| Scaffold system is a lattice structure constructed with slender members through specific connection,and the mechanical behavior of which exhibits complex under influence of various factors.Mesnwhile,the relationships between stochastic responses and random parameters of scaffold system effected by stochastic factors are usually highly nonlinear implicit functions,which leads to great difficulty for reliability analysis.The response surface methods(RSM)which adopt the explicit approximation function in the probability space to describe the relationship between target stochastic response and random parameters have been developed for reliability analysis of large complex structural systems.However,the traditional response surface methods are of low computational efficiency for the multi-objective structural reliability analysis,since their response surfaces should be reconstructed for different structural response quantities.Meanwhile,the Traditional response surface methods are of low computational accuracy and efficiency for the reliability problem with non-gaussian random variables.Focusing on the two above problems,the main objective of this paper is to propose some efficient response surface methods for analysis of structural reliability of scaffold system.The main work of this paper are listed as follows:(1)The Quadratic response surface method(QRSM)and the Hermite response surface method(HRSM)need to reconstruct the response surfaces repeatly for the structural reliability of the scaffold system,since the traditional response surface methods are scalar type and are location-dependence in structure space and/or probability spaces.Hence,the root cause of the location-independent feature and the computational efficiency problem of the traditional response surface method were discussed and verified in this paper.The results show that the QRSM can only approximate the limit state function well in the neighborhood of the design point(i.e.,it is location-dependent in the probability space).Therforce,the QRSM needs to produce the quadratic response surfaces repeatedly with the time-consuming iterative finite element analysis to search for the design point,which leads to low computational efficiency for structural reliability analysis.Different from the QRSM,the HRSM represents the target uncertain quantity with the Hermite polynomial chaos expansion.Duo to the solid mathematic foundation of the Hermite polynomial chaos expansion,the HRSM can fit the limit state function well in the whole probability space(i.e.,it is location-independent in the probability space).Hence,the HRSM is more efficient and accurate than the QRSM for structural reliability analysis.Furthermore,both the HRSM and the QRSM are scalar-type response surfaces that only can expand one structural response quantity attached the specified position of structural systems each time,which means that the unknown coefficients of response surface changed with the position of response quantity in the structure space.Therefore,both the HRSM and the QRSM are of location-dependent in the structure space,which need to reconstruct response surfaces for different target structural response quantities,leading to low computational efficiency for the multi-objective structural reliability analysis.(2)Traditional scalar-type response surface methods(i.e.,QRSM and HRSM)are location-dependence in structure and/or probability spaces and are inefficient for computing the structural reliability.In order to deal with this problem,a novel vector-type full-spaces response surface method(FRSM)that is of location-independence in probability and physical spaces was proposed for structural reliability analysis.Firstly,a vector type response surface was developed by expanding the stochastic nodal displacement vector along the Krylov basis vectors defined by the global stiffness matrix and the force vector.Then the effective collocation points were picked out of the candidate ones according to the linear independence of the combining row vector.Finally,the unknown coefficients of the proposed response surface were determined by means of the regression analysis.Analysis shows that the proposed FRSM is of location-independence in both structure and probability spaces,which overcomes the disadvantages of location-dependence of traditional scalar-type response surface methods.The proposed FRSM requires much fewer effective collocation points and times of finite element analysis,achieving much higher computational efficiency by comparing with the traditional response surface methods.The proposed FRSM also provides an efficient way to sovle the the curse of dimensionality problem arsing from the large number of the input variables.(3)In order to widen the applicion of the FRSM,a novel full-spaces response surface method(FRSM)was presented for the multi-objective structural reliability analysis with correlated non-Gaussian random variables.Firstly,the correlated non-Gaussian random variables were mapped into the independent standard non-Gaussian random variables based on the Cholesky decomposition.Then the full-space response surface was expanded by the Krylov basis vectors defined with the global stiffness matrix and load vector in the independent standard non-Gaussian probability space.Meanwhile,a small number of effective collocation points in the independent standard Gaussian probability space were established based on the linear independent of combining row vectors,which subsequently were transformed into the non-Gaussian probability space(i.e.,correlated non-Gaussian probability space and independent standard non-Gaussian probability space)based on the Nataf transformation and the Cholesky decomposition.Finally,the unknown coefficients of full-space response surface were determined by means of the regression analysis,and then the multi-objective reliability analysis can be proceed simultaneously with the explicit full-space response surface.Analysis shows that the proposed FRSM provides an efficient approach for the multi-objective structural reliability analysis with correlated non-Gaussian random variables.(4)In order to overcome the disadvantages that the traditional Hermite response surface method(HRSM)is of low accuracy and efficiency for the structural stochastic analysis involving non-Gaussian variables,a generalized polynomial chaos response surface method(GRSM)for structural stochastic analysis involving non-gaussian variables was proposed.Firstly,the univariate generalized polynomial chaos corresponding to the probability distribution types of input random variables were determined,and then the target response was expanded by the hybrid generalized polynomial response surface which were constructed by the tensor product of the univariate generalized polynomial chaos.Then the hybrid probabilistic collocation points were constructed with the next-higher order univariate generalized polynomial chaos,and the optimal collocation points were determined based on the full row rank criterion of collocation point matrix.Finally,the unknown coefficients of generalized polynomial response surface were determined by solving the linear algebraic equations.Analysis results show the proposed GRSM is much more accuracy and efficiency than the HRSM and the Montel Carlo simulation(MCS)for the structural stochastic analysis,since it can expand the target response with low expansion order(i.e.,2rd-3th order).(5)In order to determine the structural reliability of the scaffold system under the influence nonlinear factor and random factor,an integrated approach for calculating the reliability of scaffold system was proposed,which combining the GRSM with nonlinear numerical analysis model and buckling analysis model of scaffold system.Based the beam-column method,a nonlinear numerical analysis model and a buckling analysis model of scaffold system considering the second-order effect and the nonlinear semi-rigid connection was proposed,then the reliability analysis and the global sensitivity analysis can be conducted by combining the GRSM and these models.The analysis shows the the proposed method can efficiently evaluate the probabilistic safety of scaffold system effected by the coupling of the stochastic parameter and the nonlinear factor;and the GRSM is of high computational efficiency to determine the reliability and the global sensitivity index of scaffold system. |
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