摘要
工程结构中存在很多缺口件,缺口件的疲劳寿命或强度控制着结构的疲劳寿命或强度,完善而准确的缺口件疲劳寿命分布信息是结构疲劳可靠性分析的基础。缺口件型式繁多、尺寸多样,通过试验获得其寿命分布的成本很大,而光滑件的疲劳寿命分布容易获得。因此通过光滑件疲劳寿命试验数据预测缺口件疲劳寿命分布具有非常重要的理论意义和工程实际价值。
缺口件疲劳寿命分散性有很多来源,本文主要考虑局部应力应变的分散性和材料微观结构的不均匀性,其中后者引起的寿命分散性可由光滑件疲劳寿命试验数据获得,局部应力应变的分散性是本文的主要研究内容。
首先,基于最弱环节理论并结合场强法的思想,提出了疲劳有效应力这一新的局部应力应变量,它能在概率意义上更好地反映缺口的疲劳严重程度,根据计算出的疲劳有效应力查取光滑件的疲劳寿命试验数据可以方便地得到缺口件疲劳寿命,该方法可以同时考虑到应力梯度和试件尺寸对缺口件疲劳寿命的影响。
其次,研究了基于随机有限元的局部应力应变分散性计算,包括四方面的内容。一是,利用弹性模量和硬度的联系,通过对两种金属材料的平板试件表面各点洛氏硬度值的测量,并以相关函数为工具对测量数据进行统计分析,完成了弹性模量随机场相关特性的测定。二是,研究了随机场离散的KL展开Galerkin法,并选择用单元形函数作为基函数来实现该方法,给出了随机场网格为四边形单元时的具体实现过程,编制了相应的算法程序。算例表明该程序离散精度很高,满足工程计算要求。三是,研究了基于变分原理的摄动随机有限元法,编制了二维平面应力问题的随机有限元计算程序,该程序的计算精度比较令人满意。四是,提出了一种求解缺口根部一点处应力概率分布的方法,该方法将影响应力概率分布的因素分为载荷和材料性能参数两部分,且材料性能参数部分可近似等效为一个正态随机变量,从而缺口根部一点处应力可表示为载荷随机变量和材料性能参数随机变量的线性组合,容易计算得到其分布函数,算例表明这种近似比较精确和有效。
最后,建立了基于随机有限元方法的缺口件疲劳寿命分布计算模型,该模型将由随机有限元方法计算得到的疲劳有效应力分散性同微观结构不均匀性引起的寿命分散性结合起来,最终计算出缺口件寿命分布。应用该模型进行了缺口件寿命分布对材料、载荷和几何特性三种不确定性设计变量的敏度分析,分析结果表明它们对缺口件寿命分布影响程度的大小顺序依次为载荷、几何和材料性能,并且大小在同一量级上。
There are numerous notched components in engineering structures. Their fatigue life or strengthdetermine that of structures. So complete and accurate data of fatigue life distribution of notchedspecimens is the basis for the fatigue reliability analysis of structures. Due to the variety of types anddimensions of notched specimens, the cost involved in getting their fatigue life distribution throughexperiments is very high. On the contrary, the cost for the fatigue life distribution of smoothspecimens is quite small. So the research in predicting the fatigue life distribution of notchedspecimens from fatigue life test data of smooth specimens has significant value in both theory andpractical engineering.
There are many factors that cause the randomness of fatigue life of notched specimens. Two ofthem are considered in this work. They are the dispersion of local stress and strain, and thenon-homogeneity of microstructure. The randomness of fatigue life induced by the latter can beobtained from the fatigue life test data of smooth specimens. So the research emphasis is placed onthe dispersion of local stress and strain.
In the beginning, based on the weakest-link theory and the stress field intensity approach, a newlocal stress and strain parameter, fatigue effective stress, is proposed. It can express the fatigueseverity of notch in the sense of probability. Employing this parameter to refer to the fatigue life testdata of smooth specimens would lead to the fatigue life of notched specimens. This model reflects theinfluence of both stress gradient and component size on the notched specimen fatigue life.
Then, the calculation for the dispersion of local stress and strain using the stochastic finite elementmethod(SFEM) is studied, including four parts. Firstly, based on the relation between the elasticmodulus and the hardness,the correlation characteristics of the elastic modulus random fields for twometals are determined, by experimental measurements of Rockwell hardness values at a series ofpoints on the specimen and a statistical analysis of the test result utilizing the correlation function.Secondly, the Galerkin-type procedure for Karhunen-Loève expansion of a random field isinvestigated. The shape function of the random field element is chosen to be the basis to implementthis procedure. Details regarding how to realize the procedure when the random field mesh iscomposed of quadrilaterals are given. The corresponding algorithmic routine is compiled. A exampleshows that the accuracy of this routine is high, meeting the requirement of engineering computation.Thirdly, the perturbation stochastic finite element method based on the variational principle is studied.According to the idea of this method, a stochastic finite element calculation program for two dimensional plane stress problem is developed. The precision of this program is moderately satisfying.Fourthly, A method for the probability distribution of the stress at a single point at notch root isproposed. In this method, the factors which affect the probability distribution of the stress areseparated into two parts, the load and the material. The contribution of the material is approximatelyequivalent to be a normal random variable. Then the stress can be expressed as a linear combinationof the random variable of the load and the random variable of the material. Thus the probabilitydistribution of the stress can be easily obtained. This treatment is accurate and effective.
Finally, the calculation model based on SFEM for fatigue life distribution of notched specimens isestablished. In this model, the dispersion of the fatigue effective stress worked out by SFEM isorganically combined with the randomness of fatigue life induced by the non-homogeneity ofmicrostructure to eventually obtain the fatigue life distribution of notched specimens. Making use ofthis calculation model, a sensitivity analysis for fatigue life distribution of notched specimens withregard to three types of design variables with uncertainty (material, load and geometry) is carried out.The result shows that the load has the largest influence on the fatigue life distribution of notchedspecimens, then the geometry, lastly the material. However, the values of sensitivity of these threefactors are at the same magnitude.
引文
[1]高镇同,熊峻江.疲劳可靠性.北京:北京航空航天大学出版社,2000.
[2] Cheung MS, Li WC. Probabilistic fatigue and fracture analyses of steel bridges. StructuralSafety,2003,25(3):245-262.
[3]姚卫星.结构疲劳寿命分析.北京:国防工业出版社,2003.
[4]李锋,孟广伟,周振平,等.结构疲劳寿命稳健性优化设计.机械工程学报,2010,46(2):155-158.
[5]魏建锋,郑修麟,吕宝桐. LY12CZ铝合金切口件的P-S-N曲线表达式及寿命估算.机械工程材料,1997,21(1):7-10.
[6]魏建锋,郑修麟,丁召荣.变幅载荷下疲劳寿命预测及其模拟结果.机械强度,1999,21(1):66-68.
[7] Zheng X, Li Z, Baotong L. Prediction of probability distribution of fatigue life of15MnVNsteel notched elements under variable-amplitude loading. International Journal of Fatigue,1996,18(2):81-86.
[8] Yan JH, Zheng XL, Zhao K. Prediction of fatigue life and its probability distribution of notchedfriction welded joints under variable-amplitude loading. International Journal of Fatigue,2000,22(6):481-494.
[9] Neuber H. Theory of Stress Concentration for Shear-Strained Prismatical Bodies With ArbitraryNonlinear Stress-Strain Law. Journal of Applied Mechanics,1961,28(4):544-550.
[10] Yu MT, DuQuesnay DL, Topper TH. Notch fatigue behaviour of SAE1045steel. InternationalJournal of Fatigue,1988,10(2):109-116.
[11] Ting JC, Jr FVL. A crack closure model for predicting the threshold stresses of notches. Fatigue&Fracture of Engineering Materials&Structures,1993,16(1):93-114.
[12]赵永翔,王金诺,高庆.概率循环应力-应变曲线及其估算方法.机械工程学报,2000,36(8):102-106.
[13] Zhao YX. A methodology for strain-based fatigue reliability analysis. Reliability Engineeringand System Safety,2000,70(2):205-213.
[14] J Z, J T, C WH. A reliability assessment method in strain-based fatigue life analysis. Journal ofPressure Vessel Technology ASME,1998,120:99-104.
[15] Zhao J, Tang J, Wu HC. A Generalized Random Variable Approach for Strain-Based FatigueReliability Analysis. Journal of Pressure Vessel Technology,2000,122(2):156-161.
[16] Wu Y-T, Wirsching PH. Advanced Reliability Method for Fatigue Analysis. Journal ofEngineering Mechanics,1984,110(4):536-553.
[17]唐俊星,陆山.某涡轮盘低循环疲劳概率寿命数值模拟.航空动力学报,2006,21(4):706-710.
[18]王立彬,靳慧,徐步青.概率局部应力应变法.工程力学,2003,20(4):188-191.
[19] Liao M, Shi G, Xiong Y. Analytical methodology for predicting fatigue life distribution offuselage splices. International Journal of Fatigue,2001,23(1001):177-185.
[20]孟广伟,李锋,周振平.疲劳裂纹扩展寿命的可靠性及灵敏度分析.计算力学学报,2007,24(6):876-879.
[21] Peterson RE. Notch Sensitivity. Sines G, Metal Fatigue, New York: McGraw-Hill,1959:293-306.
[22] Heywood RB. Stress Concentration Factor Relating Theoretical and Practical Factors in FatigueLoading. Engineering,1955,179:146-148.
[23] Buch A. Analytical approach to size and notch-size effects in fatigue of aircraft materialspecimens. Materials Science and Engineering,1974,15(1):75-85.
[24] Miller KJ. Metal Fatigue-Past, Current and Future. Proceedings of the Institution of MechanicalEngineers, Part C: Journal of Mechanical Engineering Science,1991,205(5):291-304.
[25] Sheppard SD. Field Effects in Fatigue Crack Initiation: Long Life Fatigue Strength. Journal ofMechanical Design,1991,113(2):188-194.
[26] Yao WX. Stress field intensity approach for predicting fatigue life. International Journal ofFatigue,1993,15(3):243-246.
[27]尚德广,王大康,李明,等.随机疲劳寿命预测的局部应力应变场强法.机械工程学报,2002,38(1):67-70.
[28] Der Kiureghian A, Ke J-B. The stochastic finite element method in structural reliability.Probabilistic Engineering Mechanics,1988,3(2):83-91.
[29] Popescu R, Deodatis G, Nobahar A. Effects of random heterogeneity of soil properties onbearing capacity. Probabilistic Engineering Mechanics,2005,20(4):324-341.
[30] Vanmarcke EH, Asce M. Probabilistic modeling of soil profiles. Journal of the GeotechnicalEngineering Division,1977:1227-1246.
[31]刘春原,阎澍旺.岩土参数随机场特性及线性预测.岩土工程学报,2002,24(5):588-591.
[32]闫澍旺,朱红霞,刘润.关于随机场理论在土工可靠度计算中应用的研究.岩土工程学报,2006,28(12):2053-2059.
[33]张征,刘淑春,鞠硕华.岩土参数空间变异性分析原理与最优估计模型.岩土工程学报,1996,18(4):40-47.
[34] Abrahamsen P. A review of Gaussian random fields and correlation functions. Technical Report917, Norwegian Computing Center, Oslo;,1997.
[35] Mabit L, Bernard C. Assessment of spatial distribution of fallout radionuclides throughgeostatistics concept. Journal of Environmental Radioactivity,2007,97(2-3):206-219.
[36] Qishlaqi A, Moore F, Forghani G. Characterization of metal pollution in soils under two landusepatterns in the Angouran region, NW Iran; a study based on multivariate data analysis. Journalof Hazardous Materials,2009,172(1):374-384.
[37] Rackwitz R. Reviewing probabilistic soils modelling. Computers and Geotechnics,2000,26(3-4):199-223.
[38] Meek DW. A semiparametric method for estimating the scale of fluctuation. Computers&Geosciences,2001,27(10):1243-1249.
[39] Vanmarcke E, Shinozuka M, Nakagiri S, et al. Random fields and stochastic finite elements.Structural Safety,1986,3:143-166.
[40] Schueller GI. A state-of-the-art report on computational stochastic mechanics. ProbabilisticEngineering Mechanics,1997,12(4):197-321.
[41] Sudret B, Der Kiureghian A. Stochastic finite element methods and reliability, A state-of-the-artreport. Berkeley: University of California,2000.
[42] Liu WK, Belytschko T, Mani A. Random field finite elements. International Journal forNumerical Methods in Engineering,1986,23(10):1831-1845.
[43] Liu WK, Belytschko T, Mani A. Probabilistic finite elements for nonlinear structural dynamics.Computer Methods in Applied Mechanics and Engineering,1986,56(1):61-81.
[44] Matthies HG, Brenner CE, Bucher CG, et al. Uncertainties in probabilistic numerical analysis ofstructures and solids-Stochastic finite elements. Structural Safety,1997,19(3):283-336.
[45] Li C-C, Der Kiureghian A. Optimal Discretization of Random Fields. Journal of EngineeringMechanics,1993,119(6):1136-1154.
[46] Qin Q, Lin DJ. Some improvements in reliability stochastic finite element methods.ICOSSAR'05-The9th international conference on structural safety and reliability Rome, Italy,2005.
[47] Vanmarcke E, Grigoriu M. Stochastic Finite Element Analysis of Simple Beams. Journal ofEngineering Mechanics,1983,109(5):1203-1214.
[48]朱位秋,任永坚.随机场的局部平均与随机有限元.航空学报,1986,7(6):604-611.
[49] Deodatis G. Weighted Integral Method. I: Stochastic Stiffness Matrix. Journal of EngineeringMechanics,1991,117(8):1851-1864.
[50] Deodatis G. Bounds on Response Variability of Stochastic Finite Element Systems. Journal ofEngineering Mechanics,1990,116(3):565-585.
[51] Takada T. Weighted integral method in stochastic finite element analysis. ProbabilisticEngineering Mechanics,1990,5(3):146-156.
[52] Takada T. Weighted integral method in multi-dimensional stochastic finite element analysis.Probabilistic Engineering Mechanics,1990,5(4):158-166.
[53] Ghanem RG, Spanos PD. Spectral stochastic finite element formulation for reliability analysis.Journal of Engineering Mechanics, ASCE,1991,117(10):2351-2372.
[54] Zhang J, Ellingwood B. Orthogonal Series Expansions of Random Fields in ReliabilityAnalysis. Journal of Engineering Mechanics,1994,120(12):2660-2677.
[55] Miroslav V. Simulation of simply cross correlated random fields by series expansion methods.Structural Safety,2008,30(4):337-363.
[56] Ghanem RG, Spanos PD. Stochastic finite elements: A spectral approach. New York: SpringerVerlag,1991.
[57]史良胜,杨金忠,李少龙,等.基于KL-Galerkin解法的地下水流动随机分析.四川大学学报(工程科学版),2005,37(5):31-35.
[58]史良胜,杨金忠,陈伏龙,等. Karhunen-Loeve展开在土性各向异性随机场模拟中的应用研究.岩土力学,2007,28(11):2303-2308.
[59] Xiu D, Em Karniadakis G. Modeling uncertainty in steady state diffusion problems viageneralized polynomial chaos. Computer Methods in Applied Mechanics and Engineering,2002,191(43):4927-4948.
[60]黄斌.一种新的谱随机有限元方法.武汉理工大学学报,2004,26(5):42-45.
[61]林道锦,秦权.有限元可靠度分析中随机场离散方法.清华大学学报(自然科学版),2002,42(6):839-842.
[62] Astill CJ, Imosseir SB, Shinozuka M. Impact Loading on Structures with Random Properties.Journal of Structural Mechanics,1972,1(1):63-77.
[63] Shinozuka M, Lenoe E. A probabilistic model for spatial distribution of material properties.Engineering Fracture Mechanics,1976,8(1):217-227.
[64] Cambou B. Application of first-order uncertainty analysis in the finite element method in linearelasticity.2nd International conference on applications of statistics and structural engineering,Aachen,1975:67-87.
[65] Dendrou BA, Houstis EN. An inference-finite element model for field problems. AppliedMathematical Modelling,1978,2(2):109-114.
[66] Beacher GB, Ingra TS. Stochastic FEM In Settlement Predictions. Journal of the GeotechnicalEngineering Division,1981,107(4):449-463.
[67] Handa K, Anderson K. Application of finite element methods in statistical analysis of structures.3rd international conference on structure safety and reliability, Trondheim, Norway,1981:409-417.
[68] Hisada T, Nakagiri S. Stochastic finite element method developed for structure safety andreliability.3rd International conference on structure safety and reliability, Trondheim, Norway,1981:395-408.
[69] Liu WK, Belytschko T, Mani A. Applications of Probabilistic Finite Element Methods inElastic/Plastic Dynamics. Journal of Engineering for Industry,1987,109(1):2-8.
[70] Liu WK, Besterfield G, Belytschko T. Transient probabilistic systems. Computer Methods inApplied Mechanics and Engineering,1988,67(1):27-54.
[71] Kleiber M, Antunez H, Hien TD, et al. Parameter sensitivity in nonlinear mechanics: theory andfinite element computations. New York: John Wiley&Sons,1997.
[72] Kaminski M. Generalized perturbation-based stochastic finite element method in elastostatics.Computers& Structures,2007,85(10):586-594.
[73] Hua XG, Ni YQ, Chen ZQ, et al. An improved perturbation method for stochastic finite elementmodel updating. International Journal for Numerical Methods in Engineering,2008,73(13):1845-1864.
[74] Shinozuka M, Deodatis G. Response Variability Of Stochastic Finite Element Systems. Journalof Engineering Mechanics,1988,114(3):499-519.
[75] Yamazaki F, Member A, Shinozuka M, et al. Neumann Expansion for Stochastic Finite ElementAnalysis. Journal of Engineering Mechanics,1988,114(8):1335-1354.
[76]杨杰,陈虬. Neumann随机有限元的一种推广形式.计算力学学报,2005,22(6):681-684.
[77] Papadrakakis M, Papadopoulos V. Robust and efficient methods for stochastic finite elementanalysis using Monte Carlo simulation. Computer Methods in Applied Mechanics andEngineering,1996,134(3-4):325-340.
[78] Papadopoulos V, Papadrakakis M. Stochastic finite element-based reliability analysis of spaceframes. Probabilistic Engineering Mechanics,1998,13(1):53-65.
[79] Deodatis G, Shinozuka M. Bounds on Response Variability of Stochastic Systems. Journal ofEngineering Mechanics,1989,115(11):2543-2563.
[80] Elishakoff I, Ren Y. The bird's eye view on finite element method for structures with largestochastic variations. Computer Methods in Applied Mechanics and Engineering,1999,168(1-2):51-61.
[81] Ghanem R, Brzakala W. Stochastic Finite-Element Analysis of Soil Layers with RandomInterface. Journal of Engineering Mechanics,1996,122(4):361-369.
[82] Ghanem RG. Stochastic finite elements with multiple random non-normal properties. Journal ofEngineering Mechanics, ASCE,1999,125(1):26-40.
[83] Chung DB, Gutierrez MA, Borst Rd. Object-oriented stochastic finite element analysis of fibremetal laminates. Computer Methods in Applied Mechanics and Engineering,2005,194(12-16):1427-1446.
[84] Ngah MF, Young A. Application of the spectral stochastic finite element method forperformance prediction of composite structures. Composite Structures,2007,78(3):447-456.
[85] Chen N-Z, Guedes Soares C. Spectral stochastic finite element analysis for laminatedcomposite plates. Computer Methods in Applied Mechanics and Engineering,2008,197(51-52):4830-4839.
[86]黄斌,张鹏.随机结构有限元分析的递推求解方法.计算力学学报,2005,22(6):767-770.
[87] Chung DB, Gutiérrez MA, Graham-Brady LL, et al. Efficient numerical strategies for spectralstochastic finite element models. International Journal for Numerical Methods in Engineering,2005,64(10):1334-1349.
[88] Ghanem RG, Kruger RM. Numerical solution of spectral stochastic finite element systems.Computer Methods in Applied Mechanics and Engineering,1996,129(3):289-303.
[89] Pellissetti MF, Ghanem RG. Iterative solution of systems of linear equations arising in thecontext of stochastic finite elements. Advances in Engineering Software,2000,31(8-9):607-616.
[90] Liu WK, Besterfield GH, Belytschko T. Variational Approach to Probabilistic Finite Elements.Journal of Engineering Mechanics,1988,114(12):2115-2133.
[91] Kleiber M, Hien TD. The stochastic finite element method-basic perturbation technique andcomputer implementation. New York: John Wiley&Sons,1992.
[92]张汝清,高行山.随机变量的变分原理及有限元法.应用数学和力学,1992,13(5):383-388.
[93] Ramakrishnan CV, Francavilla A. Structural Shape Optimization Using Penalty Functions.Journal of structural mechanics,1974,3(4):403-422.
[94] Wang SY, Sun YB, Gallagher RH. Sensitivity analysis in shape optimization of continuumstructures. Computers& Structures,1985,20(5):855-867.
[95] Pandey PC, Bakshi P. Analytical response sensitivity computation using hybrid finite elements.Computers& Structures,1999,71(5):525-534.
[96] Cheng G, Gu Y, Zhou Y. Accuracy of semi-analytic sensitivity analysis. Finite Elements inAnalysis and Design,1989,6(2):113-128.
[97] Pedersen P, Cheng GD, Rasmussen J. On Accuracy Problems for Semi-Analytical SensitivityAnalyses. Mechanics of Structures and Machines,1989,17(3):373-384.
[98] Cheng G, Olhoff N. Rigid body motion test against error in semi-analytical sensitivity analysis.Computers& Structures,1993,46(3):515-527.
[99] Olhoff N, Rasmussen J, Lund E. A Method of "Exact" Numerical Differentiation for ErrorElimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses. In:Mechanics of Structures and Machines, Vol.21: Taylor&Francis,1993:1-66.
[100] Haftka RT, Adelman HM. Recent developments in structural sensitivity analysis. Structural andMultidisciplinary Optimization,1989,1(3):137-151.
[101] Sergeyev O, Mroz Z. Sensitivity analysis and optimal design of3D frame structures for stressand frequency constraints. Computers&Structures,2000,75(2):167-185.
[102] Moita JS, Infante Barbosa J, Mota Soares CM, et al. Sensitivity analysis and optimal design ofgeometrically non-linear laminated plates and shells. Computers&Structures,2000,76(1-3):407-420.
[103] Parente E, Vaz LE. Improvement of semi-analytical design sensitivities of non-linear structuresusing equilibrium relations. International Journal for Numerical Methods in Engineering,2001,50(9):2127-2142.
[104] Liu C, Wang TL, Qin Q. Study on Sensitivity of Modal Parameters for Suspension Bridges.Structural Engineering and Mechanics,1999,8(5):453-464.
[105] Zhang QW, Chang TYP, Chang CC. Finite-Element Model Updating for the Kap Shui MunCable-Stayed Bridge. Journal of Bridge Engineering,2001,6(4):285-293.
[106] Song D, Chen S, Qiu Z. Stochastic sensitivity analysis of eigenvalues and eigenvectors.Computers& Structures,1995,54(5):891-896.
[107] Hien TD, Kleiber M. Stochastic structural design sensitivity of static response. Computers&Structures,1991,38(5-6):659-667.
[108] Hien TD, Kleiber M. Stochastic design sensitivity in structural dynamics. International Journalfor Numerical Methods in Engineering,1991,32(6):1247-1265.
[109] Bhattacharyya B, Chakraborty S. Sensitivity Statistics of3D Structures under ParametricUncertainty. Journal of Engineering Mechanics,2001,127(9):909-914.
[110]屠义强,江克斌,胡业平,等.基于随机有限元方法的结构响应灵敏度分析.解放军理工大学学报(自然科学版),2001,2(2):78-81.
[111]颜永年,俞新陆,郑楚鸿.疲劳绝对尺寸效应的统计分析.机械工程学报,1986,22(4):73-87.
[112]王少华.疲劳损伤场统计处理和疲劳缺口效应分析.西南交通大学学报,1992,6(6):19-26.
[113]陆山,张鸿,唐俊星,等.考虑尺寸效应的轮盘应力疲劳概率寿命分析方法.航空动力学报,2011,26(9):2039-2043.
[114] Makkonen M. Notch size effects in the fatigue limit of steel. International Journal of Fatigue,2003,25(1):17-26.
[115] Wormsen A, Sjodin B, Harkegard G, et al. Non-local stress approach for fatigue assessmentbased on weakest-link theory and statistics of extremes. Fatigue&Fracture of EngineeringMaterials&Structures,2007,30(12):1214-1227.
[116] Shepherd DP. The Prediction of Fatigue Life Distributions from the Analysis of Plain SpecimenData. Joumal of ASTM International,2004,1(8):30-45.
[117] Shirani M, Harkegard G. Fatigue life distribution and size effect in ductile cast iron for windturbine components. Engineering Failure Analysis,2011,18(1):12-24.
[118] Schweiger G, Heckel K. Size effect in randomly loaded specimens. International Journal ofFatigue,1986,8(4):231-234.
[119] Heckel K, K hler J, München T. Experimentelle Untersuchung des statistischenGr eneinflusses im Dauerschwingversuch an ungekerbten Stahlproben. Materialwissenschaftund Werkstofftechnik,1975,6(2):52-54.
[120] Bertsche B, Schropel H, Seifried A. Statistical size effect and failure-free time in dimensioningof machine components. Journal of Engineering Design,2006,17(3):259-270.
[121]姚卫星.金属材料疲劳行为的应力场强法描述.固体力学学报,1997,18(1):38-48.
[122] Makkonen M. Statistical size effect in the fatigue limit of steel. International Journal of Fatigue,2001,23(5):395-402.
[123] Norberg S, Olsson M. The effect of loaded volume and stress gradient on the fatigue limit.International Journal of Fatigue,2007,29(12):2259-2272.
[124]高镇同,傅惠民.疲劳强度的频率分布和疲劳强度特征函数.机械强度,1985,(2):27-34.
[125] Wang Y-Y, Yao W-X. Evaluation and comparison of several multiaxial fatigue criteria.International Journal of Fatigue,2004,26(1):17-25.
[126]高镇同等.疲劳性能试验设计和数据处理:直升机金属材料疲劳性能可靠性手册.北京:北京航空航天大学出版社,1999.
[127] Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by instrumentedindentation: Advances in understanding and refinements to methodology. Journal of MaterialsResearch,2004,19(1):3-20.
[128] Oliver WC, Pharr GM. An Improved technique for determining hardness and elastic modulususing load and displacement sensing indentation experiments. Journal of Materials Research,1992,7(6):1564-1583.
[129] Bao YW, Wang W, Zhou YC. Investigation of the relationship between elastic modulus andhardness based on depth-sensing indentation measurements. Acta Materialia,2004,52(18):5397-5404.
[130] Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesqproblem for a punch of arbitrary profile. International Journal of Engineering Science,1965,3(1):47-56.
[131] Pharr GM, Oliver WC, Brotzen FR. On the generality of the relationship among contactstiffness, contact area, and elastic modulus during indentation. Journal of Materials Research,1992,7(3):613-617.
[132]盛骤,谢式千,潘承毅.概率论与数理统计.北京:高等教育出版社,1989.
[133]闰澎旺,贾晓黎,郭怀志,等.土性剖面随机场模型的平稳性和各态历经性验证.岩土工程学报,1995,17(3):1-9.
[134]王建锋.用地质统计学计算土层相关尺度.土木工程学报,2005,38(10):102-107.
[135]孙洪泉.地质统计学及其应用.徐州:中国矿业大学出版社,1990.
[136] Jankowski R, Wilde K. A simple method of conditional random field simulation of groundmotions for long structures. Engineering Structures,2000,22(5):552-561.
[137] Liu XM, Xu JM, Zhang MK, et al. Application of geostatistics and GIS technique tocharacterize spatial variabilities of bioavailable micronutrients in paddy soils. EnvironmentalGeology,2004,46(2):189-194.
[138]陈虬,刘先斌.随机有限元法及其工程应用.成都:西南交通大学出版社,1993.
[139]苏成,徐瑞,范学明.二维随机场离散的曲边单元局部平均法.华南理工大学学报(自然科学版),2008,36(3):104-107.
[140] Lawrence MA. Basis random variables in finite element analysis. International Journal forNumerical Methods in Engineering,1987,24(10):1849-1863.
[141]秦权,林道锦,梅刚.结构可靠度随机有限元:理论及工程应用.北京:清华大学出版社,2006.
[142] Jordan DW, Smith P. Nonlinear ordinary differential equations. Oxford: Oxford university press,1977.
[143] Nayfeh AH. perturbation methods. New York: John Wiley&Sons,1973.
[144] Shinozuka M, Jan CM. Digital simulation of random processes and its applications. Journal ofSound and Vibration,1972,25(1):111-128.
[145] Huang SP. A collocation-based spectral stochastic finite element analysis-stochastic responsesurface approach. Chinese Journal of Computational Mechanics,2007,24(2):173-180.
[146]邓建.结构可靠性分析的多项式数值逼近法.计算力学学报,2002,19(2):212-216.
[147]南宫自军.结构可靠性计算的一种新方法.西北工业大学学报,1998,16(4):599-603.
[148] Helton JC, Davis FJ, Johnson JD. A comparison of uncertainty and sensitivity analysis resultsobtained with random and Latin hypercube sampling. Reliability Engineering& SystemSafety,2005,89(3):305-330.
[149]吴学仁.飞机结构金属材料力学性能手册第1卷.北京:航空工业出版社,1997.
[150]姚卫星,顾怡.结构可靠性设计.北京:航空工业出版社,1997.
[151]张伟.结构可靠性理论与应用.北京:科学出版社,2008.
[152] Haug EJ, Arora JS.实用最优设计:机械系统与结构系统(郁永熙).北京:科学出版社,1985.
[153]张旭明,王德信.结构灵敏度分析的解析方法.河海大学学报,1998,26(5):44-49.