Module 10 · Lesson 10.2
Lumped and consistent mass
Two ways to describe where the mass is, both approximations, missing the exact answer from opposite sides — which makes the gap between them a mesh check that needs no exact solution.
Why this matters
For a building, mass really is concentrated at the floors, and lumping it there is not an approximation at all.
For a chimney, a bridge deck, a long-span floor beam or a mast, the mass is distributed along the member itself, and where you put it in the model is a decision with consequences. This lesson is about that decision, and about a check it makes available.
By the end of this lesson you should be able to
- Derive the consistent mass matrix of an axial element
- Explain what the off-diagonal terms of a mass matrix mean physically
- Say why lumped and consistent mass bracket the exact answer
- Choose between them for a given problem
From first principles
The consistent mass matrix of an axial element
We want to show: Derive m = (ρAL/6)[[2,1],[1,2]] for a two-node bar element, from the kinetic energy of the distributed mass and the same shape functions used for its stiffness.
The mass of a bar is spread along its length, and every part of it is moving. If we assume the displacement varies linearly between the two ends — which is exactly what we assumed to get the element's stiffness — then the velocity varies linearly too. Adding up the kinetic energy of every slice, and writing the total in terms of the two end velocities, gives a matrix. It comes out with off-diagonal terms because a point in the middle of the bar is being moved by BOTH ends at once.
Try it
Lumped mass or consistent mass?
Both are approximations. Compared against a member whose exact frequency is known, they miss from opposite sides.
Member
- Exact continuous solution
- 1350 rad/s (216 Hz)
- Consistent mass
- 1490 rad/s
- Lumped mass
- 1220 rad/s
- Gap between the two
- 20.23%
+10.27% — a consistent-mass model is stiff and light, so it reads high
-9.97% — lumping puts more mass at the moving end, so it reads low
A useful mesh-adequacy check that needs no exact solution: if the two disagree, the mesh is too coarse.
- Consistent mass
- Lumped mass
- Exact
Consistent mass matrix(kg)
| 157.0 |
Lumped mass matrix(kg)
| 235.5 |
Look at the off-diagonal entries. The consistent matrix has them and the lumped one does not — that is the whole difference. Physically, the consistent matrix says that accelerating one end of a member generates an inertia force at the other, because the material between them is being accelerated too.
Which to use
- For a lumped-mass building model where the mass really is at the floors, lumping is not an approximation at all — it is the correct description.
- For a member whose own distributed mass matters — a long-span floor beam, a chimney, a bridge deck — consistent mass converges faster and is the better choice per element.
- Lumped mass gives a diagonal matrix, which makes explicit integration cheap. That is why explicit codes for impact and blast use it.
- With enough elements both converge to the same answer, which is the real reassurance: neither is a different theory, only a different discretisation.
What this shows: Lumping discards the inertia coupling between the ends of a member. Consistent mass keeps it, at the price of a full matrix — and the two bracket the true answer, so the gap between them measures the mesh.
Why they bracket, and what that buys
A finite-element model is generally too STIFF, because constraining the structure to deform in the assumed shapes is a restriction, and restrictions add stiffness. That pushes frequencies up.
Consistent mass is derived from the same assumed shapes, so it inherits the same stiffness bias and reads high.
Lumped mass moves material away from the interior towards the nodes. For a fixed–free bar that puts MORE mass on the free end than the consistent distribution does, which pulls the frequency down — and for a coarse mesh it overshoots, reading low.
So the two straddle the exact answer, and that gives a check available with no exact solution at all:
Run the same model with lumped and with consistent mass. If the two frequencies differ appreciably, the mesh is too coarse. If they agree, the answer is bracketed and the mesh is adequate.
For a single axial element the two are about 10% apart on either side. At four elements the gap is under 1%.
Which to use
| Situation | Use |
|---|---|
| Building floors, plant items, tanks | Lumped — the mass really is at the nodes, so it is exact |
| Beam, chimney, mast, bridge deck: distributed mass matters | Consistent — converges faster per element |
| Explicit integration for blast or impact | Lumped — a diagonal M makes the method trivial |
| Rotational inertia matters | Consistent — lumping has no rotational terms at all |
| Checking mesh adequacy | Both — the gap between them measures it |
The rotational blind spot
A lumped mass matrix has no rotational entries. Put half the element mass at each node as a point mass and it has no moment of inertia about that node.
Usually harmless. It matters when rotational inertia genuinely participates: a heavy plant item on a flexible bracket, a mass on a cantilever arm, a floor diaphragm rotating about a vertical axis. In the last case the omission does not degrade the answer; it removes the mode, which is the subject of the next lesson.
Worked example
One element or four? Using the bracket as a mesh check
Given
- A steel mast, fixed at the base, free at the top: 8 m tall
- Uniform section, A = 0.012 m², ρ = 7 850 kg/m³, E = 210 GPa
- Axial vibration is required (a rare case, chosen because the exact answer is known)
Find
The first axial frequency by both mass models, and whether one element is adequate.
Assumptions
- Uniform prismatic member, axial motion only
Practice
A bar element has ρAL = 900 kg. What is the leading diagonal entry of its consistent mass matrix, in kg?
Practice
For that same bar, what is the diagonal entry of the LUMPED mass matrix, in kg?
Practice
A fixed–free bar is 5 m long with E/ρ = 2.675 × 10⁷ m²/s². What is the exact first axial circular frequency, in rad/s?
Check yourself
What do the off-diagonal terms of a consistent mass matrix represent physically?
Summary
- Consistent mass uses the same shape functions as the stiffness — hence the name
- m = (ρAL/6)[[2,1],[1,2]] for an axial bar, from the kinetic-energy integral
- Off-diagonals mean accelerating one end produces inertia force at the other
- Lumping puts half the mass at each node and discards that coupling
- Both conserve the total mass; they distribute it differently
- They bracket the exact frequency, so their gap measures the mesh with no exact solution
- Lumped is exact where mass really is at the nodes, and is required for explicit integration
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