On Selecting Composite Functions Based on Polynomials for Responses Describing Extreme Magnitudes of Structures
Source abstract
Abstract
The main aim of this work was to investigate a numerical error in determining limit state functions, which describe the extreme magnitudes of steel structures with respect to random variables. It was assisted here by the global version of the response function method (RFM). Various approximations of trial points generated on the basis of several hundred selected reference composite functions based on polynomials were analyzed. The final goal was to find some criterion—between approximation and input data—for the selection of the response function leading to relative a posteriori errors less than 1%. Unlike the classical problem of curve fitting, the accuracy of the final values of probabilistic moments was verified here as they can be used in further reliability calculations. The use of the criterion and the associated way of selecting the response function was demonstrated on the example of steel diagrid grillages. It resulted in quite high correctness in comparison with extended FEM tests.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Probabilistic and Robust Engineering Design · Geotechnical Engineering and Analysis · Structural Health Monitoring Techniques
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Department of Structural Mechanics, Faculty of Civil Engineering, Architecture and Environmental Engineering, Łódź University of Technology, 90-924 Łódź, Poland
References (85)
Show source-provided references
1. Meyer Boake, T. (2013). Understanding Steel Design, Birkhäuser.
2. Moon, K. (2009, January 2–4). Design and Construction of Steel Diagrid Structures. Proceedings of the 11th Nordic Steel Construction Conference-NSCC 2009, Malmö, Sweden.
3. A probability-based structural code
4. First-order third-moment reliability method10.1016/0167-4730(94)00021-H ↗
5. Development of High Order Moment Standardization Method into structural design and its efficiency
6. Fourth-Moment Standardization for Structural Reliability Assessment10.1061/(ASCE)0733-9445(2007)133:7(916) ↗
7. An improved fourth-order moment reliability method for strongly skewed distributions10.1007/s00158-020-02546-y ↗
8. Second-Order Fourth-Moment Method for Structural Reliability10.1061/(ASCE)EM.1943-7889.0001199 ↗
9. Ghanem, R.G., and Spanos, P.D. (1991). Stochastic Finite Elements: A Spectral Approach, Springer.10.1007/978-1-4612-3094-6 ↗
10. Monte Carlo techniques in computational stochastic mechanics10.1007/BF02736747 ↗
11. An APDM-based method for the analysis of systems with uncertainties10.1016/j.cma.2014.06.014 ↗
12. Matching the principal deformation mode method with the probability transformation method for the analysis of uncertain systems10.1002/nme.6018 ↗
13. Comparative studies of metamodelling techniques under multiple modelling criteria10.1007/s00158-001-0160-4 ↗
14. Approximation methods in multidisciplinary analysis and optimization: A panel discussion10.1007/s00158-004-0389-9 ↗
15. Metamodels for computer-based engineering design: Survey and recommendations10.1007/PL00007198 ↗
16. Santner, T.J., Williams, B.J., and Notz, W.I. (2003). The Design and Analysis of Computer Experiments, Springer.10.1007/978-1-4757-3799-8 ↗
17. Draper, N.R., and Smith, H. (1998). Applied Regression Analysis, Wiley.10.1002/9781118625590 ↗
18. Myers, R.H., and Montgomery, D.C. (2002). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Wiley. [2nd ed.].
19. Metamodeling: What is it good for?10.1109/MDT.2009.63 ↗
20. Fang, K.-T., Li, R., and Sudjianto, A. (2005). Design and Modeling for Computer Experiments, Chapman and Hall/CRC.10.1201/9781420034899 ↗
21. Efficient computation of global sensitivity indices using sparse polynomial chaos expansions10.1016/j.ress.2010.06.015 ↗
22. Adaptive sparse polynomial chaos expansion based on least angle regression10.1016/j.jcp.2010.12.021 ↗
23. Smith, M. (1993). Neural Networks for Statistical Modeling, Von Nostrand Reinhold.
24. Multivariate Adaptive Regression Splines
25. Regularized orthogonal least squares algorithm for constructing radial basis function networks10.1080/00207179608921659 ↗
26. A multiwavelet support vector regression method for efficient reliability assessment10.1016/j.ress.2014.12.002 ↗
27. Design and analysis of computer experiments
28. Various response functions in lattice domes reliability via analytical integration and finite element method10.2478/ijame-2018-0026 ↗
29. Generalized stochastic finite element method in elastic stability problems10.1016/j.compstruc.2010.08.009 ↗
30. On semi-analytical probabilistic finite element method for homogenization of the periodic fiber-reinforced composites10.1002/nme.3097 ↗
31. A new look at the response surface approach for reliability analysis10.1016/0167-4730(93)90003-J ↗
32. Effect of response surface parameter variation on structural reliability estimates10.1016/S0167-4730(02)00013-9 ↗
33. High-order limit state functions in the response surface method for structural reliability analysis10.1016/j.strusafe.2006.10.003 ↗
34. An improvement of the response surface method10.1016/j.strusafe.2011.02.001 ↗
35. Metamodels of optimal quality for stochastic structural optimization10.1016/j.probengmech.2017.09.003 ↗
36. A fast and efficient response surface approach for structural reliability problems10.1016/0167-4730(90)90012-E ↗
37. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation10.1016/j.cma.2004.02.026 ↗
38. Hierarchical parallelisation for the solution of stochastic finite element equations10.1016/j.compstruc.2004.11.014 ↗
39. Kleiber, M., and Hien, T. (1992). The Stochastic Finite Element Method, Wiley.
40. Random field finite elements10.1002/nme.1620231004 ↗
41. Stochastic finite element analysis of composite structures based on mesoscale random fields of material properties10.1016/j.cma.2017.08.002 ↗
42. On influence of the response functions on the diagrid and orthogonal grillages reliability by the stochastic iterative perturbation-based finite element method10.1063/1.5019164 ↗
43. Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., and Watt, S.M. (1992). First Leaves: A Tutorial Introduction to Maple V, Springer.10.1007/978-1-4615-6996-1 ↗
44. On polynomial response surfaces and Kriging for use in structural optimization of crashworthiness10.1007/s00158-004-0487-8 ↗
45. Reliability-based design optimization of structural systems under hybrid probabilistic and interval model10.1016/j.compstruc.2015.08.009 ↗
46. A comparison of approximate response functions in structural reliability analysis10.1016/j.probengmech.2007.12.022 ↗
47. Transformed perturbation stochastic finite element method for static response analysis of stochastic structures10.1016/j.finel.2013.10.003 ↗
48. Response-Surface Approach for Reliability Analysis10.1061/(ASCE)0733-9399(1989)115:12(2763) ↗
49. Change-of-variable interval stochastic perturbation method for hybrid uncertain structural-acoustic systems with random and interval variables10.1016/j.jfluidstructs.2014.07.005 ↗
50. Recursive approach for random response analysis using non-orthogonal polynomial expansion10.1007/s00466-009-0375-6 ↗
51. A polynomial dimensional decomposition for stochastic computing10.1002/nme.2394 ↗
52. Performance evaluation of metamodelling methods for engineering problems: Towards a practitioner guide10.1007/s00158-019-02352-1 ↗
53. Bendat, J.S., and Piersol, A.G. (1971). Random Data: Analysis and Measurement Procedures, Wiley.
54. Feller, W. (1965). An Introduction to Probability Theory and Its Applications, Wiley.
55. Vanmarcke, E. (1983). Random Fields: Analysis and Synthesis, MIT Press.
56. Kottegoda, N.T., and Rosso, R. (2008). Applied Statistics for Civil and Environmental Engineers, Blackwell.
57. Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects10.1007/s00707-018-2174-7 ↗
58. Kamiński, M.M. (2013). The Stochastic Perturbation Method for Computational Mechanics, Wiley.10.1002/9781118481844 ↗
59. Hughes-Hallett, D., Gleason, A.M., and McCallum, W.G. (2019). Calculus: Single and Multivariable, Wiley. [7th ed.].
60. Optimization of the truss-type structures using the generalized perturbation-based Stochastic Finite Element Method10.1016/j.finel.2012.08.002 ↗
61. The least squares stochastic finite element method in structural stability analysis of steel skeletal structures10.1515/ijame-2015-0020 ↗
62. Björck, Å. (1996). Numerical Methods for Least Squares Problems, SIAM.10.1137/1.9781611971484 ↗
63. Wolberg, J. (2005). Data Analysis Using the Method of Least Squares: Extracting the Most Information from Experiments, Springer.
64. On the dual iterative stochastic perturbation-based finite element method in solid mechanics with Gaussian uncertainties10.1002/nme.4976 ↗
65. Dual probabilistic homogenization of the rubber-based composite with random carbon black particle reinforcement10.1016/j.compstruct.2016.01.038 ↗
66. Kamiński, M.M., and Pokusiński, B.M. (2015, January 8–11). Reliability of some axisymmetric shell structure by the response function method and the generalized stochastic perturbation technique. Proceedings of the Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues—3rd Polish Congress of Mechanics, PCM 2015 and 21st International Conference on Computer Methods in Mechanics, CMM 2015, Gdansk, Poland.
67. On sequentially coupled thermo-elastic stochastic finite element analysis of the steel skeletal towers exposed to fire10.1016/j.euromechsol.2016.11.008 ↗
68. Dual Probabilistic Analysis of the Transient Heat Transfer by the Stochastic Finite Element Method with Optimized Polynomial Basis
69. Über Empirische Funktionen und die Interpolation Zwischen Äquidistanten Ordinaten
70. A New Look at the Statistical Model Identification10.1109/TAC.1974.1100705 ↗
71. Estimating the Dimension of a Model10.1214/aos/1176344136 ↗
72. Application of model-selection criteria to some problems in multivariate analysis10.1007/BF02294360 ↗
73. Steiger, J.H., and Lind, J.C. (1980, January 28). Statistically based tests for the number of common factors. Proceedings of the Annual Meeting of the Psychometric Society: Structural Equation Modeling, Iowa City, IA, USA.
74. Graphs in statistical analysis10.1080/00031305.1973.10478966 ↗
75. European Committee for Standardization (2002). EN 1990: Eurocode—Basis of Structural Design, European Committee for Standardization.
76. European Committee for Standardization (2002). EN 1991-1-1: Eurocode 1: Actions on Structures—Part 1-1: General Actions—Densities, Self-Weight, Imposed Loads for Buildings, European Committee for Standardization.
77. European Committee for Standardization (2006). EN 1993-1-1: Eurocode 3: Design of Steel Structures—Part 1-1: General Rules and Rules for Buildings, European Committee for Standardization.
78. European Committee for Standardization (2000). EN 13474-2: Glass in Building—Design of Glass Panes—Part 2: Design for Uniformaly Distributed Loads, European Committee for Standardization.
79. European Committee for Standardization (2008). EN 13474-3: Glass in Building—Determination of the Strength of Glass Panes—Part 3: General Method of Calculation and Determination of Strength of Glass by Testing, European Committee for Standardization.
80. European Committee for Standardization (2013). EN 16612: Glass in Building—Determination of the Load Resistance of Glass Panels by Calculation and Testing, European Committee for Standardization.
81. Oden, J.T. (1972). Finite Elements of Nonlinear Continua, McGraw-Hill.
82. Owen, D.R.J., and Hinton, E. (1980). Finite Elements in Plasticity—Theory and Practice, Pineridge Press.
83. Zienkiewicz, O.C., Taylor, R.L., and Fox, D.D. (2014). The Finite Element Method for Solid and Structural Mechanics, Elsevier. [7th ed.].
84. Schittich, C., Staib, G., Balkow, D., Schuler, M., and Sobek, W. (2007). Glass Construction Manual, Birkhäuser. [2nd ed.].10.11129/detail.9783034615549 ↗
85. An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics10.1016/j.probengmech.2004.06.001 ↗