Module 9 · Lesson 9.3
Solving, and recovering what you actually wanted
Displacements first, then reactions from the struck-out rows, then member forces from the displacements — in that order, and never any other.
Why this matters
The finite element method computes displacements. Everything an engineer actually asks for — moments, axial forces, stresses, reactions — is recovered from them afterwards.
That ordering has a consequence people are surprised by: the displacements are the most accurate thing the model produces, and everything derived from them is less accurate. Stress is a derivative of displacement, and differentiating a numerical field always costs accuracy. It is why a mesh that gives good deflections can give poor stresses, and why Module 12 refuses to let you judge a mesh by its deflected shape.
By the end of this lesson you should be able to
- State the order in which quantities are recovered and why it cannot be reversed
- Recover reactions from the rows that were struck out
- Recover member force from the change in length of a bar
- Explain why displacements are more accurate than the stresses derived from them
The order
- 1.Solve the reduced system for the free displacements: Kff uf = ff.
- 2.Scatter those back into a full-length displacement vector, with zeros at the restrained degrees of freedom.
- 3.Recover the reactions by multiplying the full stiffness matrix by that vector and subtracting any applied load, reading off the restrained rows — the rows that were struck out.
- 4.Recover member forces from each element's own end displacements.
- 5.Check equilibrium: applied loads plus reactions must vanish.
Step 3 is the one worth dwelling on. The struck-out rows were removed because their displacements were known. But those rows are still perfectly good equations — they say what force is needed at that degree of freedom to hold it still. That force is the reaction. Nothing extra is computed; the information was there all along.
Member force from displacement
For a bar, take the two end displacements in global axes, project them onto the bar's own direction to get the extension, and multiply by k:
e = c(u₂ₓ − u₁ₓ) + s(u₂ᵧ − u₁ᵧ), and F = k·e
Positive means the bar got longer, which means tension. That sign convention is a choice, and different software makes it differently — which is why Module 12 has a case about a whole truss reported in compression.
Why the residual matters more than anything else on the page
After recovery, sum every applied load and every reaction, direction by direction. In exact arithmetic the sum is zero. In floating point it is a small number.
How small? In a healthy model, many orders of magnitude below the applied load — a residual of a few newtons on a few thousand kilonewtons. A residual you can see is not rounding. It means load is being lost, and there are only two ways that happens: something is not connected, or the matrix is so ill-conditioned that the solve has thrown away the digits that carried the answer.
Total loads against total reactions costs one subtraction and catches a large fraction of everything that can go wrong. Run it on every model, and again after every change.
What it calculates: The forces the supports must supply
- K
- The FULL global stiffness matrix, before reduction (kN/m)
- u
- Full displacement vector, zero at restrained degrees of freedom (m)
- f
- Applied load vector (kN)
- R
- Reactions, non-zero only at restrained degrees of freedom (kN)
This assumes
- The displacements have already been solved for — reactions cannot be found first
- Any load applied directly at a restrained degree of freedom is subtracted, or it will be double-counted
In plain terms: The full matrix is used, not the reduced one. The struck-out rows were never wrong; they were merely not needed to find the displacements. Now they are exactly what is needed to find the reactions.
Conditioning: how many of the printed digits are real
A 64-bit floating-point number carries about sixteen significant decimal digits. A solve does not preserve all of them. The loss is governed by the condition number of the reduced matrix — the ratio of its largest to its smallest eigenvalue.
A useful rule, to be used as a rule and not trusted as a law:
A condition number of 10ᵏ costs roughly k of your sixteen digits.
So a well-conditioned frame at 10³ leaves thirteen digits — far more than the two a structural result deserves. A model with a 5 mm stub element among 6 m members can reach 10¹², leaving four, and the noise starts to show in results you were relying on.
This is the arithmetic behind advice that otherwise sounds like fussiness. Do not use very short, very stiff elements. Do not model a hundred kilometres from the origin. Do not mix stiffnesses spanning a million. Each of those spends the same finite budget of precision, and Module 12 has a case for each.
Worked example
Reading the health of a solve
Given
- A truss model with 240 elements
- Total applied vertical load: 1 840 kN
- Total vertical reaction reported: 1 803 kN
- The model contains four link elements 8 mm long, between members averaging 4 m
Find
Whether these results can be used, and what to do next
Practice
A model applies 2 400 kN of vertical load and reports 2 397 kN of vertical reaction. Express the residual as a percentage of the applied load.
Practice
A reduced stiffness matrix has a condition number of 10⁹. Using the rule that a condition number of 10ᵏ costs about k of the sixteen available significant digits, how many significant digits survive the solve?
Check yourself
Why are the displacements from a finite element analysis more accurate than the stresses derived from them?
Check yourself
The equilibrium residual on a healthy model is about 10⁻¹⁴ of the applied load. What does a residual of 0.05 % mean?
Check yourself
Why are member forces recovered from displacements rather than solved for directly?
Summary
- Displacements are solved first; everything else is recovered from them
- Reactions come from the struck-out rows, using the full matrix
- Member force comes from the projection of end displacements onto the member
- Stresses are less accurate than displacements because they are a derivative
- The equilibrium residual is the gate, not a formality — a visible residual is a finding
- A condition number of 10ᵏ costs roughly k of your sixteen digits
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