CIV5136 Chap.1 Degrees of Freedom and Structural Idealisation
Degrees of Freedom and Structural Idealisation
Degrees of freedom are the independent joint translations and rotations chosen to describe a structural idealisation. The element type supplies candidate movements, supports remove restrained movements, and any additional kinematic assumption must be stated.
The final coordinate order controls the load vector, displacement vector and every row and column of the global stiffness matrix.
A useful idealisation starts by separating the physical structure from the mathematical model. A plane truss retains joint translations because its pin-connected members carry axial action.
A beam model retains transverse movement and rotation, while a plane frame adds axial translation to bending movements. The coordinate set therefore follows the element behaviour that has been assumed, not every motion that a real three-dimensional object could display.
Support symbols then remove movement. A fixed support suppresses the plane-frame translations and rotation.
A pin suppresses translation but permits rotation, and a roller suppresses movement normal to its surface. Reactions are conjugate to restrained movements, so a reaction force does not make the corresponding displacement an active unknown. Prescribed settlement is also a known displacement, though its value need not be zero.
Numbering completes the model.
Every force entry, displacement entry and stiffness address must refer to the same arrow on the sketch. A singular matrix should trigger a search for an unrestrained rigid-body mode, an internal mechanism, a missing connection or duplicated coordinate. Numerical regularisation is not a structural repair. Identify the zero-energy movement and decide whether it is physically intended before changing the model.
What this chapter covers
- 01
Structural idealisation from physical member to nodes and elements
- 02
Plane truss, beam and frame movement sets
- 03
Support restraints and prescribed movements
- 04
Global positive directions and coordinate numbering
- 05
Active and restrained coordinate partitions
- 06
Rigid-body modes, mechanisms and singularity
- 07
Conjugate nodal forces and support reactions
- 08
DOF sketches as a matrix audit trail
Worked example · free
Count the active coordinates of a portal frame
- +1A rigid plane-frame joint has horizontal translation, vertical translation and rotation.
- +1Each fixed base removes its two translations and rotation from the active set.
- +2Number the left top joint uB, vB, thetaB and the right top joint uC, vC, thetaC.
- +1Write d = [uB, vB, thetaB, uC, vC, thetaC] transpose.
- +1Keep the same sequence for applied loads and the global matrix addresses.
Key terms
- Degree of freedom
- An independent joint movement used as a coordinate of the structural model.
- Structural idealisation
- The chosen representation of a physical structure by nodes, members, supports and constitutive assumptions.
- Active coordinate
- A displacement coordinate whose value is unknown and solved from the reduced equilibrium system.
- Restrained coordinate
- A prescribed joint movement, commonly zero at a support, whose conjugate action is a reaction.
- Rigid-body mode
- A movement that produces no member deformation and therefore no elastic restoring force.
Degrees of Freedom and Structural Idealisation FAQ
Why do degrees of freedom come before the stiffness matrix?
They define what the structure is allowed to do and provide the address system for loads, displacements and stiffness coefficients. A wrong list makes every later calculation describe a different physical model. It also changes which support reactions remain outside the active displacement solution. Verify the list against a free-motion sketch. That same diagram identifies which conjugate reactions will be recovered.
Does a pin support remove rotation from a frame joint?
A pin restrains translation but permits rotation. The rotational coordinate may remain in a frame model even though the support carries no reaction moment, depending on the connected element formulation.
Why can a stiffness matrix become singular?
A free rigid-body movement, an internal mechanism, a missing restraint or inconsistent coordinate definition can leave a movement with no restoring force. Identify that motion before changing any matrix value.
Can a valid coordinate solve to zero?
Yes. Symmetry or a particular load case can make an active displacement numerically zero. Existence of the coordinate depends on kinematics, not on the result of one loading pattern.
How are forces paired with displacement coordinates?
Each translation is paired with a force in the same direction, and each rotation is paired with a moment about the same axis. This conjugate relationship keeps the virtual work and stiffness equations dimensionally consistent. Label the force arrow beside its movement arrow before assembling the load vector, especially when global and member axes differ.
What should I show when counting movements?
Show the element type, support restraints, any extra kinematic constraint and the numbered positive arrows. A bare count cannot reveal whether a roller direction was misread or a frame rotation was omitted. The sketch should let another reader rebuild the displacement vector and identify the support reactions without seeing the later matrix.
Exam move
Practise DOF selection on small sketches before doing matrix arithmetic. For each structure, name the element type, draw every candidate joint movement, cross out the restrained movements and number what remains. Then write the conjugate load beside each arrow. When a later matrix is singular, return to the sketch and look for a zero-energy movement instead of forcing an inverse.
This chapter is the cheapest place to catch a model error because nothing has yet been assembled. Add one stability test to every sketch: imagine the unloaded structure moving in each proposed coordinate and decide which members deform. If a movement creates no strain anywhere, either the model contains a mechanism or the supports are incomplete. Keep a short list of these failure modes and use it before inspecting algebra.
