# Numerical Simulation

Principles, Methods and Models
• C. C. Baniotopoulos
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 419)

## Abstract

Aim of the present chapter is to present certain basic principles, numerical techniques and algorithmic models applied to the simulation of the structural response of steel semi-rigid connections. For the numerical treatment of this problem, the Finite Element Method is applied. Nonlinearities introduced by unilateral contact and friction over the connection interfaces, as well as material nonlinearities including yielding have been taken into consideration in all of the herein presented models. Within this framework, a variational inequality and a quadratic programming approach have been applied to the unilateral contact problem in steel connections and the corresponding sensitivity analysis has been performed. Based on this methodology, an effective 2-D numerical model for steel connections including contact and plasticity is first presented and, in the sequel, a parametric analysis of the response of steel base plate connections is carried out. Mesh size, choice of finite elements and other parameters affecting the accuracy and the numerical behaviour of the models are also discussed. In the last paragraphs of this chapter a parametric analysis of the response of hollow section joints by means of an effective 3-D model has also been performed giving rise to interesting numerical results that describe with accuracy any possible failure mode of the joint under investigation.

## Keywords

Variational Inequality Base Plate Quadratic Programming Problem Quadratic Optimization Problem Unilateral Contact
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Abdalla, K. M. (1988). Application of the Theory of Unilateral Problems to the Calculation of Connections of Steel Structures. Ph.D. Dissertation, Aristotle University, Thessaloniki.Google Scholar
2. Abdalla, K. M., and Baniotopoulos, C. C. (1991). Design sensitivity investigations of column splices in steel structures. In Proceedings of the First Greek Conference on Steel Structures. Athens: M.S.R.S., 120–129.Google Scholar
3. Abdalla, K. M., and Stavroulakis, G. E. (1989). Zur rationalen Berechnung des “Prying-Actions”Phänomens in Schraubenverbindungen. Stahlbau 58: 233–238.Google Scholar
4. AISC (1981). Manual of Steel Structures. Chicago: AISC, 8th Edition.Google Scholar
5. Ballio, G. and Mazzolani, F. M. (1977): Theory and Design of Steel Structures. Bristol: J. W. Arrowsmith.Google Scholar
6. Baniotopoulos, C. C. (1994). On the numerical assessment of the separation zones in semi-rigid column base plate connections. Structural Engineering and Mecanics 2: 1–15.Google Scholar
7. Baniotopoulos, C. C., (1995): On the separation process in bolted steel splice plates. Journal of Constructional Steel Research 32: 15–35.
8. Baniotopoulos, C. C., Karoumbas, G., and Panagiotopoulos, P. D. (1992). A contribution to the analysis of steel connections by means of quadratic programming techniques. In Proceedings of the First European Conference on Numerical Methods in Engineering. Amsterdam, London, New York, Tokyo: Elsevier, 519–525.Google Scholar
9. Baniotopoulos, C. C., and Abdalla, K. M. (1993). Steel column-to-column connections under combined load: A quadratic programming approach. Computers and Structures 46: 13–20.
10. Baniotopoulos, C. C., Sokol, Z., and Wald, F. (1999). Column base connections. in COST CI Report of WG6-Numerical simulation. Brussels-Luxemburg: European Commission, 32–47.Google Scholar
11. Bendsoe, M. P., Olhoff, N., and Sokolowski, J. (1985). Sensitivity analysis of problems of Elasticity with unilateral constraints. Journal of Structural Mechanics 13:201–222.Google Scholar
12. Bendsoe, M. P. and Sokolowski, J. (1988). Design sensitivity analysis for elastic-plastic analysis problems. Mechanics, Structures and Machines 16: 81–102.Google Scholar
13. Bogdan, H., Kuzmanovic, O., and Willems, N. (1977). Steel Design of Structural Engineers. New Jersey: Prentice Hall.Google Scholar
14. Bortman, J., and Szabo, B.A. (1992). Nonlinear models for fastened structural connections. Computers and Structures 43: 909–923.
15. Bufler, H. (1968): Zur Theorie diskontinuierlichen und kontinuierlichen Verbindungen. Ingenieur Archiv 37: 176–188.
16. Butler, H., and Müßigmann, F. (1977). Kraftübertragung bei diskontinuierlichen Scher-verbindungen. Stahlbau 46: 12–20.Google Scholar
17. CECM (1978). European Recommendation for Steel Construction. Brussels: ECCS.Google Scholar
18. Chen, W. F., and Patel, K. V. (1981). Static behavior of beam-to-column moment. ASCE Journal of Structural Division 107: 1815–1838.Google Scholar
19. Chen, W. F., and Lui, E. M. (1986). Steel beam-to-column moment connections. Part I: Flange moment connections. Solid Mechanics Archives 11:257–316.Google Scholar
20. Cook, R. A., and Klinger, R. E. (1990). Ductile multiple-anchor steel-to-concrete connections.:ASCE Annul! of Structural Division 118: 1645–1665.Google Scholar
21. Dewol, J. ‘I’. (1978). Axially loaded column base plates. AS(’E Journal of Structural Division 104: 781–794.Google Scholar
22. Dewolfe, J. T., and Sansley, E. F. (1983). Column base plates with axial loads and moments. ASCE Journal of Structural Division 106: 2167–2184.Google Scholar
23. Eurocode 3- Part 1, Annex K (normative) (1994). Hollow section lattice girder connections,ENV 19931–1: 1992/A1, Brussels.Google Scholar
24. Fichera, G. (1972). Boundary value problems in Elasticity with unilateral constraints. In Encyclopedia of Physics. Berlin: Springer, VIa/2: 391–424.Google Scholar
25. Fisher, J. W., and Struik, J. H. A. (1978). Guide to Design Criteria for Bolted and Riveted Joints. New York: John Wiley and Sons.Google Scholar
26. Fletcher, R. (1987). Practical Optimization Methods. N. York: John Wiley Sons.Google Scholar
27. Fling, R. S. (1970). Design of steel bearing plates. AISC Engineering Journal 7: 37–40.Google Scholar
28. Hyer, M. W., and Klang, E. C. (1985). Contact stresses in pin-loaded orthotropic plates. International Journal of Solids Structures 21: 957–975.
29. Jaspart, J.-P. (1994): Numerical simulation of a T-stub. Experimental Data. COST CI Numerical simulation working group Cl WD6/94/09:l-9.Google Scholar
30. Jaspart, J.-P. and Bursi, O. (1997). Calibration of a finite element model for bolted endplate steel connections. Journal of Constructional Steel Research 44: 225–262.
31. Kato, B. and McGuire, W. (1973). Analysis of T-stub flange to column connection. ASCE Journal of Structural Division 99: 865–888.Google Scholar
32. Kontoleon, M., Mistakidis, E. S., Baniotopoulos, C. C., and Panagiotopoulos, P. D. (1999). Parametric analysis of the structural response of steel base-plate connections. Computers and Structures 71: 87103.Google Scholar
33. Korol, R., and Packer, J. (1992). Behaviour of RHS Gap K-joints at Service Loads. University of Toronto, Toronto.Google Scholar
34. Koskimaki, M., and Niemi, E. (1990). Finite Element Studies on the Behavior of Rectangular Hollow Section K-joints. Lapeenranta University of Technology, Rovaniemi.Google Scholar
35. Krishnamurthy, N., and Graddy, D. (1976). Correlation between 2-and 3-dimensional finite element analysis of steel bolted end plate connections. Computers and Structures 6: 381–389.
36. Krishnamurthy, N. (1978), A fresh look at bolted steel end-plate behavior and design. AISC Engineering Journal 15: 39–49.Google Scholar
37. Künzi, H., and Krelle, W. (1962). Nichtlineare Programmierung. Berlin: Springer.
38. Kukreti, A. R., Murray, T. M., and Abolmaali, A. (1987). End plate connection moment-rotation relationship. Journal of Constructional Steel Research 8: 137–157.
39. Maier, G. (1968). A quadratic programming approach for certain classes of non-linear structural problems. Meccanica, 2: 121–130.
40. Mistakidis, E. S., Baniotopoulos, C. C., and Panagiotopoulos, P.D. (1996a): A numerical method for the analysis of semi-rigid base plate connections. In Proceedings of ECCOMAS 96, Paris: J. Wiley Sons Ltd, 842–848.Google Scholar
41. Mistakidis, E. S., Baniotopoulos, C. C., Bisbos, C. D., and Panagiotopoulos, P. D. (1996b). A 2-D numerical method for the analysis of steel T-stub connections. In Proceedings of the 2“`’ Greek Conference on Computational Mechanics, Chania: GRACM, 777–748.Google Scholar
42. Mistakidis, E. S., Baniotopoulos, C. C. and Panagiotopoulos, P. D. (1998). An effective two-dimensional numerical method for the analysis of a class of steel connections. Computational Mechanics 21: 363–371.
43. Mistakidis, E. S., Baniotopoulos, C. C., Bisbos, C. D., and Panagiotopoulos, P.D. (1997). Steel T-stub Connections Under Static Loading: an Effective 2-D Numerical Model. Journal of Constructional Steel Research 44: 51–67.
44. Moreau, J. J., Panagiotopoulos, P. D., and Strang, G. (eds) (1988). Topics in Nonsmooth Mechanics. Basel, Boston: Birkhäuser.
45. Moreau, J. J., and Panagiotopoulos, P.D. (eds) (1988). Nonsmooth Mechanics and Applications. CISM Lecture Notes 302. Wien, New York: Springer.Google Scholar
46. Murray, T.M. (1983), Design of lightly loaded steel column base plates. AISC Engineering Journal 20: 143–152.Google Scholar
47. Necas, J., Jarusek, J., and Haslinger, J. (1980). On the solution of the variational inequality to the Signorini problem with small friction. Bulletino U.M.I. 17B: 796–811.
48. Paker, S. A., and Morris, L.J. (1977). A limit state design method for tension of bolted column connections. Structural Engineer 55: 876–889.Google Scholar
49. Paker, J. A., Wardenier, J., Kurobane, Y., Dutta, D. and Yeomans, N. (1992). Design Guide for Rectangular Section (RHS) Joints under Predominantly Static Loading. CIDECT, Verlag TUV Reinland, Kologne.Google Scholar
50. Panagiotopoulos, P. D. (1975). A nonlinear programming approach to the unilateral contact and friction boundary value problem in the theory of Elasticity. Ingenieur Archiv 44: 421–432.
51. Panagiotopoulos, P. D. (1976). Convex analysis and unilateral static problems. Ingenieur Archiv 45: 55–68.
52. Panagiotopoulos, P. D. (1985). Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Basel, Boston: Birkhäuser ( (1989), Russian translation, Moscow: Mir).Google Scholar
53. Panagiotopoulos, P. D. (1988). Hemivariational Inequalities. Applications in Mechanics and Engineering. Boston-Basel: Birkhäuser.Google Scholar
54. Panagiotopoulos, P. D., and Talaslidis, D. (1980). A linear analysis approach to the solution of certain classes of variational inequality problems in Structural Analysis. International Journal of Solids and Structures 16: 991–1005.
55. Panagiotopoulos, P. D., Baniotopoulos, C. C. and Avdelas, A. V. (1984). Certain propositions on the activation of yield modes in Elastoplasticity and their applications to deterministic and stochastic problems. Zeitschrift fuer Angewandte Mathematik und Mechanik 64: 491–501.
56. Panagiotopoulos, P. D., and Strang, G. (1988). Topics in Nonsmooth Mechanics. Boston-Basel: Birkhäuser.
57. Raffa, F., and Strona, P. (1984). Boundary Element Method application to bolted joint analysis. Engineering Analysis 1:78–89.Google Scholar
58. Sakellariadou, H. I., Bisbos, C. D., Thomopoulos, K., and Panagiotopoulos, P. D. (1996). A 3-D numerical study of the T-stub problem taking into account the interfacial unilateral contact effects. In Proceedings of the Conference in Advances in Computational Methods for Simulation. Civil Comp Press, Edinburgh: 115–124.Google Scholar
59. Soh, A., and Soh, C. (1994). On the Compatibility Conditions for Modeling Tubular Joints Using Different Types of Elements. Nanyang University of Singapore, Singapore.Google Scholar
60. Stockwell, Jr., F. W. (1975). Preliminary base plate selection. AISC Engineering Journal 12: 92–99.Google Scholar
61. Mistakidis, E. S., Baniotopoulos, C. C., Bisbos, C. D., and Panagiotopoulos, P.D. (1997). Steel T-stub Connections Under Static Loading: an Effective 2-D Numerical Model. Journal of Constructional Steel Research 44: 51–67.
62. Moreau, J. J., Panagiotopoulos, P. D., and Strang, G. (eds) (1988). Topics in Nonsmooth Mechanics. Basel, Boston: Birkhäuser.
63. Moreau, J. J., and Panagiotopoulos, P.D. (eds) (1988). Nonsmooth Mechanics and Applications. CISM Lecture Notes 302. Wien, New York: Springer.Google Scholar
64. Murray, T.M. (1983), Design of lightly loaded steel column base plates. AISC Engineering Journal 20: 143–152.Google Scholar
65. Necas, J., Jarusek, J., and Haslinger, J. (1980). On the solution of the variational inequality to the Signorini problem with small friction. Bulletino U.M.I. 17B: 796–811.
66. Paker, S. A., and Morris, L.J. (1977). A limit state design method for tension of bolted column connections. Structural Engineer 55:876–889.Google Scholar
67. Paker, J. A., Wardenier, J., Kurobane, Y., Dutta, D. and Yeomans, N. (1992). Design Guide for Rectangular Section (RHS) Joints under Predominantly Static Loading. CIDECT, Verlag TUV Reinland, Kologne.Google Scholar
68. Panagiotopoulos, P. D. (1975). A nonlinear programming approach to the unilateral contact and friction boundary value problem in the theory of Elasticity. Ingenieur Archiv 44: 421–432.
69. Panagiotopoulos, P. D. (1976). Convex analysis and unilateral static problems. Ingenieur Archiv 45: 5568.Google Scholar
70. Panagiotopoulos, P. D. (1985). Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Basel, Boston: Birkhäuser ( (1989), Russian translation, Moscow: Mir).Google Scholar
71. Panagiotopoulos, P. D. (1988). Hemivariational Inequalities. Applications in Mechanics and Engineering. Boston-Basel: Birkhäuser.Google Scholar
72. Panagiotopoulos, P. D., and Talaslidis, D. (1980). A linear analysis approach to the solution of certain classes of variational inequality problems in Structural Analysis. International Journal of Solids and Structures 16: 991–1005.
73. Panagiotopoulos, P. D., Baniotopoulos, C. C. and Avdelas, A. V. (1984). Certain propositions on the activation of yield modes in Elastoplasticity and their applications to deterministic and stochastic problems. Zeitschrift fuer Angewandte Mathematik und Mechanik 64: 491–501.
74. Panagiotopoulos, P. D., and Strang, G. (1988). Topics in Nonsmooth Mechanics. Boston-Basel: Birkhäuser.
75. Raffa, F., and Strona, P. (1984). Boundary Element Method application to bolted joint analysis. Engineering Analysis 1: 78–89.
76. Sakellariadou, H. I., Bisbos, C. D., Thomopoulos, K., and Panagiotopoulos, P. D. (1996). A 3-D numerical study of the T-stub problem taking into account the interfacial unilateral contact effects. In Proceedings of the Conference in Advances in Computational Methods for Simulation. Civil Comp Press, Edinburgh: 115–124.Google Scholar
77. Soh, A., and Soh, C. (1994). On the Compatibility Conditions for Modeling Tubular Joints Using Different Types of Elements. Nanyang University of Singapore, Singapore.Google Scholar
78. Stockwell, Jr., F. W. (1975). Preliminary base plate selection. AISC Engineering Journal 12: 92–99.Google Scholar
79. Talaslidis, D., and Panagiotopoulos, P. D. (1982). A linear finite element approach to the solution of variational inequalities arising in contact problems of Structural Dynamics. Inernational Journal of Numerical Methods in Engineering 18: 1505–1520.
80. Thambiratnam, D. P., and Paramasivam, P. (1986). Base plates under axial loads and moments. ASCE Journal of Structural Division 112: 1166–1181.
81. Thambiratnam, D. P., and Krishnamurthy, N. (1989). Computer analysis of column base plates. Computers and Structures 33: 839–850.
82. Thomopoulos, K. (1985). Improvement of the design method for steel column base plates via an inequality approach. Civil Engineering for Practicing and Design Engineers 4: 923–933.Google Scholar
83. Thomopoulos, K. (1986). A new method for rectangular hollow section splices via contact analysis. Civil Engineering for Practicing and Design Engineers 5: 443–452.Google Scholar
84. Vogt, F. (1947). The Load Distribution in Bolted or Riveted Joints in Light-Alloy Structures. Technical Report TM 1135. Washington DC: NACA.Google Scholar
85. Wald, F. (1993). Column Base Connections. A Comprehensive State of the Art. Prague: CVUT.Google Scholar
86. Wald, F. (1995). Patky Sloupu, Column Bases. Prague: CVUT.Google Scholar
87. Wardenier, J. (1984). Hollow Section Joints. Delft: Delft University Press.Google Scholar