How this calculator works
An element stiffness matrix maps nodal displacements and rotations to conjugate forces and moments. A truss carries axial force only and is rotated into global x-y coordinates. A beam uses local transverse displacements and counterclockwise rotations. A planar frame adds axial stiffness and rotates the full matrix into global coordinates. The selected unit system scales translational and rotational coefficients separately; their units are listed per coefficient. These are unrestrained element matrices and are intentionally singular before assembly and support constraints. Shear deformation, geometric stiffness and connection releases are excluded.
Formula
Truss k = (EA/L) d d^T, d = [cos theta, sin theta, -cos theta, -sin theta]. Beam coefficients include 12EI/L^3, 6EI/L^2, 4EI/L and 2EI/L. Frame K_global = T^T K_local T.
Worked example
For a horizontal truss with E = 200 GPa, A = 0.001 m2 and L = 2 m, the axial entries are +/-100000000 N/m. For a beam with EI = 200000 N m2 and L = 2 m, k11 = 300000 N/m and k12 = 300000 N.
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Frequently asked questions
Why can I not invert the matrix directly?
An isolated unrestrained element has rigid-body modes. Assemble a structure and apply sufficient supports before solving.
Do all matrix entries have N/m units?
No. Rotation columns and moment rows change dimensions. The coefficient table gives the correct unit for every entry.
Which way are rotations positive?
Positive nodal rotations are counterclockwise. Beam mode is local; frame and truss modes are expressed in the global x-y system.