reliability analysis of one dimentional structures using explicit probability density function of static response in stochastic finite element

Document Type : Original Research

Authors
R. choobdarian 1
A. yazdani 2
H. Dabbagh 3
1 student
2 Associate Professor of Kurdistan University - Department of Civil Engineering
3 Assistant Professor, University of Kurdistan - Civil Engineering Department
Abstract
Reliability analysis, with considering the randomness of geometry, materials and loading variables, is a good guide for structural design in engineering. The stochastic finite element method is used to analyzing the structural systems, with regard to uncertainty in random parameters of structures. Solving the equations of stochastic finite element leads to the calculation of all possible structural responses taking into account all the random variables of the structural system. Due to the uncertainty effect on the response of structural systems, reliability analysis is essential. However, due to the limitations of the classical methods of reliability analysis, there is a need to calculate the reliability index based on the probability density function of response in structures. In this study, by combining perturbation method and change – of – variable method and without the limitation of the statistical type of random distribution of random variables, the probability density function of response is calculated. By calculating the probability density function of the static response of the structures, the probability of failure and the reliability index are obtained directly.

It is obvious that the accuracy of the result of the stochastic finite element analysis depends on the random field element meshes. For this purpose, the distributed random field is discretized over the number of elements of equal length in structural members for each random variable.

In stochastic finite element method, due to the uncertainty of the characteristics of random variables in the structure, it is necessary to define a correlation function between a random variable in different elements. The reliability index is considered as a measure of convergence by considering the scale of fluctuations and the correlation function of random variables in adjacent elements in structure. In each number of elements, the structural reliability index is calculated using the explicit probability density function of response and the structural resistance function to converge on a certain number of elements. In this study, the stochastic finite element analysis is performed in linear static mode for a simple beam and a cantilever column and the variables of geometry, materials and loading are considered randomly with real statistical distributions according to the literature review. As can be expected from the deterministic finite element method, as the number of elements increase and the meshing is smaller, the reliability index increases.Considering the lower scale of fluctuations for random variables makes the reliability index converge to a larger value. However, on a larger scale of fluctuations convergence occurs in a smaller number of elements due to the greater correlation of random variables in adjacent elements. Also, the results of the probability density function of the response with the explicit method compared to the Monte Carlo simulation method show a good match in result. The advantage of using the reliability index as a measure of convergence in meshing, the configuration of limited random components is to consider all possible structural responses instead of using the average structural response in the design of structures.

Keywords

Subjects

1- F. Casciata, I. Negri, R. Rackwitze, 1991, Geometrical variability in structural members and systems, Joint Committee on Structural Safety.
2- R. Lu, Y. Luo, J. P. Conte, 1995, Reliability evaluation of reinforced concrete beams, Structural Safety 14, 277-298.
3- A. Yazdani, N. Eftekhari, 2015, The effect of structural properties and ground motion variables on the global response of structural systems, Civil Engineering and Environmental Systems 32(3), 1-14
4- H. Contreras, 1980, The stochastic finite element method, Computers and. Structures, 12(3), 341-348.
5- M. Kleiber, T.D. Hien, 1992, The Stochastic Finite Element Method, Wiley Online Library, Volume 10, Issue 4
6- G.I, Schuëller, 2006, Developments in stochastic structural mechanics, Applied Mechanics 75, 755-773.
7- G. Stefanou, 2009, The stochastic finite element method: past, present and future, Computer Method In Applied Mechanics And Engineering 198 ,1031–1051.
8- J.E. Hurtado, A.H. Barbat, 1998, Monte Carlo techniques in computational stochastic mechanics, Computational Method In Engineering 5(1), 3–29.
9- H. Zhang, R.L, Mullen, R. Muhanna, 2010, Interval Monte Carlo methods for structural reliability, Structural Safety 32, 183-190
10- F. Yamazaki, M. Shinozuka, 1988 Neumann expansion for stochastic finite element analysis, Engineering Mechanics 114(8), 35- 54.
11- WK .Liu, T. Belytschko, A. Mani, 1986 Random field finite elements, Numerical methods in Engineering 23, 1831-45.
12- E. Vanmarcke, M. Grigoriu, 1983 Stochastic finite element analysis of simple beams, Engineering Mechanics ASCE 109, 1203-1214
13- C. Papadimitriou, L.S. Katafygiotis, J.L. Beck, 1995, Approximate analysis of response variability of uncertain linear systems, Probabilistic Engineering Mechanics 10(4), 251–264.
14- A.M. Freudental, 1947, The safety of structures, Transactions, ASCE 112, 125-159.
15- A.Der Kiureghian , J.B. Ke , 1988, The stochastic finite element method in structural reliability ,Probabilistic Engineering Mechanics 3(2), 83–91.
16- P.L. Liu, A.Der Kiureghian, 1991, Finite element reliability of geometrically non linear uncertain structures, Engineering Mechanics 117(8), 1806–1825.
17- B. Sudret , A. Der Kiureghian, 2000 Stochastic finite elements and reliability: a state-of-the-art report, Structural Engineering, Mechanics And Materials, Report No UCB/SEMM/08
18- S. Huang, S. Mahadevan, R. Rebba, 2007, Collocation-based stochastic finite element analysis for random field problems, Probablistic Engineering Mechanics 22(2), 194-205
19- S. Mahadevan, A. Halder, 1991, Practical Random Field Discretization In Stochastic Finite Element Analysis, Structural Safety 9, 283 – 304.
20- R.G. Ghanem, P.D. Spanos, 1991 Spectral stochastic finite-element formulation for reliability analysis, Engineering Mechanics 117(10), 2351–2371.
21- E. H. Vanmarcke, 1994, Stochastic finite elements and experimental measurements, Probabilistic Engineering Mechanics 9(2), 103-114.
22- R. Hambli, 2009, Statistical damage analysis of extrusion processes using finite element method and neural networks simulation, Finite Elements in Analysis and Design 45(10), 640 – 649.
23- M. Jalalpour, J.K. Guest, T. Igusa, 2013, Reliability – based topology optimization of trusses with stochastic stiffness, Structural Safety 43, 41-49.
24- H.A. Jesen, F. Mayorga, M.A. Valdebenito, 2015, Reliability sensitivity estimation of nonlinear structural systems under stochastic excitation: A simulation-based approach, Computer Methods In Applied Mechanics And Engineering 289, 1-23.
25- A. Der Kiureghian, 1996, Structural reliability methods for seismic safety assessment: a review, Engineering Structures 18(6), 412-424.
26- D.M. Tartakovsky, D. Xiu, 2006, Stochastic analysis of transport in tubes with rough walls, Computational Physics 217(1), 248–259.
27- M.Aldosary, J.Wang and C.Li, 2018, Structural reliability and stochastic finite element methods: State-of-the-art review and evidence-based comparison, Engineering Computations 35(6), 2165-2214
28- W. Gao, D. Wu , K.Gao, X.Chen, F. Tin-Loi; 2018, Structural reliability analysis with imprecise random and interval fields, Applied Mathematical modelling 55, 49-67
29- A. Sofi, E. Romeo, 2017, A Unified Response Surface Framework For The Interval And Stochastic Finite Element Analysis Of Structures With Uncertain Parameters, Probabilistic Engineering Mechanics (54), 25-36.
30- D. Li, Zh. Zheng, et al, 2017, Stochastic nonlinear vibration and reliability of orthotropic membrane structure under impact load, Thin-Walled Structures (119), 247-255.
31- P. Ni, J. Li, et al, 2020, Reliability analysis and design optimization of nonlinear structures, Reliability Engineering and System Safety (198), 1010-1016
32- . Liu, X. Ruan, 2019, Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions, Structuers And Infrastructure Engineering (15), 1573-2479
33- F.N. Momin, H.R. Millwater, R.W. Osborn, M.P. Enright, 2010, A non-intrusivemethod to add finite element-based random variables to a probabilistic design code, Finite Element In Analysis And Design 46(3), 280–287.
34- D. Straub, A.D. Kiureghian, 2010, Bayesian network enhanced with structural reliability methods: application, Engineering Mechanics 136(10), 1259–1270.
35- X.G. Hua, Y.Q. Ni, Z.Q. Chen, J.M. Ko, 2007, An improved perturbation method for stochastic finite element model updating, Numerical Methods In Engineering 73(13), 1845–1864.
36- A. Culla, A. Carcaterra, 2007, Statistical moments predictions for a moored floating body oscillating in random waves, Sound And Vibration 308(2), 44–66.
37- L .J. M. Aslett, T. Nagapetyan, S. J. Vollmer, 2017, Multilevel Monte Carlo for Reliability Theory, Reliability Engineering System And Safety 165, 188-196.
38- S. Huang, S. Mahadevan, R. Rebba, 2007, Collocation – based stochastic finite element analysis for random field problems, Probabilistic Engineering Mechanics 22(2), 194-205.
39- L. Pichler, H.J. Pradlwarter, 2009, Evolution of probability densities in the phase for reliability analysis of non-linear structures, Structural Safety 31(4), 316 – 324.
40- A. Yazdani, T. Takada, 2011, Probabilistic study of the influence of ground motion variables on response spectra, Structural Engineering and Mechanics 39(6), 877-893.
41- D. Xia, J. L. Yun, 2014, Transformed perturbation stochastic finite element method for static response analysis of stochastic structures, Finite Elements in Analysis and Design 79, 9-21.
42- D. Xia, 2013, Response Probability Analysis of Random Acoustic Field Based on Perturbation Stochastic Method and Change-of-Variable Technique, Vibration and Acoustics 135(5), 1312-1333.
43- B. Rong, X. Rui, L.Tao, 2012, Perturbation Finite Element Transfer Matrix Method for Random Eigenvalue Problems of Uncertain Structures, Applied Mechanics 79(2), 1005-1013.
44- B. Xia, D. Yu, X. Han, C. Jiang, 2014, Unified response probability distribution analysis of two hybrid uncertain acoustic fields, Computer Methods in Appied Mechanics and Engineering 276, 20–34.
45- B. Xia, D. Yu, 2014, An interval random perturbation method for structural-acoustic system with hybrid uncertain parameters, Numerical Methods in Engineering 97(3), 181–206.
46- E.H. Vanmarcke, 1983, Random Fields: Analysis and Synthesis, Advancing Earth and Space Science 64(37).
47- Y. Shuwang, G. Linping, 2015, Calculation of Scale of Fluctuation and Variance Reduction Function, Transactions of Tianjin University 21(1), 41-49
48- JCSS Probabilistic Model Code PART2: Load Models, 2001, 99CON-DYN/M0098.
49- T. Vrouwenvelder, M. Holicky, J. Markova, JCSS Probabilistic Model Code, Example Applications,
50- R. Lu, Y. Luo, J. P. Conte, 1994, Reliability evaluation of reinforced concrete beams, structural safety 14(4), 277-298.
51- F. F. Udoeyo, P. Ugbem , 1995, Dimensional Variables in Reinforced Concrete Members, Structural Engineering, 1865-1867
52- N. B. Hossain, M. G. Stewart, 2001, Probabilistic Models Of Damaging Deflections For Floor Elements, Performance Of Constructes Facilities 15(4),
53- P. Gardoni1; A. D Kiureghian, 2002, Probabilistic Capacity Models and Fragility Estimates for Reinforced Concrete Columns based on Experimental Observations, Engineering Mechanics 128(10), 383-393.
Modares Civil Engineering journal
Volume 21, Issue 2
May and June 2021
Pages 49-62