- Email: [email protected]
Design of a steam-heated sterilizer based on finite element method stress analysis
International Journal of Pressure Vessels and Piping 78 (2001) 627±635
www.elsevier.com/locate/ijpvp
Design of a steam-heated sterilizer based on ®nite element method stress analysis R.M. Natal Jorge*, A.A. Fernandes Department of Mechanical Engineering and Industrial Management, Faculty of Engineering, University of Porto, Rua Dr Roberto Frias, 4200-465 Porto, Portugal Received 28 November 2000; revised 1 August 2001; accepted 31 August 2001
Abstract A steam-heated sterilizer is a pressure vessel of rectangular cross section externally reinforced by members welded to the ¯at surfaces of the vessel. These members are pressure vessels too with rectangular cross section. The ASME code provides alternative rules (Division 2) for the design of pressure vessels based on a Design By Analysis route. The stresses on the sterilizer were computed by the ®nite element method followed by the calculation of the stress ®elds according to the classi®cation established in the ASME code. Structural members with shell intersections (as in the present case) present dif®culties due to problems of linearisation and categorisation. In the present work the shell stress resultants were used instead of smoothed stresses. The operating conditions of the vessel involve cyclic application of loads requiring design based on fatigue analysis. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Finite element; Vessel of non-circular cross section; Design by analysis
1. Introduction Pressure vessel design has been historically based on Design By Formula. Standard vessel con®gurations are designed using a series of simple formulae and charts. In addition to the Design By Formula route, many national codes and standards for pressure vessel and boiler design, also provide for a Design By Analysis (DBA) route, where the admissibility of a design is checked, or proven, via a detailed investigation of the structure's behaviour under the external loads to be considered. DBA procedures do not specify particular implementation tools. However, the most widely used technique in contemporary pressure vessel design is the ®nite element method (FEM), a powerful technique allowing the detailed modelling of complex vessels [1]. All the DBA routes in the major codes and standards of pressure equipment are based on the rules proposed in the ASME pressure vessel and boiler code [2], which was formulated in the late 1950's. All these routes lead to the same wellknown problems, especially the stress categorisation problem (see for example Refs. [3,4]). In general, this does not present a problem in cases where the analysis uses thin shells. However, for the analysis of structural members with shell intersections, where the calculated nodal stresses cannot be * Corresponding author. Tel.: 1351-22-508-1713; fax: 1351-22-508-1584. E-mail address: [email protected] (R.M. Natal Jorge).
easily identi®ed as bottom, middle or top, the problems of linearisation and categorisation become more dif®cult. In fact, when a structural member has shell intersections, the nodal stress tensors, for example, the stress tensors of bottom points, cannot be smoothed. In the present work the use of nodal shell stress resultants are proposed as variables to be smoothed before the procedures of linearisation. With these smoothed nodal shell stress resultants for all the elements, the shell stress resultants can be interpolated to the Gauss points. In the Gauss points, the stress tensors can be carried out to the top, middle and bottom of the shell. Thus, these stress tensors must be extrapolated to the nodes and no smoothing is made. Finally, for each node and at the element level, the linearisation is computed based on Kroenke's procedure [5]. 2. Design by analysis Most of the DBA guidelines given in the codes relate to design based on elastic analysis Ð this is the so-called elastic route [6]. In this approach, based on the ASME code [2], the designer is required to classify the calculated stress into primary, secondary and peak categories and apply speci®ed allowable stress limits. The magnitudes of the allowable values assigned to the various stress categories re¯ect the nature of their associated failure mechanisms. Therefore it is essential that the categorisation
0308-0161/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(01)00078-3
628
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
stress distribution at Gauss points in one element
a)
b) Fig. 1. Plane problems: unsmoothed (a) and smoothed stresses (b).
average has no problem. However, in shell elements more than one point through the thickness is present, usually named: bottom, middle and top. For example, at node i (Fig. 2) the stress tensor for bottom, middle and top points can be de®ned. These stress tensors are obtained from the element T, element L and element R. For one single point, at node i, there will be the contribution of each element (three in this example). Obviously, in this case the stress tensors cannot be extrapolated from the Gauss points to the nodes and then make an average. The reason is simple: in element L and element R at node i the bottom point, the middle point and the top point are the same, but in the element T the bottom point, for example, has nothing to do with the points from elements L and R. In the work being reported, the shell stress resultants are used. In this case, for each element the shell stress resultants in the Gauss point in the local referential system are calculated. In the same manner working with the stress components, the shell stress resultants are extrapolated to the nodes (Fig. 3). This fact implies that, in general, for each shell stress resultant (STR) the equilibrium equations are not veri®ed, since the extrapolation is independent from element to element, i.e.:
procedure is performed correctly. The designer is required to decompose the elastic stress ®eld into these three categories and apply the appropriate stress limits. Stress categorisation is probably the most dif®cult aspect of the DBA procedure and, paradoxically, the problem has become more dif®cult as the stress analysis techniques have improved. In structural members with shell intersections, the problem caused by the dif®culty associated with the smoothed stresses is even more complex. 2.1. Smoothed procedure Analysis involving numerically integrated elements such as isoparametric elements, have shown that the integration points are the best stress sampling points. The nodes, which are the most useful output locations for stresses, appear to be the worst sampling points. Reasons for this phenomenon can be found in Ref. [7]. So the most popular procedure consists of calculating the nodal stresses (each component) by extrapolation based on integration points. In this step, each component has several nodal values, because they are obtained at the element level (Fig. 1a). Taking into account these nodal values, a single value can be obtained for each component, by the application of some kind of average (Fig. 1b). This procedure, very popular for plane stress and plane strain situations, presents some problems when applied to structural shells. In plane stress (or plane strain) elements there is one single point through the thickness, so the nodal
D STR
n X i1
STRi ± 0
element T node m
T
m
R
L bottom
top
j
i
top element L
element R node i
node j
bottom
node i
node i
bottom
node k
Fig. 2. Shell elements de®nition of top and bottom side.
k
1
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635 element T
Q Ty
N Txy
element L L N xy
N Ly
N Lx Q xL
Q Ly
element T
Q Tx element R
N Ty N T x N Rx
N Ry
629
element L
Q Rx N Rxy
M Tx
M Txy
M Txy
M Ty
M Rx
M Lxy M Rxy L
Q Ry
M Ly
M Lxy
element R
Mx
M Ry
M Rxy
Fig. 3. Shell stress resultants.
where n is the number of elements meeting at the node under consideration. For each node, in order to verify the equilibrium equations it is proposed that the shell stress resultants should be distributed among the elements meeting at the joint. These distributed shell stress resultants in each element 2
1 6 e 6 8 9 6 6 60 > s xx t > > > 6 > > > > 6 > > 6 > > s t > > yy 6 > > < = 60 txy t 6 6 6 > > > > 6 > > > > 6 t t > > xz 60 > > > > > > : t t ; 6 6 6 yz 6 6 4 0
0
0
12t 2 3 e
1 e
0
0
0
1 e
0
2
shell stress resultants can be interpolated to the Gauss points. From shell theory [9] and based on the shell stress resultants the stress components at a Gauss point are calculated at any point through the thickness at a distance from the middle point of t according to (see Fig. 4):
0
0
0
0
12t e3
0
0
0
12t e3
0
0
0
2
2
0
0
0
0
0
6t 3 2 3 2e e
0
0
0
0
0
0
are proportional to their relative original values. Inspired by the idea of the moment-distribution method of analysis of framed structures, the ratio of the shell stress resultant in an element to the equilibrium equation can be called the distribution factor (DF) [8]. It follows that the DF for an element is equal to the original value of shell stress resultant divided by the sum of the original values of shell stress resultant of the elements meeting at the joint, that is, u STRi u DFi X n u STRj u
2
j1
where i refers to the element considered and n refers to the elements meeting at the joint. The ®nal shell stress resultants can now be obtained by STRi 2D STR £ DFi 1 STRi
3
This procedure works like a smoothed shell stress resultant, where the original values are taken into account in the same way as a distributor of rigidity. Finally, with all the nodes in equilibrium, making use of the same shape functions that are used to evaluate the displacement vector in any point inside an element, the
! 0 6t2 2 e3
38 9 Nx > > > 7> > 7> > > 7> > > Ny > > 7> > > 7> > > 7> > > N 7> xy > > > 7> > > 7> < Mx > = 7> 7 7 7> My > > 7> > > > 7> > > 7> > > > M 7> > xy > 7> > > 7 > > > ! 7> > Q x > > > 7 > 3 5> > > : ; Qy 2e
4
In Eq. (4), for the middle point t 0; for the top point t 1e=2 and for the bottom point t 2e=2: Subsequently the three stress tensors are extrapolated at bottom, middle and top surfaces to the nodes. 2.2. Stress categorisation As mentioned in Section 2.1 the three stress tensors are carried out in the nodal local co-ordinates at every node, speci®cally: ² stress tensor with their components at bottom of the shell: s ijb ; ² stress tensor with their components at middle of the shell: s ijm ; ² stress tensor with their components at top of the shell: s ijt : Based on these three stress tensors a stress function is needed to calculate every stress component at any point through the thickness. This distribution can be approximated by a parabolic function (see Fig. 4): s ij t
s ijt 2 s ijb 2 t b m 2 t 1 s ijm 5 t s 1 s 2 2s 1 ij ij ij e e2
630
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
stress (Pm); local primary membrane stress (PL); primary bending stress (Pb); secondary stress (Q); peak stress (F). The membrane and bending components are then categorised depending upon the location of the supporting line segment. In this work the stress categories taken into account are:
top surface middle surface
bottom surface supporting line segment
² effective membrane stress: mem PL s Imem 2 s III
9
² effective bending stress: Fig. 4. Stress distribution through thickness.
ben Q s Iben 2 s III
Following the work of Kroenke, at any node, each component of the stress tensor can be decomposed into three subsets [5]: s ijmem
² stress tensor of membrane stresses : is the tensor whose components, constant along the supporting line segment, are equal to the average of the stresses along this supporting line segment; ² stress tensor of bending stresses s ijben : is the tensor whose components, varying linearly across the thickness of the shell, are calculated based on the beam bending theory (assuming a rectangular cross section with unit width); ² stress tensor of non-linear stresses s ijnln : is the tensor whose components are equal to: s ijnln s ij 2 s ijmem 2 s ijben
6
According to these de®nitions, the stress tensors of membrane and bending stresses as a function of the stress tensors s ijb s ijm and s ijt can be obtained: 1 Z1 e=2 1 t s ij 1 s ijb 2 2s ijm 1 s ijm 7 s ijmem s ij t dt e 2 e=2 6 s ijben
s ijt 2 s ijb 6 Z1 e=2 s tt dt ij 2 e2 2 e=2
8
It is important to note that the nodal membrane and nodal bending stresses can be calculated directly from expression (4). Obviously, in this case and for several components of stresses tensor the non-linear term cannot be obtained, since those stresses are constants, or depend linearly on t (it is the case for s xx ; s yy and txy ). 2.3. Assessment criteria According to the ASME code [2], one requirement for the acceptability of a design is that the calculated stress intensities shall not exceed speci®ed allowable limits. These limits depend of the stress category from which the stress intensity is derived. The ASME code establishes the following classi®cation of stresses: general primary membrane
10
² effective non-linear stress: nln F s Inln 2 s III
11
nIn mem where s InIn ; s IInIn ; s III ; s Imem ; s IImem ; s III and s Iben ; ben ben s II ; s III are the eigenvalues of the stress tensors de®ned in Eqs. (6)±(8), it corresponds to the description of the principal stresses. In agreement with the values of effective stresses calculated by Eqs. (9)±(11) and according to the ASME code the following evaluation criteria were used:
PL # 1:5Sm
12
PL 1 Q # 3Sm
13
PL 1 Q 1 F # S a
14
where Sm and Sa are the allowable stress intensities, for static and fatigue evaluations. In fatigue analysis if there are two or more types of stress cycle which produce signi®cant stresses, their cumulative effect shall be evaluated. According to the ASME code, it is necessary to specify the number of times each type of stress cycle will be repeated during the life of the vessel. In determining the number of the cycles for each stress cycle type, consideration shall be given to the superposition of cycles of various origins, which produce a total stress difference range greater than the stress difference ranges of the individual cycles [2]. 3. Example: analysis of a steam sterilizer In order to assess the proposed methodology, an analysis of a steam-heated sterilizer is carried out. Based on the classi®cation of Weiû, this kind of pressure vessel is not typical [10]. The steam-heated sterilizer that has been studied has the geometry shown in Fig. 5. In this ®gure, only one eighth of the component is represented corresponding to the ®nite element model used. The sterilizer has
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
631
steam jacket closure collar
thk = 5 2 1
thk = 6
guide thk = 6
thk = 5
Thicknesses: - Chamber: 5 mm - Steam jacket 1: 6 mm - Steam jacket 2: 5 mm - Closure collar: 6 mm - Guide: 25 mm
chamber beam elements (studs)
Fig. 5. Steam-heated sterilizer: geometry and ®nite element model.
four-steam jackets, which work as reinforcing members, two-closure collars and two doors. The doors have an ascending or descending vertical sliding movement in two guides. These guides are bolted rigidly to studs welded to the external steam jackets (see Fig. 6). The operation of the sterilizer can be summarised in the following way: the door is slid to the closed position; pressure is introduced in the closure collars in order to maintain suf®cient pressure on the silicon gasket to ensure a tight joint with the door and prevent leakage of steam. The sterilizer is then subjected to internal pressure, maintaining the pressure in the closure collars. The load combinations indicated in Table 1 re¯ect this operation procedure in detail and were de®ned on the basis of service load spectra supplied by the user, illustrated in Fig. 7 (for the chamber only). The calculation route can be summarised as follows: evaluate by FEM for each load combination, the nodal
values on the guides due to the loads exerted by the door on the guides; combine these forces with the loads present in the body. The guides were modelled with shell elements and the studs with beam elements. Due to the diameter of these studs (40 mm) the connection stud/steam jacket was modelled through rigid elements. The material elastic properties are: modulus of elasticity E 210 GPa (at 208C), E 196 GPa (at 1378C) and Poisson's ratio n 0:3: The allowable stresses for static evaluation are: Sm 208C 130 MPa; Sm 1378C 110 MPa: The results presented in this paper were calculated when the doors are closed. The static stress ®elds obtained correspond to the critical load combination (10th combination Ð see Table 1). The acceptance criteria are given by expressions (12) and (13). The sterilizer is subjected in service to both pressure and temperature (as illustrated in Fig. 7) variations inducing
Thicknesses: - Lateral bar: 12 mm - Door: 6 mm
steam jacket
beam elements (studs)
rigid elements guide
lateral bar
Fig. 6. Steam-heated sterilizer: ®nite element model of the doors.
studs
632
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
Table 1 Combinations of loads Combination
Pressure on steam jackets and closure collar (Pa)
Pressure on chamber (Pa)
Global temperature (8C)
1st Combination 2nd Combination 3rd Combination 4th Combination 5th Combination 6th Combination 7th Combination 8th Combination 9th Combination 10th Combination
1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000
273,300 273,300 273,300 273,300 2100,000 2100,000 2100,000 1110,000 1220,000 1240,000
150 180 1121 1137 150 180 1137 1137 1137 1137
cyclic loads requiring a fatigue analysis. The values of PL, Q and F correspond to the Tresca criterion de®ned according to Eqs. (9)±(11). These values should be calculated for all load combinations. The allowable fatigue stress should be in accordance with criterion (14). From the working cycles of the chamber, steam jacket and closure collar ten combinations of pressure and temperature were selected (see Table 1). As each load combination has their maximum values, these occur at different points. So, for each load combination the maximum resultant stress can be expressed by S1 2 S3=2 PL 1 Q 1 F=2
15
Pressure xE5 [Pa]
The identi®ed critical points and their positions are shown in Fig. 8, for a static evaluation. Table 2 presents the values of maximum resultant stress PL 1 Q=2 for each load combination. As can be seen the 10th combination
corresponds to the most critical load factor. This is true for the three components of the sterilizer: closure collar (P1), chamber (P5) and steam jacket 1 (P9). The general primary membrane stress intensities were not included in Table 2 since their magnitudes are very low. The design of the sterilizer is controlled by local primary membrane stresses (PL) and secondary stresses (Q). The local membrane stress intensities calculated satis®ed the evaluation criterion PL # 1:5Sm : The highest stress intensity calculated (at point P9 for a service temperature of 1378C), as shown in Table 2, satis®es the criterion PL 1 Q # 3Sm 2 £ 62 MPa , 3Sm 3 £ 110: The fatigue strength of a pressure vessel is usually governed by the fatigue strength of details, in particular the stress concentration created by such details. The sterilizer under evaluation is subjected in service to load spectra illustrated in Fig. 7 for one working loading/ unloading cycle. The vessel is operated eight times per day,
2.4 1.9 1.4 0.9 0.4 -0.1
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
-0.6 -1.1
Temperature [ºC]
200 180 160 140 120 100 80 60 40 20 0
Fig. 7. Service load spectra for the chamber.
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
633
Table 2 Maximum resultant stresses PL 1 Q=2 on steam-heated sterilizer (MPa) Combination
Closure collar
1st Combination 2nd Combination 3rd Combination 4th Combination 5th Combination 6th Combination 7th Combination 8th Combination 9th Combination 10th Combination
Chamber
Steam jacket 1
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
1 1 1 1 3 3 3 22 38 39
17 17 17 17 18 18 18 2 5 5
6 6 6 6 6 6 6 26 30 31
28 28 28 28 35 35 35 6 13 15
4 4 4 4 19 19 19 40 55 57
31 31 31 31 32 32 32 3 6 7
22 22 22 22 23 23 13 7 10 11
19 19 19 19 26 25 25 14 29 32
13 25 33 39 13 24 38 50 61 62
3 31 40 47 19 32 48 44 43 43
which gives 38400 cycles during the lifetime of the vessel (15 years). Each operating cycle induces several stress cycles at each point, which can be evaluated by means of an adequate counting technique. In the present paper the reservoir method was used [11]. The stress history for the selected points is illustrated in Fig. 9. Thus the speci®ed number of times each type of stress cycle of types 1, 2, 3 etc. is repeated during the life of the vessel is n1, n2, n3 etc. respectively. Table 3 illustrates the total number of service cycles (ni) for each stress range type at the ten critical points (P1¼P10). From the applicable ASME design fatigue curve the maximum number of repetitions (Ni) which would be allowable is calculated. For each type of stress cycle the damage factors Ui were calculated: Ui ni =Ni
16
The cumulative damage factor U is derived: X Ui U
Table 3 summarises the calculations for all the critical points. As can be seen the total damage is less than 1, which means that the fatigue criterion is satis®ed. 4. Conclusions A methodology for stress categorisation in pressure vessels analysis containing structural members with shell intersections was proposed. This methodology is based on smoothed shell stress resultants in contrast to the traditional smoothed stresses. With this proposed procedure some of the problems related to the traditional classi®cation of bottom, middle and top surfaces are avoided. The methodology was
P2 P4
P5
P1
P3
17
i
P7
P6 P8
P9
P10 Fig. 8. Position of the points with maximum stresses.
40
(S1-S3)/2
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635 (S1-S3)/2
634
P1
30
20 16 12
20
8
10
4
0
P2
0 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
35
P3
30
time (S1-S3)/2
(S1-S3)/2
time
25 20 15 10 5 0 0
2
4
6
40 35 30 25 20 15 10 5 0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
P4 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
70
P5
60
time (S1-S3)/2
(S1-S3)/2
time
50 40 30 20 10 0 0
2
4
6
40 35 30 25 20 15 10 5 0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
P6 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
25
P7
20
time (S1-S3)/2
(S1-S3)/2
time 35 30 25
15
20
10
15 10
5
5
P8
0
0 0
2
4
6
0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
time (S1-S3)/2
(S1-S3)/2
time 70
P9
60 50
50 40
40
30
30
20
20
10
10
P10
0
0 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
time
Fig. 9. Stress history at the critical points.
time
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
635
Table 3 Cumulative damage based on the graphs of Fig. 9 Localisation
Stress range (Ds )
Number of cycles (n)
Number of cycles from design fatigue curve (N)
Damage U n=N
P1
40 21 3 2 18 15 12 31 20 1 35 22 15 2 19 36 54 2 36 15 2 23 16 4 26 5 13 3 63 37 2 43 41
1 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400 2 £ 38400 9 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 2 £ 38400 1 £ 38400 5 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400
1 £ 10 11 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 107 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011
3:840 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1027 1:920 £ 1026 3:840 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026 7:680 £ 1027 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026 3:840 £ 1026 7:680 £ 1027 3:840 £ 1027 1:920 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 3:840 £ 1023 1:920 £ 1026 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026
P2 P3 P4
P5
P6 P7 P8
P9 P10
applied to the design of a real sterilizer. Static and fatigue assessments were carried out. Acknowledgements The ®nancial support of the ®rm Jose dos Santos Monteiro, Lda through project `Esterilizador JSM 490/ 2PD' is gratefully acknowledged. References [1] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum (1996±1998). Int J Pressure Vessels Piping 1999;76:461±85. [2] ASME-CODE, Section VIII Division 2, The American Society of Mechanical Engineers, 1998. [3] Roche RL. Practical procedure for stress classi®cation. Int J Pressure Vessels Piping 1989;37:27±44. [4] Hechmer JL, Hollinger GL. Considerations in the calculations of the
[5]
[6]
[7]
[8] [9] [10]
[11]
P Cumulative damage ( Ui)
6.528 £ 10 26
3:072 £ 1026 6:144 £ 1026 6:912 £ 1026
6:526 £ 1026
6:144 £ 1026 3:072 £ 1026 6:912 £ 1026
3:846 £ 1023 2:304 £ 1026
primary plus secondary stress intensity range for code stress classi®cation. In: Seshadri R, editor. Codes and standards and applications for design and analysis of pressure vessel and piping components, vol. 136. New York: ASME PVP, 1988. Kroenke WC. Classi®cation of ®nite element stresses according to ASME Section III stress categories. Proceedings of 94th ASME Winter Annual Meeting, 1973. Design-by-Analysis Manual, European Pressure Equipment Research Council, European Commission, DG-JRC/IAM, Petten, The Netherlands, 1999. Hinton E, Campbell JS. Local and global smoothing of discontinuos ®nite element functions using a least squares method. Int J Numer Meth Engng 1974;8:461±80. Ghali A, Neville AM. Foundations structural analysis. 3rd ed. London: Chapman & Hall, 1989. Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. Singapore: McGraw-Hill, 1970. Weiû E, Rauth M. FEM-integrated concept for the detailed proof of fatigue strength of nozzle-to-vessel connections. Int J Pressure Vessels Piping 2000;77:215±25. BS 5500, Speci®cation for un®red fusion welded pressure vessels, British Standard Institution, 1997.
www.elsevier.com/locate/ijpvp
Design of a steam-heated sterilizer based on ®nite element method stress analysis R.M. Natal Jorge*, A.A. Fernandes Department of Mechanical Engineering and Industrial Management, Faculty of Engineering, University of Porto, Rua Dr Roberto Frias, 4200-465 Porto, Portugal Received 28 November 2000; revised 1 August 2001; accepted 31 August 2001
Abstract A steam-heated sterilizer is a pressure vessel of rectangular cross section externally reinforced by members welded to the ¯at surfaces of the vessel. These members are pressure vessels too with rectangular cross section. The ASME code provides alternative rules (Division 2) for the design of pressure vessels based on a Design By Analysis route. The stresses on the sterilizer were computed by the ®nite element method followed by the calculation of the stress ®elds according to the classi®cation established in the ASME code. Structural members with shell intersections (as in the present case) present dif®culties due to problems of linearisation and categorisation. In the present work the shell stress resultants were used instead of smoothed stresses. The operating conditions of the vessel involve cyclic application of loads requiring design based on fatigue analysis. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Finite element; Vessel of non-circular cross section; Design by analysis
1. Introduction Pressure vessel design has been historically based on Design By Formula. Standard vessel con®gurations are designed using a series of simple formulae and charts. In addition to the Design By Formula route, many national codes and standards for pressure vessel and boiler design, also provide for a Design By Analysis (DBA) route, where the admissibility of a design is checked, or proven, via a detailed investigation of the structure's behaviour under the external loads to be considered. DBA procedures do not specify particular implementation tools. However, the most widely used technique in contemporary pressure vessel design is the ®nite element method (FEM), a powerful technique allowing the detailed modelling of complex vessels [1]. All the DBA routes in the major codes and standards of pressure equipment are based on the rules proposed in the ASME pressure vessel and boiler code [2], which was formulated in the late 1950's. All these routes lead to the same wellknown problems, especially the stress categorisation problem (see for example Refs. [3,4]). In general, this does not present a problem in cases where the analysis uses thin shells. However, for the analysis of structural members with shell intersections, where the calculated nodal stresses cannot be * Corresponding author. Tel.: 1351-22-508-1713; fax: 1351-22-508-1584. E-mail address: [email protected] (R.M. Natal Jorge).
easily identi®ed as bottom, middle or top, the problems of linearisation and categorisation become more dif®cult. In fact, when a structural member has shell intersections, the nodal stress tensors, for example, the stress tensors of bottom points, cannot be smoothed. In the present work the use of nodal shell stress resultants are proposed as variables to be smoothed before the procedures of linearisation. With these smoothed nodal shell stress resultants for all the elements, the shell stress resultants can be interpolated to the Gauss points. In the Gauss points, the stress tensors can be carried out to the top, middle and bottom of the shell. Thus, these stress tensors must be extrapolated to the nodes and no smoothing is made. Finally, for each node and at the element level, the linearisation is computed based on Kroenke's procedure [5]. 2. Design by analysis Most of the DBA guidelines given in the codes relate to design based on elastic analysis Ð this is the so-called elastic route [6]. In this approach, based on the ASME code [2], the designer is required to classify the calculated stress into primary, secondary and peak categories and apply speci®ed allowable stress limits. The magnitudes of the allowable values assigned to the various stress categories re¯ect the nature of their associated failure mechanisms. Therefore it is essential that the categorisation
0308-0161/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(01)00078-3
628
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
stress distribution at Gauss points in one element
a)
b) Fig. 1. Plane problems: unsmoothed (a) and smoothed stresses (b).
average has no problem. However, in shell elements more than one point through the thickness is present, usually named: bottom, middle and top. For example, at node i (Fig. 2) the stress tensor for bottom, middle and top points can be de®ned. These stress tensors are obtained from the element T, element L and element R. For one single point, at node i, there will be the contribution of each element (three in this example). Obviously, in this case the stress tensors cannot be extrapolated from the Gauss points to the nodes and then make an average. The reason is simple: in element L and element R at node i the bottom point, the middle point and the top point are the same, but in the element T the bottom point, for example, has nothing to do with the points from elements L and R. In the work being reported, the shell stress resultants are used. In this case, for each element the shell stress resultants in the Gauss point in the local referential system are calculated. In the same manner working with the stress components, the shell stress resultants are extrapolated to the nodes (Fig. 3). This fact implies that, in general, for each shell stress resultant (STR) the equilibrium equations are not veri®ed, since the extrapolation is independent from element to element, i.e.:
procedure is performed correctly. The designer is required to decompose the elastic stress ®eld into these three categories and apply the appropriate stress limits. Stress categorisation is probably the most dif®cult aspect of the DBA procedure and, paradoxically, the problem has become more dif®cult as the stress analysis techniques have improved. In structural members with shell intersections, the problem caused by the dif®culty associated with the smoothed stresses is even more complex. 2.1. Smoothed procedure Analysis involving numerically integrated elements such as isoparametric elements, have shown that the integration points are the best stress sampling points. The nodes, which are the most useful output locations for stresses, appear to be the worst sampling points. Reasons for this phenomenon can be found in Ref. [7]. So the most popular procedure consists of calculating the nodal stresses (each component) by extrapolation based on integration points. In this step, each component has several nodal values, because they are obtained at the element level (Fig. 1a). Taking into account these nodal values, a single value can be obtained for each component, by the application of some kind of average (Fig. 1b). This procedure, very popular for plane stress and plane strain situations, presents some problems when applied to structural shells. In plane stress (or plane strain) elements there is one single point through the thickness, so the nodal
D STR
n X i1
STRi ± 0
element T node m
T
m
R
L bottom
top
j
i
top element L
element R node i
node j
bottom
node i
node i
bottom
node k
Fig. 2. Shell elements de®nition of top and bottom side.
k
1
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635 element T
Q Ty
N Txy
element L L N xy
N Ly
N Lx Q xL
Q Ly
element T
Q Tx element R
N Ty N T x N Rx
N Ry
629
element L
Q Rx N Rxy
M Tx
M Txy
M Txy
M Ty
M Rx
M Lxy M Rxy L
Q Ry
M Ly
M Lxy
element R
Mx
M Ry
M Rxy
Fig. 3. Shell stress resultants.
where n is the number of elements meeting at the node under consideration. For each node, in order to verify the equilibrium equations it is proposed that the shell stress resultants should be distributed among the elements meeting at the joint. These distributed shell stress resultants in each element 2
1 6 e 6 8 9 6 6 60 > s xx t > > > 6 > > > > 6 > > 6 > > s t > > yy 6 > > < = 60 txy t 6 6 6 > > > > 6 > > > > 6 t t > > xz 60 > > > > > > : t t ; 6 6 6 yz 6 6 4 0
0
0
12t 2 3 e
1 e
0
0
0
1 e
0
2
shell stress resultants can be interpolated to the Gauss points. From shell theory [9] and based on the shell stress resultants the stress components at a Gauss point are calculated at any point through the thickness at a distance from the middle point of t according to (see Fig. 4):
0
0
0
0
12t e3
0
0
0
12t e3
0
0
0
2
2
0
0
0
0
0
6t 3 2 3 2e e
0
0
0
0
0
0
are proportional to their relative original values. Inspired by the idea of the moment-distribution method of analysis of framed structures, the ratio of the shell stress resultant in an element to the equilibrium equation can be called the distribution factor (DF) [8]. It follows that the DF for an element is equal to the original value of shell stress resultant divided by the sum of the original values of shell stress resultant of the elements meeting at the joint, that is, u STRi u DFi X n u STRj u
2
j1
where i refers to the element considered and n refers to the elements meeting at the joint. The ®nal shell stress resultants can now be obtained by STRi 2D STR £ DFi 1 STRi
3
This procedure works like a smoothed shell stress resultant, where the original values are taken into account in the same way as a distributor of rigidity. Finally, with all the nodes in equilibrium, making use of the same shape functions that are used to evaluate the displacement vector in any point inside an element, the
! 0 6t2 2 e3
38 9 Nx > > > 7> > 7> > > 7> > > Ny > > 7> > > 7> > > 7> > > N 7> xy > > > 7> > > 7> < Mx > = 7> 7 7 7> My > > 7> > > > 7> > > 7> > > > M 7> > xy > 7> > > 7 > > > ! 7> > Q x > > > 7 > 3 5> > > : ; Qy 2e
4
In Eq. (4), for the middle point t 0; for the top point t 1e=2 and for the bottom point t 2e=2: Subsequently the three stress tensors are extrapolated at bottom, middle and top surfaces to the nodes. 2.2. Stress categorisation As mentioned in Section 2.1 the three stress tensors are carried out in the nodal local co-ordinates at every node, speci®cally: ² stress tensor with their components at bottom of the shell: s ijb ; ² stress tensor with their components at middle of the shell: s ijm ; ² stress tensor with their components at top of the shell: s ijt : Based on these three stress tensors a stress function is needed to calculate every stress component at any point through the thickness. This distribution can be approximated by a parabolic function (see Fig. 4): s ij t
s ijt 2 s ijb 2 t b m 2 t 1 s ijm 5 t s 1 s 2 2s 1 ij ij ij e e2
630
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
stress (Pm); local primary membrane stress (PL); primary bending stress (Pb); secondary stress (Q); peak stress (F). The membrane and bending components are then categorised depending upon the location of the supporting line segment. In this work the stress categories taken into account are:
top surface middle surface
bottom surface supporting line segment
² effective membrane stress: mem PL s Imem 2 s III
9
² effective bending stress: Fig. 4. Stress distribution through thickness.
ben Q s Iben 2 s III
Following the work of Kroenke, at any node, each component of the stress tensor can be decomposed into three subsets [5]: s ijmem
² stress tensor of membrane stresses : is the tensor whose components, constant along the supporting line segment, are equal to the average of the stresses along this supporting line segment; ² stress tensor of bending stresses s ijben : is the tensor whose components, varying linearly across the thickness of the shell, are calculated based on the beam bending theory (assuming a rectangular cross section with unit width); ² stress tensor of non-linear stresses s ijnln : is the tensor whose components are equal to: s ijnln s ij 2 s ijmem 2 s ijben
6
According to these de®nitions, the stress tensors of membrane and bending stresses as a function of the stress tensors s ijb s ijm and s ijt can be obtained: 1 Z1 e=2 1 t s ij 1 s ijb 2 2s ijm 1 s ijm 7 s ijmem s ij t dt e 2 e=2 6 s ijben
s ijt 2 s ijb 6 Z1 e=2 s tt dt ij 2 e2 2 e=2
8
It is important to note that the nodal membrane and nodal bending stresses can be calculated directly from expression (4). Obviously, in this case and for several components of stresses tensor the non-linear term cannot be obtained, since those stresses are constants, or depend linearly on t (it is the case for s xx ; s yy and txy ). 2.3. Assessment criteria According to the ASME code [2], one requirement for the acceptability of a design is that the calculated stress intensities shall not exceed speci®ed allowable limits. These limits depend of the stress category from which the stress intensity is derived. The ASME code establishes the following classi®cation of stresses: general primary membrane
10
² effective non-linear stress: nln F s Inln 2 s III
11
nIn mem where s InIn ; s IInIn ; s III ; s Imem ; s IImem ; s III and s Iben ; ben ben s II ; s III are the eigenvalues of the stress tensors de®ned in Eqs. (6)±(8), it corresponds to the description of the principal stresses. In agreement with the values of effective stresses calculated by Eqs. (9)±(11) and according to the ASME code the following evaluation criteria were used:
PL # 1:5Sm
12
PL 1 Q # 3Sm
13
PL 1 Q 1 F # S a
14
where Sm and Sa are the allowable stress intensities, for static and fatigue evaluations. In fatigue analysis if there are two or more types of stress cycle which produce signi®cant stresses, their cumulative effect shall be evaluated. According to the ASME code, it is necessary to specify the number of times each type of stress cycle will be repeated during the life of the vessel. In determining the number of the cycles for each stress cycle type, consideration shall be given to the superposition of cycles of various origins, which produce a total stress difference range greater than the stress difference ranges of the individual cycles [2]. 3. Example: analysis of a steam sterilizer In order to assess the proposed methodology, an analysis of a steam-heated sterilizer is carried out. Based on the classi®cation of Weiû, this kind of pressure vessel is not typical [10]. The steam-heated sterilizer that has been studied has the geometry shown in Fig. 5. In this ®gure, only one eighth of the component is represented corresponding to the ®nite element model used. The sterilizer has
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
631
steam jacket closure collar
thk = 5 2 1
thk = 6
guide thk = 6
thk = 5
Thicknesses: - Chamber: 5 mm - Steam jacket 1: 6 mm - Steam jacket 2: 5 mm - Closure collar: 6 mm - Guide: 25 mm
chamber beam elements (studs)
Fig. 5. Steam-heated sterilizer: geometry and ®nite element model.
four-steam jackets, which work as reinforcing members, two-closure collars and two doors. The doors have an ascending or descending vertical sliding movement in two guides. These guides are bolted rigidly to studs welded to the external steam jackets (see Fig. 6). The operation of the sterilizer can be summarised in the following way: the door is slid to the closed position; pressure is introduced in the closure collars in order to maintain suf®cient pressure on the silicon gasket to ensure a tight joint with the door and prevent leakage of steam. The sterilizer is then subjected to internal pressure, maintaining the pressure in the closure collars. The load combinations indicated in Table 1 re¯ect this operation procedure in detail and were de®ned on the basis of service load spectra supplied by the user, illustrated in Fig. 7 (for the chamber only). The calculation route can be summarised as follows: evaluate by FEM for each load combination, the nodal
values on the guides due to the loads exerted by the door on the guides; combine these forces with the loads present in the body. The guides were modelled with shell elements and the studs with beam elements. Due to the diameter of these studs (40 mm) the connection stud/steam jacket was modelled through rigid elements. The material elastic properties are: modulus of elasticity E 210 GPa (at 208C), E 196 GPa (at 1378C) and Poisson's ratio n 0:3: The allowable stresses for static evaluation are: Sm 208C 130 MPa; Sm 1378C 110 MPa: The results presented in this paper were calculated when the doors are closed. The static stress ®elds obtained correspond to the critical load combination (10th combination Ð see Table 1). The acceptance criteria are given by expressions (12) and (13). The sterilizer is subjected in service to both pressure and temperature (as illustrated in Fig. 7) variations inducing
Thicknesses: - Lateral bar: 12 mm - Door: 6 mm
steam jacket
beam elements (studs)
rigid elements guide
lateral bar
Fig. 6. Steam-heated sterilizer: ®nite element model of the doors.
studs
632
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
Table 1 Combinations of loads Combination
Pressure on steam jackets and closure collar (Pa)
Pressure on chamber (Pa)
Global temperature (8C)
1st Combination 2nd Combination 3rd Combination 4th Combination 5th Combination 6th Combination 7th Combination 8th Combination 9th Combination 10th Combination
1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000 1240,000
273,300 273,300 273,300 273,300 2100,000 2100,000 2100,000 1110,000 1220,000 1240,000
150 180 1121 1137 150 180 1137 1137 1137 1137
cyclic loads requiring a fatigue analysis. The values of PL, Q and F correspond to the Tresca criterion de®ned according to Eqs. (9)±(11). These values should be calculated for all load combinations. The allowable fatigue stress should be in accordance with criterion (14). From the working cycles of the chamber, steam jacket and closure collar ten combinations of pressure and temperature were selected (see Table 1). As each load combination has their maximum values, these occur at different points. So, for each load combination the maximum resultant stress can be expressed by S1 2 S3=2 PL 1 Q 1 F=2
15
Pressure xE5 [Pa]
The identi®ed critical points and their positions are shown in Fig. 8, for a static evaluation. Table 2 presents the values of maximum resultant stress PL 1 Q=2 for each load combination. As can be seen the 10th combination
corresponds to the most critical load factor. This is true for the three components of the sterilizer: closure collar (P1), chamber (P5) and steam jacket 1 (P9). The general primary membrane stress intensities were not included in Table 2 since their magnitudes are very low. The design of the sterilizer is controlled by local primary membrane stresses (PL) and secondary stresses (Q). The local membrane stress intensities calculated satis®ed the evaluation criterion PL # 1:5Sm : The highest stress intensity calculated (at point P9 for a service temperature of 1378C), as shown in Table 2, satis®es the criterion PL 1 Q # 3Sm 2 £ 62 MPa , 3Sm 3 £ 110: The fatigue strength of a pressure vessel is usually governed by the fatigue strength of details, in particular the stress concentration created by such details. The sterilizer under evaluation is subjected in service to load spectra illustrated in Fig. 7 for one working loading/ unloading cycle. The vessel is operated eight times per day,
2.4 1.9 1.4 0.9 0.4 -0.1
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
-0.6 -1.1
Temperature [ºC]
200 180 160 140 120 100 80 60 40 20 0
Fig. 7. Service load spectra for the chamber.
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
633
Table 2 Maximum resultant stresses PL 1 Q=2 on steam-heated sterilizer (MPa) Combination
Closure collar
1st Combination 2nd Combination 3rd Combination 4th Combination 5th Combination 6th Combination 7th Combination 8th Combination 9th Combination 10th Combination
Chamber
Steam jacket 1
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
1 1 1 1 3 3 3 22 38 39
17 17 17 17 18 18 18 2 5 5
6 6 6 6 6 6 6 26 30 31
28 28 28 28 35 35 35 6 13 15
4 4 4 4 19 19 19 40 55 57
31 31 31 31 32 32 32 3 6 7
22 22 22 22 23 23 13 7 10 11
19 19 19 19 26 25 25 14 29 32
13 25 33 39 13 24 38 50 61 62
3 31 40 47 19 32 48 44 43 43
which gives 38400 cycles during the lifetime of the vessel (15 years). Each operating cycle induces several stress cycles at each point, which can be evaluated by means of an adequate counting technique. In the present paper the reservoir method was used [11]. The stress history for the selected points is illustrated in Fig. 9. Thus the speci®ed number of times each type of stress cycle of types 1, 2, 3 etc. is repeated during the life of the vessel is n1, n2, n3 etc. respectively. Table 3 illustrates the total number of service cycles (ni) for each stress range type at the ten critical points (P1¼P10). From the applicable ASME design fatigue curve the maximum number of repetitions (Ni) which would be allowable is calculated. For each type of stress cycle the damage factors Ui were calculated: Ui ni =Ni
16
The cumulative damage factor U is derived: X Ui U
Table 3 summarises the calculations for all the critical points. As can be seen the total damage is less than 1, which means that the fatigue criterion is satis®ed. 4. Conclusions A methodology for stress categorisation in pressure vessels analysis containing structural members with shell intersections was proposed. This methodology is based on smoothed shell stress resultants in contrast to the traditional smoothed stresses. With this proposed procedure some of the problems related to the traditional classi®cation of bottom, middle and top surfaces are avoided. The methodology was
P2 P4
P5
P1
P3
17
i
P7
P6 P8
P9
P10 Fig. 8. Position of the points with maximum stresses.
40
(S1-S3)/2
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635 (S1-S3)/2
634
P1
30
20 16 12
20
8
10
4
0
P2
0 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
35
P3
30
time (S1-S3)/2
(S1-S3)/2
time
25 20 15 10 5 0 0
2
4
6
40 35 30 25 20 15 10 5 0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
P4 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
70
P5
60
time (S1-S3)/2
(S1-S3)/2
time
50 40 30 20 10 0 0
2
4
6
40 35 30 25 20 15 10 5 0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
P6 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
25
P7
20
time (S1-S3)/2
(S1-S3)/2
time 35 30 25
15
20
10
15 10
5
5
P8
0
0 0
2
4
6
0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
time (S1-S3)/2
(S1-S3)/2
time 70
P9
60 50
50 40
40
30
30
20
20
10
10
P10
0
0 0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
time
Fig. 9. Stress history at the critical points.
time
R.M. Natal Jorge, A.A. Fernandes / International Journal of Pressure Vessels and Piping 78 (2001) 627±635
635
Table 3 Cumulative damage based on the graphs of Fig. 9 Localisation
Stress range (Ds )
Number of cycles (n)
Number of cycles from design fatigue curve (N)
Damage U n=N
P1
40 21 3 2 18 15 12 31 20 1 35 22 15 2 19 36 54 2 36 15 2 23 16 4 26 5 13 3 63 37 2 43 41
1 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400 2 £ 38400 9 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 2 £ 38400 1 £ 38400 5 £ 38400 2 £ 38400 5 £ 38400 1 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400 10 £ 38400 1 £ 38400 5 £ 38400
1 £ 10 11 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 107 1 £ 1011 1 £ 1011 1 £ 1011 1 £ 1011
3:840 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1027 1:920 £ 1026 3:840 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026 7:680 £ 1027 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026 3:840 £ 1026 7:680 £ 1027 3:840 £ 1027 1:920 £ 1026 7:680 £ 1027 1:920 £ 1026 3:840 £ 1027 3:840 £ 1026 3:840 £ 1023 1:920 £ 1026 3:840 £ 1026 3:840 £ 1027 1:920 £ 1026
P2 P3 P4
P5
P6 P7 P8
P9 P10
applied to the design of a real sterilizer. Static and fatigue assessments were carried out. Acknowledgements The ®nancial support of the ®rm Jose dos Santos Monteiro, Lda through project `Esterilizador JSM 490/ 2PD' is gratefully acknowledged. References [1] Mackerle J. Finite elements in the analysis of pressure vessels and piping, an addendum (1996±1998). Int J Pressure Vessels Piping 1999;76:461±85. [2] ASME-CODE, Section VIII Division 2, The American Society of Mechanical Engineers, 1998. [3] Roche RL. Practical procedure for stress classi®cation. Int J Pressure Vessels Piping 1989;37:27±44. [4] Hechmer JL, Hollinger GL. Considerations in the calculations of the
[5]
[6]
[7]
[8] [9] [10]
[11]
P Cumulative damage ( Ui)
6.528 £ 10 26
3:072 £ 1026 6:144 £ 1026 6:912 £ 1026
6:526 £ 1026
6:144 £ 1026 3:072 £ 1026 6:912 £ 1026
3:846 £ 1023 2:304 £ 1026
primary plus secondary stress intensity range for code stress classi®cation. In: Seshadri R, editor. Codes and standards and applications for design and analysis of pressure vessel and piping components, vol. 136. New York: ASME PVP, 1988. Kroenke WC. Classi®cation of ®nite element stresses according to ASME Section III stress categories. Proceedings of 94th ASME Winter Annual Meeting, 1973. Design-by-Analysis Manual, European Pressure Equipment Research Council, European Commission, DG-JRC/IAM, Petten, The Netherlands, 1999. Hinton E, Campbell JS. Local and global smoothing of discontinuos ®nite element functions using a least squares method. Int J Numer Meth Engng 1974;8:461±80. Ghali A, Neville AM. Foundations structural analysis. 3rd ed. London: Chapman & Hall, 1989. Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. Singapore: McGraw-Hill, 1970. Weiû E, Rauth M. FEM-integrated concept for the detailed proof of fatigue strength of nozzle-to-vessel connections. Int J Pressure Vessels Piping 2000;77:215±25. BS 5500, Speci®cation for un®red fusion welded pressure vessels, British Standard Institution, 1997.