Transformation matrix for 3D frame element

Hi! I have constructed the local matrices K and M(12x12 matrices) for my frame elements in 3D, but I dont know how to continue to the global system with transformation matrix. I found a lot about 2D problems but in 3D I dont know how to build the matrix in matlab or anything else. If someone knows how to do it, please give me some help. Thank you!!

Answer 1

[UDAYA PEDDIRAJU]

Hi Christos,

I see you're looking for help with the transformation matrix for 3D frame elements in MATLAB.

Since you've already built the local stiffness (K) and mass (M) matrices (12x12 each) for your elements, transforming them to the global system using a transformation matrix is the next step. Here's how you can achieve that:

1. Building the Transformation Matrix:

The transformation matrix (T) relates the local displacements and forces at the element nodes to the global ones. In 3D, it's a 4x4 matrix. This function helps you create T based on the global coordinates of two nodes (node_1 and node_2) of the element:

function T = create_transformation_matrix(node_1, node_2)
  % Calculate direction cosines
  dx = node_2(1) - node_1(1);
  dy = node_2(2) - node_1(2);
  dz = node_2(3) - node_1(3);
  L = sqrt(dx^2 + dy^2 + dz^2);

  cos_theta_x = dx / L;
  cos_theta_y = dy / L;
  cos_theta_z = dz / L;

  % Direction cosines matrix
  direction_cosines_matrix = [
      cos_theta_x, cos_theta_y, cos_theta_z, 0;
      -cos_theta_y, cos_theta_x, 0, 0;
      -cos_theta_z, 0, cos_theta_x, 0;
      0, 0, 0, 1;
  ];

  T = direction_cosines_matrix;
end

2. Transforming Local Matrices:

With the transformation matrix T, you can transform the local stiffness (K) and mass (M) matrices to the global system:

% Global stiffness matrix
K_global = T' * K * T;

% Global mass matrix
M_global = T' * M * T;

Explanation:

• The create_transformation_matrix function calculates the direction cosines based on nodal coordinates, representing the cosines between local and global axes.
• These direction cosines populate the transformation matrix (T).
• The local stiffness (K) and mass (M) matrices are multiplied by the transpose (T') of T and then by T again to transform them to the global system (K_global and M_global).

This should give a clear idea!