In this lecture, we'll cover:

• How to compute nodal displacements by inverting the reduced structure stiffness matrix
• How to implement the force–displacement solution in Python using NumPy
• How to construct the full global displacement vector to obtain reaction forces
• How to calculate member axial forces using transformation matrices and local coordinates

In this lecture, we move from forming the reduced structure stiffness matrix to actually solving for the unknown nodal displacements. We take the reduced 2$\times$2 stiffness matrix, invert it using NumPy, and multiply it by the external force vector to obtain the horizontal and vertical displacements at node two. Along the way, we address practical implementation details in Python, such as converting an array to a NumPy matrix in order to access the inverse operation, formatting outputs, and ensuring realistic material properties are included to obtain meaningful displacement values.

We then use the computed nodal displacements to determine the reaction forces by reconstructing the full global displacement vector, including the zero displacements at the restrained nodes, and multiplying it by the primary stiffness matrix. Finally, we calculate the axial forces in each member by forming the appropriate transformation matrices, converting global displacements to local coordinates, and applying the axial force relationship,

By the end of the lecture, we have obtained nodal displacements, support reactions, and internal member forces, completing the full structural analysis for the example problem.

Next up:

In the next lecture, we visualise the structural response by plotting the undeformed and deformed shapes of the truss using Matplotlib.