Computational Engineering / Finite-element building blocks
Freedoms and families
What a degree of freedom is, what each element family carries, and what happens where two families meet.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
- Review the DOF explorer and solver handling of loaded zero-stiffness freedoms; elimination is valid only for genuinely unused unloaded variables.
- Validate beam-solid and shell coupling examples against the actual formulation, local axes and connection stiffness.
- The end-condition table reports midspan deflection; verify lab output labels if they claim a maximum.
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 notesWhy this matters
Element formulations define the displacement variables and deformation mechanisms represented by a model. Check those variables, their directions and the connection constraints when different element families meet. A compatible connection must transfer the actions needed by the physical detail.
By the end of this lesson you should be able to
- Define a degree of freedom in terms the model actually uses
- State how many freedoms each element family has per node
- Predict what a joint between two families transfers
- Recognise a freedom the model has given no stiffness
What you should already know
- Module 6's abstraction ladder — the element families are the rungs, made concrete
- Module 9 assembles these into a stiffness matrix; this module is what goes into it
Degrees of freedom
A degree of freedom is an independent variable describing motion in the chosen formulation. A conventional 3D frame node has three translations and three rotations. A displacement-based solid node usually has three translations; rotations follow from the displacement field.
Counts depend on the formulation and dimensional setting. A planar truss has two translations per node; a space truss has three. Membranes may use local in-plane variables or three global translations, without linear out-of-plane bending stiffness. Shells commonly have five or six variables per node. Springs and rigid constraints can act on selected translational or rotational variables.
An omitted variable and an active variable with zero stiffness are different. An active unsupported freedom may create a singular system; an excluded variable cannot represent the corresponding displacement directly.
Connecting element families
Sharing a single node between a conventional beam and a displacement-based solid shares translations but does not directly transfer the beam-end moment into the solid. Transfer a moment through an appropriate coupling to a group of solid nodes, so that a force couple represents the connection action. Check the stiffness and distribution introduced by the coupling.
Similarly, a single-node beam-to-membrane connection does not automatically represent a moment connection. A coupling over several nodes can transfer moments that the membrane formulation supports. It cannot supply out-of-plane bending stiffness absent from the wall model. Shell-to-beam connections also need checks of local axes and rotational capabilities; counting variables alone is insufficient.
Zero-stiffness freedoms
A released beam end may leave an unused nodal rotation with no associated stiffness. If it is genuinely uncoupled and unloaded, the solver can eliminate it from the active system and report the action. A loaded unsupported rotation or a mechanism cannot safely be resolved by holding the freedom at zero. Investigate the cause and distinguish an unused variable from a missing physical restraint.
Try it
Degrees-of-freedom explorer
Join two element families and see what the joint transfers. The verdict is arithmetic on the freedoms each side has.
One side of the joint
The other side
- Beam (frame) freedoms per node
- 6
- Solid freedoms per node
- 3
- Shared at the joint
- 3
- Transfers moment?
- no
- Model size if all beam (frame)
- 840 freedoms
A beam (frame) meeting a solid at one node shares 3 freedoms and transfers no moment. If the real joint is continuous, the model is wrong and it will not say so.
Beam (frame) carries
- Axial
- Bending about both axes
- Shear
- Torsion
Beam (frame) ignores
- Stress distribution within the section
- Local behaviour at holes, copes and connections
- Warping, unless the formulation includes it
It reports a member force, not a stress. Asking it about a detail returns a number that describes the whole section.
What this shows: A joint transfers only what both sides have a freedom for — which is the whole of the beam-into-solid problem.
Worked example
A beam framing into a solid-meshed corbel
Given
- A beam is modelled with line elements and connected to a solid mesh at one node
- The physical detail is a full-strength welded connection
Find
What the model actually transfers
Practice
A three-dimensional frame model has 140 nodes. How many degrees of freedom does it have?
Practice
In the simplified planar membrane example, how many in-plane translations are shared with a compatible beam node?
Check yourself
A wall is meshed with membrane elements and then loaded by wind acting perpendicular to it. What does the model report?
Worked example
Sizing a model before building it
Given
- A three-dimensional frame of a 10-storey building on a 5 × 4 bay grid
- One node at every column line at every floor
Find
The number of degrees of freedom, and what that implies
Worked example
What a joint between two families can carry
Given
- A beam element frames into a wall meshed with membrane elements
- The real detail is a moment connection
Find
What is transferred, and what to do
Check yourself
Why does a solid element have no rotational degrees of freedom?
Check yourself
A node's rotation has no stiffness at all. What is the right response from a solver?
Check yourself
What is the risk in a chain of rigid links?
Check yourself
A bar element carries axial force only. What follows for a model made entirely of bars?
Summary
- A degree of freedom is a direction the model may move in, and the unknown solved for
- A model can only resist, and only fail, in directions it has freedoms for
- A solid element has three freedoms per node and no rotations at all
- A single shared beam-solid node does not transfer beam rotation; a suitable multi-node coupling can transfer moment as a force couple
- The fix for a moment connection into a solid is a force couple, not a setting
- A freedom with no stiffness should be reported, not silently restrained