Module 7 · Lesson 7.1
Freedoms and families
What a degree of freedom is, what each element family carries, and what happens where two families meet.
Why this matters
Every finite element model is built from the same short list of parts, and almost every modelling error is a wrong choice among them rather than anything to do with the mathematics.
The useful way to hold the list is not by name but by what each part can carry. A joint transfers what both sides have a freedom for, and nothing else — which turns several notorious modelling problems into arithmetic you can do before building anything.
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
A degree of freedom
A degree of freedom is a direction in which the model is permitted to move, and it is the unknown the solver actually solves for. A node in a three-dimensional frame has six: three translations and three rotations.
Two consequences follow immediately, and both matter more than the definition:
A model can only resist what it has freedoms for. A membrane element has two freedoms per node, both in-plane. A wall meshed with membrane elements has no stiffness whatever against load perpendicular to it — not a small stiffness, none.
A model can only fail in directions it has freedoms in. This is why the freedoms with no stiffness matter so much. A node whose rotation nothing restrains is a mechanism in that direction, and a solver's response to it varies from a clear refusal to a very large number reported without comment.
The families
| Family | Freedoms per node | Carries | Ignores |
|---|---|---|---|
| Bar (truss) | 3 | Axial only | Bending, shear, torsion |
| Beam (frame) | 6 | Axial, bending both ways, shear, torsion | Stress within the section, local detail |
| Spring | 3 | A stated stiffness in a stated direction | Everything else, including the other directions |
| Rigid link | 6 | A kinematic relation | Any flexibility at all |
| Plane stress / membrane | 2 | In-plane direct and shear stress | Out-of-plane bending entirely |
| Plate / shell | 6 | Membrane action, two-way bending, twist | Through-thickness variation |
| Solid | 3 | The full stress state | Rotational freedoms — it has none |
The last row is the one worth staring at. A solid element has three freedoms per node, the same as a bar. It has no rotations. That is not a limitation of any particular software; a solid continuum has no rotational degree of freedom to offer.
What a joint transfers
A joint transfers what both sides have a freedom for. Since moment requires a rotational freedom on both sides:
| Joint | Shared freedoms | Transfers moment? |
|---|---|---|
| Beam to beam | 6 | Yes |
| Beam to plate | 6 | Yes |
| Beam to membrane | 2 | No |
| Beam to solid | 3 | No |
| Beam to bar | 3 | No |
This is the whole of the 'beam framing into a solid mesh' problem, and it is arithmetic rather than judgement. A beam attached to a solid mesh at a single node is pinned, whatever the drawing shows, because the solid has nothing to receive a moment with.
The fix is not a setting. It is to spread the connection over several nodes so that the moment is carried as a couple of forces — which is what the real detail does too.
Freedoms with no stiffness
A related case, and a common one: release every member framing into a node and that node's rotation has nothing holding it. A propped cantilever modelled as a single member with a released end does exactly this.
It is not an error. It is a freedom the model was never given any stiffness in, and the right response is to report it rather than either crash or silently restrain it. A solver that quietly holds such a freedom has changed the model without saying so.
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
How many degrees of freedom are shared at a joint between a beam element and a membrane element?
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 joint transfers what both sides have a freedom for — beam to solid transfers no moment
- 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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint