Computational Engineering / module 9
How the stiffness method works
The glass box: element stiffness, transformation, assembly by addition, boundary conditions by striking out rows, and the recovery of everything else.
What you will understand
- Write the axial stiffness of a bar and say which of its three inputs go wrong most often
- Explain the two-node bar matrix from equilibrium rather than by quoting it
- Show that the global matrix is the local one seen from a rotated viewpoint
- Assemble a global stiffness matrix by adding element contributions
- Apply boundary conditions by striking out rows and columns, and say what that means physically
- Recover reactions from the struck-out rows and member forces from the displacements
- Use the equilibrium residual and the condition number as health checks on a solve
Lessons
The element and its transformation
Where the 4×4 bar matrix comes from, why every row sums to zero, and why the global form contains no new physics.
Assembly and boundary conditions
Adding element contributions into the global matrix, and what striking out a row physically means.
Solving, and recovering what you actually wanted
Solve the free displacements, then recover reactions and member forces from the complete element and global equations.
Module summary
- Everything the method does reduces to f = ku, written once per degree of freedom
- Assembly is addition; a disconnected node leaves a zero row because nothing added to it
- The unreduced global matrix is always singular — it does not know where the structure is
- Striking out a restrained row is the arithmetic of 'the structure is held here'
- Reactions come from the rows that were struck out, which is why they are recovered last
- Member forces come from displacements, not the other way round
- The equilibrium residual is the cheapest check in structural computing and it is never optional
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Check yourself
Why is the assembled global stiffness matrix singular before any restraint is applied?
Practice
Two bars share a node. Bar A has k = 60 000 kN/m, bar B has k = 20 000 kN/m, and they act in parallel along the same direction. A 40 kN load is applied at the shared node. What force does bar A carry, in kN?
Practice
A model has 340 nodes in three dimensions, with 6 degrees of freedom each, and 27 restrained degrees of freedom. How many equations are in the reduced system?