悬臂梁matlab有限元编程

以下是基于Matlab的悬臂梁四节点/八节点四边形单元有限元编程的示例代码：

% 定义悬臂梁的参数
L = 1; % 悬臂梁长度
b = 0.02; % 悬臂梁宽度
h = 0.04; % 悬臂梁高度
E = 2e11; % 弹性模量
rho = 7800; % 密度
nu = 0.3; % 泊松比
P = 1000; % 集中力

% 定义节点坐标
x = [0, L, L, 0, L/2, L/2, 0, L, L, 0, L/2, L/2];
y = [0, 0, h, h, 0, 0, h, h, 0, h/2, h/2, h/2];

% 定义单元节点
element = [1, 2, 5, 6; 2, 3, 6, 7; 1, 5, 4, 8; 2, 6, 3, 9; 5, 6, 10, 11; 6, 7, 11, 12; 4, 8, 5, 10; 6, 9, 7, 12];

% 定义单元材料属性
D = E*h^3/(12*(1-nu^2));
Ke = zeros(8, 8, 8);
for i = 1:8
    for j = 1:8
        Ke(:,:,i) = Ke(:,:,i) + [12, 6*L, -12, 6*L; 6*L, 4*L^2, -6*L, 2*L^2; -12, -6*L, 12, -6*L; 6*L, 2*L^2, -6*L, 4*L^2]/(E*h^3/(12*(1-nu^2)))*b*h/2;
    end
end

% 定义全局刚度矩阵和质量矩阵
K = zeros(12, 12);
M = zeros(12, 12);
for i = 1:8
    for j = 1:8
        K(element(i,:), element(j,:)) = K(element(i,:), element(j,:)) + Ke(:,:,i);
        if i == j
            M(element(i,:), element(j,:)) = M(element(i,:), element(j,:)) + rho*b*h*L/8*[2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0; 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0; 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0; 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1; 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 0; 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0; 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0; 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2]/L;
        end
    end
end

% 定义边界条件
K(1,:) = 0;
K(:,1) = 0;
K(1,1) = 1;
K(4,:) = 0;
K(:,4) = 0;
K(4,4) = 1;

% 求解位移和应力
U = K\P';
sigma = zeros(8, 1);
for i = 1:8
    sigma(i) = D*[12/L^3, 6/L^2, -12/L^3, 6/L^2]*[U(element(i,:), 1); U(element(i,:), 2); U(element(i,:), 3); U(element(i,:), 4)];
end

% 输出结果
disp('节点位移：')
disp(U)
disp('单元应力：')
disp(sigma)