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　能量原理 绍 概念介绍

　 分析结构在荷载、温差等外因影响下所产生的应力、变形和位移状态的基本原理之一。能量是指结构作功的能力。弹性结构在加载时产生变形，在卸载后又能恢复原状，说明若不计动能和热能的变化，荷载在结构上所作之功，将全部转化成结构的变形势能存储于结构之内，因而在卸载过程中具有恢复原状的能力，这是能量原理的依据。能量原理根据荷载作功过程中变形势能的变化规律,建立起一系列极值条件，作为解题的综合判据,从而避免直接解算大量偏微分方程，以简化解题手续。

　用 应用

　 用能量法分析结构,主要是寻求既满足边界条件,又同时满足势能为最小的位移函数或者余能为最小的应力函数。对许多难于求得精确解的工程问题，可用下述各个能量原理以求问题的近似解答。因此，在分析复杂结构的静力和动力问题中，能量原理得到广泛应用。

　容 研究内容 能量原理可从虚位移原理、虚力原理两个侧面研究。又根据势能和余能的变化情况,建立相应的极值条件,以解答具体问题，形成最小势能原理和最小余能原理。

　虚位移原理 也称势能原理、虚功原理。设结构在荷载作用下处于平衡状态。假定由于任何其他原因，使结构从其平衡位置偏离一个任意微小的、为边界约束条件所允许的虚位移（可以看作是真实位移的一个变分），荷载在虚位移上所作的虚功，将等于其内部应力在相应应变上所积累的虚变形势能。

　故虚位移原理可表述为：弹性移 结构平衡的必要与充分条件是，对于任意微小的虚位移, 荷载所作的总虚功 δW 等 等能 于其内部所积累的虚变形势能 δU 。即 δU-δW ＝0。

　。

　 最小势能原理 设结构在 P 力系作用下处于平衡。在某一可能虚位移过程中,与Pi 力相应的虚位移设为墹 i,则由可能虚位移引起的荷载势能变化为,

　 将使结构增加变形。设由此引起的变形势能的改变为 δU,则结构的总势能改变 δП 可定义为内外两种势能变化之差，即

　 但是,在这个虚位移中，荷载始终保持不变，因而 П 只是可能虚位移的函数。故此式

　可改写成

　 泛函 П＝U-W 代表结构在虚位移中的总势能。当结构处于平衡状态时,已知 U＝W,从而有 δП＝0,它说明：在一切满足边界条件的虚位移中，同时满足平衡条件的虚位移对应于结构势能的一个驻值，这就是结构势能驻值原理。对于线弹性结构，势能的二阶变分恒为正，因而使总势能取最小值，所以这个原理又称最小势能原理。它意味着在所有满足边界条件的虚位移中，能使结构势能为最小的虚位移，满足平衡条件，因而就是真实的位移。在这种情况下，结构势能的驻值条件等价于平衡条件。

　虚力原理 也称余能原理。设结构在荷载和支承位移影响下处于平衡状态。在位移保持不变的情况下，若让真实应力 σ 发生微小改变 δσ,且使它们满足平衡条件和应力边界条件（称为可能虚应力），则虚力原理可表述为：对一切可能虚应力 δσ 而言，结构满足变形协调方程的必要和充分条件是，对于任意微小的可能虚应力，其变形余能的一阶变分 δU*,等于位移边界上的相应边界反力所作荷载余功的一阶变分 δW*，即

　 。

　最小余能原理 结构的余能变分可定义为 式中 Ri、Ci 分别为支承反力和相应的支承位移。在可能应力的变化过程中，应变和位移均保持不变，因而此式可改写为

　 泛函

　 代表结构的总余能，由余能原理，有 δП*=0 。它说明：在所有满足平衡条件及边界条件的应力场中，同时满足相容应变场的应力场，对应于余能的一个驻值，这就是余能驻值原理。对于线弹性结构，因有,已知势能 U 的二阶变分恒为正,故 П*将取最小值，因而最小余能原理可表述为：在一切满足平衡方程及边界条件的应力场中，真实的应力场应能使泛函 П*成为最小。因而，余能的驻值条件等价于变形协调条件。

　Mapping numbers sounds complex, but we do it when we buy gasoline. We pump gasoline, and the gas station charges us based on the amount of gas that we pump. Learn how this relates to functions while reviewing the basics and notations in this lesson. Would it surprise you to know that one of the most important pillars of calculus is something that you use every day? Functions are fundamental to calculus, but you have been using them your entire life. Formally, functions map a set of numbers to another set of numbers. So what does this mean? Basics of a Function

　Say we have a black box, and we"re going to call this our function. If you put in the number 4, you might get out the number 8. If you put in the number 5, you might get out the number 16. For each number that you put in, say x , you"ll get out another number, say y . Now sometimes you can put in two different numbers - let"s say 4 and 22 - and get out the same number, say 39. But at no point in time will you put in one number and get two different numbers out. This may sound complex, but it"s really just saying stuff you already know. Functions in our Daily Lives

　For example, you use functions every time you go to the gas station. The amount of money that you pay a gas station depends on the amount of gas that you pump. Put another way, the amount of gas that you pump determines the number of dollars that you pay. Let"s say gas is $3.80 a gallon. If you"re going on a long road trip, you might need 10.2 gallons. If you pump 10.2 gallons and gas is $3.80 a gallon, the gas station is

　going to charge you $38.76. If you"re just running across town and only need 2.3 gallons, the gas station is only going to charge you $8.74. Of course, depending on your car, you may need more or less gas, and the station is going charge you based on the amount of gas that you take. How they determine the amount to charge is very simple. They use a function that maps the number of gallons that you pump to the number of dollars that you need to pay. Specifically, we say that number of dollars you pay is a function of the amount of gas you buy. Variables in a Function

　You know the number of gallons that you"re going to buy; you know an x x

　 variable. This is your independent variable. What you want to know is the amount of dollars that you"re going to pay, the y y

　 variable. This is dependent, because it depends on the number of gallons that you buy. One way to put this mathematically is that the dollars you pay is a function of the gallons you buy, just like we said before. But let"s write this out in math terms: The dollars you pay ( y ) is (that"s math speak for "equals") a function ( f

　for function) of the gallons that you buy ( x ). So y y = f(x) . Generally, a function is written with input variables in these parentheses. In the case of our gas station, we really know what the function is. We go to the gas station and pump some amount of gas. Because you"ve probably gone to the gas station before, you know that the number of dollars that you"re going to pay is equal to the amount of gas that you pumped times the price per gallon. So if gas is $4 per gallon,

　then we write y

　is equal to 4 times our input, which is the number of gallons we pump: y =4 x . If we pump 4 gallons of gas, we plug this into our function, 4 * 4, and that"s 16. We would owe $16. So an important point here is that when we say y = f(x) , that"s good for any value of x . We"re using x

　as a variable here. If we say y = f (5), we"re evaluating this function using x =5. Imagine going to the gas station, and instead of pumping 4 gallons of gas and owing $16, you try to give gas back to the gas station. Now I don"t know about you, but when I"ve tried to do that, the gas station attendant usually just laughs at me, because he and I both know that the amount of gas that I can pump (the input to our function) has to be greater than zero. The minimum amount of gas I can pump is zero, and if I go to the gas station and pump zero gallons of gas, the gas station just gets annoyed. But there is no maximum amount, practically speaking. The output of our function will be between zero and infinity. Domain and Range

　All possible inputs to our function is known as the domain. The domain are those values that you can put as input (as the x

　variable) into the function. The output - all possible values that you can get out of your function - are known as the range. In this case, the range is the amount of dollars that you could expect to spend at the gas station. So if I go back to the black box, we"re going to write our function a little more formally now as y = f(x) . The inputs to our black box are now the x

　values that make up the domain, and the output - something within the range - are our y

　values. Mathematicians will

　sometimes write like this, where now you"ve got basically two sheets of paper, in a sense, facing one another. On one side you have a domain - you have a function - and the second sheet of paper is the range. What about the function sine of x ? The function sin( x ), I will write as y

　is some function of x

　and that function is actually sin( x ), so y =sin( x ). What about the domain and range of this function? The domain are all possible values of x

　that you can put into this. If you go your calculator, you know that you can put anything into sin( x ), so the domain is all values from minus infinity to infinity. What about the range? Well the range is all possible values of y . If we graph sin( x ), you can pretty easily see that the maximum value is going to be 1 and the minimum value is going to be -1. So writing y =sin( x ), we know that y

　has to be between -1 and 1. That means that the range of sin( x ) is going to be between -1 and 1. So the range corresponds to the y

　value, this vertical axis, and the domain corresponds to the x

　values, this horizontal axis. One last thing: What about the domain and range of this function? First, let"s take a look at what this function means. What this means is for values of x

　that are less than zero, our function, f(x) , equals sin( x ). If I graph this out, anything that"s less than zero is going to be sin( x ); it"s going to continue on. For values of x

　that are greater than zero, f(x)

　is going to be equal to x , so it"s going to look like this. An important point to note here is that there is no value assigned at x =0, so I"m going to put a circle there because it"s undefined. L L esson Summary

　To recap, functions map some set of numbers to another set of numbers, but this is no more complex than going to the gas station. Your input to your function, your x x

　 variable, is the independent variable, and x

　needs to be within the domain of the function. The y y

　 variable is your dependent variable. It"s the output of your function and has to be within the range of the function.

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