matlab 有限元计算

1、内容简介

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2、内容说明

3、仿真分析

% =====================================================================
clear;
clc;
close all
%% part1
E = 100e9; % elastic modulus
NU = 0.3; % possion ratio
th = 2e-3; % thickness
w = BilinearQuadElementStiffness(E,NU,th,0,0,0.03,0,0.03,0.01,0,0.01,2)
% 这个w就是第一问的刚度矩阵
%% part2
w1 = BilinearQuadElementStiffness(E,NU,th,0,0,0.01,0,0.01,0.03,0,0.03,2);
w2 = BilinearQuadElementStiffness(E,NU,th,0,0.03,0,0.04,0.03,0.04,0.03,0.03,2);
K = zeros(14,14,'double');
K = BilinearQuadAssemble(K,w1,1,2,3,4);
K = BilinearQuadAssemble(K,w2,4,5,6,7);
delta = pi/2;
amp = 200e6*2e-3*10e-3/2;
Fx1 = amp*sin(delta+pi);
Fy1 = amp*cos(delta+pi);
Fx2 = amp*sin(delta+pi);
Fy2 = amp*cos(delta+pi);
% ===================== 2 elements =================
k = [K(3:8,3:8) K(3:8,10:14);K(10:14,3:8) K(10:14,10:14)];
F = zeros(11,1);
F(8) = Fx1;F(9) = Fy1;
F(10) = Fx2;F(11) = Fy2;
U = k\F
U_2 = [0;0;U(1:6);0;U(7:end)]
F_2 = K*U_2;

u1 = U_2(1:8);
sigma1 = BilinearQuadElementStresses(E,NU,0,0,0.01,0,0.01,0.03,0,0.03,2,u1);
y2_1 = BilinearQuadElementPStresses(sigma1)/E % strain
u2 = U_2(7:14);
sigma2 = BilinearQuadElementStresses(E,NU,0,0.03,0,0.04,0.03,0.04,0.03,0.03,2,u2);
y2_2 = BilinearQuadElementPStresses(sigma2)/E % strain

%% 4个elements
w1 = BilinearQuadElementStiffness(E,NU,th,0,0,0.01,0,0.01,0.015,0,0.015,2);
w2 = BilinearQuadElementStiffness(E,NU,th,0.01,0.015,0,0.015,0,0.03,0.01,0.03,2);
w3 = BilinearQuadElementStiffness(E,NU,th,0,0.03,0.01,0.03,0.01,0.04,0,0.04,2);
w4 = BilinearQuadElementStiffness(E,NU,th,0.01,0.03,0.01,0.04,0.03,0.04,0.03,0.03,2);
K_4 = zeros(20,20,'double');
K_4 = BilinearQuadAssemble(K_4,w1,1,2,3,4);
K_4 = BilinearQuadAssemble(K_4,w2,3,4,5,6);
K_4 = BilinearQuadAssemble(K_4,w2,5,6,7,8);
K_4 = BilinearQuadAssemble(K_4,w2,6,7,10,9);

k = [K_4(3:14,3:14) K_4(3:14,16:20);K_4(16:20,3:14) K_4(16:20,16:20)];
F = zeros(17,1);
F(14) = Fx1;F(15) = Fy1;
F(16) = Fx2;F(17) = Fy2;
U = k\F
U_4 = [0;0;U(1:12);0;U(13:end)]
F_4 = K_4*U_4

u1 = U_4(1:8);
sigma1 = BilinearQuadElementStresses(E,NU,0,0,0.01,0,0.01,0.015,0,0.015,2,u1);
y4_1 = BilinearQuadElementPStresses(sigma1)/E % strain
u2 = U_4(5:12);
sigma2 = BilinearQuadElementStresses(E,NU,0.01,0.015,0,0.015,0,0.03,0.01,0.03,2,u2);
y4_2 = BilinearQuadElementPStresses(sigma2)/E % strain
u3 = U_4(9:16);
sigma3 = BilinearQuadElementStresses(E,NU,0,0.03,0.01,0.03,0.01,0.04,0,0.04,2,u3);
y4_3 = BilinearQuadElementPStresses(sigma3)/E % strain
u4 = [U_4(11:14);U_4(13:14);U_4(17:18)];
sigma4 = BilinearQuadElementStresses(E,NU,0.01,0.03,0.01,0.04,0.03,0.04,0.03,0.03,2,u4);
y4_4 = BilinearQuadElementPStresses(sigma4)/E % strain

%%
figure
x1 = [0 0.01 0.01 0 0];
y1 = [0 0 0.015 0.015 0];
line(x1,y1)
hold on
x2 = [0.01 0.01 0 0];
y2 = [0.015 0.03 0.03 0.015];
line(x2,y2)
hold on
x3 = [0.01 0.01 0 0];
y3 = [0.03 0.04 0.04 0.03];
line(x3,y3)
hold on
x4 = [0.01 0.03 0.03 0.01];
y4 = [0.03 0.03 0.04 0.04];
line(x4,y4)
hold on
xlim([-0.01 0.04])
ylim([-0.004 0.045])

figure
x1 = [0 0.01+U_4(3) 0.01+U_4(5) 0+U_4(7) 0];
y1 = [0 0+U_4(4) 0.015+U_4(6) 0.015+U_4(8) 0];
line(x1,y1)
hold on
x2 = [0.01+U_4(5) 0.01+U_4(11) 0+U_4(9) 0+U_4(7)];
y2 = [0.015+U_4(6) 0.03+U_4(12) 0.03+U_4(10) 0.015+U_4(8)];
line(x2,y2)
hold on
x3 = [0.01+U_4(11) 0.01+U_4(13) 0+U_4(15) 0+U_4(9)];
y3 = [0.03+U_4(12) 0.04+U_4(14) 0.04+U_4(16) 0.03+U_4(10)];
line(x3,y3)
hold on
x4 = [0.01+U_4(11) 0.03+U_4(17) 0.03+U_4(19) 0.01+U_4(13)];
y4 = [0.03+U_4(12) 0.03+U_4(18) 0.04+U_4(20) 0.04+U_4(14)];
line(x4,y4)
hold on
xlim([-0.01 0.04])
ylim([-0.004 0.045])

figure
x1 = [0 0.01 0.01 0 0];
y1 = [0 0 0.015 0.015 0];
line(x1,y1)
hold on
x2 = [0.01 0.01 0 0];
y2 = [0.015 0.03 0.03 0.015];
line(x2,y2)
hold on
x3 = [0.01 0.01 0 0];
y3 = [0.03 0.04 0.04 0.03];
line(x3,y3)
hold on
x4 = [0.01 0.03 0.03 0.01];
y4 = [0.03 0.03 0.04 0.04];
line(x4,y4)
hold on
x1 = [0 0.01+U_4(3) 0.01+U_4(5) 0+U_4(7) 0];
y1 = [0 0+U_4(4) 0.015+U_4(6) 0.015+U_4(8) 0];
line(x1,y1,'LineWidth',2,'color','red')
hold on
x2 = [0.01+U_4(5) 0.01+U_4(11) 0+U_4(9) 0+U_4(7)];
y2 = [0.015+U_4(6) 0.03+U_4(12) 0.03+U_4(10) 0.015+U_4(8)];
line(x2,y2,'LineWidth',2,'color','red')
hold on
x3 = [0.01+U_4(11) 0.01+U_4(13) 0+U_4(15) 0+U_4(9)];
y3 = [0.03+U_4(12) 0.04+U_4(14) 0.04+U_4(16) 0.03+U_4(10)];
line(x3,y3,'LineWidth',2,'color','red')
hold on
x4 = [0.01+U_4(11) 0.03+U_4(17) 0.03+U_4(19) 0.01+U_4(13)];
y4 = [0.03+U_4(12) 0.03+U_4(18) 0.04+U_4(20) 0.04+U_4(14)];
line(x4,y4,'LineWidth',2,'color','red')
hold on
xlim([-0.01 0.04])
ylim([-0.004 0.045])

4、参考论文

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