Research Papers: Design and Analysis

Minimum Weight Design for Toroidal Shells With Strengthening Component

[+] Author and Article Information
Vu Truong Vu

Faculty of Civil Engineering,
Ho Chi Minh City University of Transport,
No. 2, D3 Street,
Binh Thanh District,
Ho Chi Minh City, Vietnam
e-mail: vutruongvu@gmail.com

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received May 1, 2015; final manuscript received August 18, 2015; published online October 6, 2015. Assoc. Editor: Kunio Hasegawa.

J. Pressure Vessel Technol 138(2), 021202 (Oct 06, 2015) (7 pages) Paper No: PVT-15-1081; doi: 10.1115/1.4031445 History: Received May 01, 2015; Revised August 18, 2015

This article presents an approach to the minimum weight design for toroidal shells with strengthening component subject to internal pressure. The optimal shape is obtained by adjusting the geometry and wall thickness of the cross section associated with the thickness and position of the strengthening component. Constraints include first yield pressure, plastic pressures, plastic instability pressure, and internal volume of toroid. The weight saving can reach over 70% in some toroid configurations. The comparison of two optimization methods shows that differential evolution (DE) slightly outperforms particle swarm optimization (PSO) in the majority of investigated cases.

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Vu, V. T. , 2010, “ Minimum Weight Design for Toroidal Pressure Vessels Using Differential Evolution and Particle Swarm Optimization,” Struct. Multidiscip. Optim., 42(3), pp. 351–369. [CrossRef]
Steele, C. R. , 1965, “ Toroidal Pressure Vessels,” J. Spacecr. Rockets, 2(6), pp. 937–943. [CrossRef]
Marketos, J. D. , 1963, “ Optimum Toroidal Pressure Vessel Filament Wound Along Geodesic Lines,” AIAA J., 1(8), pp. 1942–1945. [CrossRef]
Mandel, G. , 1983, “ Torus-Shaped Pressure Vessel,” UK Patent No. GB 2,110,566A.
Li, S. , and Cook, J. , 2002, “ An Analysis of Filament Overwound Toroidal Pressure Vessels and Optimum Design of Such Structures,” ASME J. Pressure Vessel Technol., 124(2), pp. 215–222. [CrossRef]
Vu, V . T. , 2013, “ Optimum Shape of Constant Stress Toroidal Shells,” ASME J. Pressure Vessel Technol., 135(2), p. 024501.
Blachut, J. , 1995, “ Plastic Loads for Internally Pressurised Torispheres,” Int. J. Pressure Vessels Piping, 64(2), pp. 91–100. [CrossRef]
Storn, R. , and Price, K. , 1997, “ Differential Evolution—A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces,” J. Global Optim., 11(4), pp. 341–359. [CrossRef]
Kennedy, J. , and Eberhart, R. , 1995, “ Particle Swarm Optimization,” IEEE International Conference on Neural Networks (ICNN), Perth, WA, Nov. 27–Dec. 1, pp. 1942–1948.
Clerc, M. , and Kennedy, J. , 2002, “ The Particle Swarm-Explosion, Stability, and Convergence in a Multidimensional Complex Space,” IEEE Trans. Evol. Comput., 6(1), pp. 58–73. [CrossRef]


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Fig. 1

Geometry of toroid

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Fig. 2

Correlation between R/r and σvmax/σvmin in toroids

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Fig. 3

Shapes of benchmark toroids

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Fig. 4

Definition of pv1 and pv2 and their relative positions with pyp and ppi in a toroid under internal pressure

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Fig. 5

Positions of the strengthening component in toroid

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Fig. 6

Toroid’s cross section with the strengthening component

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Fig. 7

Typical finite-element model of a toroid with strengthening

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Fig. 9

Optimal geometry of circular, constant thickness cross sections with loading constraint types

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Fig. 8

Optimal geometry of elliptical, variable thickness cross sections with loading constraint types



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