懸臂梁MATLAB有限元算例注釋

1、用有限元法對懸臂梁分析的算例算例:h = 10mm。如下圖所示的懸臂梁，受均布載荷q = 1N/ mm2作用。E = 2. 1 x 105n/mm2,卩=0.3厚度現(xiàn)用有限元法分析其位移及應(yīng)力。jt5X | = 555l、52、開梁可視為平面應(yīng)力狀態(tài)，先按圖示尺寸劃分為均勻的三角形網(wǎng)格，共有8X 10= 80個(gè)單元,個(gè)節(jié)點(diǎn)，坐標(biāo)軸以及單元與節(jié)點(diǎn)的編號如圖。將均布載荷分配到各相應(yīng)節(jié)點(diǎn)上，把有約束的節(jié)點(diǎn) 53、54、55視作固定鉸鏈，建立如圖所示的離散化計(jì)算模型。程序計(jì)算框圖J （續(xù)左）處理根部約束，修改【K】【Q】程序中的函數(shù)功能介紹及源代碼1. LinearTriangleElementSti

2、ffness(E,NU,t,xi,yi,xj,yj,xm,ym) 該函數(shù)用于計(jì)算平面應(yīng)力情況下彈性模量為E、泊松比為NU厚度為t、第一個(gè)節(jié)點(diǎn)坐標(biāo)為(xi,yi)、第二個(gè)節(jié)點(diǎn)坐標(biāo)為(xj,yj)、第三個(gè)節(jié) 點(diǎn)坐標(biāo)為(xm, ym)時(shí)的線性三角形元的單元?jiǎng)偠染仃?該函數(shù)返回6X 6的單位剛度矩陣k.2. LinearTriangleAssemble(K,k,i,j,m) 該函數(shù)將連接節(jié)點(diǎn) i,j,m 的線性三角形元的單元?jiǎng)偠染仃?k 集成到整體剛度矩陣 K。每集成一個(gè)單元，該函數(shù)都將返回2NX 2N的整體剛度矩陣K.3. LinearTriangleElementStresses(E,NU,t,x

3、i,yi,xj,yj,xm,ym,u)-該函數(shù)計(jì)算在平面應(yīng)力情況下 彈性模量為E、泊松比為NU厚度為t、第一個(gè)節(jié)點(diǎn)坐標(biāo)為(xi,yi) 第二個(gè)節(jié)點(diǎn)坐標(biāo)為(xj,yj)、第三個(gè)節(jié)點(diǎn)坐標(biāo) 為(xm, ym)以及單元位移矢量為 u時(shí)的單元應(yīng)力。該函數(shù)返回單元應(yīng)力矢量。函數(shù)源代碼：function y = LinearTriangleElementStiffness(E,NU,t,xi,yi,xj,yj,xm,ym)A = (xi*(yj-ym) + xj*(ym-yi) + xm*(yi-yj)/2;%三角形單元面積,單元節(jié)點(diǎn)應(yīng)該按逆時(shí)針排序,保證每個(gè)三角形單元的面積都為正值(也可作為一個(gè)小函數(shù)：

4、LinearTriangleElementArea ) betai = yj-ym;betaj = ym-yi;betam = yi-yj;gammai = xm-xj;gammaj = xi-xm;gammam = xj-xi;B = betai 0 betaj 0 betam 0 ;0 gammai 0 gammaj 0 gammam ;gammai betai gammaj betaj gammam betam/(2*A);%B為應(yīng)變矩陣，其中 betai=yi-ym,betaj=ym-yi,betam=yi-yj.gammai=xm-xj, gammaj=xi-xm, gammam=x

5、j-xi.D = (E/(1-NU*NU)*1 NU 0 ; NU 1 0 ; 0 0 (1-NU)/2;%助彈性矩陣，分為平面應(yīng)力問題和平面應(yīng)變問題對于平面應(yīng)力問題 D = (E/(1-NU*NU)*1 NU 0 ; NU 1 0 ; 0 0 (1-NU)/2;對于平面應(yīng)變問題 E1=E/(1-NU*NU)， NU1=NU(/ 1-NU)y = t*A*B'*D*B;%單元?jiǎng)偠染仃噁unction y = LinearTriangleAssemble(K,k,i,j,m)K(2*i-1,2*i-1) = K(2*i-1,2*i-1) + k(1,1); K(2*i-1,2*i) =

6、K(2*i-1,2*i) + k(1,2);K(2*i-1,2*j-1) = K(2*i-1,2*j-1) + k(1,3); K(2*i-1,2*j) = K(2*i-1,2*j) + k(1,4);K(2*i-1,2*m-1) = K(2*i-1,2*m-1) + k(1,5); K(2*i-1,2*m) = K(2*i-1,2*m) + k(1,6);K(2*i,2*i-1) = K(2*i,2*i-1) + k(2,1);K(2*i,2*i) = K(2*i,2*i) + k(2,2);K(2*i,2*j-1) = K(2*i,2*j-1) + k(2,3);K(2*i,2*j) =

7、K(2*i,2*j) + k(2,4);K(2*i,2*m-1) = K(2*i,2*m-1) + k(2,5); K(2*i,2*m) = K(2*i,2*m) + k(2,6);K(2*j-1,2*i-1) = K(2*j-1,2*i-1) + k(3,1); K(2*j-1,2*i) = K(2*j-1,2*i) + k(3,2);K(2*j-1,2*j-1) = K(2*j-1,2*j-1) + k(3,3); K(2*j-1,2*j) = K(2*j-1,2*j) + k(3,4);K(2*j-1,2*m-1) = K(2*j-1,2*m-1) + k(3,5); K(2*j-1,2

8、*m) = K(2*j-1,2*m) + k(3,6);K(2*j,2*i-1) = K(2*j,2*i-1) + k(4,1);K(2*j,2*i) = K(2*j,2*i) + k(4,2);K(2*j,2*j-1) = K(2*j,2*j-1) + k(4,3);K(2*j,2*j) = K(2*j,2*j) + k(4,4);K(2*j,2*m-1) = K(2*j,2*m-1) + k(4,5); K(2*j,2*m) = K(2*j,2*m) + k(4,6);K(2*m-1,2*i-1) = K(2*m-1,2*i-1) + k(5,1); K(2*m-1,2*i) = K(2*

9、m-1,2*i) + k(5,2);K(2*m-1,2*j-1) = K(2*m-1,2*j-1) + k(5,3); K(2*m-1,2*j) = K(2*m-1,2*j) + k(5,4);K(2*m-1,2*m-1) = K(2*m-1,2*m-1) + k(5,5); K(2*m-1,2*m) = K(2*m-1,2*m) + k(5,6);K(2*m,2*i-1) = K(2*m,2*i-1) + k(6,1); K(2*m,2*i) = K(2*m,2*i) + k(6,2);K(2*m,2*j-1) = K(2*m,2*j-1) + k(6,3); K(2*m,2*j) = K(

10、2*m,2*j) + k(6,4);K(2*m,2*m-1) = K(2*m,2*m-1) + k(6,5); K(2*m,2*m) = K(2*m,2*m) + k(6,6);y = K; %對號入座，如前所述，每集成一次都將返回2NX 2N的整體剛度矩陣 K.此題為110 X 110function y = LinearTriangleElementStresses(E,NU,t,xi,yi,xj,yj,xm,ym,u)A = (xi*(yj-ym) + xj*(ym-yi) + xm*(yi-yj)/2;betai = yj-ym;betaj = ym-yi;betam = yi-yj;

11、gammai = xm-xj;gammaj = xi-xm;gammam = xj-xi;B = betai 0 betaj 0 betam 0 ;0 gammai 0 gammaj 0 gammam ;gammai betai gammaj betaj gammam betam/(2*A);D = (E/(1-NU*NU)*1 NU 0 ; NU 1 0 ; 0 0 (1-NU)/2;%平面應(yīng)力和平面應(yīng)變問題兩種情況y = D*B*u; %單元應(yīng)力計(jì)算主程序源代碼%選取 2 個(gè)基本單%外部荷載，此處不E=21e7;NU=0.3;t=0.01; stifflike5=LinearTriangl

12、eElementStiffness(E,NU,t,0.4,0.08,0.36,0.08,0.36,0.06,1) 元，調(diào)用M文件 stifflike1=LinearTriangleElementStiffness(E,NU,t,0.4,0.08,0.36,0.06,0.4,0.06,1) K=sparse(110,110); %creat a xishu matrix for total stiff 創(chuàng)建一個(gè)稀疏矩陣 for i=1:49if rem(i,5)%模取余，bool 型變量，非零即為真j=i;K=LinearTriangleAssemble(K,stifflike5,j,j+5,j

13、+6);%節(jié)點(diǎn)編號K=LinearTriangleAssemble(K,stifflike1,j,j+6,j+1);enden d%將每個(gè)單元?jiǎng)偠染仃嚰傻娇倓傊蠯=full(K);%轉(zhuǎn)化稀疏矩陣 k=K(1:100,1:100);k=K,zeros(100,10);zeros(10,100),eye(10); k=sparse(k);%利用邊界條件簡化基本方程Q=sparse(2:10:92,1,-200,-400,-400,-400,-400,-400,-400,-400,-400,-400,110,1); 包括約束條件，通過形函數(shù)確定，是不是可以理解為梁的兩端為中間的一半呢？ d=kQ;

14、 %高斯消元法，比克萊姆法則在計(jì)算速度上有絕對的優(yōu)勢！ x=0:0.04:0.4;plot(x,d(106:-10:6)%基本繪圖命令grid %帶網(wǎng)格 y=zeros(80,3); q=0;for i=1:49switch rem(i,5)case 1j=2*i;u=d(j-1) d(j) d(j+11) d(j+12) d(j+1) d(j+2);u=u'xl=0.4;yl=0.08;xm=0.36;ym=0.06;xn=0.4;yn=0.06; y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)&#

15、39; xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 2j=2*i;u=d(j-1) d(j) d(j+11) d(j+12) d(j+1) d(j+2);u=u'xl=0.4;yl=0.06;xm=0.36;ym=0.04;xn=0.4;yn=0.04; y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)' xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 3j=2*i;u=d(j-1) d(j) d(j+11) d(j+12) d(j+1

16、) d(j+2);u=u'xl=0.4;yl=0.04;xm=0.36;ym=0.02;xn=0.4;yn=0.02; y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)' xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 4j=2*i;u=d(j-1) d(j) d(j+11) d(j+12) d(j+1) d(j+2);u=u'xl=0.4;yl=0.02;xm=0.36;ym=0;xn=0.4;yn=0;y(i+q,:)=LinearTriangleElem

17、entStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)'xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;otherwiseq=q+3;endendq=4;for i=1:49switch rem(i,5)case 1j=2*i;u=d(j-1) d(j) d(j+9) d(j+10) d(j+11) d(j+12);u=u'xl=0.4;yl=0.08;xm=0.36;ym=0.08;xn=0.36;yn=0.06;y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,

18、yn,u)'xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 2j=2*i;u=d(j-1) d(j) d(j+9) d(j+10) d(j+11) d(j+12);u=u'xl=0.4;yl=0.06;xm=0.36;ym=0.06;xn=0.36;yn=0.04; y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)'xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 3j=2*i;u=d(j-1) d(j) d(j+9) d(j+10)

19、 d(j+11) d(j+12);u=u'xl=0.4;yl=0.04;xm=0.36;ym=0.04;xn=0.36;yn=0.02;y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)'xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;case 4j=2*i;u=d(j-1) d(j) d(j+9) d(j+10) d(j+11) d(j+12);u=u'xl=0.4;yl=0.02;xm=0.36;ym=0.02;xn=0.36;yn=0;y(i+q,:)=LinearTriangleElementStresses(E,NU,t,xl,yl,xm,ym,xn,yn,u)'xl=xl-0.04;xm=xm-0.04;xn=xn-0.04;otherwis