A powerful forcebased approach for the limit analysis of threedimensional frames
 Konstantinos V. Spiliopoulos,
 Nikos G. Dais
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
For the estimation of the strength of a structure, one could avoid detailed elastoplastic analysis and resort, instead, to direct limit analysis methods that are formulated within linear programming. This work describes the application of the force method to the limit analysis of threedimensional frames. For the limit analysis of a framed structure, the force method, being an equilibriumbased approach, is better suited than the displacement method and results, generally, to faster solutions. Nevertheless, the latter has been used mostly, since it has a better potential for automation. The difficulty for the direct computerization of the force method is to automatically pick up the structure’s redundant forces. Graph theory concepts may be used to accomplish this task, and a numerical procedure was proposed for the optimal plastic design of plane frames. An analogous approach is developed herein for the limit analysis of space frames which is computationally more cumbersome than the limit analysis of plane frames. The proposed procedure results in hypersparse matrices, and in conjunction with the kinematic upper bound linear program which is solved by a sparse solver, the proposed method appears computationally very efficient. It is also proved that it is much more effective than any displacementbased formulation. The robustness and efficiency of the approach are testified by numerical examples for grillages and multistorey frames that are included.
 Orbison, J.G., McGuire, W., Abel, J.F. (1982) Yield surface applications in nonlinear steel frame analysis. Comp. Methods Appl. Mech. Eng. 33: pp. 557573 CrossRef
 Liew, J.Y.R., Chen, H., Shanmugan, N.E., Chen, W.F. (2000) Improved nonlinear plastic hinge analysis of space frame structures. Eng. Struct. 22: pp. 13241338 CrossRef
 Kim, SE., Park, MH., Choi, SH. (2001) Direct design of threedimensional frames using practical advanced analysis. Eng. Struct. 23: pp. 14911502 CrossRef
 Long, H.V., Nguyen, D.H. (2008) Secondorder plastichinge analysis of 3D steel frames including strain hardening effects. Eng. Struct. 30: pp. 35053512 CrossRef
 Neal, B.G. (1970) The Plastic Method of Structural Analysis. Chapman & Hall, London
 Cohn, M.Z., Ghosh, S.K., Parimi, S.R. (1972) Unified approach to theory of plastic structures. Eng. Mech. (ASCE) 98: pp. 11331158
 Maier, G., Pastor, J., Ponter, A.R.S., Weichert, D.: Direct methods of limit and shakedown analysis. In: de Borst, R., Mang, H.A. (eds.) Comprehensive Structural Integrity—Fracture of Materials from Nano to Macro, : Numerical and Computational Methods, vol. 3, pp 637–684. Elsevier (2003)
 Dantzig, G. (1963) Linear Programming and Extensions. Princeton University Press, Princeton
 Karmarkar, N. (1984) A new polynomial time algorithm for linear programming. Combinatorica 4: pp. 373395 CrossRef
 Andersen, K.D., Christiansen, E., Overton, M.L. (1998) Computing limit loads by minimizing a sum of norms. SIAM J. Sci. Comp. 3: pp. 10461062 CrossRef
 Pastor, F., Loute, E. (2005) Solving limit analysis problems: an interior point method. Commun. Numer. Methods Eng. 21: pp. 631642 CrossRef
 Vanderbei, R.J. (2008) Linear Programming: Foundations and Extensions. Kluwer, Dordrecht
 Weichert, D., Ponter, A.R.S. (eds.): Limit States of Materials and Structures. Springer, Wien (2009)
 Nguyen, D.H. (1984) CEPAO—an automatic program for rigidplastic and elasticplastic analysis and optimization of frame structures. Eng. Struct. 6: pp. 3351 CrossRef
 Long, H.V., Nguyen, D.H. (2007) Limit and shakedown analysis of 3D Steel frames. Eng. Struct. 30: pp. 18952004 CrossRef
 Long, H.V., Nguyen, D.H. (2010) Plastic optimization of 3D steel frames under fixed or repeated loading: reduction formulation. Eng. Struct. 32: pp. 10921099 CrossRef
 Skordeli, MAA., Bisbos, C.D. (2010) Limit and shakedown analysis of 3d steel frames via approximate ellipsoidal yield surfaces. Eng. Struct. 32: pp. 15561557 CrossRef
 SAP2000 V14.2.4. Integrated Finite Element Analysis and Design of StructuresBasic Analysis Reference Manual. Computers and Structures Inc, Berkeley (2010)
 Kaneko, I., Lawo, M., Thierauf, G. (1982) On computational procedures for the force method. Int. J. Numer. Methods. Eng. 18: pp. 14691495 CrossRef
 Damkilde, L., Høyer, O. (1993) An efficient implementation of limit state calculations based on lowerbound solutions. Comput. Struct 49: pp. 953962 CrossRef
 Kaveh, A., Koohestani, K. (2008) An efficient graphtheoretical force method for threedimensional finite element analysis. Commun. Numer. Methods Eng. 24: pp. 15331551 CrossRef
 Spiliopoulos, K.V. (1997) On the automation of the force method in the optimal plastic design of frames. Comp. Methods Appl. Mech. Eng. 141: pp. 141156 CrossRef
 Spiliopoulos, K.V. (1999) A fully automatic force method for the optimal shakedown design of frames. Comp. Mech. 23: pp. 299307 CrossRef
 Spiliopoulos, K.V., Patsios, T.N. (2009) A quick estimate of the strength of uniaxially tied framed structures. J. Constr. Steel Res. 65: pp. 17631775 CrossRef
 Henderson, J.C., Maunder, E.A.W. (1969) A problem in applied topology: on the selection of cycles for the flexibility analysis of skeletal structures. J. Inst. Maths. Appl. 5: pp. 254269 CrossRef
 Henderson, J.C., Bickley, W.G. (1955) Statical indeterminacy of a structure. J. Aircr Eng. 27: pp. 400402
 Smith, D.L.: Plastic limit analysis. In: Smith DL (ed.) Mathematical Programming Methods in Structural Plasticity. pp. 61–82, Springer, Berlin (1990)
 Maier, G. (1969) Linear flowlaws of elastoplasticity: a unified general approach. Rendic. Acad. Naz. Lincei, Series 8 47: pp. 266
 Trillat, M., Pastor, J. (2005) Limit analysis and Gurson’s model. Eur. J. Mech. Solids 24: pp. 800819 CrossRef
 MOSEK ApS, Denmark.: the MOSEK optimization tools version 6.0 (revision 135)—user’s manual and reference (2012) Download at http://www.mosek.com
 Nicholson, T.A.J. (1966) Finding the shortest route between two points in a network. Comput. J. 26: pp. 275280 CrossRef
 Lau, H.T.: Algorithms on Graphs. Tab Books (1989)
 Heyman, J. (1996) Elements of the Theory of Structures. Cambridge University Press, Cambridge CrossRef
 Chakrabarty, J. (2006) Theory of Plasticity. Elsevier ButterworthHeinemann, Oxford
 AISC.: Load and Resistance Factor Design Specification for Steel Buildings. American Institute of Steel Construction, Chicago (1993)
 Title
 A powerful forcebased approach for the limit analysis of threedimensional frames
 Journal

Archive of Applied Mechanics
Volume 83, Issue 5 , pp 723742
 Cover Date
 20130501
 DOI
 10.1007/s0041901207145
 Print ISSN
 09391533
 Online ISSN
 14320681
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Numerical methods
 Limit analysis
 Force method
 Graph theory
 Grillages
 Multistoreyed frames
 Industry Sectors
 Authors

 Konstantinos V. Spiliopoulos ^{(1)}
 Nikos G. Dais ^{(1)}
 Author Affiliations

 1. Department of Civil Engineering, Institute of Structural Analysis and Antiseismic Research, National Technical University of Athens, Zografou, 15780, Athens, Greece