将此结构按三角形单元划分成432个三角形（X方向分成18段，Y方向分成12段），总共分成19X13=247个结点的有限元模型，具体步骤详细程序如下：
tic;
Initial_info=[0.09 0.06 18 12];
disp(该程序计算的是',num2str(Initial_info(3)+1),'X',num2str(Initial_info(4)+1),'=',...
num2str((Initial_info(3)+1)*(Initial_info(4)+1)),'个结点的有限元模型']);
LX=Initial_info(1);
LY=Initial_info(2);
nx=Initial_info(3);
ny=Initial_info(4);
ne=2*nx*ny;
np=(nx+1)*(ny+1);
for i=1:nx+1;
j=1:ny+1;
Np(i,j)=j+(i-1)*(ny+1);
end
生成节点编号矩阵Np
for i=1:nx+1;
j=1:ny+1;
XX(i,j)=(i-1)*LX/nx;
YY(i,j)=(j-1)*LY/ny;
end
XY=[reshape(XX',np,1),reshape(YY',np,1)];
nx2=nx/2;
Np1=Np(1:nx2+1,:);
Np2=Np(nx2+1:end,:);
for i=1:nx2*ny;
if rem(i,nx2)==0
xp=nx2;
yp=i/nx2;
else
xp=rem(i,nx2);
yp=fix(i/nx2)+1;
end
Dof1(i,:)=[Np1(xp,yp),Np1(xp+1,yp),Np1(xp,yp+1)];
Dof1(i+nx2*ny,:)=[Np1(xp+1,yp),Np1(xp+1,yp+1),Np1(xp,yp+1)]; Dof2(i,:)=[Np2(xp,yp),Np2(xp+1,yp),Np2(xp+1,yp+1)];
Dof2(i+nx2*ny,:)=[Np2(xp,yp),Np2(xp+1,yp+1),Np2(xp,yp+1)];
end
Dof=[Dof1;Dof2];
for i=1:ne
unit(i,:)=[XY(Dof(i,1),1),XY(Dof(i,2),1),XY(Dof(i,3),1),...
XY(Dof(i,1),2),XY(Dof(i,2),2),XY(Dof(i,3),2)];
end
disp('前处理完成');
前处理完成
单元刚度矩阵
E=2*10^11;u=0.3; 平面应力问题
D=E/(1-u^2)*[1 u 0;u 1 0;0 0 (1-u)/2];
for i=1:ne
xi=unit(i,1);
yi=unit(i,4);
xj=unit(i,2);
yj=unit(i,5);
xm=unit(i,3);
ym=unit(i,6);
ai=xj*ym-xm*yj;
aj=xm*yi-xi*ym;
am=xi*yj-xj*yi;
bi=yj-ym;
bj=ym-yi;
bm=yi-yj;
ci=-(xj-xm);
cj=-(xm-xi);
cm=-(xi-xj);
area=abs((ai+aj+am)/2);
B = [bi 0 bj 0 bm 0
0 ci 0 cj 0 cm
ci bi cj bj cm bm];
Be{i,1} = B/2/area ;
ke{i,1}=[Be{i,1}]'*D*Be{i,1}*area;
end
总刚度矩阵叠加
KK=sparse(2*np,2*np);
for ie=1:ne
a=Dof(ie,1);
b=Dof(ie,2);
c=Dof(ie,3);
DOF(1)=2*a-1;
DOF(2)=2*a;
DOF(3)=2*b-1;
DOF(4)=2*b;
DOF(5)=2*c-1;
DOF(6)=2*c;
for n1=1:6
for n2=1:6
KK(DOF(n1),DOF(n2))= KK(DOF(n1),DOF(n2))+ke{ie,1}(n1,n2);
end
end
end
单元等效节点荷载
y=(0:LY);
P=(10^7/0.03)*y-10^7; 左右受变化的三角形荷载,在如图的坐标系下
Re=sparse(ne,6);
for i=1:ne;
switch i
case num2cell(1:ne/2-nx2)
Pe=[0 0 0 0 0 0];
case num2cell(ne/2-nx2+1:ne/2)
Pe=-LX*P*[0,0,0,1,0,1]/nx/2;
case num2cell(ne/2+1:ne-nx2)
Pe=[0 0 0 0 0 0];
otherwise
Pe=-LX*P*[0,0,0,1,0,1]/nx/2;
end
Re(i,:)=Pe;
end
荷载叠加
Rr=sparse(1,2*np);
for i=1:ne
a=Dof(i,1);
b=Dof(i,2);
c=Dof(i,3);
DOF(1)=2*a-1;
DOF(2)=2*a;
DOF(3)=2*b-1;
DOF(4)=2*b;
DOF(5)=2*c-1;
DOF(6)=2*c;
for n1=1:6
Rr(DOF(n1))= Rr(DOF(n1))+Re(i,n1);
end
end
生成需处理的行列
cp=[1:nx+1];
ctype=ones(1,length(cp));
ctype(nx2+1)=2;
cp_all=(cp-1)*(ny+1)+1;
p_stake=zeros(1,2*length(cp));
for i=1:length(cp)
switch ctype(i)
case {2}
p_stake(2*i)=2*cp_all(i);
p_stake(2*i-1)=2*cp_all(i)-1;
case {1}
p_stake(2*i)=2*cp_all(i);
p_stake(2*i-1)=[];
otherwise
p_stake(2*i)=[];
p_stake(2*i-1)=2*cp_all(i)-1;
end
end
[m,j]=find(p_stake==0);
p_stake(:,j)=[];
处理对应的行列
KK_d=KK;
KK_f=KK;
KK_d(p_stake,:)=[];
KK_d(:,p_stake)=[];
KK_f(:,p_stake)=[];
KK_f=KK_f(p_stake,:);
Rr_unkown=Rr;
Rr_unkown(:,p_stake)=[];
RR=transpose(Rr_unkown);
[L,U]=lu(KK_d);
UU=U\\(L\\RR);
Rx=KK_f*UU;
数值计算部分
UU_all=UU';
for i=1:length(p_stake)
UU_all=[UU_all(:,1:p_stake(i)-1),0,UU_all(:,p_stake(i):end)];
end
for i=1:np
UU_info(i,:)=[UU_all(2*i-1),UU_all(2*i)];
end
出图部分（运行后显示）
梁尺寸及荷载图
figure;
set(gcf,'outerposition',get(0,'ScreenSize'));
set(gcf,'name','梁的尺寸及荷载');
line([-0.015,-0.01],[0.06,0.06]),hold on
line([-0.01,-0.01],[0.06,0]),hold on
line([-0.01,-0.005],[0,0]),hold on
line([-0.005,-0.015],[0,0.06]),hold on
line([0.105,0.1],[0.06,0.06]),hold on
line([0.1,0.1],[0.06,0]),hold on
line([0.1,0.095],[0,0]),hold on
line([0.095,0.105],[0,0.06]),hold on
rectangle('position',[0,0,0.09,0.06]),hold on
quiver(-0.01,0.06,-0.0055,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(-0.01,0.05,-0.0038,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(-0.01,0.04,-0.002,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(-0.01,0.02,0.002,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(-0.01,0.01,0.0038,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(-0.01,0,0.0055,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0.06,0.0055,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0.05,0.0038,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0.04,0.002,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0.02,-0.002,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0.01,-0.0038,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
quiver(0.1,0,-0.0055,0,'LineWidth',2.0,'MaxHeadSize',0.8,'color','k'),hold on
text(-0.0155,0.062,'1000N/cm^2')
text(-0.0105,-0.002,'1000N/cm^2')
text(0.1,0.062,'1000N/cm^2')
text(0.095,-0.002,'1000N/cm^2')
text(0,-0.002,'梁宽9cm，高6cm，厚1cm。E=2×10^11N/m^2，μ=0.3（此图单位：m）')
axis([-0.025 0.12 -0.015 0.07]),hold on;
网格图
figure;
set(gcf,'outerposition',get(0,'ScreenSize'));
set(gcf,'name','网格划分');
for i=1:ne
line([unit(i,1:3),unit(i,1)],[unit(i,4:6),unit(i,4)]);
end
axis([-0.025 0.12 -0.015 0.07]),hold on;
xytext=num2str([1:np]');
text(XY(:,1)+LX/nx/8,XY(:,2)+LY/ny/4,xytext);
print(gcf,'-dbitmap','model.bmp');
位移矢量图
figure;
set(gcf,'outerposition',get(0,'ScreenSize'));
set(gcf,'name','结点的位移矢量场');
quiver(XY(:,1),XY(:,2),-UU_info(:,1),UU_info(:,2)),
axis([-0.025 0.12 -0.015 0.07]),hold on;
for i=1:ne
plot([unit(i,1:3),unit(i,1)],[unit(i,4:6),unit(i,4)],'r:');
end
print(gcf,'-dbitmap','位移矢量场.bmp');
disp('梁顶面结点');
JD=13:13:247
disp('梁顶面结点位移');
WY=[-UU_info(13);-UU_info(26);-UU_info(39);-UU_info(52);-UU_info(65);-UU_info(78);... -UU_info(91);-UU_info(104);-UU_info(117);-UU_info(130);UU_info(143);UU_info(156);... UU_info(169);UU_info(182);UU_info(195);UU_info(208);UU_info(221);UU_info(234);UU_info(247);]
结点位移
梁顶面结点
JD =
13 26 39 52 65 78 91 104 117 130 143 156 169 182 208 221 梁顶面结点位移
WY =
1.0e-006 *
0.6750
0.6000
0.5250
0.4500
0.3750
0.3000
0.2250
234 247
195 0.1500
0.0750
0.0000
-0.0750
-0.1500
-0.2250
-0.3000
-0.3750
-0.4500
-0.5250
-0.6000
-0.6750
figure;
set(gcf,'outerposition',get(0,'ScreenSize'));
set(gcf,'name','结点位移');
plot(JD,WY,'r:','LineWidth',2.0),grid on
toc;
因篇幅问题不能全部显示，请点此查看更多更全内容