design optimization of rigid metal containers

Finite Elements in Analysis and Design 37 (2001) 273}286

Design optimization of rigid metal containers

Jyhwen Wang

Technology Center, Weirton Steel Corporation, 3006 Birch Drive, Weirton, WV 26062, USA

Abstract

Evaluating the structural capability of a food container has been viewed as a challenging task. With morepowerful computer hardware and software, "nite element methods are now in place for e$cient and accurateassessment of axial load and paneling performances of cans. Thus, the metal container optimization problemis tackled with FEA tools. In this paper, the structure tests and numerical simulation methods are presented.The e!ects of design parameters on the structure performance of containers are discussed. A designoptimization method is then developed based on the solution of point inclusion problem in computationalgeometry. The iterative planar subdivision algorithm generates the optimal design that meets the structurerequirements with least material consumption. The proposed method can be used to optimize the existingand new metal containers. � 2001 Elsevier Science B.V. All rights reserved.

Keywords: Container design; Optimization

1. Introduction

The development of tin can was started by Nicholas Appert in 1790s' for preserving food tosustain Napoleon's army. Appert, a Parisian chef and confectioner, collected the 12,000 francsaward for his new process of canning [1]. Today metal containers have become commonly used forfood packaging. It is estimated that 30 billion food cans are made annually in the United States.With the huge production volume, the can industry is aggressively developing new technologies toreduce costs. Cost improvements that seem minuscule per container may actually result insigni"cant savings.Resource (input material) reduction is one of the most powerful means in cost reduction for can

makers. What should not be compromised in material reduction are the structural capabilities ofthe container. There are three major structural requirements for metal food cans: axial load,paneling, and bulge and buckle requirements. These structure capabilities for can bodies and lidsare imposed by food packaging and processing conditions.

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0168-874X/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 8 7 4 X ( 0 0 ) 0 0 0 4 3 - 3

Fig. 1. Sti!eners (beads) on a metal container.

After the food is "lled into a can, the body and the lid are assembled/seamed together. During theseaming process, the can body is subjected to an axial compressive force. Also, the containers aregenerally stacked up in many layers during shipping. The cans in the bottom layer are subjected tothe combined weight and dynamic loads of the cans in the layers above. These two conditionsprescribe the axial load requirement.The sterilization process leads to the bulge/buckle and the paneling requirements.When the food

container is processed in high temperature, the internal pressure increases. The bulge/bucklerequirement ensures that there is no permanent deformation on the top and bottom of the can. Ine!ect, the lids act like diaphragms, expanding during retorting and turning to a slightly concaveshape after thermal processing [2].The container is usually under vacuum loading after cooling. That is, the internal pressure is less

than the atmospheric pressure due to condensation. The paneling requirement prevents the canwall to collapse under vacuum. It is a common practice that sti!eners (beads), as shown in Fig. 1,are added to the wall to enhance paneling performance.The axial load and the paneling requirements tend to be con#icting objectives in can design.

With no beads, a straight wall can generally has good axial load but inferior paneling performance.On the other hand, the axial load will drop and the paneling pressure will be enhanced as the beaddepth increases. Thus, the designer's ultimate objective is to maximize both the axial load and thepaneling performances; while minimizing the wall thickness to reduce material usage.Previous e!orts were mostly made in analyzing known designs using di!erent analysis methods.

Sodeik and Sauer [3] investigated the mechanical behavior of the food can during sterilization.Analytical models were developed to predict the axial load and paneling performances. Traversin

274 J. Wang / Finite Elements in Analysis and Design 37 (2001) 273}286

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et al. [4] conducted "nite element analyses and experiments to evaluate food containers. Witha special focus, the in#uence of geometrical variations has been studied. Kahrs and his colleagues[5] investigated the bead depth e!ects on axial load and paneling performances. It was shown that"nite element techniques could predict the axial load with good accuracy. However, an empiricalfactor was used to correct the di!erence in paneling between FEA and experiment. The applicabil-ity of FEA on bead forming, axial load, and paneling simulations was investigated by Jones et al.[6]. The simulations were conducted by using an explicit FEA code. The study highlights theimportance of non-axisymmetric conditions in the real world cases. It was found that none of theabove-mentioned references addressed the design issues of metal containers.In this paper, "nite element modeling techniques are used to analyze the axial load and the

paneling performances. The in#uences of the design parameters on the structure performance areinvestigated. Moreover, a systematic design strategy is presented. Given the performance require-ments and a preliminary design, the iterative design cycles will lead the search of optimal wallthickness and bead depth. The proposed methodology can be used to optimize existing and newmetal containers.

2. Container structural analyses

Traditionally, optimization of metal containers has been a tedious and costly process. Theactivities involve iterative and extensive design, tooling fabrication, can making, and performanceevaluation. Advances in computer technology and computational mechanics have resulted ine!ective and e$cient methods to predict the structure capability of cans. In this paper, two typicalcontainer testing procedures } axial load and paneling tests, are simulated using "nite elementanalysis.

2.1. Axial load analysis

The axial load test simulates the loading condition during double seaming and transport. In thistest, the sample is placed between a "xed horizontal plate and a platform as shown in Fig. 2. Duringthe test, the platform is pushed up, and sensors are attached to record the vertical compressive load.As the can sample buckles, the sudden drop of the load (force) will be registered as the axial loadperformance. The failures often take place at the radius of a bead, and circumferential post bucklingfailure has been observed. Generally, the axial load requirement ranges from 2220 to 3560N (500 to800 lbs). The material properties, the wall thickness and the bead depth are known to a!ect axialload performance.An analytical model based on bending formulation was developed [3] to predict the axial load

performance. It was found that a more realistic and accurate prediction could be obtained through3-D explicit "nite element simulation [6]. The result of explicit dynamic code is rate-dependent,and the computational cost is high due to small time step. Thus, it is suggested to use nonlinear,implicit "nite element code to simulate axial load test. In the implicit, quasi-static analysis, thedisplacement of the platform was prescribed, and the reaction force on the can was calculated.Both 2-D axisymmetric and 3-D analyses were tried using ABAQUS in this study. A 2-D model

cannot fully describe the manufacturing variations exist in the test samples. It often over-predicts

J. Wang / Finite Elements in Analysis and Design 37 (2001) 273}286 275

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Fig. 2. Axial load test.

(about 10%) the axial load capability. As the interest is in design optimization, true structurecapability should be evaluated. The less time consuming 2-D axisymmetric analysis is adopted.Fig. 3 shows a typical result of FEA axial load simulation. The model was constructed using

axisymmetric shell element. The element size ranges from 0.28 to 1.04mm. The bead geometry wasrepresented by at least four elements. Rick's method was used, and the analysis predicted the onsetof collapse (drop in reaction force) at about 2080N which agrees reasonably well with theexperiment.

2.2. Paneling analysis

The paneling test simulates a food container under vacuum loading after retorting process. Thetest is conducted in a chamber where an empty and seamedmetal container is placed on a plate. Airis pumped into the sealed chamber as shown in Fig. 4. During pressure ramp up, the top andbottom lids usually turn concave "rst, then the can body collapses. The pressure reading at thismoment is referred to as paneling performance. The can usually exhibits three or four circumferen-tial waves after buckling. Depending on the "lling and processing conditions, the panelingrequirement can range from 140 to 240 kPa (20 to 35 psi). It is known that the can size, the aspectratio, the body wall thickness, the bead geometry, and the bead pattern all in#uence panelingperformance.An analytical model based on buckling analysis of thin-walled pipe was developed in Ref. [3]. As

the can geometry is more complicated than, and the loading is di!erent from that of pipe, the modeldescribes paneling only su$ciently well. Explicit dynamic (shown in Fig. 5) and implicit quasi-staticanalyses were conducted. It was also found that both methods are not e!ective for panelingsimulation. Instead, paneling prediction is obtained from linearized buckling eigenvalue analysisusing ABAQUS. Axisymmetric shell element similar to the axial load analysis was again used forpaneling simulation. The analysis predicts the load at which the onset of buckling instability


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Fig. 3. FEA simulation result of an axial load test.

Fig. 4. Paneling test equipment.


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Fig. 5. FEA simulation result of paneling test.

occurs. Both the buckling load factors and buckled modes can be obtained. Studies showed thatpredictions are generally 8}15% higher than actual experiments.One main assumption in this type of analysis is that displacement prior to buckling is considered

small. As the lids buckle (to concave) before side wall paneling, this shape change is prescribed inthe model to eliminate the unwanted lid buckling modes.

3. E4ects of design parameters

Given a required "ll volume, engineers have a few standard can sizes to choose from. Withstandard heights and diameters, costly changeover of conveyor system in a can plant can beeliminated. Once the can height and diameter are "xed, the design parameters now include beadcon"guration (number of beads and bead spacing), bead depth, and wall thickness. The in#uence ofthese parameters on the structure performance is investigated.

3.1. Ewect of bead conxguration

The bead con"guration does not have a signi"cant in#uence on the axial load, but plays a majorrole in paneling performance. Beads are designed to sti!en the unsupported wall section to increasepaneling pressure. For a given number of beads, Fig. 6 shows that there exists an optimal beadspacing. Increasing the number of beads will increase the paneling pressure. However, the e!ectlevels o! as shown in Fig. 7. Selection of bead con"guration is not a pure engineering issue. It isrelated to the appearance of the container and marketing concerns (presentation, graphics) that areincluded in the decision-making process.


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Fig. 6. E!ect of bead spacing.

Fig. 7. E!ect of the number of beads.

3.2. Ewect of bead depth

Bead design that improves the axial load will generally lead to the deterioration of panelingperformance, and vice versa. Axial load vs. bead depth and paneling pressure vs. bead depth arecharacterized in Fig. 8. In this study, a 7-bead, 401�508 (diameter: 99.57mm, height: 136.27mm)was used. The material thickness was 0.16mm.


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Fig. 8. E!ect of bead depth.

3.3. Ewect of material thickness

It can be observed that as the wall thickness decreases, both axial load and paneling perfor-mances will su!er. Fig. 9 shows the paneling pressure and axial load as functions of the metalthickness. The same 7-bead can in the previous bead depth study was used, and the bead depth wasset at 1.27mm.

4. Design optimization

The objective of design optimization is to minimize the material usage while satisfying both theaxial load and the paneling requirements can be met. Thus, the optimization problem can berepresented as

Min ts.t.

f�(X

�)'F

��,

f�(X

�)'P

��,


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Fig. 9. E!ect of material thickness.

where t is the metal thickness, f�and f

�are the axial load and paneling performances, X

�and

X�are the sets of design parameters that in#uence axial load and paneling, and F

��and P

��are

the axial load force and paneling pressure requirements.Unfortunately, even with the trends presented earlier, there are no known functions to describe

f�and f

�exactly. Therefore, a design optimization procedure based on "nite element analysis is

developed to obtain the "nal design (as shown in Fig. 10).In this process, the can size is "rst established based on required "ll (packaging) volume. The

fully beaded structure is then constructed since it provides the best paneling performance.However, other bead con"gurations can be adopted to address the appearance issue. The lowerand upper bounds of the bead depth are [0, d

��], where 0 indicates straight wall (no bead), and

d��

is the maximum achievable depth in beading process. The thickness is between [0, t��

], wheret��

is the maximum acceptable wall thickness from cost calculation. Before entering the optimiza-tion routine, the bead depth and material thickness are assigned arbitrarily between the lower andupper bounds.The axial load and paneling capabilities of the initial design is evaluated. The iterative search

based on FEA results will lead to a converged optimal solution. Thus, the number of beads, beadspacing, the bead depth, and the metal thickness are determined.


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Fig. 10. Design optimization procedure.

4.1. FEA-based optimization

The optimization iteration involves the search for material thickness and bead depth to meet thestructural requirements. As shown in Fig. 11, the known axial load and paneling requirementsdivide the constraint (performance) space into quadrants (Q1, Q2, Q3, and Q4). Feasible solutionslie in the "rst quadrant. Before the procedure starts, tolerances for convergence are set, for example,within a% and b% of the minimum axial load and paneling requirements. A search strategy isdesigned to move the initial solution into this small `optimal solution regiona (OSR) in the "rstquadrant.Although the structure performances cannot be exactly de"ned as functions of design variables,

the trends discussed earlier can be envisaged as curves shown in Fig. 11. The distorted imaginarygrids created by the curves are used to explain the search mechanism. Here, t

�indicates constant

wall thickness with di!erent bead depth, and d�indicates constant bead depth with di!erent wall

thickness.


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Fig. 11. Optimization in the performance (constraint) space.

The following simple search strategy approaches the optimal design through iterations. Notethat `x3A++ represents x is a member of A.

1. Current solution (CS)"initial solution2. Evaluate CS.3. If CS 3 Q14. Then, If CS 3 OSR5. Then, stop search. Report optimal solution.6. Else, decrease thickness to move CS into Q1, Q2 or Q4. Go to Step 3, 7 or 11.

7. If CS 3 Q28. Then, decrease bead depth to move CS into Q1 or Q3. Go to Step 3 or 9.

9. If CS 3 Q310. Then, increase thickness to move CS into Q2 or Q4. Go to Step 7 or 11.

11. If CS 3 Q412. Then, increase bead depth to move CS into Q1 or Q3. Go to Step 3 or 9.

In fact, the optimization problem is a variation of point-location problem or point-inclusionproblem in computational geometry. The objective is to "nd a point p (optimal solution) that iscontained in region R (OSR). The Initial Solution is "rst evaluated (Step 2), and it could be in anyquadrant. In Steps 6, 8, 10, and 12, change in thickness or bead depth is made, and FEA evaluationsensure the interim solution moves into one of the adjacent quadrants. The augmentation of thesolution is based on bisection, or binary search. The solutionmoves along the grid lines, and the d}tsolution space is e!ectively reduced in iterations. As Preparata and Shamos [7] pointed out,bisection is the fastest known search method for planar subdivision.


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Fig. 12. Example problem } progressions of design optimization.

Table 1Example problem } material properties and initial design�

Material property

E � > UTS n-value200Gpa 0.28 630MPa 632MPa 0.12

Initial designDiameter Height Thickness Bead depth102mm 140mm 0.140mm 0.889mm

�E is the Young's modulus, � the Poisson's ratio, > the yield strength, UTS the ultimate tensile strength and n the workhardening exponent.

5. Example problem

A metal container with a 102mm diameter and a 140mm height is used to demonstrate thedesign optimization method. Seven-bead structure with equal spacing is selected as the beadcon"guration. The axial load and paneling requirements are 2670N (600 lbs) and 100kPa (15 psi),respectively. Table 1 shows the material properties and the initial design.The tolerance of convergence is set at 5% for both axial load and paneling such that search for

optimal solution will continue until the axial load and paneling performance fall into the ranges of[2669, 2802] N and [103, 108] kPa, respectively.FEA evaluations of the initial solution show that it had an axial load of 2340N, and paneling of

35kPa. As it is not in the OSR, the iterative search is conducted. As demonstrated in Table 2 andshown in Fig. 12, it takes four iterations to optimize the design. The solution eventually reaches theOSR with material thickness t"0.161mm and bead depth d"1.168mm, and the performance of2727N for axial load and 107 kPa for paneling.


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Table 2Example problem } design optimization process

Solution Thickness (mm) Bead depth (mm) Axial load (N) Paneling (kPa)

Iteration 11 0.140 0.889 2340 352 0.196 0.889 4804 1133 0.168 0.889 3745 74

Iteration 24 0.168 1.397 2353 1765 0.168 1.143 3038 117

Iteration 36 0.140 1.143 1993 717 0.154 1.143 2549 908 0.161 1.143 2811 102

Iteration 49 0.161 1.397 2277 15510 0.161 1.270 2491 12511 0.161 1.219 2647 11312 0.161 1.168 2727 107

6. Conclusion

In general, appropriate "nite element analyses can predict the performances within 10% of thevalues obtained from experiments. However, the imperfection of the test specimen cannot beoverlooked. Out of roundness, out of cylindricity, localized metal thinning (may occur duringbeading), dent, defective weld could all contribute to the di!erences between FEA and actual testresults. To evaluate the true performance of a design, manufacturing variations were not con-sidered in the FEA models. In summary, implicit quasi-static and linearized buckling analyses arerecommended for axial load and paneling simulations, respectively. Also in this study, a designmethod was developed to optimize food containers. The design optimization problem is trans-formed into a point inclusion problem. The FEA-based optimization utilizes bisection algorithm toaugment the solution e$ciently. With continuous planar subdivision, the search guarantees toconverge. The result is a design that meets the structure performance requirements with leastmaterial consumption. The optimization procedure is currently conducted by FEA analyst, and theprocess can be automated in the future.

Acknowledgements

The author would like to thank Dr. Leon Steiner for conducting some of the FEA simulations,and Ms. Paula Costantini for preparing the "gures.


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References

[1] J. Hanlon, Handbook of Packaging Engineering, Institute of Packaging Professionals, 1992.[2] G. Robertson, Food Packaging } Principles and Practice, Marcel Dekker, New York, 1993.[3] M. Sodeik, R. Sauer, Mechanical behaviour of food cans under radial and axial load, 3rd International Tin Plate

Conference, 1984.[4] M. Traversin, P. Magain, P. Jodogne, P. Dubreuil, P. Cochet, Mechanical behaviour optimisation of three-piece

tinplate cans using numerical simulations, "fth International Tin Plate Conference 1992.[5] J. R. Kahrs, D. J. Radakovic, S. J. Bianculli, Optimization of food can bead design by "nite element techniques, US

Steel Research, 1992.[6] I. B. Jones, D. R. J. Owen, D. Jones, A. J. L. Crook, G. Q. Liu, Applicability of Finite ElementMethod to Design and

Optimization of Food Cans, Ironmaking and Steelmaking 25 (1) 1998.[7] F.P. Preparata, M.I. Shamos, Computational Geometry } An Introduction, Springer, Berlin, 1985.


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design optimization of rigid metal containers

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