外文翻譯--優(yōu)化結(jié)構(gòu)設(shè)計(jì)

1、<p><b>　　優(yōu)化結(jié)構(gòu)設(shè)計(jì)</b></p><p>　　W. PRAGER 3</p><p>　　摘要。數(shù)學(xué)技術(shù)被應(yīng)用在典型的優(yōu)化結(jié)構(gòu)設(shè)計(jì)問(wèn)題這一領(lǐng)域。介紹一個(gè)關(guān)于一個(gè)桿的設(shè)計(jì)為了描述最大化繞度和顯示怎樣適當(dāng)?shù)碾x散化可能導(dǎo)致一個(gè)非線性的問(wèn)題，在這種情況下的復(fù)雜的程序。最優(yōu)布局已經(jīng)被討論了一段時(shí)間。一種新的建立最優(yōu)標(biāo)準(zhǔn)的方法已經(jīng)說(shuō)明了被設(shè)計(jì)一個(gè)靜不定梁

2、或一個(gè)變截面的繞度在一個(gè)單一的集中壓力下。其他的應(yīng)用這個(gè)方法被簡(jiǎn)單的討論，并且用一個(gè)多功能的設(shè)計(jì)的簡(jiǎn)單的例子來(lái)結(jié)束這份文件。</p><p><b>　　1 。導(dǎo)言</b></p><p>　　結(jié)構(gòu)最優(yōu)化的最普通的問(wèn)題或許可以表述如下：從所有的滿足某些限制的結(jié)構(gòu)設(shè)計(jì)，選擇其中一個(gè)最低成本的。注意這個(gè)聲明并不定義一個(gè)唯一的設(shè)計(jì);可能同時(shí)有幾個(gè)最優(yōu)化的設(shè)計(jì)有相同的成本。&

3、lt;/p><p>　　典型的設(shè)計(jì)將考慮滿足變形或受力的最大約束，或者最小約束的承載能力，屈服載荷，或固有頻率。單一的和多用途的結(jié)構(gòu)都要被考慮，即是受單一因素或多重因素的約束。</p><p>　　設(shè)計(jì)聲明中的花費(fèi)也許會(huì)參照到制造成本或總生產(chǎn)成本和生產(chǎn)中結(jié)構(gòu)的壽命。 在航天結(jié)構(gòu)中，燃油成本需要執(zhí)行最大的重量但最小的質(zhì)量是他們?cè)O(shè)計(jì)的唯一目標(biāo)，這個(gè)觀點(diǎn)將要使用在下面文章。</p>&

4、lt;p>　　在這個(gè)文獻(xiàn)的第一部分，優(yōu)化設(shè)計(jì)的典型問(wèn)題將用已經(jīng)應(yīng)用這個(gè)方面的數(shù)學(xué)技術(shù)來(lái)說(shuō)明第二部分將要關(guān)注有廣泛應(yīng)用的有很大前途的最近發(fā)展的技術(shù)</p><p>　　這整個(gè)文章，它強(qiáng)調(diào)具有最優(yōu)整體結(jié)構(gòu)是必須被仔細(xì)的定義沒(méi)有意義的方案是要避免的。</p><p>　　事實(shí)上還要強(qiáng)調(diào)指出某些直觀的最優(yōu)準(zhǔn)則對(duì)工程師來(lái)說(shuō)不一定提供真正的最優(yōu)解。為了更清晰的介紹設(shè)計(jì)原則，大多數(shù)例子是關(guān)于單一約

5、束的結(jié)構(gòu)盡管多約束的結(jié)構(gòu)是具有更大的實(shí)際意義。</p><p><b>　　2 。離散</b></p><p>　　去探索具有數(shù)學(xué)性質(zhì)的最優(yōu)化結(jié)構(gòu)問(wèn)題，這是經(jīng)常有用的用一個(gè)分立模擬取代連續(xù)問(wèn)題。考慮，例如，簡(jiǎn)直彈性梁在圖。 1最大偏轉(zhuǎn)所產(chǎn)生的給予負(fù)荷6P不會(huì)超過(guò)給定值。對(duì)于離散性問(wèn)題，用一系列用彈性鉸鏈連接的剛性棒取代梁。</p><p>　　

6、圖1.分立模擬彈性梁。</p><p>　　在圖1中，緊緊三個(gè)鉸鏈已經(jīng)被介紹了。但是，為了得到真實(shí)的結(jié)果，這個(gè)離散取決于鉸鏈的數(shù)目。彎矩可以轉(zhuǎn)換鉸鏈數(shù)目i和角度的關(guān)系為 = (1)</p><p>　　其中是彈性剛度的鉸鏈。由于是靜定梁， 在鉸鏈上的彎矩獨(dú)立的剛度；因此，</p><p>　　=5Ph=, =3Ph=, =Ph=. (2)</p

7、><p>　　接下來(lái)，彎曲角度將被視為最小。在一個(gè)直角坐標(biāo)系的實(shí)際空間中,i=1,2,3,這個(gè)非負(fù)性質(zhì)的彎曲角度和在鉸鏈上的繞度定義凸可行域。</p><p><b>　　，，0,</b></p><p>　　5+3+-6/h0,</p><p>　　3+9-3-6/h0, （3）</p>

8、<p>　　+3+5-6/h0，</p><p>　　作為接下來(lái)將要講的例子， 一組質(zhì)量的剛度假定達(dá)到剛度的一定比例。這個(gè)設(shè)計(jì)即是 ++=Min 或，通過(guò)公式（2）</p><p>　　5/+3/+1/=Min (4)</p><p>　　值得注意的是 通過(guò)公式（3）-（4），一個(gè)局部?jī)?yōu)化對(duì)于整體優(yōu)化必須的。</p>&l

9、t;p>　　這句話很重要因?yàn)閯倓傞_(kāi)始的設(shè)計(jì)對(duì)于滿足所有約束的相鄰設(shè)計(jì)是沒(méi)有什么實(shí)際價(jià)值的。也要注意到 優(yōu)化總體上 不會(huì)對(duì)應(yīng)到位于一個(gè)邊的或恰好位于一個(gè)頂點(diǎn)區(qū)域的一個(gè)點(diǎn)的空間設(shè)計(jì)。這句話直觀的表明沖突的約束不一定是有用的。假如，舉個(gè)例子來(lái)說(shuō)，設(shè)計(jì),,條件<<=.假如是剛度的最小變量，設(shè)計(jì)+,-,,擁有相同的質(zhì)量的理想繞度為，， 滿足<,<<=而且三個(gè)剛度降低一定的比例直到第一個(gè)鉸鏈的繞度是。假如這個(gè)探

10、討是正確的降低結(jié)構(gòu)只來(lái)那個(gè)的過(guò)程能夠被重復(fù)直到鉸鏈1和2有相同的質(zhì)量&。接下來(lái)設(shè)計(jì)的更改和都有想同的少量的增加而則降低兩倍目的是保持質(zhì)量常數(shù)。用這種方式，可能有爭(zhēng)論關(guān)于；優(yōu)化設(shè)計(jì)必須對(duì)應(yīng)一個(gè)邊上的一個(gè)點(diǎn)或者可行性區(qū)域上的頂點(diǎn)，由于優(yōu)化設(shè)計(jì)，兩三個(gè)不平等的約束就必須列方程。這沖突的約束通常會(huì)出現(xiàn)在工程界，顯然用手是不能完成的。具有不平等約束繞度的最小重量梁的設(shè)計(jì)近期已經(jīng)已經(jīng)討論了被Haug and Kirmser（見(jiàn)1）較早前調(diào)查

11、（見(jiàn)，例如，參2-4 ）在某一特定點(diǎn)所涉及的不等式約束對(duì)撓度，舉例來(lái)說(shuō)，在載荷集中在一個(gè)點(diǎn)上。在特殊情況下該點(diǎn)的最大繞度位置是已知的，舉例來(lái)說(shuō)，從對(duì)稱的考慮，一個(gè)約束擁有最大繞</p><p><b>　　3 。布局優(yōu)化</b></p><p>　　在前面的示例，類型和布局結(jié)構(gòu)（簡(jiǎn)單支持，直梁）被給予并且一些某些地方的參數(shù)（剛度值）是設(shè)計(jì)師選擇的。一個(gè)更有挑戰(zhàn)性的問(wèn)題

12、就是類型和/或布局也必須選擇最佳的。</p><p>　　數(shù)字顯示，由桁架支持的給出點(diǎn)的應(yīng)力載荷P和Q ，即連接桿組成的結(jié)構(gòu)，布局就是要去盡量減少重量。為了簡(jiǎn)化分析，Dorn, Gomory,and Greenberg（見(jiàn)5 ）通過(guò)劃分網(wǎng)格其橫向間距L和垂直間距的h描述這個(gè)問(wèn)題（圖2 a ）優(yōu)化是接下來(lái)發(fā)現(xiàn)需要解決的線性規(guī)劃。優(yōu)化布局取決于質(zhì)量的比例h/L和P/Q=0,0.5,和2.0.</p>

13、<p>　　圖.2 .優(yōu)化布置的桁架根據(jù)多恩，戈莫里，格林伯格（見(jiàn)5 ） 。</p><p>　　因?yàn)閔/L=1和P/Q是一個(gè)給定數(shù)，優(yōu)化值是唯一的除開(kāi)某些臨界值P/Q，其中優(yōu)化布局的變化，舉例來(lái)說(shuō)，從圖2c到踢2d。接下來(lái)例子，然而，承認(rèn)一個(gè)無(wú)限大的優(yōu)化布局是所有相關(guān)的擁有同樣重量的結(jié)構(gòu)重量。</p><p>　　三個(gè)同樣大小的作用力P，彼此之間成120 °角，已經(jīng)給

14、出的點(diǎn)成等邊三角形（圖3a）。這些連接點(diǎn)連接的構(gòu)架用最小質(zhì)量設(shè)計(jì)。當(dāng)上界約束提供軸向應(yīng)力在任何桿。數(shù)據(jù)3b和3c是可行的布局。這些力作用在靜定機(jī)構(gòu)的桿上之后從平衡的角度考慮,每個(gè)桿件的橫截面都會(huì)有一個(gè)大小的橫向應(yīng)力。</p><p>　　接下來(lái)討論Maxwell的觀點(diǎn)（見(jiàn)6，PP第175-177 ）表明兩個(gè)設(shè)計(jì)有相同的質(zhì)量。設(shè)想飛機(jī)都是用相同的材料組成的，單位平面產(chǎn)生的張力達(dá)到e對(duì)所有的線性元素。通過(guò)虛擬的規(guī)律，

15、這個(gè)桿件P上所有的點(diǎn)的位移所產(chǎn)生的虛擬功等價(jià)于內(nèi)部虛擬功=F 每個(gè)桿件受力為F 力在桿方向的虛擬位移為,如果桿件的橫截面積是A長(zhǎng)度是L，則有F=A 和=L則有</p><p>　　=AL=V （5）</p><p>　　V是使用的所有材料的體積。現(xiàn)在得到功取決于載荷和所有點(diǎn)的虛擬位移除開(kāi)獨(dú)立布置</p><p>　　圖。 3 。選擇最優(yōu)設(shè)計(jì)。&l

16、t;/p><p>　　的桿件；他等價(jià)于兩個(gè)機(jī)構(gòu)如果下面=和（5）這兩個(gè)構(gòu)架使用相同數(shù)量的材料。</p><p>　　如果兩個(gè)構(gòu)架的橫截面積都減半， 每個(gè)新的構(gòu)架能夠驅(qū)動(dòng)滿載荷強(qiáng)度P/2并且不違反設(shè)計(jì)約束.按圖3d的方式疊加桿件另外用相同質(zhì)量的構(gòu)建按圖3d和3c疊加所有構(gòu)架加載滿載荷P。</p><p>　　圖。 4 。替代解決問(wèn)題，在圖。第3 a </p&

17、gt;<p>　　圖4顯示的另一個(gè)解決問(wèn)題的方法。所有重桿件的中心線是圓弧的。每個(gè)桿件的軸向力和他們的軸向應(yīng)力有關(guān)</p><p>　　其他輕些的桿件。他們也根據(jù)軸向拉伸應(yīng)力，除開(kāi)桿件AO,BO和CO，組成圓錐。正常情況下桿件的邊緣區(qū)域是受力的密集區(qū)域。如果緊緊是有限的數(shù)目被使用就像圖4并且這些邊緣是多變形而不是圓弧 ，這就是重量稍稍重一點(diǎn)點(diǎn)的結(jié)果。 首先申明，然而，如果桿件連接件（節(jié)點(diǎn)板和鉚釘或焊

18、接）的質(zhì)量被考慮其中這個(gè)就不是有效的。</p><p>　　在圖4中的桿件也許可以被有厚度統(tǒng)一材質(zhì)均勻的桿件替代。然而質(zhì)量是取勝之本，設(shè)計(jì)也是這樣的，然而，設(shè)計(jì)構(gòu)架的時(shí)候遇到的狹隘的問(wèn)題要被排除。在這種情況下，被排除的設(shè)計(jì)將不會(huì)比其他的設(shè)計(jì)的質(zhì)量更輕。然而，除開(kāi)這一類對(duì)一個(gè)最優(yōu)的進(jìn)行有足夠廣度定義的，或許緊緊對(duì)一系列降低質(zhì)量的設(shè)計(jì)進(jìn)行融合一個(gè)最優(yōu)的這不是考慮的范圍之內(nèi)</p><p>　　

19、圖。 5 。優(yōu)化結(jié)構(gòu)轉(zhuǎn)遞周邊荷載至中環(huán)環(huán)的桁架而非磁盤狀 圖5對(duì)這句話進(jìn)行了說(shuō)明。 在周邊的離散的徑向載荷等價(jià)于中央形成一個(gè)環(huán)狀的小質(zhì)量的構(gòu)件。</p><p>　　如果這個(gè)聲明的結(jié)構(gòu)將要被表達(dá)磁場(chǎng)形狀的連續(xù)變厚度所取代，優(yōu)化后的結(jié)構(gòu)如圖5要為排除。注意清楚看看圖5他所顯示的緊緊是質(zhì)量大的成員。</p><p>　　這些之間，質(zhì)量輕些的成員之間關(guān)系是稠密的，他們之間是以螺線形狀相交

20、的。</p><p>　　這個(gè)問(wèn)題在圖3中已經(jīng)有一個(gè)解決方案，每個(gè)構(gòu)件都緊緊是包含受力的桿件。圖6說(shuō)明了一個(gè)問(wèn)題既要使用沒(méi)有受力的也要使用受力的并且只有唯一解。上方的數(shù)據(jù)是橫向載荷P會(huì)產(chǎn)生彎曲，底部的剛性結(jié)構(gòu)可以看為是無(wú)限小的質(zhì)量，在桿件上的應(yīng)力應(yīng)該在-和之內(nèi)。</p><p>　　這個(gè)最優(yōu)的構(gòu)架邊緣桿件的質(zhì)量較大；質(zhì)量大的構(gòu)件中間的構(gòu)件的質(zhì)量較輕，由圖6表達(dá)。注意在位移密集的桿件連接處定

21、義一個(gè)位移區(qū)域他的的基點(diǎn)固定。</p><p>　　一個(gè)移動(dòng)的受力區(qū)域都擁有這一規(guī)律即 =/ E和=-/E 其中E是彈性模量。事實(shí)上，如果u和v是位移分量類似于直角坐標(biāo)系中的x和y，那么+就是個(gè)常量有以下的關(guān)系即</p><p>　　+=0， （6）</p><p>　　其中x和y顯示著不同的坐標(biāo)關(guān)系。類

22、似的，事實(shí)上最大的主應(yīng)變e1擁有連續(xù)的線性關(guān)系</p><p>　　4*-(+)( +)=-4 （7）</p><p>　　從公式（6）中可以看出，其中存在函數(shù)如下</p><p>　　=,=- （8）</p><p>　　把 公式（8）代入公式（7）中則

23、有</p><p>　　4 +=4 (9)</p><p>　　沿著根部弧有，==0,則可以推到出</p><p>　　=0， =0 （10）</p><p>　　其中是沿著根部弧的微變量。</p><p>　　微分方程（9）是一個(gè)雙曲

24、線，其特點(diǎn)主應(yīng)變是線性變化的。柯西條件在公式（10）中元素在根部是是獨(dú)特的，并且和公式（8）位移也有關(guān)系 </p><p>　　圖。 6 。在傳輸載荷P下彎曲和剛性壁的獨(dú)特的優(yōu)化結(jié)構(gòu)</p><p>　　這些位移現(xiàn)在將使用作為真正位移在虛功原理在一個(gè)任意的結(jié)構(gòu)上其載荷P傳達(dá)到基座弧（圖6）并且每個(gè)連桿都在一個(gè)軸向應(yīng)力為@o之下使用Maxwell公式則有

25、，可以得到== 其中｜｜=A并且｜｜因?yàn)槊總€(gè)單位的拉伸或者壓縮量超過(guò)/E就不是線性變化了，</p><p>　　=∑｜F｜｜｜ (/E)V, (11)</p><p>　　其中V是所有材料的總體積。</p><p>　　接下來(lái)，設(shè)想第二中結(jié)構(gòu)它是由有規(guī)律線性應(yīng)變的的連桿組成并且他要考慮到虛擬的移動(dòng)區(qū)域和底部相應(yīng)的應(yīng)變 涉及到結(jié)構(gòu)的質(zhì)量將要用星號(hào)

26、標(biāo)記。就像前面所講的那樣運(yùn)用虛擬原理，最有=,但是*=并且=</p><p>　　則有 == (12)</p><p>　　則可以看出=,比較表達(dá)式（11）和（12）則可以看出第二種方案的結(jié)構(gòu)使用的材料要比第一種方案少。剛剛介紹的觀點(diǎn)來(lái)自于Michell（見(jiàn)7），然而，是一個(gè)純粹的靜態(tài)的邊界條件，因此不能達(dá)到一個(gè)獨(dú)特的優(yōu)化結(jié)構(gòu)。對(duì)一個(gè)獨(dú)特的優(yōu)化涉及來(lái)說(shuō)最重要的

27、是運(yùn)用運(yùn)動(dòng)學(xué)邊界條件已經(jīng)被作者指出（見(jiàn)8）</p><p>　　圖。 7 。幾何布局優(yōu)化。</p><p>　　圖7 說(shuō)明了一個(gè)重要的具有幾何性質(zhì)的在有規(guī)律的應(yīng)變和無(wú)規(guī)律的應(yīng)變組成的區(qū)域種的正交曲線應(yīng)變 讓由ABC和DEF組成的兩個(gè)固定曲線。角度是由一個(gè)曲線上的切線和另外一個(gè)曲線上點(diǎn)的切線相交的夾角。在平移的理論下，正交的曲線他的幾何性質(zhì)可以表明他最大的剪應(yīng)力（滑移線）的方向在這個(gè)背景下

28、，它們通常后來(lái)被Hencky (見(jiàn). 9) and Prandtl (見(jiàn). 10)命名；它們的結(jié)果已經(jīng)被廣泛的應(yīng)用（見(jiàn)，例如，見(jiàn)。11-13）</p><p>　　圖。 8 。優(yōu)化布置時(shí)，可用空間范圍內(nèi)垂直通過(guò)A和B 。</p><p>　　圖8顯示了最優(yōu)空間的布局即可用的結(jié)構(gòu)空間是垂直連線A和B之間的范圍 。因?yàn)檫@個(gè)固定支座弧是一個(gè)直線部分，在三角形ABC中間沒(méi)有連桿。再次顯示，他的邊緣

29、的桿件的質(zhì)量重，其他的連桿緊緊一些并且質(zhì)量輕。這些桿件不布局有些類似于人類的骨架的結(jié)構(gòu)(see, for instance,ReL 14, p. 12, Fig. 6)。Michell結(jié)構(gòu)給出了更深一步的理解，參照。15-16.</p><p>　　4 。新方法，建立優(yōu)化準(zhǔn)則</p><p>　　圖。 9 。梁展不斷截面。</p><p>　　在圖9中的梁是建立在A

30、上面的并且B和C只是給予了簡(jiǎn)單的支撐。</p><p>　　承受載荷P的這一點(diǎn)的繞度的值是。這個(gè)梁有一個(gè)核心部分他的寬度是B并且它的高度是H。這個(gè)梁他的寬度是B并且它們連續(xù)的厚度滿足《H和《H在和上這樣的目的是盡量減少這個(gè)結(jié)構(gòu)梁的質(zhì)量由于他的核心面的尺寸已經(jīng)被定義了，盡量減少這個(gè)梁的質(zhì)量也就意味著要盡量減少制約質(zhì)量的尺寸。</p><p>　　此外，由于厚度為橫截面積的抗彎曲彈性剛度，其中

31、i=1,2,有,其中E是楊氏模量，</p><p><b>　　(13)</b></p><p>　　這個(gè)可能被視為盡量減少質(zhì)量的方程。</p><p>　　使得是從桿件上典型橫截面到桿件最左邊的距離，并且在這個(gè)橫截面上的曲率和彎矩分別是和則的表達(dá)式可以寫為如下</p><p>　　==

32、(14)</p><p>　　就是在Li進(jìn)行微積分。</p><p>　　在此框架內(nèi)的問(wèn)題，設(shè)計(jì)一個(gè)梁就是確定的值，i=1，2.假如和都滿足設(shè)計(jì)的約束（給出只），并且和假設(shè)載荷下給出的曲率，根據(jù)（14）公式</p><p>　　= （15）</p><p>　　此外，曲率是約束變動(dòng)的（即滿足繞度的）根據(jù)，根據(jù)最小勢(shì)能原理根

33、據(jù)即</p><p><b>　　(16)</b></p><p>　　約去兩邊的在式子（16）中在根據(jù)式子（15）可得</p><p><b>　　（17）</b></p><p>　　這里 （18）</p><p>　　

34、則就是每個(gè)單位平方米的曲率在上。假如</p><p><b>　　(19)</b></p><p>　　從公式（17）和（13）得到其滿足這設(shè)計(jì)另外設(shè)計(jì)的約束不能比剛剛滿足約束的設(shè)計(jì)更重。因此條件（19）是最優(yōu)的，這個(gè)條件也是下面所要講到的。</p><p>　　應(yīng)用這個(gè)定義則有 （20）</p><p

35、>　　設(shè)計(jì)的條件不應(yīng)該比設(shè)計(jì)條件是的質(zhì)量更重由下面公式得到</p><p><b>　?。?1）</b></p><p>　　換個(gè)方面說(shuō)，不等式（17）從最小勢(shì)能原則得到</p><p><b>　　（22）</b></p><p>　　, 和，將作為和的載體像坐標(biāo)系。</p>

36、<p>　　這個(gè)不等式（21）中不能位于第二和第四象限，并且這個(gè)不等式要求和是個(gè)非負(fù)的。</p><p>　　現(xiàn)在，優(yōu)化設(shè)計(jì)和他的平均曲率都是未知的但是是唯一的。換個(gè)方面來(lái)講，是受的值所限，因此當(dāng)?shù)姆较虮贿x擇時(shí)其等級(jí)也所確定。此外，在這個(gè)最優(yōu)化設(shè)計(jì)中的，他的結(jié)構(gòu)質(zhì)量將最接近最小質(zhì)量。接近于邊緣空間的相應(yīng)的一半將被不等式（21）確定。假如和的坐標(biāo)是非負(fù)的，那么和的坐標(biāo)必須位于正常的一半空間內(nèi)，因此，（19

37、）是最優(yōu)化的必要條件，這是根據(jù)Sheu and Prager (見(jiàn). 17).</p><p><b>　　5。多用途的設(shè)計(jì)</b></p><p>　　圖。 11 。多用途的設(shè)計(jì)。</p><p>　　圖11說(shuō)明了一個(gè)多用途的設(shè)計(jì)的問(wèn)題。在第一個(gè)原因下，張度為L(zhǎng)下的伸長(zhǎng)率不會(huì)超過(guò)值。在第二個(gè)條件下，在中央給定的載荷T下的繞度不會(huì)超過(guò)給定的值；

38、并且，</p><p>　　在第三個(gè)條件下，他的屈曲載荷至少是B。注意設(shè)計(jì)的約束是個(gè)不等式的形式，以為最優(yōu)設(shè)計(jì)或許是一個(gè)或是兩個(gè)。</p><p>　　下面的掃個(gè)因素是相互制約的。正如第四部分，取得下面的不等式</p><p>　　，， （23）</p><p>　　其中是縱向位移在這個(gè)模型中

39、，且 和是梁和柱上的繞度。由公式（23）可以得到</p><p>　　， ， （24） </p><p>　　其中是常數(shù)。很容易看到這些最優(yōu)條件是不兼容的。因?yàn)樨?fù)荷L上的縱向應(yīng)變U……被認(rèn)為是第一最優(yōu)條件，但是負(fù)載T下的曲率將不滿足第二最優(yōu)條件。 因此不等式（23）不能左右相加，他們乘積得</

40、p><p><b>　　（25） </b></p><p><b>　　這個(gè)不等式表明</b></p><p>　　=Const （26）</p><p>　　是一個(gè)充分條件。這個(gè)條件也是必要的。 他可以變成另外一個(gè)形式</p>&

41、lt;p>　?。?7） </p><p>　　其中 ,和是面應(yīng)力是分別在配合，梁和柱上。其他的例子和理論，參見(jiàn)。32-33</p><p><b>　　6。結(jié)束語(yǔ)</b></p><p>　　總的概括而言，應(yīng)該強(qiáng)調(diào)指出設(shè)最典型的計(jì)約束主要討論在第四部分，不是只是緊緊只說(shuō)建立最優(yōu)準(zhǔn)則的方法。事實(shí)上，最優(yōu)化準(zhǔn)則在繼續(xù)發(fā)展。舉例

42、來(lái)說(shuō)，標(biāo)準(zhǔn)（ 31 ） 優(yōu)化設(shè)計(jì)給出了動(dòng)態(tài)偏轉(zhuǎn)已第一次出現(xiàn)在文件上，這里沒(méi)有已經(jīng)被解決的例子4。在優(yōu)化設(shè)計(jì)中給出了剛度參見(jiàn)35.同樣，這里已經(jīng)簡(jiǎn)單的討論了限制性的優(yōu)化梁的設(shè)計(jì)，但是但不是必需的。</p><p>　　JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 6, No. I. 1970</p><p>　　SURVEY PA

43、PER</p><p>　　Optimization of Structural Design I.~</p><p>　　W. PRAGER 3</p><p>　　Abstract. Typical problems of optimal structural design are discussed to indicate mathematical techn

44、iques used in this field. An introductory example(Section 2) concerns the design of a beam for prescribed maximal deflection and shows how suitable discretization may lead to a problem of nonlinear programming, in this c

45、ase, convex programming. The problem of optimal layout of a truss (Section 3) is discussed at some length. A new method of establishing optimality criteria (Section 4) is illustrat</p><p>　　1. Introduction&l

46、t;/p><p>　　The most general problem of structural optimization may be stated as follows: from all structural designs that satisfy certain constraints, select one of minimal cost. Note that this statement does n

47、ot necessarily define a unique design; there may be several optimal designs of the same minimal cost.</p><p>　　Typical design constraints that will be considered in the following specify upper bounds for def

48、ormations or stresses, or lower bounds for load-carrying capacity, buckling load, or fundamental natural frequency. Both singlepurpose and multipurpose structures will be considered, that is, structures that are respecti

49、vely subject to a single design constraint or a multiplicity of constraints.</p><p>　　The term cost in the statement of the design objective may refer to the manufacturing cost or to the total cost of manufa

50、cture and operation over the expected lifetime of the structure. In aerospace structures, the cost of the fuel needed to carry a greater weight frequently overshadows the cost of manufacture to such an extent that minima

51、l weight becomes the sole design objective. This point of view will be adopted in the following.</p><p>　　In the first part of this paper, typical problems of optimal design will be discussed to illustrate m

52、athematical techniques that have been used in this field. The second part will be concerned with a promising technique of wide applicability that has been developed recently. Throughout the paper, it will be emphasized t

53、hat the class of structures within which an optimum is sought must be carefully defined if meaningless solutions are to be avoided. The fact will also be stressed that certain int</p><p>　　2. Discretization&

54、lt;/p><p>　　To explore the mathematical character of a problem of structural optimization, it is frequently useful to replace the continuous structure by a discrete analog. Consider, for instance, the simply-su

55、pported elastic beam in Fig. 1. The maximum deflection produced by the given load 6P is not to exceed a given value To discretize the problem, replace the beam by a sequence of rigid rods that are connected by elastic h

56、inges. In Fig. 1, only</p><p>　　Fig. 1. Discrete analog of elastic beam.</p><p>　　three hinges have been introduced; but, to furnish realistic results, the discretization would have to use a muc

57、h greater number of hinges. The bending moment transmitted across the ith hinge is supposed to be related to the angle of flexure by</p><p><b>　　= (1) </b></p><p>　　where is the e

58、lastic stiffness of the hinge. Since the beam is statically determinate, the bending moments at the hinges are independent of the stiffnesses ; thus,</p><p>　　=5Ph=, =3Ph=, =Ph=. (2)</p><

59、p>　　In the following, the angles of flexure , will be treated as small. In a design space with the rectangular Cartesian coordinates, i = 1, 2, 3, the nonnegative character of the angles of flexure and the constraints

60、 on the deflections at the hinges define the convex feasible domain</p><p><b>　　，，0,</b></p><p>　　5+3+-6/h0,</p><p>　　3+9-3-6/h0, （3）</p><p>　　+3+

61、5-6/h0，</p><p>　　As will be shown in connection with a later example, the cost (in terms of weight) of providing a certain stiffness may be assumed to be proportional to this stiffness. The design objective

62、thus is ++=Min or, by (2),</p><p>　　5/+3/+1/=Min (4)</p><p>　　Note that, for the convex program (3)-(4), a local optimum is necessarily a global optimum. This remark is important becau

63、se a design that can only be stated to be lighter than all neighboring designs satisfying the constraints is of little practical interest. Note also that the optimum will not, in general, correspond to a point of design

64、space that lies on an edge or coincides with a vertex of the feasible domain. This remark shows that the intuitively appealing concept of competing constraint</p><p>　　Minimum-weight design of beams with ine

65、quality constraints on deflection has recently been discussed by Haug and Kirmser (Ref. 1). Earlier investigations (see, for instance, Refs. 2-4) involved inequality constraints on the deflection at a specific point, for

66、 instance, at the point of application of a concentrated load. In special cases, where the location of the point of maximum deflection is known a priori, for instance, from symmetry considerations, a constraint on the ma

67、ximum deflection can </p><p>　　3. Optimal</p><p>　　In the preceding example, the type and layout of the structure (simply supported, straight beam) were given and only certain local parameters (

68、stiffness values) were at the choice of the designer. A much more challenging problem arises when type and/or layout must also be chosen optimally.</p><p>　　Figure 2a shows the given points of application of

69、 loads P and Q that are to be transmitted to the indicated supports by a truss, that is, a structure consisting of pin-connected bars, the layout of which is to be determined to minimize the structural weight. To simplif

70、y the analysis, Dorn, Gomory, and Greenberg (Ref. 5) discretized the problem by restricting the admissible locations of the joints of the truss to the points of a rectangular grid with horizontal spacing l and vertical s

71、pacing h (</p><p>　　Fig. 2. Optimal layout of truss according to Dorn, Gomory, and Greenberg (Ref. 5).</p><p>　　on the values of the ratios h/l and P/Q. Figures 2b through 2d show optimal layout

72、s for h/l = 1 and P/Q = O, 0.5, and 2.0.</p><p>　　For h/l = 1 and a given value of P/Q, the optimal layout is unique except for certain critical values of P/Q, at which the optimal layout changes, for instan

73、ce, from the form in Fig. 2c to that in Fig. 2d. The next example, however, admits an infinity of optimal layouts that are all associated with the same structural weight.</p><p>　　Three forces of the same in

74、tensity P, with concurrent lines of action that form angles of 120 ° with each other, have given points of application that form an equilateral triangle (Fig. 3@ A truss that connects these points is to be designed

75、for minimal weight, when an upper bound is prescribed for the magnitude of the axial stress in any bar.</p><p>　　Figures 3b and 3c show feasible layouts. After the forces in the bars of these statically det

76、erminate trusses have been found from equilibrium considerations, the cross-sectional areas are determined to furnish an axial stress of magnitude in each bar. </p><p>　　The following argument, which is due

77、 to Maxwell (Ref. 6, pp. 175-177), shows that the two designs have the same weight.</p><p>　　Imagine that the planes of the trusses are subjected to the same virtual, uniform, planar dilatation that produces

78、 the constant unit extension e for all line elements. By the principle of virtual work, the virtual external work of the loads P on the virtual displacements of their points of application</p><p>　　Fig. 3. A

79、lternative optimal designs.</p><p>　　equals the virtual internal work =Fof the bar forces F on the virtual elongations ~ of the bars. If cross-sectional area and length of the typical bar are denoted by A an

80、d L, then F=A and =L. Thus,</p><p>　　=AL=V （5）</p><p>　　where V is the total volume of material used for the bars of the truss. Now, depends only on the loads and the virtual displaceme

81、nts of their points of application but is independent of the layout of the bars; therefore, it has the same value for both trusses. If follows from=and (5) that the two trusses use the same amount of material.</p>

82、<p>　　If all cross-sectional areas of the two trusses are halved, each of the new trusses will be able to carry loads of the common intensity P/2 without violating the design constraint. Superposition of these trus

83、ses in the manner shown in Fig. 3d then results in an alternative truss for the full load intensity P that has the same weight as the trusses in Figs. 3b and 3c.</p><p>　　Fig. 4. Alternative solution to prob

84、lem in Fig. 3a.</p><p>　　Figure 4 shows another solution to the problem. The center lines of the heavy edge members are circular arcs. The axial force in each of these members has constant magnitude correspo

85、nding to the tensile axial stress . The other bars are comparatively light. They are also under the tensile axial stress and are prismatic, except for the bars AO, BO, and CO, which are tapered.</p><p>　　Th

86、e bars that are normal to the curved edge members must be densely packed. If only a finite number is used, as in Fig. 4, and the edge members are made polygonal rather than circular, a slightly higher weight results. Thi

87、s statement, however, ceases to be valid when the weight of the connections between bars (gusset plates and rivets or welds) is taken into account.</p><p>　　The interior bars in Fig. 4 may also be replaced b

88、y a web of uniform thickness under balanced biaxiat tension. While fully competitive as to weight, this design has, however, been excluded by the unnecessarily narrow formulation of the problem, which called for the desi

89、gn of a truss. In this case, the excluded design does not happen to be lighter than the others. However, unless the class of structures within which an optimum is sought is defined with sufficient breadth, it may only fu

90、rnish a se</p><p>　　Figure 5 illustrates this remark. The discrete radial loads at the periphery are to be transmitted to the central ring by a structure of minimal weight.</p><p>　　If the word

91、structure in this statement were to be replaced by the expression</p><p>　　Fig. 5. Optimal structure for transmitting peripheral loads to central ring is truss rather than disk</p><p>　　disk of

92、continuously varying thickness, the optimal structure of Fig. 5 would be excluded. Note that Fig. 5 shows only the heavy members. Between these, there are densely packed light members along the logarithmic spirals that i

93、ntersect the radii at </p><p>　　The problem indicated in Fig. 3a has an infinity of solutions, each of which contains only tension members. Figure 6 illustrates a problem that requires the use of compression

94、 as well as tension members and has a unique solution. The horizontal load P at the top of the figure is to be transmitted to the curved, rigid foundation at the bottom by a trusslike structure of</p><p>　　F

95、ig. 6. Unique optimal structure for transmission of load P to curved, rigid wall.</p><p>　　minimal weight, the stresses in the bars of which are to be bounded by- and . The optimal truss has heavy edge membe

96、rs; the space between them</p><p>　　is filled with densely packed, light members, only a few of which are shownin Fig. 6. Note that the displacements of the densely packed joints of thestructure define a dis

97、placement field that leaves the points of the foundation fixed. A displacement field satisfying this condition wilt be called kinematically admissible.</p><p>　　There is a kinematically admissible displaceme

98、nt field that everywhere has the principal strains =/ E and =-/E, where E is Young's modulus. Indeed, if u and v are the (infinitesimal) displacement components with respect to rectangular axes x and y, the fact that

99、 the invariant + vanishes furnishes the relation</p><p>　　+=0， （6）where the subscripts x and y indicate differentiation with respect to the coordinates. Similarly, the fact that the maximum principal strain

100、 has the constant value e1 yields the relation</p><p>　　4*-(+)( +)=-4 （7）In view of (6), there exists a function such that</p><p>　　=,=- （8）Substituti

101、on of (8) into (7) finally furnishes</p><p>　　4 +=4 (9)Along the foundation are, u = v = O, which is equivalent to</p><p>　　=0， =0 （10）where is the der

102、ivative of T along the normal to the foundation are.</p><p>　　The partial differential equation (9) is hyperbolic, and its characteristics are the lines of principal strain. The Cauchy conditions (10) on the

103、 foundation arc uniquely determine the function , and hence the displacements (8), in a neighborhood of this arc.</p><p>　　These displacements will now be used as virtual displacements in the application of

104、the principle of virtual work to an arbitrary trusslike structure that transmits the load P to the foundation are (Fig. 6) and in which each bar is under an axial stress of magnitude %. With the notations used above in t