有限元程序課程設(shè)計(jì)

重慶大學(xué)本科學(xué)生課程設(shè)計(jì)任務(wù)書(shū)課程設(shè)計(jì)題目有限元程序設(shè)計(jì)學(xué)院資源及環(huán)境科學(xué)學(xué)院專(zhuān)業(yè)工程力學(xué)年級(jí)2010級(jí)參數(shù)和設(shè)計(jì)要求：1.獨(dú)立完成有限元程序設(shè)計(jì)。2.獨(dú)立選擇計(jì)算算例，并能通過(guò)算例判斷程序的正確性。3.獨(dú)立完成程序設(shè)計(jì)報(bào)告，報(bào)告內(nèi)容包括理論公式、程序框圖、程序本體、計(jì)算算例，算例結(jié)果分析、結(jié)論等。學(xué)生應(yīng)完成的工作：復(fù)習(xí)掌握有限單元法的根本原理。掌握彈性力學(xué)平面問(wèn)題3節(jié)點(diǎn)三角形單元或4節(jié)點(diǎn)等參單元有限元方法的計(jì)算流程，以及單元?jiǎng)偠染仃?、等效?jié)點(diǎn)載荷、節(jié)點(diǎn)應(yīng)變、節(jié)點(diǎn)應(yīng)力和高斯積分等的計(jì)算公式。用Fortran語(yǔ)言編寫(xiě)彈性力學(xué)平面問(wèn)題3節(jié)點(diǎn)三角形單元或4節(jié)點(diǎn)等參單元的有限元程序。在VisualFortran程序集成開(kāi)發(fā)環(huán)境中完成有限元程序的編輯和調(diào)試工作。利用編寫(xiě)的有限元程序，計(jì)算算例，分析計(jì)算結(jié)果。撰寫(xiě)課程設(shè)計(jì)報(bào)告。目前資料收集情況〔含指定參考資料〕：1.王勖成，有限單元法，北京：高等教育出版社，2002。2.O.C.Zienkiewicz,R.L.Taylor,FiniteElementMethod,5thEition,McGraw-HallBookCompanyLimited,2000。3.張汝清，董明，結(jié)構(gòu)計(jì)算程序設(shè)計(jì)，重慶：重慶大學(xué)出版社，1988。課程設(shè)計(jì)的工作方案：第1周星期一上午：教師講解程序設(shè)計(jì)方法，程序設(shè)計(jì)要求和任務(wù)安排。第1周星期一至星期二完成程序框圖設(shè)計(jì)。第1周星期三至第2周星期四完成程序設(shè)計(jì)。第2周星期五完成課程設(shè)計(jì)報(bào)告。任務(wù)下達(dá)日期2013年6月6日完成日期2013年07月03日指導(dǎo)教師〔簽名〕學(xué)生〔簽名〕一、前言有限單元法是在當(dāng)今工程分析中獲得最廣泛應(yīng)用的數(shù)值計(jì)算方法，由于其通用性和有效性，受到工程技術(shù)界的高度重視。伴隨著計(jì)算機(jī)科學(xué)和技術(shù)的快速開(kāi)展，現(xiàn)已成為計(jì)算機(jī)輔助設(shè)計(jì)和計(jì)算機(jī)輔助制造的重要組成局部。由于有限元法是通過(guò)計(jì)算機(jī)實(shí)現(xiàn)的，因此有限元程序的編制及相關(guān)軟件的研發(fā)就變得尤為重要。從二十世紀(jì)五十年代以來(lái)，有限元軟件的開(kāi)展按目的和用途可分為專(zhuān)用軟件和大型通用商業(yè)軟件，而且軟件往往集成了網(wǎng)格自動(dòng)劃分，結(jié)果分析和顯示等前后處理功能，而且隨著時(shí)間的開(kāi)展，大型通用商業(yè)軟件的功能由線性擴(kuò)展到非線性，由結(jié)構(gòu)擴(kuò)展到非結(jié)構(gòu)等等，這一系列強(qiáng)大功能的實(shí)現(xiàn)與運(yùn)用都要求我們對(duì)有限元法的根底理論知識(shí)有較為清楚的認(rèn)識(shí)以及對(duì)程序編寫(xiě)的根本能力有較好掌握。1.元分析程序的理論根底作為彈性力學(xué)微分方程的等效積分形式，虛位移原理和虛應(yīng)原理分別是平衡方程與力的邊界條件和幾何方程與位移邊界條件的等效積分形式。將物理方程引入虛位移原理和虛應(yīng)力原理可以分別導(dǎo)出最小位能原理和最小余能原理，它們本質(zhì)上和等效積分的伽遼金“弱”形式相一致，這是建立彈性力學(xué)有限原方程一般表達(dá)格式的理論根底。對(duì)于通過(guò)彈性力學(xué)變分原理建立的彈性力學(xué)問(wèn)題有限元方法，其未知場(chǎng)變量是位移，以節(jié)點(diǎn)位移為根本未知量，并以最小位能原理為根底建立的有限單元為位移元。彈性力學(xué)平面問(wèn)題有限元分析表達(dá)格式的建立步驟一般為：a．的模型進(jìn)行單元離散，一般采用三角形單元或四邊形單元，不過(guò)由于四邊形單元具有更高精度，應(yīng)用更為普遍，一般而言一次或二次單元已足以滿足精度要求。離散后對(duì)單元進(jìn)行編號(hào)，并且給定單元各節(jié)點(diǎn)的整體編碼以及局部坐標(biāo)系下按逆時(shí)針局部編碼。b．坐標(biāo)系下對(duì)各單元構(gòu)造形函得到由單元各節(jié)點(diǎn)位移表示的單元位移形式，進(jìn)而得到單元?jiǎng)偠染仃?，利用等參元性質(zhì)和雅閣比矩陣進(jìn)行組集，建立整體剛度矩陣。c．等效結(jié)點(diǎn)載荷列陣并組集成結(jié)構(gòu)節(jié)點(diǎn)等效載荷列陣得到單元各節(jié)點(diǎn)位移與結(jié)構(gòu)節(jié)點(diǎn)等效載荷列陣的線代方程組，求解得到位移，進(jìn)而可得應(yīng)力應(yīng)變。根據(jù)以上一般步驟，編制相應(yīng)的計(jì)算機(jī)程序并采用數(shù)值積分方法處理有關(guān)數(shù)學(xué)計(jì)算就可以得到完整的彈性力學(xué)問(wèn)題有限元分析程序。問(wèn)題有限元分析教學(xué)程序本程序可對(duì)二維彈性力學(xué)問(wèn)題行有限元分析計(jì)算。計(jì)算采用的單元形式為四邊形四節(jié)點(diǎn)單元或者四邊形八節(jié)點(diǎn)單元，對(duì)于對(duì)稱和非對(duì)稱矩陣采用變帶寬存儲(chǔ)方法，最終輸出結(jié)果為單元各節(jié)點(diǎn)的位移。二、平面4、8節(jié)點(diǎn)有限元公式及計(jì)算原理2.1根本公式有限元平衡方程單元?jiǎng)偠染仃嚒?〕單元等效結(jié)點(diǎn)載荷2.2單元位移插值及坐標(biāo)變換通過(guò)Serendipity四邊形單元格式構(gòu)造插值函數(shù)。對(duì)于4節(jié)點(diǎn)單元，插值函數(shù)為：對(duì)于8節(jié)點(diǎn)單元，插值函數(shù)為：其中：〔i=1,2,3,4〕位移插值坐標(biāo)變換2.3單元?jiǎng)偠鹊挠?jì)算〔1〕彈性矩陣平面應(yīng)力問(wèn)題：；平面應(yīng)變問(wèn)題：；式中E和v分別為材料Young氏模量和Poisson比。應(yīng)變矩陣形狀函數(shù)對(duì)局部坐標(biāo)偏導(dǎo)：，形狀函數(shù)對(duì)整體坐標(biāo)偏導(dǎo)：Jacobi矩陣：Jacobi矩陣的逆：其中：?jiǎn)卧獎(jiǎng)偠染仃囉?jì)算單元?jiǎng)偠染仃嘒auss積分：?jiǎn)卧獎(jiǎng)偠染仃囎泳仃嚕骸瞚=1，2〕2.4等效結(jié)點(diǎn)載荷集中載荷直接施加在結(jié)點(diǎn)上體積力產(chǎn)生的等效結(jié)點(diǎn)載荷向量體積力產(chǎn)生的等效結(jié)點(diǎn)載荷Gauss積分：體積力產(chǎn)生的等效結(jié)點(diǎn)載荷子向量：其中：假設(shè)體積力作用方向于整體坐標(biāo)系的y軸正方向角度，那么：2.5單元應(yīng)變和應(yīng)力計(jì)算單元應(yīng)變單元中任意點(diǎn)的應(yīng)變：?jiǎn)卧腥我恻c(diǎn)的應(yīng)變：其中可作為Gauss積分點(diǎn)或單元結(jié)點(diǎn)的整體坐標(biāo)，其于局部坐標(biāo)間的關(guān)系為：,單元應(yīng)力三、程序結(jié)構(gòu)〔1〕有限元程序系統(tǒng)的組成及分析過(guò)程我們本次設(shè)計(jì)主要針對(duì)二維彈性力學(xué)靜力學(xué)問(wèn)題。有限元程序系統(tǒng)通常包括處理、有限元程序本體和后處理三局部。有限元軟件系統(tǒng)的組成及分析過(guò)程如下列圖所示：CAD模型CAD模型、材料參數(shù)、邊界條件、分析類(lèi)型定義、網(wǎng)格劃分等。有限元分析本體程序前處理后處理核心：各種計(jì)算方法的實(shí)現(xiàn)。以圖形、曲線和表格等方式表達(dá)、分析計(jì)算結(jié)果?！?〕有限元計(jì)算程序框圖輸入離散模型數(shù)據(jù)輸入離散模型數(shù)據(jù)型數(shù)據(jù)計(jì)算單元?jiǎng)偠汝嚱M集結(jié)構(gòu)剛度矩陣計(jì)算單元等效結(jié)點(diǎn)載荷組集結(jié)構(gòu)結(jié)點(diǎn)載荷列陣引入位移邊界條件求解線性方程組其它輔助計(jì)算輸出結(jié)果結(jié)束單元循環(huán)形成K形成P消除K的奇異求解Ka=P得結(jié)點(diǎn)位移a計(jì)算應(yīng)力、應(yīng)變等〔3〕有限元程序主要編程技巧a.結(jié)構(gòu)模塊化主程序：組織調(diào)用子程序，過(guò)程清晰，不涉及復(fù)雜計(jì)算。子模塊：實(shí)現(xiàn)特定功能，可用多個(gè)子程序組成。子程序：實(shí)現(xiàn)簡(jiǎn)單功能，便于調(diào)用和擴(kuò)展。b.覆蓋技術(shù)在主程序中定義一個(gè)整數(shù)組和一個(gè)實(shí)型數(shù)組，分別存放整型和實(shí)型邊界數(shù)組。設(shè)計(jì)動(dòng)態(tài)數(shù)組表。動(dòng)態(tài)數(shù)組覆蓋技術(shù)?！?〕有限元程序主程序翻開(kāi)輸入數(shù)據(jù)，調(diào)用PMESH，調(diào)用PROFIL計(jì)算K的對(duì)角元地址和方程號(hào)，分配數(shù)組大小，調(diào)用ASSEM組集整體K，調(diào)用PLOAD組集載荷列向量，調(diào)用DATRI對(duì)剛度矩陣進(jìn)行三角分解，調(diào)用DASOL解方程，調(diào)用PRTDIS輸出結(jié)點(diǎn)位移，調(diào)用TRESSS計(jì)算GAUSS點(diǎn)和結(jié)點(diǎn)應(yīng)變和應(yīng)力?！?〕變量說(shuō)明NUMNP：節(jié)點(diǎn)總數(shù)NUMEL：?jiǎn)卧倲?shù)NUMMAT：材料種類(lèi)NLOAD：載荷數(shù)NDM：每個(gè)節(jié)點(diǎn)坐標(biāo)方向數(shù)NDF：節(jié)點(diǎn)自由度NEN：每個(gè)單元節(jié)點(diǎn)數(shù)。ID〔K，J〕：節(jié)點(diǎn)J的第K個(gè)自由度約束編號(hào)X〔K，J〕：節(jié)點(diǎn)J的第K個(gè)坐標(biāo)IX〔K，J〕：J單元中K節(jié)點(diǎn)的全局節(jié)點(diǎn)編號(hào)F〔K，J〕：結(jié)點(diǎn)J第K方向上的集中載荷F〔NDF，NLOAD〕：集中載荷X〔NDM，NUMNP〕：結(jié)點(diǎn)坐標(biāo)IX〔NEN，NUMEL〕：?jiǎn)卧xF(K,J)：節(jié)點(diǎn)J在K方向上集中載荷大小ALFL=TRUE：非對(duì)稱矩陣集合ALFL=FALSE：對(duì)稱矩陣集合EE：楊氏彈性模量XNU：泊松比ITYPE：?jiǎn)栴}類(lèi)型Ｓ：?jiǎn)卧獎(jiǎng)偠染仃嚕校狠d荷和內(nèi)力向量AD：對(duì)角元素AU：上三角元素AL：下三角元素JP：上〔下〕三角元素每行〔列〕最后元素LD：一個(gè)單元中各自由度方程數(shù)JD：列高ID：邊界條件，=0：無(wú)約束；=1：固定約束。〔6〕數(shù)據(jù)輸入格式整數(shù)：I5格式小數(shù)：F10.4格式〔7〕程序本體〔見(jiàn)附錄〕程序驗(yàn)證算例算例：平面板受軸向拉伸長(zhǎng)3m寬0.4m的平面板左端鉸接固定，右端受到水平方向向右、大小為10N的均布載荷，模型如下列圖所示：F=10NF=10NF=10N1284519F=10NF=10NF=10N1284519其中結(jié)構(gòu)參數(shù)：F=10N,L=3m,h=0.4m材料參數(shù)：E=200GPa,ν=0.31、建立坐標(biāo)系，將板劃分成6個(gè)如上圖所示的四邊形四節(jié)點(diǎn)單元，進(jìn)行單元編號(hào)，均布載荷處理為右端三節(jié)點(diǎn)每個(gè)節(jié)點(diǎn)受水平向右大小10N的力，豎直方向不受力。1.1程序input文件中輸入：example(NEN=4)126132241110.00000.00002001.00000.00003002.00000.00004003.00000.00005110.00000.20006001.00000.20007002.00000.20008003.00000.20009110.00000.400010001.00000.400011002.00000.400012003.00000.400011265223763348745610956711106781211200.0E+90.30001410.00000.0000810.00000.00001210.00000.00001.2調(diào)試程序得到結(jié)果為：nodaldisplacements10.0000E+000.0000E+0020.3638E-090.2813E-1030.7369E-090.2030E-1040.1126E-080.2437E-1050.0000E+000.0000E+0060.3609E-09-0.2981E-2370.7429E-09-0.1095E-2280.1104E-08-0.2132E-2290.0000E+000.0000E+00100.3638E-09-0.2813E-10110.7369E-09-0.2030E-10120.1126E-08-0.2437E-102、建立坐標(biāo)系，將板劃分成6個(gè)如上圖所示的四邊形八節(jié)點(diǎn)單元，進(jìn)行單元編號(hào)，均布載荷處理為右端三節(jié)點(diǎn)每個(gè)節(jié)點(diǎn)受水平向右大小10N的力，豎直方向不受力。其中結(jié)構(gòu)參數(shù)：F=10N,L=3m,h=0.4m材料參數(shù)：E=200GPa,ν=0.32.1程序input文件中輸入：example(NEN=8)296132281110.00000.00002001.00000.00003002.00000.00004003.00000.00005110.00000.20006001.00000.20007002.00000.20008003.00000.20009110.00000.400010001.00000.400011002.00000.400012003.00000.400013000.50000.000014001.50000.000015002.50000.000016110.00000.100017001.00000.100018002.00000.100019003.00000.100020000.50000.200021001.50000.200022002.50000.200023110.00000.300024001.00000.300025002.00000.300026003.00000.300027000.50000.400028001.50000.400029002.50000.400011265131720162237614182117334871519221845610920242723567111021252824678121122262925200.0E+90.30001410.00000.0000810.00000.00001210.00000.00002.2調(diào)試程序得到結(jié)果為：nodaldisplacements10.0000E+000.0000E+0020.3734E-090.2492E-1030.7549E-090.4768E-1040.1185E-080.6785E-1050.0000E+000.0000E+0060.3770E-09-0.1562E-2270.7371E-09-0.3614E-2280.1086E-08-0.7600E-2290.0000E+000.0000E+00100.3734E-09-0.2492E-10110.7549E-09-0.4768E-10120.1185E-08-0.6785E-10130.1849E-090.2673E-10140.5567E-090.1616E-10150.9277E-090.1568E-11160.0000E+000.0000E+00170.3737E-090.1232E-10180.7517E-090.2430E-10190.1114E-080.3177E-10200.1810E-09-0.5215E-23210.5559E-09-0.2565E-22220.9462E-09-0.5238E-22230.0000E+000.0000E+00240.3737E-09-0.1232E-10250.7517E-09-0.2430E-10260.1114E-08-0.3177E-10270.1849E-09-0.2673E-10280.5567E-09-0.1616E-10290.9277E-09-0.1568E-113.數(shù)據(jù)分析其理論計(jì)算過(guò)程如下，對(duì)于平面應(yīng)力問(wèn)題有：QUOTEQUOTE其中QUOTE：代入計(jì)算得任一距原點(diǎn)水平距離為x的點(diǎn)其位移為：理論解與數(shù)值解比照?qǐng)D如下：可見(jiàn)接近自由端處兩曲線很接近，當(dāng)X=3m時(shí)，u=15x10E-9m，對(duì)應(yīng)輸出文件中的節(jié)點(diǎn)3、7、11的輸出值和3、7、11點(diǎn)有限元解較為接近，誤差為很小。同時(shí)可以發(fā)現(xiàn)靠近固定端處理論解與數(shù)值解誤差變大。分別對(duì)x=2處的理論解和有限元解進(jìn)行比擬發(fā)現(xiàn)誤差較小。考慮到實(shí)際情況固定端的鉸支約束條件對(duì)實(shí)際位移的影響以及離散單元數(shù)目較少等可能造成的誤差，可以認(rèn)為該程序是正確的。另外數(shù)值由可以看出同的對(duì)稱性，符合根本力學(xué)理論結(jié)果。程序可用于平面問(wèn)題的求解，可獲得較好的位移分布。程序適用于平面應(yīng)力、平面應(yīng)變狀態(tài)結(jié)構(gòu)4、8節(jié)點(diǎn)單元有限元分析，可求出結(jié)構(gòu)的位移、應(yīng)力以及應(yīng)變。五、算例1、算例：選擇超靜定梁中點(diǎn)受集中載荷作用的結(jié)構(gòu)，如下列圖所示：其中結(jié)構(gòu)參數(shù)：F=10kN,L=4m,h=0.4m材料參數(shù)：E=200GPa,ν=0.32、單元網(wǎng)格劃分及規(guī)定對(duì)上述結(jié)構(gòu)采用10×1,10×2三種單元網(wǎng)格劃分形式，分別采用四結(jié)點(diǎn)和八結(jié)點(diǎn)四邊形單元。規(guī)定單元編號(hào)先沿y正方向，再沿x正方向。為了方便比擬，所有編號(hào)位置固定，例如第12結(jié)點(diǎn)，不管是10×1單元四結(jié)點(diǎn)，還是10×1單元八結(jié)點(diǎn)，其位置不變。其不同之處在于八結(jié)點(diǎn)增加的結(jié)點(diǎn)編號(hào)在四結(jié)點(diǎn)的根底上插入，而不是替換原有的結(jié)點(diǎn)編號(hào)。特殊點(diǎn)如下所示：1161121722116112172212331162817123221233116281712322(1).10X1單元四結(jié)點(diǎn)input文件中輸入：WSJ_10X1(NEN=4)2210112241110.00000.00002000.40000.00003000.80000.00004001.20000.00005001.60000.00006002.00000.00007002.40000.00008002.80000.00009003.20000.000010003.60000.000011114.00000.000012114.00000.400013003.60000.400014003.20000.400015002.80000.400016002.40000.400017002.00000.400018001.60000.400019001.20000.400020000.80000.400021000.40000.400022110.00000.400011221222232021334192044518195561718667161777815168891415991013141010111213200.0E+90.30001170.0000-10.0E+310X1單元四結(jié)點(diǎn)input文件中輸出：nodaldisplacements10.0000E+000.0000E+002-0.2029E-06-0.2673E-063-0.3048E-06-0.8380E-064-0.3055E-06-0.1509E-055-0.2059E-06-0.2083E-0560.2994E-21-0.2330E-0570.2059E-06-0.2083E-0580.3055E-06-0.1509E-0590.3048E-06-0.8380E-06100.2029E-06-0.2673E-06110.0000E+000.0000E+00120.0000E+000.0000E+0013-0.2015E-06-0.2671E-0614-0.3018E-06-0.8376E-0615-0.3012E-06-0.1510E-0516-0.1986E-06-0.2077E-0517-0.1853E-21-0.2365E-05180.1986E-06-0.2077E-05190.3012E-06-0.1510E-05200.3018E-06-0.8376E-06210.2015E-06-0.2671E-06220.0000E+000.0000E+00(2).10X1單元八結(jié)點(diǎn)input文件中輸入：WSJ_10X1(NEN=8)5310112281110.00000.00002000.40000.00003000.80000.00004001.20000.00005001.60000.00006002.00000.00007002.40000.00008002.80000.00009003.20000.000010003.60000.000011114.00000.000012114.00000.400013003.60000.400014003.20000.400015002.80000.400016002.40000.400017002.00000.400018001.60000.400019001.20000.400020000.80000.400021000.40000.400022110.00000.400023000.20000.000024000.60000.000025001.00000.000026001.40000.000027001.80000.000028002.20000.000029002.60000.000030003.00000.000031003.40000.000032003.80000.000033114.00000.200034003.60000.200035003.20000.200036002.80000.200037002.40000.200038002.00000.200039001.60000.200040001.20000.200041000.80000.200042000.40000.200043110.00000.200044000.20000.400045000.60000.400046001.00000.400047001.40000.400048001.80000.400049002.20000.400050002.60000.400051003.00000.400052003.40000.400053003.80000.40001122122234244432232021244145423341920254046414451819263947405561718273848396671617283749387781516293650378891415303551369910131431345235101011121332335334200.0E+90.30001170.0000-10.0E+310X1單元八結(jié)點(diǎn)input文件中輸出：nodaldisplacements10.0000E+000.0000E+002-0.2896E-06-0.3882E-063-0.4446E-06-0.1198E-054-0.4464E-06-0.2169E-055-0.2954E-06-0.2984E-056-0.1067E-19-0.3336E-0570.2954E-06-0.2984E-0580.4464E-06-0.2169E-0590.4446E-06-0.1198E-05100.2896E-06-0.3882E-06110.0000E+000.0000E+00120.0000E+000.0000E+0013-0.2881E-06-0.3880E-0614-0.4419E-06-0.1198E-0515-0.4415E-06-0.2170E-0516-0.2928E-06-0.2992E-05170.1160E-19-0.3406E-05180.2928E-06-0.2992E-05190.4415E-06-0.2170E-05200.4419E-06-0.1198E-05210.2881E-06-0.3880E-06220.0000E+000.0000E+0023-0.1582E-06-0.1219E-0624-0.3874E-06-0.7546E-0625-0.4638E-06-0.1684E-0526-0.3906E-06-0.2617E-0527-0.1666E-06-0.3254E-05280.1666E-06-0.3254E-05290.3906E-06-0.2617E-05300.4638E-06-0.1684E-05310.3874E-06-0.7546E-06320.1582E-06-0.1219E-06330.0000E+000.0000E+00340.7195E-09-0.3635E-06350.1556E-08-0.1195E-05360.2019E-08-0.2174E-05370.3347E-08-0.3008E-05380.7962E-21-0.3388E-0539-0.3347E-08-0.3008E-0540-0.2019E-08-0.2174E-0541-0.1556E-08-0.1195E-0542-0.7195E-09-0.3635E-06430.0000E+000.0000E+00440.1574E-06-0.1217E-06450.3852E-06-0.7543E-06460.4599E-06-0.1684E-05470.3863E-06-0.2615E-05480.1608E-06-0.3252E-0549-0.1608E-06-0.3252E-0550-0.3863E-06-0.2615E-0551-0.4599E-06-0.1684E-0552-0.3852E-06-0.7543E-0653-0.1574E-06-0.1217E-06(3).10X2單元四結(jié)點(diǎn)input文件中輸入：WSJ_10X2(NEN=4)3320112241110.00000.00002000.40000.00003000.80000.00004001.20000.00005001.60000.00006002.00000.00007002.40000.00008002.80000.00009003.20000.000010003.60000.000011114.00000.000012114.00000.400013003.60000.400014003.20000.400015002.80000.400016002.40000.400017002.00000.400018001.60000.400019001.20000.400020000.80000.400021000.40000.400022110.00000.400023110.00000.200024000.40000.200025000.80000.200026001.20000.200027001.60000.200028002.00000.200029002.40000.200030002.80000.200031003.20000.200032003.60000.200033114.00000.2000112242322325243342625445272655628276672928778302988931309910323110101133321123242122122425202113252619201426271819152728171816282916171729301516183031141519313213142032331213200.0E+90.30001170.0000-10.0E+310X2單元四結(jié)點(diǎn)input文件中輸出：nodaldisplacements10.0000E+000.0000E+002-0.2089E-06-0.2816E-063-0.3174E-06-0.8637E-064-0.3188E-06-0.1558E-055-0.2117E-06-0.2146E-056-0.8065E-20-0.2403E-0570.2117E-06-0.2146E-0580.3188E-06-0.1558E-0590.3174E-06-0.8637E-06100.2089E-06-0.2816E-06110.0000E+000.0000E+00120.0000E+000.0000E+0013-0.2074E-06-0.2813E-0614-0.3146E-06-0.8635E-0615-0.3136E-06-0.1558E-0516-0.2072E-06-0.2139E-05170.5719E-20-0.2439E-05180.2072E-06-0.2139E-05190.3136E-06-0.1558E-05200.3146E-06-0.8635E-06210.2074E-06-0.2813E-06220.0000E+000.0000E+00230.0000E+000.0000E+0024-0.7077E-09-0.2647E-0625-0.1592E-08-0.8609E-0626-0.1668E-08-0.1561E-0527-0.5226E-08-0.2158E-0528-0.1099E-20-0.2429E-05290.5226E-08-0.2158E-05300.1668E-08-0.1561E-05310.1592E-08-0.8609E-06320.7077E-09-0.2647E-06330.0000E+000.0000E+00(4).10X2單元八結(jié)點(diǎn)input文件中輸入：WSJ_10X2(NEN=8)8520112281110.00000.00002000.40000.00003000.80000.00004001.20000.00005001.60000.00006002.00000.00007002.40000.00008002.80000.00009003.20000.000010003.60000.000011114.00000.000012114.00000.400013003.60000.400014003.20000.400015002.80000.400016002.40000.400017002.00000.400018001.60000.400019001.20000.400020000.80000.400021000.40000.400022110.00000.400023110.00000.200024000.40000.200025000.80000.200026001.20000.200027001.60000.200028002.00000.200029002.40000.200030002.80000.200031003.20000.200032003.60000.200033114.00000.200034000.20000.000035000.60000.000036001.00000.000037001.40000.000038001.80000.000039002.20000.000040002.60000.000041003.00000.000042003.40000.000043003.80000.000044114.00000.100045003.60000.100046003.20000.100047002.80000.100048002.40000.100049002.00000.100050001.60000.100051001.20000.100052000.80000.100053000.40000.100054110.00000.100055000.20000.200056000.60000.200057001.00000.200058001.40000.200059001.80000.200060002.20000.200061002.60000.200062003.00000.200063003.40000.200064003.80000.200065114.00000.300066003.60000.300067003.20000.300068002.80000.300069002.40000.300070002.00000.300071001.60000.300072001.20000.300073000.80000.300074000.40000.300075110.00000.300076000.20000.400077000.60000.400078001.00000.400079001.40000.400080001.80000.400081002.20000.400082002.60000.400083003.00000.400084003.40000.400085003.80000.400011224233453555422325243552565333426253651575244527263750585155628273849595066729283948604977830294047614888931304146624799103231424563461010113332434464451123242122557476751224252021567377741325261920577278731426271819587179721527281718597080711628291617606981701729301516616882691830311415626783681931321314636684672032331213646585662.00E+110.30001170.0000-10.0E+310X2單元八結(jié)點(diǎn)input文件中輸出：nodaldisplacements10.0000E+000.0000E+002-0.3021E-06-0.3948E-063-0.4523E-06-0.1231E-054-0.4533E-06-0.2217E-055-0.3066E-06-0.3052E-056-0.6288E-19-0.3413E-0570.3066E-06-0.3052E-0580.4533E-06-0.2217E-0590.4523E-06-0.1231E-05100.3021E-06-0.3948E-06110.0000E+000.0000E+00120.0000E+000.0000E+0013-0.3006E-06-0.3946E-0614-0.4493E-06-0.1231E-0515-0.4483E-06-0.2217E-0516-0.2990E-06-0.3059E-05170.5240E-19-0.3488E-05180.2990E-06-0.3059E-05190.4483E-06-0.2217E-05200.4493E-06-0.1231E-05210.3006E-06-0.3946E-06220.0000E+000.0000E+00230.0000E+000.0000E+0024-0.7495E-09-0.3869E-0625-0.1469E-08-0.1227E-0526-0.2015E-08-0.2219E-0527-0.7048E-09-0.3060E-0528-0.6477E-20-0.3443E-05290.7048E-09-0.3060E-05300.2015E-08-0.2219E-05310.1469E-08-0.1227E-05320.7495E-09-0.3869E-06330.0000E+000.0000E+0034-0.1663E-06-0.1340E-0635-0.3955E-06-0.7793E-0636-0.4707E-06-0.1723E-0537-0.3970E-06-0.2668E-0538-0.1670E-06-0.3315E-05390.1670E-06-0.3315E-05400.3970E-06-0.2668E-05410.4707E-06-0.1723E-05420.3955E-06-0.7793E-06430.1663E-06-0.1340E-06440.0000E+000.0000E+00450.1448E-06-0.3846E-06460.2216E-06-0.1228E-05470.2224E-06-0.2219E-05480.1473E-06-0.3061E-0549-0.3274E-19-0.3433E-0550-0.1473E-06-0.3061E-0551-0.2224E-06-0.2219E-0552-0.2216E-06-0.1228E-0553-0.1448E-06-0.3846E-06540.0000E+000.0000E+0055-0.3552E-09-0.1062E-0656-0.1107E-08-0.7656E-0657-0.1899E-08-0.1724E-0558-0.2739E-08-0.2682E-0559-0.7310E-08-0.3343E-05600.7310E-08-0.3343E-05610.2739E-08-0.2682E-05620.1899E-08-0.1724E-05630.1107E-08-0.7656E-06640.3552E-09-0.1062E-06650.0000E+000.0000E+0066-0.1433E-06-0.3845E-0667-0.2186E-06-0.1228E-0568-0.2181E-06-0.2219E-0569-0.1418E-06-0.3065E-05700.2054E-19-0.3470E-05710.1418E-06-0.3065E-05720.2181E-06-0.2219E-05730.2186E-06-0.1228E-05740.1433E-06-0.3845E-06750.0000E+000.0000E+00760.1655E-06-0.1338E-06770.3933E-06-0.7791E-06780.4670E-06-0.1723E-05790.3924E-06-0.2668E-05800.1669E-06-0.3311E-0581-0.1669E-06-0.3311E-0582-0.3924E-06-0.2668E-0583-0.4670E-06-0.1723E-0584-0.3933E-06-0.7791E-0685-0.1655E-06-0.1338E-063.變形圖用ANSYS軟件劃分網(wǎng)格繪制變形圖如下：A.10X1單元.10X1單元四結(jié)點(diǎn)(2).10X1單元八結(jié)點(diǎn)B.10X2單元，同理可以畫(huà)出變形圖4.有限元程序Fortran計(jì)算結(jié)果比擬10X1單元四結(jié)點(diǎn)、10X1單元八結(jié)點(diǎn)、10X2單元四結(jié)點(diǎn)、10X2單元八結(jié)點(diǎn)的output比擬如下〔選取局部結(jié)點(diǎn)比擬〕：(1).10X1單元四結(jié)點(diǎn)和10X1單元八結(jié)點(diǎn)比擬〔選取結(jié)點(diǎn)6和結(jié)點(diǎn)17〕10X1單元四結(jié)點(diǎn)：60.2994E-21-0.2330E-0517-0.1853E-21-0.2365E-0510X1單元八結(jié)點(diǎn)：6-0.1067E-19-0.3336E-05170.1160E-19-0.3406E-05(2).10X1單元四結(jié)點(diǎn)和10X2單元四結(jié)點(diǎn)比擬〔選取結(jié)點(diǎn)6和結(jié)點(diǎn)17〕10X1單元四結(jié)點(diǎn)：60.2994E-21-0.2330E-0517-0.1853E-21-0.2365E-0510X2單元四結(jié)點(diǎn)：6-0.8065E-20-0.2403E-05170.5719E-20-0.2439E-05(3).10X1單元四結(jié)點(diǎn)和10X2單元八結(jié)點(diǎn)比擬〔選取結(jié)點(diǎn)6和結(jié)點(diǎn)17〕10X1單元四結(jié)點(diǎn)：60.2994E-21-0.2330E-0517-0.1853E-21-0.2365E-0510X2單元八結(jié)點(diǎn)：6-0.6288E-19-0.3413E-05170.5240E-19-0.3488E-05求解理論解此算例的彎矩圖如下列圖，橫截面尺寸如下列圖：由材料力學(xué)知識(shí)可以得到彎矩圖，由圖乘計(jì)算可以求出l/2處位移為：其中：，，，，，代入可求得：分析因結(jié)構(gòu)是兩端固定，所以在中間位移變形最大，取梁中間上下兩點(diǎn)〔分別是17結(jié)點(diǎn)和6結(jié)點(diǎn)〕進(jìn)行結(jié)果分析。分別用Fortran、ANSYS、理論解表示。由于17結(jié)點(diǎn)和6結(jié)點(diǎn)兩點(diǎn)處于中間位置，對(duì)稱，所以X方向位移為零。結(jié)點(diǎn)X位移類(lèi)型10X1-410X1-810X2-410X2-817〔Fortran〕-0.2365E-05-0.3406E-05-0.2439E-05-0.3488E-0517〔ANSYS〕-0.236E-05-0.341E-0517〔理論解〕-0.3125E-05-0.3125E-05-0.3125E-05-0.3125E-05由上表可看出，F(xiàn)ortran和ANSYS計(jì)算結(jié)果幾乎一樣，所以只要分析Fortran的計(jì)算結(jié)果和理論解之間的誤差。百分誤差如下表：10X1-410X1-810X2-410X2-817〔Fortran〕-0.2365E-05-0.3406E-05-0.2439E-05-0.3488E-0517理論解-0.3125E-05-0.3125E-05-0.3125E-05-0.3125E-0517結(jié)點(diǎn)百分誤差24.3%8.99%21.9%11.6