Module 7 · Lesson 7.2
Restraints and releases
How the model is held to the world, and what happens inside it — with the moments and deflections for every case.
Why this matters
Restraints and releases are two of the shortest entries in any modelling manual and between them they cause more wrong answers than the whole of the solver.
The reason is that they are both easy to set and hard to see. A wrong restraint and a right one look identical on screen; the difference appears only in numbers that seem perfectly reasonable. This lesson computes all of them.
By the end of this lesson you should be able to
- Distinguish a restraint from a release
- Compute the moments and deflections for three sets of end conditions
- Show that a release is a spring of zero stiffness
- Recognise which restraint mistakes stop the model and which do not
Two different things
A restraint joins the model to the world. It is a statement about what is outside the model, and it produces a reaction.
A release describes what happens inside the model, between two members at a joint. It produces no reaction and transfers no force in the released direction.
Confusing them is common because both are set on the same screen in most software. The test is whether the thing being described is outside the model or inside it.
What end conditions do
One 8 m span, 12 kN/m, EI = 21 000 kN·m²:
| End conditions | Support moment | Midspan moment | Deflection |
|---|---|---|---|
| Simply supported | 0 | 96.0 kN·m | 30.48 mm |
| Propped cantilever | 96.0 kN·m | 48.0 kN·m | 12.19 mm |
| Fixed both ends | 64.0 kN·m | 32.0 kN·m | 6.10 mm |
Three numbers worth carrying:
- Fixing both ends divides the deflection by exactly five — 5wL⁴/384EI against wL⁴/384EI.
- The peak moment moves from midspan to the supports, and it falls, from wL²/8 to wL²/12.
- The propped cantilever's fixed end takes wL²/8 — more than either end of the fully fixed beam.
That last one surprises people. Removing restraint from one end increases the moment at the other.
A release is a spring of zero
It is tempting to treat 'released', 'pinned' and 'sprung' as three different settings. They are one parameter.
A rotational spring of stiffness s between the member end and the node reduces exactly to a release as s → 0, and to full continuity as s → ∞. There is no discontinuity anywhere between them, and real connections live in the middle. Module 8 sweeps that parameter and finds the middle is where all the action is.
One consequence is worth stating plainly. A release should be implemented by condensing the freedom out, not by inserting a very small stiffness. A condensed release carries exactly zero moment; a small-stiffness release carries a small moment, and 'small' is relative to numbers you have not checked.
Four ways to get the supports wrong
The same 8 m beam, meant to be simply supported:
| Model | Deflection | What happens |
|---|---|---|
| Pin and roller | 30.48 mm | Correct |
| Pinned at both ends | 30.48 mm | Solves. Identical in a linear analysis |
| Rotation held at both ends | 6.10 mm | Solves. Exactly one fifth of the deflection |
| Vertical only at both ends | — | Mechanism. Will not solve |
Three of the four solve. Only one of them is caught.
The third is the dangerous one. 'Fully restrained' is a phrase that means laterally restrained to a designer and rotation held to a modelling dialogue, and the consequence is a beam five times stiffer than the one being built. Nothing warns, and 6 mm is an entirely plausible deflection for an 8 m beam.
The second is worth understanding too. Adding horizontal restraint at both ends changes nothing at all in a linear analysis, because a transversely loaded beam develops no axial force in linear theory. It is not harmless — under a large deflection or a temperature change it becomes very significant indeed — but a linear run will never show it.
What it calculates: The factor by which a model over-states a beam's stiffness when rotation is held at both ends by mistake
- w
- uniformly distributed load (kN/m)
- L
- span (m)
- EI
- flexural rigidity (kN·m²)
This assumes
- A prismatic member under a uniform load, linear elastic
In plain terms: The factor is exactly five because both expressions share wL⁴/384EI and differ only by the 5. It is worth carrying as the size of the error a mis-set restraint produces.
Try it
Support and release laboratory
Three sets of end conditions, and four ways to get the supports wrong. Only one of the four refuses to run.
EI = 21 000 kN·m² throughout.
| Ends | Support M | Midspan M | Deflection |
|---|---|---|---|
| simple | 0.00 | 96.00 | 30.48 mm |
| propped | 96.00 | 48.00 | 12.19 mm |
| fixed | 64.00 | 32.00 | 6.10 mm |
Moments in kN·m. Fixing both ends divides the deflection by exactly five.
| Model | Deflection | vs correct | Solves? |
|---|---|---|---|
| Pin and roller | 30.48 mm | × 1.000 | yes |
| Pinned at both ends | 30.48 mm | × 1.000 | yes |
| Rotation held at both ends | 6.10 mm | × 0.200 | yes |
| Vertical only at both ends | — | — | no — mechanism |
Three of the four solve. The one that reports a fifth of the true deflection raises nothing at all, and ‘fully restrained’ means one thing to a designer and another to a dialogue box.
Freedoms the model gave no stiffness
- simple: L rz, R rz
- propped: R rz
- fixed: none
What this shows: Fixing both ends divides the deflection by exactly five, and three of the four common restraint mistakes solve cleanly.
Worked example
The cost of 'fully restrained'
Given
- An 8 m secondary beam, 12 kN/m, EI = 21 000 kN·m²
- Meant to be simply supported; the modeller ticks 'fully restrained' at both ends
Find
What the model reports and what is actually built
Practice
A uniformly loaded span deflects 30.48 mm when simply supported. What does it deflect when both ends are fully fixed?
Practice
An 8 m span carries 12 kN/m. What is the fixed-end moment at the built-in end if the other end is propped?
Predict first
A simply supported beam model has horizontal restraint accidentally left on at both ends. What does a linear analysis report?
Summary
- A restraint joins the model to the world; a release acts inside it
- Fixing both ends divides the deflection by exactly five
- A propped cantilever's fixed end takes wL²/8, more than a fully fixed beam's wL²/12
- A release is a spring of zero stiffness — condense it, do not fake it with a small number
- Three of the four common restraint mistakes solve cleanly
- 'Fully restrained' means two different things to a designer and a dialogue box
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