.7105 0.7105
Weight = 22.745N Carbon fiber-epoxy resin; d/D = l/l0 3.0 1.0 12 30 5.291
0.193
25
89.7
Glass fiber-epoxy resin; d/D = l/S 6.0 6.0 12 45 6.769 3.0 1.0 12 30 6.802
0.872 0.876
25 25
89.9 89.9
0.6413 0.6389
0.6407 0.6386
Weight = 40.35 N No, of design variables = 4; No. of system constraints = 6; No. of system goals = 2.
(i) Carbon fiber-epoxy resin; d/D = l/5. The template was exercised using three different initial or starting designs. The final solution converged to one point from three different starting points. The deflection match at the interface, which is the principal criterion for reducing stress concentration, is seen to be very good at this solution point; the difference in deflections between the spherical portion and the cylindrical portion is 0.3 pm. The bond break failure criterion in the middle layer in the cylindrical section is seen to be the limiting constraint in this case. The winding angle in the hemispherical section tends to lie at the upper limit of 25”. The spherical shell construction is most efficient at a 45” angle as the stresses in the membrane are uniform. This is also indicated in the parametric study results [21]. Since the upper bound of this angle is taken to be 25”, the solution point tries to be as close to this condition as possible. (ii) Carbon fiber-epoxy resin; d/D = l/10. No change in solution was observed from the preceding solution. This is expected because of the nature of the fo~ulation. The d/D ratio does not appear any where and hence the value of the ratio is not relevant to the solution of this formulation. (iii) Glass fiber-epoxy resin; d/D = l/5. As expected the weight of the pressure vessel is seen to be increased because of the greater thicknesses of the layers and the higher density of this material. The thicknesses are higher because of the lower strength
and stiffness of this composite material system. The deflection match is again seen to be very good. The difference in deflections between the spherical portion and the cylindrical portion is again 0.3 pm. Co~pre~e~i~e case. To test the ~omprehensiveness of the template, three different designs, hereafter called scenarios, are used as starting solutions. The final solution is presented in Table 4 and it can be seen that they are nearly identical. This shows that the template is complete enough to converge to the same solution for widely varying scenarios. It can be observed that the fiber angle in the outer layers of the cylindrical section converges to its upper bounds, 90”, consistently in all the scenarios. This implies that there is an overwrap of material in the hoop direction in the cylindrical section of the pressure vessel. This is required because of the biaxial force field in that section of the pressure vessel. Yet another observation regarding the behavior of the template is made and it is shown that it agrees with intuition. It is observed that one of the constraints representing bond break failure criterion is always active throughout the solution process for all the three scenarios and the fiber break constraints never become active. This is expected because the fiber strength of the composite is much higher than the bond strength of the composite and hence they cannot be limiting constraints by themselves. The defIection match is again very good, 0.1% at worst, and this is indicative of the
Table 4. Comprehensive ease: solution Initial design (t, R, r, L-mm; a-degrees) T,
‘I 10 1.0 3.115
a2 25 12 17.7
aI 90 0 54.4
R
Final design (t, R, r, L-mm; a-degrees) r
L
to
t1
a2
al
20 300 60.0 1000 5.569 0.203 25 90 1.0 70.0 10.0 140 5.569 0.2027 25 4.115 90.0 10.0 500 5.569 0.2033 25 No. of design variables = 7; No. of system ~onstmints = 12; No. of system goals = 4. t wrph= 0.7479, wq, = 0.7478 mm, weight = 26.5345N. = 0.7488 mm, weight = 26.512N. SWsph= 0.7477, weYl = 0.7476mm, weight = 26.5165N. § W,pll= 0.7477, wcYl
R
r
84.21
16.84 16.84 16.84
L
505.3t 504.9$ SOS.O$
HARSHAVARDHAN KARANDIKAR
1474
appropriateness system goal.
of using the deflection criterion as a
Parametric studies. A plot of the results obtained from Parametric Study A is shown in Fig. 2. It is observed that the performance factor increases with an increase in d/D ratio. Since the pressure vessel is checked for failure only at the interface between the hemispherical and cylindrical sections, the increase in boss diameter, d, does not affect the solution. However, the increase in boss diameter reduces the weight of the pressure vessel and this leads to an increase in performance factor [see eqn (ll)]. When the upper bound of the fiber angle, c+, is increased to 45” from 25” (see Sec. 2.3), the solution converges close to this value. This is accompanied by a reduction in thickness of the middle layer. When the fiber angle is increased to 45”, the hoop strength of the middle layer in the cylindrical section (see Fig. 1) increases and hence a smaller thickness is able to resist the hoop stress. A logarithmic plot of the system variables at different target volumes obtained from Parametric Study B is shown in Fig. 3. It is observed that the chamber diameter and the length of the pressure vessel increase in order to effect the desired increase in volume. As the chamber diameter increases, the pressure load on the pressure vessel increases [see eqns (Al), (A2) and (Al l)]. This leads to an increase in the thickness as seen in Fig. 3. It is seen that the behavior of the template for the two parametric studies is justifiable. These parametric studies illustrate the ease with which such studies can be conducted using the template. Commentary on cases. The exploratory case dealing primarily with weight minimization is extensively reported in the literature, as stated in [l], and it is shown in this paper that such simple problems can be solved using the compromise DSP formulation. The comprehensive case established the effectiveness of using the compromise DSP for larger problems with multiple objectives, namely the performance factor, volume and weight. This case is much closer to the real-world than the exploratory case. Finally, to illustrate the effectiveness of the template in the designing for concept phase of composite pressure vessel design, two parametric studies are conducted and the behavior of the template is justified. As mentioned earlier, the emphasis here is on the method and not the numbers per se. It is evident from the results that the compromise DSP template can be effectively used in designing for concept of composite pressure vessels. This work sets the stage for combining material selection and manufacture with design, the ultimate aim of this project.
4. CLOSURE
In this paper we report on the usefulness of an approach using compromise Decision Support Problems in “designing for concept” of a pressure
et
al.
vessel made of composite materials. The dimensional synthesis of the pressure vessel is modeled as a compromise Decision Support Problem; with system variables corresponding to the dimensions of the pressure vessel; system constraints modeling the bonding break and fiber break failure criteria and some geometric constraints and system goals including the performance factor, deflection match at the interface between the cylindrical and the hemispherical shells, weight and the volume of the pressure vessel. The capability of the compromise Decision Support Problem template to handle multiple objectives helps the designer in modeling the design closer to the real life situation than the traditional single objective formulations with which other researchers seek to “optimize” their designs. The formulation of the template provides the means to perform parametric studies with ease to obtain information about the various aspects of the design. This capability is useful in the designing for concept phase of the design process because it saves time and effort. Our first aim in this paper was to demonstrate the effectiveness of using a novel criterion to reduce stress concentration at the “knuckle” region of the pressure vessel. In designing for concept where the design is likely to be revised, it is not profitable to use sophisticated analytical tools like finite element analysis. This novel criterion makes possible use of membrane theory (it involves a simple calculation) in designing for concept of a pressure vessel with different shell sections and thence this enables us to achieve the first aim of the paper. A preliminary indication of the efficacy of this criterion in reducing the stress concentration is reported in [2]. Our other aim is to examine the efficacy of the compromise DSP in the designing for concept phase for composite pressure vessels. This is illustrated through a set of case studies and parametric studies. An initial case, namely the exploratory case, is solved to learn about the behavior of the template. This is followed by a more complex case, namely the comprehensive case, which establishes the efficacy of using a compromise DSP template for solving realistic composite pressure vessel design problems. Two parametric studies are conducted to demonstrate the ease with which such studies can be conducted using the template. Further work can be done by investigating the effect of changing the priorities assigned to various system goals and repeating the study with a different dome contour. It is reported by Lark [33] that the hemispherical contour is the least efficient. Another improvement is to include the economic efficiency of the pressure vessel and investigate the trade-off between technical efficiency as represented by the performance factor and economic efficiency. As indicated earlier efforts are under way to integrate material selection and aspects of manufacture in the template and also make provision for a non-load bearing liner as is used in industrial practise.
Compromise: design of composite material pressure vessels Acknowledgements-We gratefully acknowledge the financial contribution made by the Shell Development Company, Houston, Texas to initiate this project in 1986. We gratefully acknowledge the financial contribution made by our corporate sponsor, the B. F. Goodrich Company, Brecksville, Ohio for the continuation of this project. The computer costs were underwritten by the University of Houston. Support for Ramesh Srinivasan was provided by the Department of Mechanical Engineering, University of Houston, Houston, Texas.
REFERENCES 1.W. K. Fuchs, H. M. Karandikar, F. Mistree and H. Eschenauer, Compromise: an effective approach for designing composite conical shell structures. Prac. 1988 Design Aufomation Conference, Orlando, September (1988). 2. W. J. Fuchs, H. M. Karandikar, R. Srinivasan and F. Mistree, Preliminary design of a filament wound pressure vessel: a coupled decision support problem. Proc. Conf on Discretization Methods and Structural Optimization: Procedures and Applications, Siegen,
F.R.G., October 5-7 (1988). 3. W. Fuchs, Struktur~a~yse und Optimierung anjsotro~r Schaien aus F~e~erbundwerksto~ ~~fr~~f~rai Analysis and Optimum Design of Anisotropic Fiber Remforced Shells). Doctoral Thesis, University of Siegen (1986)
[In German]. 4. W. Schnell and H. Eschenauer, Elastizitcitstheorie II; Schalen. Mannheim. Vienna: Bibliogranhisches Institut _ _ (1984). 5. N. Kuppuraju and F. Mistree, Compromise: an effective
6.
7.
8.
9.
10. 11. 12.
13.
approach for solving multiobjective structural design problems. Cornput. Struct. 22, 857-865 (1986). J. A. Shupe and F. Mistree, Compromise: an effective approach for the design of damage tolerant structural systems. Comput. Struct. 27, 407-415 (1987). J. A. Shupe, F. Mistree and J. S. Sobieski, Compromise: an effective approach for design of hierarchical structural systems Comput. Struct.-26, 102771037 (1987). F. Mistree, H. M. Karandikar. J. A. Shuae and E. Bascaran, Computer-based Des&n Synthesis&An Aoproach to Problem Solving. Systems Design Laboratory Report, Department of Mechanical Engineering, University of Houston, January (1989). F. Mistree, Designing for concept using decision sup port problems: a conceptual exposition. Proc. 1st int. Appl. Mech. Syst. Con& with Tutorial Workshops, Paper No. T4. Nashville, Tennessee, June 11-14 (1989). _ T. D. Lyon and F. Mistree, A computer-based method for the preliminary design of ships. J. Ship Res. 29, 251-269 (1985). W. F. Smith and F. Mistree, The influence of hierarchical decisions on ship design. Marine Technol. 24, 131-142 (1987). W. F. Smith, T. D. Lyon and B. Robson, AUSEVALA systems approach for the preliminary design of naval ships. Proc. Bicentennial Maritime Symposium. Sydney, Australia, pp. l-29. January 18-20 (1988). S. Mudali, Dimensional Synthesis of Mechanical Linkages
using Compromise
Decision
Support
Problems.
MS. Thesis, Department of Mechanical Engineering, University of Houston, October (1987). 14. N. Nguyen and F. Mistree, A computer-based method for the rational design of horizontal pressure vessels. ASME J. Meehanism~ Transm~sjons Design 108, 203-210 (1986).
and Automation
in
1.5. A. J&an and F. Mistree, ‘A computer-based method for designing statically loaded helical compression springs. ASME 1 lth Design Automation Conference, Cincinnati, Ohio, September 10-13. ASME Paper Number 85DET-75 (1985). CAS3316-3
1475
16. F. Mistree, S. Marinopoulos, D. Jackson and J. A. Shupe, The Design of Aircraft using the Decision Support Problem Technique, NASA Contractor Report 4134, April (1988). 17. S. Ma~nopoulos, D. Jackson, J. Shupe and F. Mistree, Compromise: An Eflective Approach for Conceptual Aircraft Design, Paper No. AIAA-87-2965. 18. Q.-J. Zhou, The Compromise Decision Support Problem: A Fuzzy Formulation. MS. Thesis, Department of
Mechanical Engineering, University of Houston, May (1988). 19. E. Bascaran, F. Mistree and R. B. Bannerot, Compromise: an effective approach for solving multi-objective thermal design problems. Engng Opt~~zation 12, 175-189
(1987).
20. E. Bascaran, R. B. Bannerot and F. Mistree, The conceptual development of a method for solving multiobjective hierarchical thermal design problems. 1987 National Heat Transfer Conference, Paper No. ASME 87-HT-62, Pittsburgh, August (1987). 21. F. Mistree, W. J. Fuchs, H. M. Karandikar and R. Srinivasan, Design of Composite Material Systems using the Decision Support Problem Technique. Systems Design Laboratory Report, Department of Mechanical Engineering, Houston,. October (1987). 22. N. Kuppuraju, S. Ganesan, F. Mistree and J. S. Sobieski, Hierarchical decision making in system design. Engng Optimization 8, 223-252 (1985). 23. W. F. Smith. The Devetooment of AVSEVAL: An Automated Ship Evaluation System. MS. Thesis, Department of Mechanical Engineering, University of Houston, December (1985). 24. E. Bascaran, R. B. Bannerot and F. Mistree, Hierarchical selection decision support problems in conceptual design. Engng Optim~af~on 14, 207-238 (1989). 25. F. Mistree and 0. F. Hughes, Adaptive Linear Programming: An Algorithm for Solving Muit~-objective Decision Support Problems. Systems Design Laboratory Report,
University of Houston, November (1986). 26. F. Mistree, 0. F. Hughes and H. B. Phuoc, An optimization method for the design of large, highly constrained, complex systems. Engng Optimization 5, 141-144 (1981). 27, J. P. Ignizio, Generalized goal programming: an overview. Comput. Operations Res. 10,277-289 (1983). 28. ti. N. Vanderplaats, ~~er~~aI Opt~ization Tec~jques fir Engineering Design: With Application. McGrawHill, New York (1984). 29. S. Z. Kamal. H. M. Karandikar. F. Mistree and D. Muster, Knowledge representation for disciplineindependent decision making. In Expert Systems in CAD (Edited by J. S. Gero), pp. 289-321. Elsevier, Amsterdam (1987). 30, E. Rich, Artificial Intelligence, Ch. 7. McGraw-Hill, New York (1983). 31. S. W. Tsai, A survey of macroscopic failure criteria for composite materials. J. Reinforced Plastics Composites 3 (1984).
32. A. Puck and W. Schneider, On the failure mechanisms and failure criteria of filament-wound glass-fibres in composites. Plastics and Polymers 33-44 (1969). 33. R. F. Lark, Recent advances in lightweight, filamentwound composite pressure vessel technology: composites in pressure vessels and piping. Proc. of The Energy Technology Conference, ASME, Houston, Texas (Edited by S. V. Kulkarni and C. H. Zweben), pp. 17-49, September (1977). 34. S. J. Darms,- Optimum Design for Filament-Wound Rocket-Motor Cases, Technical Documentary Report No. ASD-TDR-63-396, Directorate of Materials and Processes, Aeronautical Systems Division, WrightPatterson AFB, April (1963). 35. P. J. Conlisk and T. J. Fowler, Design of glass fiber
HARSHAVARDHAN KARANDIICAR et al.
1476
reinforced tanks and vessels: composites in pressure vessels and piping. Proc. of The Energy Technology Conference, ASME, Houston, Texas (Edited by S. V. Kulkarni and C. H. Zweben), pp. 1-15, September (1977). 36. J. R. Vinson and T. W. Chou, Composite Materials and Their Use in Struciures, pp. 314-345. Applied Science (1975). 37. J. R. Vinson, Structural Mechanics: The Behavior of Plates and Shells, pp. 109- 145. Wiley-Interscience
(1974). 38. F. Mistree and S. Z. Kamal, DSIDES: Decision Support In rhe Design of Engineering Systems. Systems Design Laboratory Report, University of Houston, December (1988).
APPENDIX: STRUCTURAL ANALYSIS OF COMPOSITE PRESSURE VESSEL The stresses and deflections due to the pressure and temperature loadings on the pressure vessel are used in the compromise DSP template [see eqns (1) (2) and (9)]. These are calculated using membrane theory and classical laminate analysis as shown in this appendix. Structural analysis of a cylindrical shell [2,3,21,36,37]
The longitudinal and circumferential stress resultants in the cylinder due to the pressure loading are computed using membrane theory as follows: N,, = p0 (R 12) (force/unit length)
(Al)
N,, = p0 R
(A2)
(force/unit length).
Thus. (A3)
The reduced stiffnesses (Q,) for each layer in the cylinder are obtained from the material properties. These stiffnesses are transformed using the fiber angles of the laminate to determine the stiffnesses for each layer in the principal directions of the cylindrical shell, (G), or Q, the global reduced stiffness matrix. The extensional stiffness matrix for the whole cylindrical shell structure, A, is determined from the global reduced stiffnesses of the various layers using the following equation:
k=l
where k = 1,2, , n is the number of layers and i, j represent principal directions. The stress resultant matrix due to the temperature loading, N,, can then be determined from the following equation: N, =
C te,,), ‘1, . a~,.AT. k=,
The stress resultants due to the pressure and the temperature loading and strains are related as shown below: N=Af--N,
646)
From eqn (A6), strains and deflections are computed which are shown in eqns (A7) and (A8). c = A-‘(N + NT).
(A7)
The deflection of the cylindrical shell along the hoop direction which is used in the deflection goal [see eqn (9)] can be expressed as below: Wcyl= t,, A.
(‘48)
The global stress matrix, d, is then calculated as shown below: 6”’ = ?j(‘)(E-
AT)
K,(k)
649)
where br is the matrix of temperature coefficients. The global stress matrix, 8, is reduced to the local stress matrix, n, using a transformation matrix, T. These are used in formulating the constraints on failure criteria [see eqns (1) and (2)l a(k) = T&k’,
(AlO)
The stress matrix, o, consists of three terms, [o,, , CT*?, ~~~1~. Since the stresses and displacements are independent of the radial distance of the layers from the axis, the outer layer and the inner layer in the cylindrical section can be combined. Thus the cylindrical section of the pressure vessel can be considered to consist of the following two layers: k = 1 The combined outer and inner layers of the cylindrical section. k = 2 The middle layer of the cylindrical section. Struclural analysis of hemispherical shell [2,20,36,38]
The stress resultant along the meridian direction and the circumferential direction due to the pressure loading is computed from membrane theory of shells, as indicated below: N,, = No0= Rp,/2.
(All)
The local reduced stiffness matrix, Q, the global reduced stiffness matrix, Q, and the extensional stiffness matrix, A, are obtained assuming a 2D orthotropic material. The coupling stiffness matrix B is absent due to the basic assumption that the shell is formed of a symmetric laminate. The thermal stress resultants along the meridian and circumferential direction are calculated using eqn (AS). The general solution for the deflection of the spherical shell is derived from the general solution for the deflection of the membrane shell. The specific solution for the deflections, u(@) and w(a), of this spherical shell is then obtained from the general solution for the spherical shell using the material law in terms of stress resultants due to pressure loading and temperature loading. The deflection normal to the mid-surface, w(G), is calculated at the interface between the hemispherical and cylindrical shell sections, i.e., Q = n/2, and is used in formulating the deflection goal [see eqn (9)]. The global strains are derived from these deflections, u(0) and w(e). t ,@= (l/R)(w + du/d@) C&,= (l/R)(w + u cot @J).
6412)
The global strain matrix, C, is then obtained as
where is the global strain matrix for the shell.
and
ye0 =O.
6413)
Compromise: design of composite material pressure vessels From the global strain matrix, C; the local strain matrix, L, is computed using the transformation matrix, T. The local stress matrix, u, can be expressed as, u =Q(c
-a
AT)
(A14)
1477
where u = [a,, , uz2,r12]*. These are used in checking for bond break failure and fiber break failure [see eqns (1) and (2)]. The hemispherical portion is checked for faiture only at # = 90” since it is the thinnest section. Note that the hemisphe~cal portion consists of only one layer.
1465-1477,
COMPROMISE: DESIGN
1989 0
MM57949/89 $3.00 + 0.00 1989 Pergamon Press plc
AN EFFECTIVE APPROACH FOR THE OF PRESSURE VESSELS USING COMPOSITE MATERIALS
HARSHAVARDHAN KARANDIKAR,t
RAMESH SRINIVASAN,~ FARROKH MISTREE~$ and WILL] J. FUCHS$
tSystems Design Laboratory, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A. §Lemmerz-Werke KGaA, Postfach 1120, D-5330, Konigswinter 1, Federal Republic of Germany (Received
8
November 1988)
Abstract-The
factors involved in designing structures made of metals and composite materials differ significantly because of the inherent non-isotropy associated with the latter. In this paper, a compromise Decision Support Problem template for the design of pressure vessels made of composite materials is presented. Membrane theory and classical laminate analysis have been utilized for the structural analysis. A novel criterion for reducing the stress concentration at the “knuckle” region of the pressure vessel has been used. To support the design methodology, a mathematical template for the design of the pressure vessel is detailed and solved. The results are examined and insights noted and discussed. The emphasis, however, is on the method and its capabilities and not on numbers obtained as a result of solving the
NOTATION boss diameter of the pressure vessel chamber diameter of the pressure vessel boss radius of the pressure vessel chamber radius of the pressure vessel length of the cylindrical section of the pressure vessel fiber orientations in each layer of the pressure vessel thicknesses in each layer of the cylindrical section
thickness in the hemispherical section internal pressure in the pressure vessel temperature difference across the walls of the pressure vessel underachievement of ith system goal, i = 1,2,...,6 overachievement of ith system goal, i = 1,2, . . ,6 meridian angle in the hemispherical section circumferential angle in the hemispherical section local stress along the fiber direction local stress perpendicular to the fiber direction local shear stress local stress matrix global stress matrix bond break fiber break strain in the longitudinal direction strain in the hoop direction strain in the meridian direction strain in the circumferential direction local strain matrix deflection of the hemisuherical shell normal to midsurface in the “knuckle” region deflection of the cylindrical shell normal to midsurface in the “knuckle” region deflection along the meridian direction stress resultant along the axial direction of the cylindrical section due to pressure load stress resultant along the hoop direction of the cylindrical section due to pressure load stress resultant along the circumferential direction of the hemispherical section
stress resultant along the meridian direction of the hemispherical section stress resultant vector due to pressure load temperature stress resultant vector reduced stiffness matrix for each layer of the pressure vessel stiffness matrix for the whole structure density of the composite material acceleration due to gravity Young’s modulus along the fiber direction of the composite material Young’s modulus perpendicular to the fiber direction of the composite material Poisson’s ratio of the composite material shear modulus of the composite material thermal coefficient in the principal directions (global)
N
2 A P g 4 E2
VI2
G %
1. INTRODUCTION TO THE DESIGN OF COMPOSITE PRESSURE VESSELS Pressure of
vessels are mostly
standard
conical).
geometries
The use of these standard
the
stress
The
difficulty
between
and
should be addressed.
deformation
of calculating
cylindrical geometries
under
hence
bending
important.
similar
loading,
This is due
of the different shell are not the same
and shear at the connection
The
high
stresses
easier.
at connections
different shell types still remains.
types,
and makes
calculations stresses
to the fact that the deformations
and
become
at the connection
de-
crease rapidly away from it, so that for the most part of
the shell the membrane
Thus,
to design
the membrane The bending
$ To whom correspondence
built using shell sections
(spherical,
tures is much
1465
stresses
a shell structure theory theory
more
are important.
one must
and the bending for composite
complicated
use both
theory [l].
material
struc-
than for structures
1466
HARSHAVARDHAN
made of homogeneous and isotropic materials and therefore involves time-consuming analysis. In the earlier stages of design, when rapid prototyping is desirable, we believe that it is not cost-effective to use these in-depth and time-consuming methods of analysis. In a recent paper [2], we show that it is possible to obtain a design of a class of shell structures made of composite materials using membrane theory alone. We postulate that by requiring the strains (or deformations) at the connection between two shell geometries be the same and then “optimally” tailoring the material we should be able to use a simplified theory for structural analysis. In this paper, the efficacy of this postulate is established. We use as an example a cylindrical pressure vessel with hemispherical end closures. The pressure vessel is subjected to an internal pressure load and a constant temperature difference across its thickness. Classical laminate analysis coupled with membrane theory for shells is used here for the structural analysis. Use of classical laminate analysis ensures that temperature loads can be taken into consideration unlike the more popular netting analysis. The deformations and stresses of the shell are calculated with the basic equations of the membrane theory for composite material shells [3,4]. This approach is appropriate for use in the earlier stages of design. The design is obtained by formulating and solving a compromise Decision Support Problem template. The efficacy of using a compromise Decision Support Problem to model and solve different aspects of structural design problems has appeared in this journal. The efficacy of using a compromise Decision Support Problem to solve multiobjective structural design problems was illustrated by Kuppuraju and Mistree [S]. Subsequently, Shupe and Mistree [6] have shown how to include reserve strength, a feature of damage tolerance, in design of structures and Shupe et al. [7] have shown how to approach the hierarchical design of structures. These papers provide a basis for the current development. We have written this paper with two aims in mind. Our first aim is to demonstrate the effectiveness of using a novel criterion to reduce stress concentration at the “knuckle” region of the pressure vessel. Our second aim is to examine the efficacy of using the compromise DSP in the “designing for concept” phase of composite pressure vessels design. 1.1. Overview of a compromise Problem
Decision Support
We make a distinction between “designing for concept” and “designing for manufacture”. We use the term designing for concept in the early stages of project initiation and the term designing for manufacture in the later stages. The difference stems primarily from the observation that in the earlier stages of project initiation the design specifications and knowledge about the product are incomplete and
KARANDIKAR et al.
the work presented in this paper falls clearly into the designing for concept category. The compromise Decision Support Problem, hereafter referred to as the compromise DSP, is an important element in the Decision Support Problem Technique under development at the University of Houston [S]. A detailed description of these terms and the role of compromise in designing for concept is given in [9]. Compromise DSPs refer to a class of constrained, multiobjective optimization problems that have widespread engineering applications. To date, ship design has been the largest single application of the compromise DSP formulation [lo- 121. Applications involving the design of damage tolerant structural [6] and mechanical systems [13- 151, the design of aircraft [16, 171, mechanisms [13, 181, a solar-thermalpowered agricultural-water pumping system [19,20], design using composite materials [l, 211 and data compression [ 181have also been reported. DSPs have been developed for hierarchical design: selectioncompromise [7, 11,221, compromise-compromise [23] and selection-selection [24]. The compromise DSP is solved using a unique optimization scheme called Adaptive Linear Programming. This scheme is described in [25,26]. The terms compromise DSP and Goal Programming [27] are synonymous to the extent that they refer to multiobjective optimization models; they both share the concept of deviation variables to model and evaluate the “goodness” of the solution with respect to the target values of goals. What distinguishes the compromise DSP formulation is the fact that it is tailored to handle common engineering design situations in which physical limitations manifest themselves as system constraints (mostly inequalities) and bounds. These constraints and bounds are handled separately from the system goals, contrary to the goal programming formulation in which everything is converted into goals. The terms compromise DSP and Mathematical Programming (for example, Vanderplaats [28]) are synonymous to the extent that they refer to system constraints that must be satisfied for feasibility. They differ in the manner in which the “goodness” of the solution is modeled and evaluated. In the compromise DSP the goodness is modeled by system goals (that are a function of both the system and the deviation variables) and a measure of the goodness is provided by the deviation function. The deviation function is always modeled using deviation variables. This is in contrast to traditional mathematical programming where multiple objectives are not explicitly modeled and a measure of goodness is provided by an objective that is modeled as a weighted function of the system variables only. In the compromise DSP formulation (unlike in traditional mathematical programming and goal programming) special emphasis is placed on the bounds of the system variables. In effect the compromise DSP is a hybrid formulation. The traditional mathematical
Compromise: design of composite material pressure vessels programming formulation is a subset of the compromise DSP and the compromise DSP is a subset (with a twist) of generalized goal programming. A compromise DSP is defined using the following descriptors: system and deviation variables, system constraints and bounds, system goals and a deviation function. The structure of the compromise DSP is given below. Given Find
Satisfy
Min~ize
A candidate alternative. The values of the system variables (they describe the physical attributes of an artifact). The values of the deviation variables (they indicate the extent to which the goals are achieved). System constraints: These must be satisfied for the solution to be feasible. System goals: These need to achieve a specified target value as far as possible. Bounds: Lower and upper limits on the system variables. An objective that quantifies the deviation of the system performance from that implied by the set of goals and their associated priority levels or relative weights.
The system descriptors have been described in [6,8, 10, 16,291 and hence will not be repeated here. 1.2. The genesis of the work This study was initiated by a grant from the Shell Development Company, Houston, U.S.A. in 1986 to examine the efficacy of using Decision Support Problems in the design of composite material structures. We used a pressure vessel made of composite materials as an example. This was completed in 1987. We have used the same example, developed for the earlier study, to understand the interaction between design and manufacturing by posing and answering the following questions. What is the failure and deflection behavior of the pressure vessel? This question is answered by conducting a parametric study to gain an understanding of the failure and deflection behavior of constituent shells of the pressure vessel. Is the use of the compromise Decision Support Problem appropriate for designing composite pressure vessels? This question is answered by designing a composite pressure vessel using a compromise DSP template (defined in the next section). We believe that material selection and dimensional synthesis have to be integrated before attempting to integrate manufacturing and design. We pose the following question: Can the processes of material selection and dimensional synthesis be integrated using a compromise DSP template? This question is answered by developing an under-
1467
standing of and modeling the interactions between material selection and dimensional synthesis using the compromise DSP template. 4. Can the processes of design and manufacturing be integrated using a compromise DSP template? This question is answered by developing an understanding of and modeling the interaction between the processes of manufacturing and design via the formulation and solution of a compromise DSP template. The first question has been addressed in [21] and we are in the process of developing answers to questions three and four. In this paper we address the second question. 2. A
COMPROMISE DECISION SUPPORT PROBLEM TEMPLATE FOR THE DJXSIGN OF A COMPOSITE MATERIAL PRESSURE VESSEL
Central to the DSP Technique and the associated software system (called DSIDES) is a scheme to represent design information in a knowledge base. This requires the conceptual categorization of knowledge in terms of representation as well as the role it plays in capturing the DSP process and domainspecific information about the product. The knowledge base includes two types of knowledge: knowledge about the process of design and knowledge about the product being designed. The former, known as procedural knowledge, is defined by Rich [30] as a set of well-defined procedures representing information about doing things. The latter, namely declarative knowledge [30], is defined as a set of facts represented (usually) according to the protocol defined by procedural knowledge. In our case, the knowledge about the process (procedural knowledge), is embodied in the Decision Support Problem Technique. A template, therefore, is a collection of the mathematical forms of a set of DSPs relevant to a specific domain. It provides the means to customize group information collected for a specific domain. Further information is provided in [29]. 2.1. Problem stutement Design a cylindrical pressure vessel with hemispherical end closures (see Fig. 1). It is subjected to internal pressure loads and a constant temperature
p = const.
AT =
-1-I Fig. 1. A schematic of the pressure vessel.
1468
HARSHAVARDHAN KARANDIKARet
load across its thickness. The volume of the pressure vessel should be as close as possible to 10’ mm’. The performance factor of the pressure vessel is to be maximized. There is experimental evidence to suggest that it is advantageous to keep the ratios of the boss diameter to the chamber diameter between l/l0 and l/S. Needless to state that the pressure vessel should not fail under the given loading conditions. The design of the pressure vessel involves the determination of the following dimensions: boss diameter chamber diameter length of the cylind~cal section fiber orientation of each layer thickness of each layer
d D L %,@2
&I,f,.
The loading and the structural configuration pressure vessel follow:
of the
Loading symmetrical internal pressure p0 (1800 psi or 12.402 MPa) constant temperature difference AT (1OOC) (AT = internal operating temperature - ambient temperature). Structural configuration Thickness distribution hemispherical shell: t2(d) = &,/sin 4 cylindrical shell: t = to+ 2t, (constant along the lon~tudinal direction) Laminate configuration hemispherical shell: one symmetrical angle-ply laminate, fiber angle CQand thickness tz(4) cylindrical shell: two symmetrical angle-ply laminates, fiber angles o,, tlr and thicknesses to, 21,. 2.2. rde~ti~cut~o~ and yodeling of system descriptors A compromise DSP template is formulated using system descriptors, namely the system variables, the deviation variables (associated with the system goals), system constraints, system goals, bounds on the system variables and the objective. The first step in developing a compromise DSP template is to examine the problem statement and identify the system variables and also the factors that influence the design. In the compromise DSP these factors are modeled as system constraints, system goals and bounds. The objective in a compromise DSP is to minimize the difference between that which is sought and that which needs to be achieved. This difference is modeled as a deviation function. System variables. The system variables for this problem are as follows: Radius of the vessel Boss radius Length of the vessel
R (mm) r (mm) L (mm)
al.
Hemispherical sections thickness at 4 = 90” fiber orientation Cylindrical section thickness of the inner and outer layers fiber orientation in the inner and outer layers
to (mm) a2 (rad) t, (mm) tl, (rad).
System constraints. The most natural system constraints that arise in structural design of the pressure vessel are the failure criteria. Since no liner was specified for the pressure vessel, first-ply failure was assumed in the computations. Failure criteria have been in use for isotropic materials for centuries [31]. They can be divided into non-interactive criteria (such as maximum stress or maximum strain) and interactive criteria (for example, quadratic approximations). Non-interactive criteria are commonly used to model failure for brittle materials whereas interactive criteria are used for ductile materials that fail by yielding. For fiber-reinforced composite materials the failure criteria, as can be expected, involve extensions of those developed for isotropic materials. In this work we have used interactive failure criteria instead of non-interactive for two reasons, namely (i) these criteria are simple to use because they involve single-valued functions in contrast to non-interactive criteria, and (ii) their use results in a smaller number of constraints being specified in the template. A variation of the Halpin-Tsai failure criterion, as reported in [32], has been used in the work reported here. This criterion includes the checking of bond break and fiber break in each layer of the laminate. In general, the choice of the specific failure criteria to be used in a compromise DSP template is left to the template developer. New criteria can be very easily substituted in these templates. The stresses on the structure due to the loading are obtained by using classical laminate analysis and membrane theory (see Appendix). The hemispherical portion is checked for failure only at 4 = 90” since this is where the section is the thinnest and the pressure load is the highest. A parametric study of the stress behavior of the hemispherical shell was conducted to confirm this conjecture and is reported in [21]. Hence, we ensure against bonding break failure by specifying, [@,,i~,WB)” + (u~/~~~~)2 f
(~,2/~12BB)21 G 1
(11
and against fiber break failure with
K~,,bHFd216 1.
(2)
Since a 2D theory is used for computing stresses, the outer layer and the inner layer in the cylindrical section can be combined for checking for failure.
Compromise: design of composite material pressure vessels Thus the cylindrical section of the pressure vessel can be considered to consist of the following two layers: The combined outer and inner layers of the cylindrical section (k = 1). The middle layer of the cylindrical section (k = 2). Since the preceding constraints are imposed for each layer of the structure there are four constraints for the cylindrical portion and two constraints for the hemispherical portion. Additional constraints are imposed to model the desired geometrical relationships between certain variables. One of them is that the chamber diameter has to be greater than the boss diameter, R-r>&
(3)
Another constraint arises due to the fact that the winding angle in the hemisphe~cal section is limited by the ratios of the boss and chamber diameters, t12- sin-‘(r/R)
3 0.
(hemispherical and cylind~cal), at the junction (the “knuckle” area) of these shells. A new approach for reducing this stress concentration is proposed here, i.e., the stress concentration will be largely eliminated if the deformations of the hemispherical and cylindrical shells at the interface are identical. This, we believe, can be accomplished by tailoring the material thereby changing the stiffnesses of the different shells that form the connection. The deformations consist of the deflections along the meridian direction, circumferential direction and normal to the mid-surface and the rotations. In this study, the stress concentration is sought to be reduced by matching the deflection normal to the mid-surface for the two shells at their interface (see Appendix for the derivation of the deflections). This criterion is used as the primary system goal and is formulated as a system goal as shown in eqn (9). Since the absolute value of the difference in deflections is being used, d; is always zero and hence only the deviation variable, d;, is to be minimized. This is included in the deviation function [see eqn (13)]
(4) I(w,~,,- w&I + d; -d:
An additional constraint is imposed due to the fact that the winding angle in the cylindrical section is limited by the ratio of the chamber diameters and the length of the section, tll - tan-‘(2R/(L
- 2R)) 2 0.
(5)
It has been determined ex~rimentally that better pressure vessel performance can be expected when the boss-to-chamber diameter ratio is between l/l0 and l/S [33, 341. Further, it is seen that many of the vessels reported in [33] have length-to-chamber diameter ratios between 1 and 3. A higher L introduces a large variation in longitu~nal fiber tension on the cylindrical section of the mandrel when using an in-plane filament winding technique. Hence the following four constraints are formulated to keep the boss-tochamber diameter ratio between l/l0 and l/.5 and length-to-chamber diameter ratio between 1 and 3: lO.r-R30 5.r
-R
(6) (7)
L-2RaO
(8)
L -6RGO.
(9)
System goals. It has been seen and reported in [33,35] that a stress concentration is induced due to differing structural response of different shell types
t The rules for fo~~lating the goals and the inclusion of the deviation variables in the deviation function are presented in [5, 7, 10, 15, 161 and are therefore not repeated here.
1469
= 0.
(IO)
Pressure vessel designs are evaluated using a number of criteria, such as weight, strength-to-density ratio, burst pressure, performance factor and the factor of safety in different directions [34]. The performance factor, defined as the (burst pressure) * (volume)/ weight, is very popular for the following reasons: (i) a variety of different pressure posite and homogeneous, can (ii) it approximates the condition safety in all directions of the
vessels, both combe compared, and of equal factor of pressure vessel.
The performance factor appears as a system goal in the compromise DSP fo~ulation [see eqn (IO)]. Since a high performance is desirable, a large target value of 10’ is set for the performance factor goal and the deviation variable, d;. is included in the deviation function and minimized,t P * Voiume/(Weight
* 10”) + d; - d+ = 1
(11)
where Volume = n * R2. [(4/3) ( R + L] Weight = 2nRpg{2.
[R . to. (n/2 - &,,i,)]
&,i, = sin-‘(r/R).
If instead of performance factor, weight alone were to be used as a criterion for evaluation then it can be modeled as a system goal as shown in eqn (I 1). Since we desire to minimize the weight, the deviation
1470
HARSHAVARDHANKARANDIKAR
variable, d,+ , is included in the deviation function and minimized, Weight + d; - d: = 0.
(12)
The volume of the pressure vessel is also represented as a system goal. In this case we have used the customer’s specification to normalize the system goal. Since we wish to achieve the customer’s specified volume both the deviation variables have to be minimized and included in the deviation function, Volume/(lO’) +- d; - d: = 1.
(13)
Numerical bounds are placed on the variables. The specific values stem from a designer’s judgment and experience. The bounds limit the feasible design space (which is computationally advantageous). The minimum layer thickness that can be wound imposes the lower bound on the thicknesses. The bounds on the winding angle in the hemispherical section are due to the limitations of the manufacturing process. The other bounds reflect a designer’s judgment and experience. Deviation function. The deviation function consists of those deviation variables that are identified for minimization while formulating the system goals, i.e., d: , d; , d: , d; , and d: The volume (being specified by the customer) and the deflection match are the most important goals and hence the deviation variables, d; , dJ , and d: , are given the highest priority for solution. The system goals corresponding to performance factor and weight are the next important and hence the deviation variables, d; and d:, are given the next highest priority. All the other deviation variables are given the lowest priority. A pre-emptive formulation of deviation function [8] follows: Bounds.
et al.
Hemispherical section thickness at 4 = 90” fiber orientation Cylindrical section thickness of the inner and outer layers fiber orientation in the inner and outer layers
to (mm) a2 (rad) t, (mm) CI,(rad).
Satisfy System constraints
Failure criteria for each layer: bonding break failure [(all !c,188)2 + h/%2BB)2
+ (~,2/~,2Ed21 G 1 (1)
break failure of the fiber [(al I lo Id21
G 1
(2)
Geometry vessel radius is greater than the boss radius R-r>0
(3)
minimum fiber angle permissible in the hemispherical section t12- sin-‘(r/R)
> 0
(4)
minimum fiber angle permissible in the cylindrical section CI,- tan-‘(2R/(L
- 2R)) > 0
(5)
boss-to-chamber diameter ratios have to be between l/l0 and l/S
Z={(d:+d;+d:),(d;+d:), (d; +d:
2.3. The mathematical formulation DSP
+d;)}.
of the compromise
Given
Loading internal pressure, P temperature difference across the thickness, AT Pressure vessel construction (see Fig. 1) thickness distribution laminate configuration Find
Dimensions
lO.r-RaO
(6)
5.r-R
(7)
(14)
length-to-chamber tween 1 and 3
diameter ratios have to be be-
L-2R>O
(8)
L-6R
(9)
System goals
Deflection match at the interface of the hemispher ical and cylindrical sections I(%,, - w&l + d; -d:
= 0
(10)
of the pressure vessel Performance factor
Radius of the vessel Boss radius Length of the vessel
R (mm) r (mm) L (mm)
P * Volume/(Weight
* 10’) + d; - d: = 1 (11)
Compromise: design of composite material pressure Weight Weight + d; - d: = 0
(12)
Volume Volume/( 10’) + d; - d: = 1.
(13)
Bounds on each system variable:
0.115 < t,, t, G20.0 0.2094 < 01~< 0.4363 0.0 < tlr < 1.5708 50.0 < R < 300 5.0
variable:
d,,d:,d;,d:,d;,d:,d;,d:~O.O. Minimize
Deviation function Z = {(d: + d; + d: ), (d; + d:), (d, + d: + d;)}.
The template is solved using System [24,25,34].
3. IMPLEMENTATION
the UH
AND DISCUSSION
(14)
DSIDES
OF RESULTS
3.1. Implementation of the compromise DSP template Based on our experience in developing complex templates such as the one presented in Sec. 2.2, we developed the template in three steps. In this context, two cases and two parametric studies were implemented and a description of each follows. Analysis of the results is presented in Sec. 3.2. The deviation function [see eqn (14)] that is given in the mathematical form of the template can be implemented in three ways.
vessels
1471
in the literature as has been indicated in [l] and presented here to establish the congruency of results. The number of system variables is 4, the number of system constraints is 6 while the number of goals is 2. While the length and diameters are fixed, the thicknesses and the orientations of the various layers are retained as variables. The constraints that are included in the template pertain to checking for bond break and fiber break in each of the three layers. Only the bounds pertaining to the system variables that are included in the template are considered. The system goals considered pertain to deflection match [eqn (lo)] and minimization of weight [eqn (12)]. Comprehensive case. This is an expanded version of the first case. The number of system variables is 7 with 13 system contraints and 4 system goals. The diameters and the length are included as variables. Seven constraints concerned with the geometry of the pressure vessel [eqns (3)-(9)] and two more goals concerned with achieving the customer specification on volume and maximizing the performance factor [eqns (11) and (13)] appear in this template. Hence, this case is much more representative of real-world problems than the exploratory case. This study was carried out with only the carbon fiber-epoxy resin system. Parametric studies. Parametric studies are very important during designing for concept which typically occurs early in the design process. Two parametric studies are presented to demonstrate the ease with which such studies can be conducted using the compromise DSP template in the designing for concept phase of the design process. In Parametric Study A, the behavior of the performance factor for changes in d/D and L/D ratio are examined. This is done by fixing d/D and L/D ratios using equality constraints [see eqns (15) and (16)] and solving the template. The performance factor at the solution is plotted against d/D at different L/D ratios in Fig. 2. It is observed that the fiber angle of the middle layer always converges to the upper bound of 25” in both the exploratory and confirmatory cases (see Tables 3 and
1. By obtaining the lexicographic minimum of the deviation function using a multiplex algorithm [27]. 2. By obtaining the minimum of the deviation function using a pseudo-pre-emptive formulation [8]. 3. By obtaining the minimum of the deviation function using an Archimedean formulation [8]. The choice of the formulation depends on the maturity of the template. The Archimedean formulation is used in all the cases and parametric studies. Exploratory case. The major dimensions of the filament wound pressure vessel, the radii and the length, are fixed in this case. This is an elementary case which has been studied and reported extensively
Fig. 2. Parametric Study A: performance factor vs d/D ratio.
1472
HARSHAVARDHAN KARANDIKAR et al.
target values for the performance factor. The results are shown in Fig. 3. d/D-x=0
wherex=O.1,0.2
,...,
L/D -y=O
where y = 1,2,3.
0.7
(15) (16)
The system variables, system constraints and goals for the three cases are summarized in Table 1. The material properties used in this study are presented in Table 2.
o.oo*+o
2SOc+6
3.2. solution
1.001,+7
7.50*+6
s.oor+6 YoMlne (mu?,
Fig. 3. Parametric Study B: changes in target volume.
4). Hence, in the next parametric study the upper bound of fiber angle of the middle layer is increased from 25” to 45”. The results for this are also shown in Fig. 2. In Parametric Study B, the behavior of the system variables at different target volumes is examined. This is done by solving the template at different
analysis
Exploratory case. The solution for this case is presented in Table 3. After complete development the solution behavior of this template is seen to be very stable. This case is solved for different material combinations and since the diameters are fixed, d/D ratios are also varied.
(i) Carbon fiber-epoxy resin at d/D = l/S (ii) Carbon fiber-epoxy resin at d/D = l/l0 (iii) Glass fiber-epoxy resin at d/D = l/S.
Table 1. The compromise DSP templates: variables, constraints and goals
System variables
Exploratory case
case, Parametric Study B
fflrti~~zr~l
&,,~,,~2,~,,R,r,~
System constraints (nos.)
(1>2)
System goals (nos.)
(10, 12)
Fixed variables
R, r, L
R r
Parametric Studv A to.
5,
a,,
5.
R,
(1,2,3,4,5,6,7,8,9)
(t,2,3,4,5,15,16)
(10, 11, 12,13) -
(10, 11,13) -
r,
L
80.0
(l/S or l/lO)R 480.0
L
Table 2. Material constants used in the analysis Material? E-glass fiberepoxy resin Density (kg/mm3) Poisson’s ratio Shear modulus (N/mm2) Young’s modulus {N/mm~) fiber direc. transverse direc. Thermal coefficient (I /OK) fiber direc. transverse direc. Bond strength (N/mm*), fiber direc., X transverse direc., Y Bond strength, shear (N/mm2), V Fiber strength (N/mm2), 2
Carbon fiberepoxy resin
Kevlar 49epoxy resin
1.55E6
G
1.94E-6 0.25 4500
0.3 4600
1.32E-6 0.25 2800
E, EZ
37232 8274
140556 9646
82000 4000
arl 872
6.3E-6 20.5B6
0.23E-6 22SE-6
- 3.6E-6 57.6&6
(t=s) (camp) (tens) (camp)
820.5 517.0 48.3 137.9
1999.5 1559.2 54.0 150.
1158.0 276.0 Il.0 83.0
(tens) (camp)
68.9 1140 1045
90.9 1620 1480
69.0 1930 174s
VI2
t The composites have 60% fiber volume fraction.
and
Compromise: design of composite material pressure vessels
1473
Table 3. Exploratory case: solution Deflection (mm)
Final design (r-mm; a-degrees)
Initial design (f-mm; a-degrees) t0 fl a2 XI fo Carbon fiber-epoxy resin; d/D = l/S 6.0 6.0 12 zt 5.29 2.0 2.0 18 5.291 3.0 1.0 12 30 5.291
‘I
aI
g2
0.193 0.193 0.193
25 25 25
Wsph
89.91 89.73 89.78
0.7105 0.7105 0.7105
WCYI
0.7104 0.7105 0.7105
Weight = 22.745N Carbon fiber-epoxy resin; d/D = l/l0 3.0 1.0 12 30 5.291
0.193
25
89.7
Glass fiber-epoxy resin; d/D = l/S 6.0 6.0 12 45 6.769 3.0 1.0 12 30 6.802
0.872 0.876
25 25
89.9 89.9
0.6413 0.6389
0.6407 0.6386
Weight = 40.35 N No, of design variables = 4; No. of system constraints = 6; No. of system goals = 2.
(i) Carbon fiber-epoxy resin; d/D = l/5. The template was exercised using three different initial or starting designs. The final solution converged to one point from three different starting points. The deflection match at the interface, which is the principal criterion for reducing stress concentration, is seen to be very good at this solution point; the difference in deflections between the spherical portion and the cylindrical portion is 0.3 pm. The bond break failure criterion in the middle layer in the cylindrical section is seen to be the limiting constraint in this case. The winding angle in the hemispherical section tends to lie at the upper limit of 25”. The spherical shell construction is most efficient at a 45” angle as the stresses in the membrane are uniform. This is also indicated in the parametric study results [21]. Since the upper bound of this angle is taken to be 25”, the solution point tries to be as close to this condition as possible. (ii) Carbon fiber-epoxy resin; d/D = l/10. No change in solution was observed from the preceding solution. This is expected because of the nature of the fo~ulation. The d/D ratio does not appear any where and hence the value of the ratio is not relevant to the solution of this formulation. (iii) Glass fiber-epoxy resin; d/D = l/5. As expected the weight of the pressure vessel is seen to be increased because of the greater thicknesses of the layers and the higher density of this material. The thicknesses are higher because of the lower strength
and stiffness of this composite material system. The deflection match is again seen to be very good. The difference in deflections between the spherical portion and the cylindrical portion is again 0.3 pm. Co~pre~e~i~e case. To test the ~omprehensiveness of the template, three different designs, hereafter called scenarios, are used as starting solutions. The final solution is presented in Table 4 and it can be seen that they are nearly identical. This shows that the template is complete enough to converge to the same solution for widely varying scenarios. It can be observed that the fiber angle in the outer layers of the cylindrical section converges to its upper bounds, 90”, consistently in all the scenarios. This implies that there is an overwrap of material in the hoop direction in the cylindrical section of the pressure vessel. This is required because of the biaxial force field in that section of the pressure vessel. Yet another observation regarding the behavior of the template is made and it is shown that it agrees with intuition. It is observed that one of the constraints representing bond break failure criterion is always active throughout the solution process for all the three scenarios and the fiber break constraints never become active. This is expected because the fiber strength of the composite is much higher than the bond strength of the composite and hence they cannot be limiting constraints by themselves. The defIection match is again very good, 0.1% at worst, and this is indicative of the
Table 4. Comprehensive ease: solution Initial design (t, R, r, L-mm; a-degrees) T,
‘I 10 1.0 3.115
a2 25 12 17.7
aI 90 0 54.4
R
Final design (t, R, r, L-mm; a-degrees) r
L
to
t1
a2
al
20 300 60.0 1000 5.569 0.203 25 90 1.0 70.0 10.0 140 5.569 0.2027 25 4.115 90.0 10.0 500 5.569 0.2033 25 No. of design variables = 7; No. of system ~onstmints = 12; No. of system goals = 4. t wrph= 0.7479, wq, = 0.7478 mm, weight = 26.5345N. = 0.7488 mm, weight = 26.512N. SWsph= 0.7477, weYl = 0.7476mm, weight = 26.5165N. § W,pll= 0.7477, wcYl
R
r
84.21
16.84 16.84 16.84
L
505.3t 504.9$ SOS.O$
HARSHAVARDHAN KARANDIKAR
1474
appropriateness system goal.
of using the deflection criterion as a
Parametric studies. A plot of the results obtained from Parametric Study A is shown in Fig. 2. It is observed that the performance factor increases with an increase in d/D ratio. Since the pressure vessel is checked for failure only at the interface between the hemispherical and cylindrical sections, the increase in boss diameter, d, does not affect the solution. However, the increase in boss diameter reduces the weight of the pressure vessel and this leads to an increase in performance factor [see eqn (ll)]. When the upper bound of the fiber angle, c+, is increased to 45” from 25” (see Sec. 2.3), the solution converges close to this value. This is accompanied by a reduction in thickness of the middle layer. When the fiber angle is increased to 45”, the hoop strength of the middle layer in the cylindrical section (see Fig. 1) increases and hence a smaller thickness is able to resist the hoop stress. A logarithmic plot of the system variables at different target volumes obtained from Parametric Study B is shown in Fig. 3. It is observed that the chamber diameter and the length of the pressure vessel increase in order to effect the desired increase in volume. As the chamber diameter increases, the pressure load on the pressure vessel increases [see eqns (Al), (A2) and (Al l)]. This leads to an increase in the thickness as seen in Fig. 3. It is seen that the behavior of the template for the two parametric studies is justifiable. These parametric studies illustrate the ease with which such studies can be conducted using the template. Commentary on cases. The exploratory case dealing primarily with weight minimization is extensively reported in the literature, as stated in [l], and it is shown in this paper that such simple problems can be solved using the compromise DSP formulation. The comprehensive case established the effectiveness of using the compromise DSP for larger problems with multiple objectives, namely the performance factor, volume and weight. This case is much closer to the real-world than the exploratory case. Finally, to illustrate the effectiveness of the template in the designing for concept phase of composite pressure vessel design, two parametric studies are conducted and the behavior of the template is justified. As mentioned earlier, the emphasis here is on the method and not the numbers per se. It is evident from the results that the compromise DSP template can be effectively used in designing for concept of composite pressure vessels. This work sets the stage for combining material selection and manufacture with design, the ultimate aim of this project.
4. CLOSURE
In this paper we report on the usefulness of an approach using compromise Decision Support Problems in “designing for concept” of a pressure
et
al.
vessel made of composite materials. The dimensional synthesis of the pressure vessel is modeled as a compromise Decision Support Problem; with system variables corresponding to the dimensions of the pressure vessel; system constraints modeling the bonding break and fiber break failure criteria and some geometric constraints and system goals including the performance factor, deflection match at the interface between the cylindrical and the hemispherical shells, weight and the volume of the pressure vessel. The capability of the compromise Decision Support Problem template to handle multiple objectives helps the designer in modeling the design closer to the real life situation than the traditional single objective formulations with which other researchers seek to “optimize” their designs. The formulation of the template provides the means to perform parametric studies with ease to obtain information about the various aspects of the design. This capability is useful in the designing for concept phase of the design process because it saves time and effort. Our first aim in this paper was to demonstrate the effectiveness of using a novel criterion to reduce stress concentration at the “knuckle” region of the pressure vessel. In designing for concept where the design is likely to be revised, it is not profitable to use sophisticated analytical tools like finite element analysis. This novel criterion makes possible use of membrane theory (it involves a simple calculation) in designing for concept of a pressure vessel with different shell sections and thence this enables us to achieve the first aim of the paper. A preliminary indication of the efficacy of this criterion in reducing the stress concentration is reported in [2]. Our other aim is to examine the efficacy of the compromise DSP in the designing for concept phase for composite pressure vessels. This is illustrated through a set of case studies and parametric studies. An initial case, namely the exploratory case, is solved to learn about the behavior of the template. This is followed by a more complex case, namely the comprehensive case, which establishes the efficacy of using a compromise DSP template for solving realistic composite pressure vessel design problems. Two parametric studies are conducted to demonstrate the ease with which such studies can be conducted using the template. Further work can be done by investigating the effect of changing the priorities assigned to various system goals and repeating the study with a different dome contour. It is reported by Lark [33] that the hemispherical contour is the least efficient. Another improvement is to include the economic efficiency of the pressure vessel and investigate the trade-off between technical efficiency as represented by the performance factor and economic efficiency. As indicated earlier efforts are under way to integrate material selection and aspects of manufacture in the template and also make provision for a non-load bearing liner as is used in industrial practise.
Compromise: design of composite material pressure vessels Acknowledgements-We gratefully acknowledge the financial contribution made by the Shell Development Company, Houston, Texas to initiate this project in 1986. We gratefully acknowledge the financial contribution made by our corporate sponsor, the B. F. Goodrich Company, Brecksville, Ohio for the continuation of this project. The computer costs were underwritten by the University of Houston. Support for Ramesh Srinivasan was provided by the Department of Mechanical Engineering, University of Houston, Houston, Texas.
REFERENCES 1.W. K. Fuchs, H. M. Karandikar, F. Mistree and H. Eschenauer, Compromise: an effective approach for designing composite conical shell structures. Prac. 1988 Design Aufomation Conference, Orlando, September (1988). 2. W. J. Fuchs, H. M. Karandikar, R. Srinivasan and F. Mistree, Preliminary design of a filament wound pressure vessel: a coupled decision support problem. Proc. Conf on Discretization Methods and Structural Optimization: Procedures and Applications, Siegen,
F.R.G., October 5-7 (1988). 3. W. Fuchs, Struktur~a~yse und Optimierung anjsotro~r Schaien aus F~e~erbundwerksto~ ~~fr~~f~rai Analysis and Optimum Design of Anisotropic Fiber Remforced Shells). Doctoral Thesis, University of Siegen (1986)
[In German]. 4. W. Schnell and H. Eschenauer, Elastizitcitstheorie II; Schalen. Mannheim. Vienna: Bibliogranhisches Institut _ _ (1984). 5. N. Kuppuraju and F. Mistree, Compromise: an effective
6.
7.
8.
9.
10. 11. 12.
13.
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using Compromise
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APPENDIX: STRUCTURAL ANALYSIS OF COMPOSITE PRESSURE VESSEL The stresses and deflections due to the pressure and temperature loadings on the pressure vessel are used in the compromise DSP template [see eqns (1) (2) and (9)]. These are calculated using membrane theory and classical laminate analysis as shown in this appendix. Structural analysis of a cylindrical shell [2,3,21,36,37]
The longitudinal and circumferential stress resultants in the cylinder due to the pressure loading are computed using membrane theory as follows: N,, = p0 (R 12) (force/unit length)
(Al)
N,, = p0 R
(A2)
(force/unit length).
Thus. (A3)
The reduced stiffnesses (Q,) for each layer in the cylinder are obtained from the material properties. These stiffnesses are transformed using the fiber angles of the laminate to determine the stiffnesses for each layer in the principal directions of the cylindrical shell, (G), or Q, the global reduced stiffness matrix. The extensional stiffness matrix for the whole cylindrical shell structure, A, is determined from the global reduced stiffnesses of the various layers using the following equation:
k=l
where k = 1,2, , n is the number of layers and i, j represent principal directions. The stress resultant matrix due to the temperature loading, N,, can then be determined from the following equation: N, =
C te,,), ‘1, . a~,.AT. k=,
The stress resultants due to the pressure and the temperature loading and strains are related as shown below: N=Af--N,
646)
From eqn (A6), strains and deflections are computed which are shown in eqns (A7) and (A8). c = A-‘(N + NT).
(A7)
The deflection of the cylindrical shell along the hoop direction which is used in the deflection goal [see eqn (9)] can be expressed as below: Wcyl= t,, A.
(‘48)
The global stress matrix, d, is then calculated as shown below: 6”’ = ?j(‘)(E-
AT)
K,(k)
649)
where br is the matrix of temperature coefficients. The global stress matrix, 8, is reduced to the local stress matrix, n, using a transformation matrix, T. These are used in formulating the constraints on failure criteria [see eqns (1) and (2)l a(k) = T&k’,
(AlO)
The stress matrix, o, consists of three terms, [o,, , CT*?, ~~~1~. Since the stresses and displacements are independent of the radial distance of the layers from the axis, the outer layer and the inner layer in the cylindrical section can be combined. Thus the cylindrical section of the pressure vessel can be considered to consist of the following two layers: k = 1 The combined outer and inner layers of the cylindrical section. k = 2 The middle layer of the cylindrical section. Struclural analysis of hemispherical shell [2,20,36,38]
The stress resultant along the meridian direction and the circumferential direction due to the pressure loading is computed from membrane theory of shells, as indicated below: N,, = No0= Rp,/2.
(All)
The local reduced stiffness matrix, Q, the global reduced stiffness matrix, Q, and the extensional stiffness matrix, A, are obtained assuming a 2D orthotropic material. The coupling stiffness matrix B is absent due to the basic assumption that the shell is formed of a symmetric laminate. The thermal stress resultants along the meridian and circumferential direction are calculated using eqn (AS). The general solution for the deflection of the spherical shell is derived from the general solution for the deflection of the membrane shell. The specific solution for the deflections, u(@) and w(a), of this spherical shell is then obtained from the general solution for the spherical shell using the material law in terms of stress resultants due to pressure loading and temperature loading. The deflection normal to the mid-surface, w(G), is calculated at the interface between the hemispherical and cylindrical shell sections, i.e., Q = n/2, and is used in formulating the deflection goal [see eqn (9)]. The global strains are derived from these deflections, u(0) and w(e). t ,@= (l/R)(w + du/d@) C&,= (l/R)(w + u cot @J).
6412)
The global strain matrix, C, is then obtained as
where is the global strain matrix for the shell.
and
ye0 =O.
6413)
Compromise: design of composite material pressure vessels From the global strain matrix, C; the local strain matrix, L, is computed using the transformation matrix, T. The local stress matrix, u, can be expressed as, u =Q(c
-a
AT)
(A14)
1477
where u = [a,, , uz2,r12]*. These are used in checking for bond break failure and fiber break failure [see eqns (1) and (2)]. The hemispherical portion is checked for faiture only at # = 90” since it is the thinnest section. Note that the hemisphe~cal portion consists of only one layer.