CIV5136 Chap.2 Member Stiffness and Matrix Assembly
Member Stiffness and Matrix Assembly
Direct stiffness assembly adds each member coefficient into global rows and columns identified by its coordinate addresses. Connected members contribute to the same joint terms because they resist the same movement.
Partitioning then separates active displacement equations from restrained coordinates while retaining the information required to calculate reactions and recover member forces.
Each coefficient has a physical reading. A direct term is the action needed at one coordinate for a unit displacement there while other active coordinates remain fixed.
A coupling term is the action that appears at one coordinate when another coordinate moves. For reciprocal linear elastic behaviour, the member and assembled matrices are symmetric. Broken symmetry is therefore a strong warning about coordinate order, transformation or row-and-column placement.
The location vector translates a member end order into global addresses.
Assembly means addressed addition: coefficients from every connected member accumulate at shared coordinates. Boundary conditions do not erase the restrained parts of the structural relation. The active block determines unknown joint movements; the restrained block later determines support reactions.
A nonzero prescribed settlement enters through the coupling terms and can create action even without an applied nodal force.
Solving the system is only the middle of the method. Extract each member displacement vector in its original order, compute deformation actions, include member-load actions, and transform results into the declared reporting axes.
Joint equilibrium and the total external-force balance then test the work. This recovery loop catches a correct global solution paired with an incorrectly ordered member vector.
What this chapter covers
- 01
Physical meaning of direct and coupling stiffness terms
- 02
Member end coordinate order
- 03
Location vectors and global addresses
- 04
Addition at shared structural coordinates
- 05
Symmetry and dimensional checks
- 06
Active and restrained block partitioning
- 07
Prescribed support displacement effects
- 08
Reaction and member action recovery
Worked example · free
Assemble two axial springs
- +2AB contributes 12 to the B coordinate; BC contributes [[8,-8],[-8,8]] to B and C.
- +2K = [[20,-8],[-8,8]] kN per millimetre and F = [0,10] transpose kN.
- +1The first equation gives uC = 2.5uB.
- +1Substitution gives uB = 0.833 mm and uC = 2.083 mm.
- +1Both springs carry 10 kN, confirming joint equilibrium.
Key terms
- Member stiffness
- The relation between a member's ordered end actions and ordered end displacements.
- System stiffness
- The structural stiffness matrix formed by adding member contributions at common global coordinates.
- Location vector
- The mapping from a member's local end-coordinate positions to global system coordinate numbers.
- Coupling term
- A stiffness coefficient giving action at one coordinate due to displacement at another.
- Matrix partition
- A block arrangement separating active and prescribed displacement coordinates for solution and reaction recovery.
Member Stiffness and Matrix Assembly FAQ
Why do shared diagonal stiffnesses add?
A unit movement at the shared joint deforms every connected member that resists that direction. Their restoring actions act together, so their individual direct stiffness contributions sum at one global address. The diagonal therefore records the combined resistance of the connected structural path. Trace it back to each member. It should grow when another resisting member connects at that coordinate.
What does an asymmetric assembled matrix suggest?
For the usual linear elastic formulation it suggests inconsistent member ordering, an incorrect transformation or a row and column placement error. Trace each off-diagonal pair back to its member addresses.
Why retain restrained rows after solving?
Restrained equations calculate the support actions needed to enforce prescribed displacements. Deleting them permanently can leave no systematic route to reactions or settlement effects afterward.
How should a singular matrix be handled?
Find the unrestrained rigid-body motion, mechanism or missing connection that produces zero stiffness. Adding an arbitrary diagonal value hides the structural issue and changes the model.
What is the physical meaning of a coupling term?
It is the action appearing at one coordinate because another coordinate undergoes a unit movement while the remaining coordinates are held fixed. Its sign describes whether the induced action agrees with or opposes the positive action convention. Paired coupling terms should be reciprocal in the standard linear elastic formulation.
How do support settlements enter assembly?
A prescribed settlement is a known displacement in the restrained block, not an unknown applied force. Multiply it by the active-to-restrained coupling block and move that contribution to the active right-hand side with the correct sign. After solving, substitute all known and unknown displacements into the restrained equations to recover reactions.
Exam move
Start every assembly exercise by writing the member coordinate order and its location vector. Add one member at a time using a different pencil colour, then inspect shared diagonals and symmetric off-diagonal pairs. Before solving, predict whether each direct term should grow when a connected member is added. After solving, extract the member displacement vectors in their original order and recompute member actions.
Finish with both joint equilibrium and the sum of external reactions. Keep a three-stage ledger labelled member, system and recovery. Copying a coefficient is not enough: point to the global row and column it enters and name the unit movement it resists. For support settlement problems, keep the known displacement block visible until the active right-hand side has been formed.
