四点高斯求积法计算四边形单元的刚度矩阵matlab

以下是使用四点高斯求积法计算四边形单元的刚度矩阵的 Matlab 代码：

% 定义四个高斯积分点及其对应的权重
gauss_points = [-0.8611363116, -0.3399810436, 0.3399810436, 0.8611363116];
gauss_weights = [0.3478548451, 0.6521451549, 0.6521451549, 0.3478548451];

% 定义本地坐标系下的节点坐标
x1 = 0; y1 = 0;
x2 = 1; y2 = 0;
x3 = 1; y3 = 1;
x4 = 0; y4 = 1;

% 定义材料参数和厚度
E = 210e9; % 弹性模量
nu = 0.3; % 泊松比
t = 0.01; % 厚度

% 定义形函数的导数
syms r s
N1 = (1-r)*(1-s)/4; N2 = (1+r)*(1-s)/4;
N3 = (1+r)*(1+s)/4; N4 = (1-r)*(1+s)/4;
dN1_dr = diff(N1, r); dN1_ds = diff(N1, s);
dN2_dr = diff(N2, r); dN2_ds = diff(N2, s);
dN3_dr = diff(N3, r); dN3_ds = diff(N3, s);
dN4_dr = diff(N4, r); dN4_ds = diff(N4, s);

% 初始化刚度矩阵和节点坐标矩阵
K = zeros(8, 8);
node_coords = [x1, y1; x2, y2; x3, y3; x4, y4];

% 进行高斯积分
for i = 1:length(gauss_points)
    for j = 1:length(gauss_points)
        r = gauss_points(i);
        s = gauss_points(j);
        w = gauss_weights(i) * gauss_weights(j);
        
        % 计算雅可比矩阵和其行列式
        J = [dN1_dr, dN2_dr, dN3_dr, dN4_dr; dN1_ds, dN2_ds, dN3_ds, dN4_ds] * node_coords;
        detJ = det(J);
        
        % 计算形函数的导数在实际坐标系下的值 
        dN_dx = inv(J) * [dN1_dr, dN2_dr, dN3_dr, dN4_dr; dN1_ds, dN2_ds, dN3_ds, dN4_ds];
        dN_dy = inv(J) * [dN1_ds, dN2_ds, dN3_ds, dN4_ds; dN1_dr, dN2_dr, dN3_dr, dN4_dr];
        
        % 计算B矩阵和刚度矩阵的贡献
        B = zeros(3, 8);
        B(1, 1:2:end) = dN_dx;
        B(2, 2:2:end) = dN_dy;
        B(3, 1:2:end) = dN_dy;
        B(3, 2:2:end) = dN_dx;
        K = K + B' * E * t * B * detJ * w;
    end
end

上述代码中，首先定义了四个高斯积分点及其对应的权重。然后定义了本地坐标系下的节点坐标和材料参数。接着使用符号计算工具定义了形函数的导数。然后初始化刚度矩阵和节点坐标矩阵。最后进行双重循环，使用高斯积分计算刚度矩阵的贡献。在每个高斯积分点上，首先计算雅可比矩阵和其行列式，然后计算形函数的导数在实际坐标系下的值，最后计算B矩阵和刚度矩阵的贡献。