径向基函数随机响应面法

针对随机响应面法对非正态分布响应与标准正态分布输入之间的复杂非线性隐函数拟合不够理想的问题,基于径向基函数在杂散数据拟合方面的优异性能,提出使用径向基函数替换Hermite多项式来解决复杂非线性隐函数拟合问题。以若干个非线性解析函数和钢管混凝土肋拱极限承载力不确定性问题作为算例,验证该方法对非正态分布响应拟合的精确性和对工程问题的适用性。算例结果表明,基于径向基函数随机响应面法对高度非线性的响应与输入隐函数拟合较好;在多参数钢管混凝土拱极限承载力不确定性问题中,精度较高,且比Hermite多项式样本点数量少。     

边坡可靠度分析的随机响应面法及程序实现

提出分析相关非正态变量可靠度计算问题的随机响应面法,采用Nataf变换成功地解决输入变量相关时随机响应面法的配点问题及可靠度计算问题。推导4～6阶Hermite随机多项式展开的解析表达式,并编写基于C#语言的随机响应面法计算程序。以岩质边坡平面滑动破坏模式为例证明随机响应面法在边坡可靠度分析中的有效性。研究结果表明,基于Nataf变换的随机响应面法能够有效分析含有相关非正态变量的边坡可靠度问题。随机响应面法的计算精度优于传统的FORM方法,其计算效率高于传统的蒙特卡罗模拟方法,其收敛性在数学意义上是有保证的。随机多项式展开的阶数几乎对边坡安全系数均值的估计没有影响,但是在边坡失效概率的计算中要选择适当的随机多项式展开的阶数。在基于随机响应面法的可靠度分析框架内,边坡安全系数计算和可靠度分析2个过程分开独立进行,同时计算安全系数和失效概率能够更加系统地进行边坡稳定性分析。研究成果为拓展随机响应面法在边坡可靠度分析中的应用奠定了一定的基础。     

Random analysis of geometrically non-linear FE modelled structures under seismic actions

In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Itô's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal similarity. By using this similarity a very effective computational procedure for evaluating response moments of any order is proposed.

Within the framework of non-Gaussian closure schemes, a technique is here presented based on a truncated Gram-Charlie expansion. To achieve this the Hermite coefficients are evaluated for multi-degree-of-freedom systems, once the multi-dimensional Hermite polynomials have been obtained in compact form.     

Zur probabilistischen Sicherheitsanalyse von Stabtragwerken nach der Theorie 2. Ordnung

Probability‐based safety analysis of bar structures according to theory of 2nd order. The probabilistic dimensioning method for bar structures follows from the deformation method according to the theory of 2nd order and maintains the practice based on the equilibrium of forces and torques. The approximation of random loads by Hermite polynomials of normal random variables leads to a linear equation system by applying a Galerkin projection. This is based on the elasticity equations of the deterministic deformation method and has to be treated iteratively. The method can be formalised and provides the failure probability, requiring reasonable efforts in the usual professional practice of a civil engineer only. By means of the cross‐section failure, it is demonstrated in which respect the probabilistic analysis expands the safety concept based on safety factors.     

结构可靠度分析的向量型层递响应面法

结构可靠度分析的传统响应面法和随机响应面法都属于标量型响应面法,为克服其局限性,利用预处理Krylov子空间法研究建立了结构可靠度分析的向量型层递响应面法。首先将随机刚度矩阵和节点荷载向量线性展开,通过选取合适的预处理器建立了预处理Krylov子空间及层递基向量,据此将随机节点位移向量进行层递展开,形成向量型的层递响应面。进而,利用混沌多项式展开层递基向量和层递响应面,根据混沌多项式确定层递响应面的样本点选取方法。在此基础上,根据可靠指标的几何含义,利用响应量的层递展开式建立了结构可靠度分析的迭代算法。算例分析表明,该方法与传统响应面法或随机响应面法相比,能够取得更高的计算效率和计算精度。     

围护结构非高斯风压极值估计的改进Hermite峰值因子法

为克服传统研究风压极值方法忽略带宽参数影响的不足和改进概率密度函数拟合效果，结合曲面拟合方法获得一种改进Hermite级数，提出了引入带宽参数的改进Hermite峰值因子法。采用改进Hermite峰值因子法和以往常用方法对开孔屋盖风压的峰值因子进行研究。结果表明：基于高斯分布的Davenport的峰值因子明显偏离非高斯峰值因子；忽略带宽参数的传统Hermite峰值因子法高估了风压的正负峰值因子；改进Hermite峰值因子法相应于传统Hermite方法和修正Hermite方法和Sadek-Simiu方法，估计的开孔屋盖风压的峰值因子最为安全准确，与观测峰值因子总体上最为接近。     

Peak factor estimation method of non-Gaussian wind pressures on long-span tri-centered cylindrical roof structures

Claddings are susceptible to damage due to underestimation of extreme wind-induced surface pressures. Commonly accepted methods for estimating the peak factor (an input used to determine cladding design loads) involve complex calculation-intensive procedures. This research develops a four-parameter unified auto-spectral model of wind pressure to simplify peak factor estimation of wind-induced surface pressure via analysis of wind tunnel wind load data on tri-centered cylindrical roofs. Values of the model parameters were identified via statistical analysis of wind tunnel wind pressure measurement on two long-span tri-centered cylindrical roof structures with different curvatures. The study identified roof regions with non-Gaussian features by inspecting probabilistic density functions of the standardized wind-induced roof pressures and the third- and fourth-order statistical moments of wind pressure time histories. The paper ultimately proposed and evaluated a simplified method for estimating the peak factors in the non-Gaussian regions, the Three-parameter Hermite Model, derived through the moment-based Hermite Model, the Revised Hermite Model, and the parameter simplification accomplished in this study. The results show that the auto-spectral model of wind-induced roof pressures can accurately estimate the zero- and second-order spectral moments, which reflects the wind pressure fluctuating characteristics and geometric features of spectral curves. Compared with the peak factors of the moment-based Hermite Model and the Revised Hermite Model, the peak factor errors estimated by the Three-parameter Hermite Model are all less than 10%. These results suggest that the Three-parameter Hermite Model simplifies the calculation with acceptable accuracy.     

非高斯风压时程峰值因子的简化计算式

采用Hermite矩模型可将非高斯时程表示为高斯时程的非线性函数,建立了非高斯时程和高斯时程之间的一一对应关系,也建立了非高斯峰值因子和高斯峰值因子之间的一一对应关系,为非高斯峰值因子、极值的计算奠定了理论基础。介绍了软化时程、硬化时程和偏斜时程的Hermite矩模型变换理论,明确了高阶矩模型的单调变换范围;在此基础上,研究了软化时程非高斯峰值因子简化计算式的理论误差。结果表明由简化计算式得到的非高斯峰值因子略大,其误差均小于20%。利用非高斯峰值因子的简化计算式,计算了平屋盖表面典型测压点的非高斯峰值因子和风压极值。分析结果表明：绝大多数测压时程样本属于软化时程,极少数样本属于硬化时程或偏斜时程;利用非高斯峰值因子的简化计算式,需要考虑测压时程的随机特性,取多个时程样本峰值因子的平均值作为非高斯峰值因子的代表值。     

Probabilistic analysis of consolidation that considers spatial variability using the stochastic response surface method

To obtain more accurate and reasonable results in the analyses of soil consolidation, the spatial variability of the soil properties should be considered. In this study, we analyzed the consolidation by vertical drains for soil improvement considering the spatial variability of the coefficients of consolidation. The coefficients for the variation in the vertical and horizontal coefficients of consolidation in Yeonjongdo, South Korea were evaluated, and the probability density function (PDF) was assumed by the Anderson–Darling goodness-of-fit test. Standard Gaussian random fields were generated based on a Karhunen–Loeve expansion, and then transformed using Hermite polynomials in the random field with the log-Gaussian PDF of the coefficient of consolidation. The average degree of consolidation was subsequently calculated using the finite difference method coupled with log-Gaussian random fields. In addition, the stochastic response surface method (SRSM) was applied for the efficient probabilistic uncertainty propagation. A sensitivity analysis was performed for the input parameters of the random field, and the spatial variability was considered using random variables from the Karhunen–Loeve expansion as the input data for the SRSM. The results indicated that when considering the spatial variability of soil properties, the probability of failure for the target degree of consolidation was smaller when the correlation distance was taken into account than when it was not. Additionally, the probability of failure decreased when the correlation distance decreased. Compared with the Monte Carlo simulation (MCS) results, the SRSM analysis can achieve results of similar accuracy to those obtained using the MCS analysis with a sample size of 100,000 (numerical runs), and a third-order SRSM expansion with only 333 numerical runs is sufficient for obtaining the probability with errors less than 0.01.     

随机结构稳定性的有效分析方法

利用非正交多项式混沌展式表达特征值和特征向量,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法求解了这些递推方程,得到了特征值的统计值。最后,用算例验证了本文方法的正确性。     

渗析法确定土水特征曲线的试验研究

土水特征曲线(SWCC)描述的是土体基质吸力与含水率之间的关系,它对于非饱和土的研究具有重要意义。在测定SWCC的各种方法中,渗析法的试验过程较简单、安全且可靠。本文以合肥地区非饱和膨胀土为研究对象,详细介绍了渗析法获得SWCC的试验过程和技术要领,并采用水浴恒温振荡器代替磁力搅拌器改进了渗析试验。结果表明:采用水浴恒温振荡器进行渗析试验是合理可行的,其优点是一次可以进行相同环境条件下多个浓度的渗析试验;使用水浴恒温振荡器对合肥地区膨胀土进行渗析试验时,其渗析平衡时间约为8天。在此基础上,采用Fredlund-Xing模型的3种型式对渗析试验结果进行了曲线拟合,得到了试验土体的SWCC及其拟合参数,可知模型3种型式的区别主要在高吸力部分。     

广义m法及在基坑支护结构计算中的应用

在基础设计中有熟悉的文克尔弹性地基模型,在桩水平力计算及支护设计中有m法,如考虑多层土层复杂情况,m法则无法求解,本文将此类问题归入广义m法,利用两结点Hermite单元形函数推导其刚度矩阵。此法便于编制计算机程序,已用于弹性地基梁、桩水平力计算、支护结构三维有限元结构计算     

高程异常曲面拟合及规则格网内插研究

介绍了部分高程异常拟合的方法,结合一个具体的实例,在2 km×2 km的小范围内采用多项式拟合的方法进行高程异常曲面的逼近,运用GPS水准获得的数据进行拟合检查,通过分析,应该采用五参数拟合的方法,并取得了3.8 cm的精度;并据此拟合多项式进行高程异常规则格网内插分析,运用MATLAB编写程序对几种插值方法进行讨论,最后得到一些实用的结论.     

土的液限塑限联合测定法的改进

按100g锥（锥角30°）进行土的液限塑限联合试验（或称联合测定）时,关键的问题是如何首先准确地确定塑限入土深度hp,然后才好从联合测定图（双对数座标）中的h-ω直线（锥入深度与含水量的关系）上读出相应于该土类的塑限值ωρ.鉴于hp是个变值,随土质而异,粉质土和粘土质土的hp值变化幅度很小,而亚砂土（或称砂质土）的hp值却变化很大,因而更关键的问题是如何准确地确定亚砂土的hp值。实践表明,对于亚砂土,用双曲线法求得的塑限值,与常规搓条法相比,稍感偏低,但根据一百多个亚砂土样品试验结果的验证,用正交三次多项式（简称三次多项式）曲线或组合曲线（由双曲线与三次多项式曲线配成）求得的塑限值则接近于搓条法塑限值。对于粉质土和粘土质土,以采用双曲线法比较合适。这样,针对不同土质情况,分别采用不同的曲线,将使测得的塑限值更符合实际。因此,三次多项式曲线的出现,是对联合测定法的进一步改进。     

High-order limit state functions in the response surface method for structural reliability analysis

The stochastic response surface method (SRSM) is a technique for the reliability analysis of complex systems with low failure probabilities, for which Monte Carlo simulation (MCS) is too computationally intensive and for which approximate methods are inaccurate. Typically, the SRSM approximates a limit state function with a multi-dimensional quadratic polynomial by fitting the polynomial to a number of sampling points from the limit state function. This method can give biased approximations of the failure probability for cases in which the quadratic response surface can not conform to the true limit state function’s nonlinearities. In contrast to recently proposed algorithms which focus on the positions of sample points to improve the accuracy of the quadratic SRSM, this paper describes the use of higher order polynomials in order to approximate the true limit state more accurately. The use of higher order polynomials has received relatively little attention to date because of problems associated with ill-conditioned systems of equations and an approximated limit state which is very inaccurate outside the domain of the sample points. To address these problems, an algorithm using orthogonal polynomials is proposed to determine the necessary polynomial orders. Four numerical examples compare the proposed algorithm with the conventional quadratic polynomial SRSM and a detailed MCS.     

Variance of the fatigue damage due to a Gaussian narrowband process

The fatigue life of a structure is inherently random when the loading is irregular. There is a vast body of literature on the analysis of the mean fatigue damage, but very few studies have been devoted to the variance. This paper presents an improved method for analyzing the variance of the damage for any narrowband Gaussian process. The field of application is not confined to the linear oscillator, unlike approaches in previous studies. The method is simple to apply; it involves a single summation for arbitrary processes, and closed form solutions are available for special cases (linear oscillator and bandpass process). The effectiveness of the method is demonstrated through case studies encompassing a variety of systems, including a realistic spectrum commonly seen in offshore engineering. Using rainflow counting of simulated time domain stresses as a benchmark, the proposed method is shown to yield a highly precise prediction of the variance. Moreover, it is significantly more accurate than an existing method for the linear oscillator system.     

Peak factors of non‐Gaussian wind forces on a complex‐shaped tall building

This paper focuses on the development of peak factor formulas of non‐Gaussian wind pressure processes after reviewing the current estimation methods of non‐Gaussian peak factors. A skewness‐dependent peak factor is proposed by accounting for the contribution of skewness and kurtosis parameters in some existing Hermite moment‐based formulas. The possible correction on the upcrossing rate used in the translation process approach is also investigated. Wind tunnel pressure data on a practical 43‐story building with unusual shape is used to validate the accuracy of the skewness‐dependent peak factor by investigating various statistical properties of wind‐induced fluctuating pressure field on the complex‐shaped tall building example. Copyright © 2012 John Wiley & Sons, Ltd.     

Buckling and geometrically nonlinear analysis of frames using one element/member

The commonly used Hermite cubic element represents the solution for the linear differential equilibrium equation of a straight beam under the Bernoulli hypothesis. However, this element grossly over-estimates the buckling load of a strut when only a single element is used to model a member. This paper presents a simple higher-order element capable of predicting the buckling load of a strut with various boundary conditions by using only a single element for each member. It is found that a direct introduction of a displacement degree of freedom to the mid-length of the cubic element will improve significantly its performance in a nonlinear analysis. The computational efficiency and formulation of this new element are very similar to those for the cubic element in spite of their discrepancies in accuracy.     

On using Legendre polynomials and amended spline transformations in the SFSM for buckling and free vibrations of plates and thin-walled beams

This paper presents two modifications to the spline finite strip method (SFSM) for plate and thin-walled beam buckling and free vibrations; these being augmenting of the flexural buckling displacements with Legendre polynomials which represent additional bubble modes, and a modification of the dedicated amended splines that are conventionally needed to handle boundary conditions. A simple matrix technique to remove the need for amended splines is described, and the efficacy of the method is demonstrated in which it is shown that inclusion of the Legendre polynomials enhances the performance of the SFSM significantly when benchmarked against other techniques.     

基于最小二乘近似构造流形方法的物理覆盖函数

流形方法可以在保持计算网格不变的前提下,通过在物理覆盖上采用任意形式的解析解级数或各阶多项式来提高数值解的精度,具有许多独特的优势。目前通常采用的、在物理覆盖上使用0阶到多阶多项式作为覆盖函数的方法,不但大幅度地增加了分析问题的总未知量数,而且给边界条件的处理带来了较多的困难。基于最小二乘近似方法来构造流形方法的物理覆盖位移函数,可以在保持总的未知量数不变的情况下构造出满足δij插值条件的高次流形单元,且可以象有限单元法一样方便地施加边界条件,克服了流形方法的上述缺点,为流形单元位移插值函数的构造提供了一条新的途径。悬臂梁、RD梁等算例验证了本文理论与方法的正确性和计算效率。