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中國礦業(yè)大學2008屆本科生畢業(yè)設(shè)計 第11頁
英文原文
Optimize the reliability of mechanical structure design
It is now generally recognized that structural and mechanical problems are nondeterministic and, consequently, engineering optimum design must cope with un-certainties，Reliability technology provides tools for formal assessment and analysis of such uncertainties，Thus, the combination of reliability-based design procedure sand optimization promises to provide a practical optimum design solution, i，e，, a de-sign having an optimum balance between cost and risk， However, reliabilty-based structural optimization programs have not enjoyed the name popularity as their deterministic counterparts， Some reasons for this are suggested， First, reliability analysis can be complicated even for simple systems， There are various methods for handling the uncertainty in similar situations (e，g，, first order second moment methods, full distribution methods)， Lacking a single method, individuals are likely to adopt separate strategies for handling the uncertainty in their particular problems， This suggests the possibility of different reliability predictions in similar structural design situations， Then, there are diverging opinions on many basic issues, from the very definition of reliability-based optimization, including the definition of the optimum solution, the objective function and the constraints, to its application in structural design practice， There is a need to formally consider these itess in the merger of present structural optimization research with reliability-based design philosophy。
In general, an optimization problem can be stated as follows,Minimize
subject to the constraints
where X is an-dimensional vector called the design vector， f(X) is called the objective function and, "k(X) and }i(X) are, respectively, the inequality and equality constraints， The number of variables n and the number of constraints， L need not be related in any way， Thus, L could be less than, equal to or greater than n in a given mathematical programming problem， In some problems, the value of L might be zero which means there are no constraints on the problem， Such type of problems are called "unconstrained" optimization problems， Those problems for which L is not equal to zero are known as "constrained" optimization problems。
Traditionally the designer assumes the loading on an element and the strength of that element to be a single valued characteristic or design value， Perhaps it is equal to some maximum (or minimum) anticipated or nominal value， Safety is assured by introducing a factor of safety, greater than one, usually applied as a reduction factor to strength。
Probabilistic design is propose: as an alternative to the conventional approach with the promise of producing "better engineered" systems， each factor in the design process can be defined and treated as a random variable， Using method-ology from probabilistic theory, the designer defines the appropriate limit state and computes the probability of failure P} of the element， The basic design requirement is that，where p f is the maximum allowable probability of failure。
Advantages of adopting the probabilistic design approach are well documented (Wu, 1984)， Basically the arguments for probabilistic design center around the fact that, relative to the conventional approach, a) risk is a more meaningful index of structural performance, and b) a reliability approach to design of a sys-tom can tend to produce an "optimum" design by ensuring a uniform risk in all components。
Optimization, which may be considered a component of operations research, is the process of obtaining the best result by finding conditions that produce the maximum or minimum value of a function， Table 1，1 illustrates area of operations research。
Mathematical programming techniques, also known as optimization methods, are useful in finding the minimum (or maximum) of a function of several variables under a prescribed set of constraints， Rao (1979) presented a definition and description of some of the various methods of mathematical programming， Stochas-tic process techniques can be used to analyze problems which are described by a set of random variables， Statistical methods enable one to analyze the experimental data and build empirical models to obtain the most accurate representations of physical behavior。
Origins of optimization theory can be traced to the days of Newton, La-grange and Cauchy in the 1800'x， The application of differential calculus to optimization was possible because of the contributions of Newton and Leibnitz， The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weirstrass， The method of optimization for constrained problems, which involves the addition of unknown multipliers became known by the name its inventor, La-grange， Cauchy presented the first application of the steepest descent method to solve minimization problems。
In spite of these early contributions, very little progress was made until the middle of the twentieth Gentry, when high-speed digital computers made the implementation of optimization procedures possible and stimulate, d further research on new methods， Spectacular advances followed, producing a m;}sssive literature on optimization techniques， This advancement also resulted in the emergence of several well-defined new areas in optimization theory。
It is interesting to note that major developments in the area of numerical methods of unconstrained optimization have been made in the TTnited Kingdom only in the 1960'x， The development of the simplex method by Dantzig (1947) for linear programming and the annunciation of the principle of optimality by Bellman (195?) for dynamic programming problems paved the wa，; f}= development of the methods of constrained optimization， The work by Kuhn and Tucker (1951) on necessary and xuflicient conditions for the optimal xolution of programming problems laid foundations for later research in nonlinear programming, the optimization area of this thesis。
Although no single technique has been found to be universally applicable for nonlinear programming, the works by Cacrol (1961)and Fiacco and McCormic (1968) suggested practical solutions by employing well-known techniques of uncon xtrained optimization， Geometric programming was developed by Dufhn, Zener and Peterson (1960)， Gomory (1963) pioneered work in integer programming, which is at this time an exciting and rapidly developing area of optimization research， Many "real-world" applications can be cast in this category of problem， Dantzig (1955) and Charnel and Cooper (1959) developed stochastic programming techniques and solved problems by assuming design parameters to be independent and normally distributed。
Techniques of nonlinear programming, employed in this study, can be categorized
1， one-dimensional minimization method
2， unconstrained multivariable minimization
A， gradient based method
B， nongradient based method
3， constrained multivariable minimization
A， gradient based method
B，gradient based method
The gradient based methods require function and derivative evaluations while the non gradient based methods require function evaluations only， In general, one would expect the gradient methods to be more effecti;re, due to the added information provided， However, if analytical derivatives are available, the question of whether a search technique should be used at all is presented， If numerical derivative approximations are utilized, the efficiency of the gradient based methods should be approximately the same as that of nongradient based methods， Gradient based methods incorporating numerical derivatives would be expected to present some numerical problems in the vicinity of the optimum, i，e，, approximations to slopes would become small， Fig， 1，1 shows the $ow chart of general iterative scheme of optimization (Rao, 1979)，
No claim is made that some methods are better than any others， A method works well on one problem may perform very poorly on another problem of the same general type， Only after much experience using all the methods can one judge which method would be better for a particular problem (Kuester snd Mize, 1973).
First attempts to apply probabilistic and statistical concepts in structural analysis date back to the beginning of this century， However, the subject aid not receive much attention until after the World War II， In October 1945, a historic paper written by A， M， Freudenthal entitled "The Safety of Structures" appeared in the proceedings of the American Society of Civil Engineers， The publication
of this paper marked the genesis of structural reliability in the U，S，A，， Professor F:eudenthal continued for many years to be in the forefront of structural reliability and risk analysis，
During the 1960's there was rapid growth of academic interest in struc-total reliability theory， Classical theory became well developed and widely known through a few influential publications such as that of Freudenthal, Garrelts, and Shi-nouzuka (1966), Pugsley (1966), Kececioglu and Cormier (1964), Ferry-Borges and Castenheta (1971, and Haugen (1968)， However, professional acceptance was low for several reasons， Probabilistic design seemed cumbersome, the theory, particularly system analysis, seemed mathematically intractible， Little data were available, and modeling error was an issue which needed to be addressed，
But there were early efforts to circumvent these limitations， Turkstra(l070) Yr}}nted structural design as a problem of decision making under uncertainty and risk， Lind, Turkstra, and Wright (1965) defined the problem of rational design of a code as finding a set of best values of the load and resistance factors， Cornell (1967) suggested the use of a second moment format, and subsequently it was demonstrated that Cornell's safety index requirement could be used to derive a set of safety factors on loads and resistances， This approach related reliability analysis to practically accepted methods of design The Cornell approach has been refined and employed in many structural standards，
Difficulties with the second moment format were uncovered 1969 when Ditlevsen and Lind independently discovered the problem of invariance， Cornell's index was not constant when certain simple problems were reformulated in a mechanically equivalent way， But the lack of invariance dilemma was overcome when Hasofer and Lind (1974) defined a generalized safety index which was invariant to mechanical formulation， This landmark paper represented a turning point in structural reliability theory， More sophisticated extensions of the Hasofer-Lind approach proposed in recent years by Rackwitz and Fiessler (1978), Chen and Lind (1982), and Wu (1984) provide accurate probability of failure estimates for complicated limit state functions，
There are many modes of failure in structural systems, depending on the configuration of the system, shapes and materials of the members, the loading conditions, etc， Lz order to perform a system reliability assessment the failure modes must be defined， However, for a large system with a high degree of redundancy it is difficult in practice to determine a priori which failure modes are probabilistically significant， The following methods have been proposed to produce approximate solutions: (a) automatic generation of safety margins, (b) the p-unzipping method, and (c) branch-and-bound method (Thoft-Christensen and Murotsu, 1986)， The state of the art in sysiems structural reliability analysis is comprehended in the works vi Bennett (1983), Ang and Tang (1984), Guenard ('1984), Ditlevsen (1986), Madsen, Krenk, and Lind (1986), and Thoft-Christensen and Murotsu (1986)， Butat this time there no general me hon for obtaining practical solutions to the system reliability problem。
中文翻譯
機械結(jié)構(gòu)的可靠性優(yōu)化設(shè)計
實際情況表明，所有的機械結(jié)構(gòu)都具有很大的不確定性，因而使得機械工程學上的優(yōu)化設(shè)計也具有不確定性，在不同的條件下所得到的優(yōu)化結(jié)果是不一樣的?？煽啃栽O(shè)計就為這種不確定性問題提供了一個很好的比較正式的評價和分析工具。因此，這種結(jié)合了可靠性設(shè)計過程的優(yōu)化設(shè)計方法就有很大實用性，也就是說，這種優(yōu)化結(jié)果同時考慮到了機構(gòu)的最優(yōu)和最安全可靠兩個設(shè)計要求。然而，基于可靠性的優(yōu)化設(shè)計方法并沒有得到廣泛地應用。造成這種情況的原因有很多方面。首先，即使對于簡單的系統(tǒng)，進行可靠性分析工作則是非常復雜的，對于比較相似的情況（例如采用完全分布法和主要分布法）有很多不同的方法用來處理這種不確定性。缺乏一種簡單的、獨立的可以相互適用的方法用來處理各種不同問題的不確定性。這就使得對于不同的但非常相似的機構(gòu)設(shè)計具有不同的可靠性設(shè)計結(jié)果。還有，在關(guān)于可靠性概念的許多基本問題上面，還存在這一些分歧，這些分歧包括了關(guān)于可靠性優(yōu)化的基本定義、優(yōu)化結(jié)果、目標函數(shù)和約束條件已經(jīng)它在結(jié)構(gòu)設(shè)計中的實際應用等多個方面?？傊?，有必要對可靠性結(jié)構(gòu)優(yōu)化設(shè)計體系進行一個系統(tǒng)的正式的研究。
一般的，一個結(jié)構(gòu)優(yōu)化問題可以歸結(jié)為如下的數(shù)學模型：
對目標函數(shù)的最小化：
（1－1）
約束條件：
（1，2，…，K） （1－2）
（，，…，L） （1－3）
其中，X是一個n維的設(shè)計變量，是目標函數(shù)，而和分別是不等式約束條件和等式約束條件。變量n的個數(shù)和約束條件L的個數(shù)與具體的應用情況有關(guān)系，不是確定不變的。因此，在一個給定的數(shù)學模型中，約束條件L的個數(shù)可能大于、等于或者小于設(shè)計變量n的個數(shù)。在一些實際問題中，約束條件L的個數(shù)為零，則表示這些問題沒有約束條件，并稱該類問題為無約束優(yōu)化問題。而另外的一些約束條件不為零的問題則稱為約束優(yōu)化問題。
在常規(guī)的設(shè)計中，認為外載荷是作用在零件上的固定單元上面，并且認為該單元的強度是一個單一的固定值或設(shè)計值。這個強度值可能是該類材料的預期或許用值的最大（最小）值。并且用安全系數(shù)來衡量零件的安全性，安全系數(shù)一般要大于一個推薦值。
采用概率設(shè)計可以在傳統(tǒng)的設(shè)計方法的基礎(chǔ)上得到相對最好的設(shè)計結(jié)果，概率設(shè)計也就是這樣的一種方法。在概率設(shè)計中，設(shè)計過程中的每個因素都需要定義并被當作隨機變量處理。利用概率論的基本方法原理，定義一個合適的限制條件并計算此條件下單元的失效概率。并且要求，其中，是最大的許用失效概率。
采用概率設(shè)計方法有許多的優(yōu)點，并且在很多的書本里面都有相關(guān)的介紹和描述（Wu，1984）。而關(guān)于概率設(shè)計方法和傳統(tǒng)設(shè)計方法的基本爭論主要有以下兩點：a）零件結(jié)構(gòu)性能方面產(chǎn)生危險破壞的可能原因有很多方面。b）如果先確定一個合適的會產(chǎn)生危險破壞的標準，那么對一個系統(tǒng)的概率設(shè)計過程往往就是一個優(yōu)化設(shè)計過程。
優(yōu)化可以被認為是一個工業(yè)研究的組成部分，主要通過利用目標函數(shù)在一定條件下的最大值或最小值條件來尋找一組最優(yōu)的結(jié)果。表1，1介紹的就是工業(yè)研究的一般領(lǐng)域。
數(shù)學處理方法，也就是所謂的優(yōu)化數(shù)學模型，常用來計算一個有有限個隨機變量的目標函數(shù)的在一定的約束條件下的最大（或最?。┲?。Rao（1979）提出了優(yōu)化的定義并且記載了一些可用來計算優(yōu)化數(shù)學模型的數(shù)學方法。隨機過程方法就可以分析一些帶有隨機變量的問題。統(tǒng)計學方法則可以用來分析一些實驗數(shù)據(jù)并建立經(jīng)驗模型以對物理行為進行最準確的描述。
至于最早的優(yōu)化方法可以追溯到17世紀時的牛頓，拉格朗日和柯西時期。由于牛頓和萊布尼茨創(chuàng)造了微分學，并得到了廣泛的應用，使得優(yōu)化過程成為可能。柏努利，歐拉，拉格朗日和魏爾斯特拉斯建立了變動的微積分學。約束優(yōu)化問題的處理方法即設(shè)計到了對設(shè)計變量的增加方法也就是所謂的拉格朗日方法，這時用該種方法的創(chuàng)造者命名的。柯西則利用最速上升法來解決最小化問題。
如果不考慮這些早期的數(shù)學家對優(yōu)化過程的所做的成果，那么直到二十世紀中期對于該類問題的處理方法都沒有取得明顯的發(fā)展和進步。此時，高速計算機已經(jīng)發(fā)明并且被用來執(zhí)行優(yōu)化處理過程，由此引發(fā)了對于新的優(yōu)化方法近一步研究和應用。隨后取得了巨大的進步，在優(yōu)化方法領(lǐng)域產(chǎn)生了大量的文獻著作。這種顯著的進步時通過對優(yōu)化理論的幾個定義明確的新領(lǐng)域的合并所得到的。
需要注意的是在二十世紀六十年代的英國已經(jīng)在無約束優(yōu)化領(lǐng)域方面取得了巨大的研究成果。丹奇克（1947）發(fā)明了單純形法用來進行計算線性規(guī)劃問題，而可以用來處理動態(tài)問題過程貝爾曼最優(yōu)原則（1957）有力的促進了約束優(yōu)化方法的發(fā)展。在庫恩（1951）的關(guān)于在必要和充分的條件下優(yōu)化問題的規(guī)劃求解的論文則為非線性問題的近一步研究奠定了基礎(chǔ)。
盡管對于非線性問題，至今并沒有發(fā)現(xiàn)一種單一的通用性的方法，但是，卡羅（1961）、費耶科和麥考密克（1968）的著作中稱應用一些比較好的方法就可以得到無約束優(yōu)化問題的實際解。達芬，奇納和彼得森（1960）發(fā)明了幾何規(guī)劃法。戈莫里（1963）創(chuàng)立了整數(shù)規(guī)劃法，并在當時的優(yōu)化研究領(lǐng)域得到了巨大的發(fā)展。在實際工程應用中的很多方面都屬于這個范疇。丹奇克（1955）和查納斯和庫珀（1959）發(fā)明了隨機規(guī)劃方法可以用來解決設(shè)計參數(shù)是相互獨立和無關(guān)的線性分布問題。
在本文中所用的非線性處理方法，可以歸結(jié)為如下幾個方面：
1 一維最小化方法。
2 無約束多變量最小化方法。
A 梯度法。
B 非梯度法。
3 約束多變量最小化方法
A 梯度法。
B 非梯度法。
梯度法需要用的目標函數(shù)的值和其微分值，而非梯度法則只需要用的目標函數(shù)的值，對目標函數(shù)是否具有微分值沒有要求。一般的來說，用的函數(shù)梯度的方法效率更高，這是因為相當于增加了已知條件。然而，如果函數(shù)解析的微分解是存在并可用的，那么就可以用到目前正在使用的一維搜尋法。如果用的的是函數(shù)的引出的數(shù)值近似方法，那么梯度法和非梯度法的效率基本相當。梯度法里面可以包括了一些數(shù)值近似法，可以用來解決一些優(yōu)化問題，其結(jié)果都是近似的，誤差非常小。圖1，1表示的就是一般的優(yōu)化流程圖。
目前，沒有相關(guān)研究成果表明那些方法會比另外一些方法更有效率，更好。一個可以很好地解決一個問題的方法可能就不能應用于同類型的另外一個問題，即每種方法都有其適用范圍和使用特點。只有對常用的方法都熟悉并有使用經(jīng)驗以后才能針對實際問題選擇一種較好的更有效率的方法（庫斯特和麥茲1973）。
在二十時間初期，就有人研究在機械結(jié)構(gòu)設(shè)計分析種應用概率和統(tǒng)計方法，這也是概率設(shè)計的最早應用。但是，這種方法并沒有受到極大的關(guān)注和認可。一直到二戰(zhàn)后，1945年10月，由弗賴登塔爾撰寫的著名論文“結(jié)構(gòu)的安全”發(fā)表在美國土木工程安全期刊上。此文的發(fā)表標志著美國結(jié)構(gòu)安全設(shè)計的起始。弗賴登塔爾教授已經(jīng)對結(jié)構(gòu)可靠形分析和危險分析的前沿問題進行了很多年的研究，并取得了相當?shù)某晒?
在二十時間六十年代可靠理論研究得到了迅速的發(fā)展，由很多學術(shù)成果。特別是Freudenthal, Garrelts, and Shinouzuka (1966), Pugsley (1966), Kececioglu and cormier (1964), Ferry-Borges and Castenheta (1971), and Haugen (1968)等人發(fā)表了一系列的相關(guān)研究成果后，經(jīng)典可靠性理論開始被人熟知并得到巨大發(fā)展。然后，在實際應用上面，由于可靠性設(shè)計實現(xiàn)起來比較麻煩、對系統(tǒng)的分析理論要用到很復雜的數(shù)學理論、可以用的的數(shù)據(jù)量較少以及數(shù)學模型錯誤引起的爭論等諸多原因，使得可靠性設(shè)計的實際應用非常少，發(fā)展比較慢。
在當時的諸多限制條件下，可靠性設(shè)計也取得了一些早期的研究成果。特克斯特拉（1970）指出了結(jié)構(gòu)設(shè)計作為一個決定性的問題應低于一定的不確定性和危險性。蘭德，特克斯特拉和賴特（1965）詳細說明了在尋找出的一系列最佳的載荷和阻力條件下一種故障模式的合理設(shè)計問題?？苾?nèi)爾（1967）則建議使用一種二階格式，并隨后在在科內(nèi)爾安全系數(shù)要求種得到了證明，該法可以推理出在載荷和阻力下的一些安全因素。這些關(guān)于可靠性分析的方法都具有一定的實際應用性。特別是科內(nèi)爾法在許多的結(jié)構(gòu)設(shè)計標準中得到了應用。這在結(jié)構(gòu)可靠性理論發(fā)展上具有里程碑的意義。在隨后的幾年中，Rackwitz 和 Fiesser （1978），Chen 和 Lind （1982）以及Wu（1984）等學者對Hasofer－Lind的方法進行了廣義的延伸和擴充，并對復雜限制狀態(tài)下目標的失效估計做了準確的分析。
當Ditlevsen和Lind分別發(fā)現(xiàn)問題的不變性特征的后，二階矩的簡單可靠度方法的局限性就被發(fā)現(xiàn)了。人們發(fā)現(xiàn)，當確定的簡單問題用另外一種等價的結(jié)構(gòu)形式表示時，科內(nèi)爾方法得到的可靠性結(jié)果不是不變的。但是，Hasofer 和Lind （1974）提出了一般的結(jié)構(gòu)可靠度定義指標，克服了上述方法的局限性。
在結(jié)構(gòu)設(shè)計系統(tǒng)中，存在著許多的失效形式，于系統(tǒng)的結(jié)構(gòu)、外形和材料的屬性、外載條件等因素有很大的關(guān)系。為了確定一個系統(tǒng)的可靠性，必須首先定義各部分的失效模式。然而，對于一個高度冗余的復雜系統(tǒng)來說，去確定各部分結(jié)構(gòu)的失效模式的優(yōu)先問題是非常困難的。利用以下的方法可以近似的得到一些結(jié)果：（a）自動產(chǎn)生安全系數(shù)。（b）β約界法。（c）分枝界限法（Thoft－Christensen 和 Murotsu, 1986）。在Bennett（1983），Ang和Tang（1984），Guenard（1984），Ditlesen（1986），Madsen，Krenk，和Lind（1986），以及Thoft-Christensen和Murotsu（1986）等人的著作中對系統(tǒng)結(jié)構(gòu)可靠性分析進行了全面詳細的介紹和分析。但是在當時的環(huán)境下，還沒有可以用來獲得系統(tǒng)可靠性問題的實際解決的一般方法。