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When Fifty Modes Are Not Enough
In modal analysis the code demands 95% of the mass. In this building, with its flexible timber diaphragm, even 50 modes do not get there. There is a rule but no way out — and in historic masonry this is the rule, not the exception.
1. The rule: 95% of the mass
The rule setting how many modes must be included in a modal analysis is explicit:
The reasoning is sound: in modal combination each mode is solved separately and the results are combined. If part of the mass participates in no mode, the inertia force that mass would generate never enters the calculation at all. The total base shear comes out short.
2. What happened in this building
Three storeys, unreinforced masonry, timber-joisted floors and a timber-framed roof. Total weight —.
| Mode | Period (s) | Frequency (Hz) | Gap from previous |
|---|---|---|---|
| computing… | |||
3. Why? The diaphragm scatters the mass
In a building with reinforced concrete floors, the slab is nearly rigid in its own plane. The whole storey moves as one piece; the mass gathers into a few global modes and the first three usually carry most of it.
A timber-joisted floor, by contrast, is flexible in its own plane. The storey does not move as one piece: each wall vibrates semi-independently together with the piece of floor above it.
Flexible diaphragm: many local modes, each carrying a small share The total mass is the same; what changes is how many pieces it is split into. Collecting 95% of a mass divided among fifty modes is far harder than collecting 95% of a mass gathered into three.
4. The signature: modal clustering
We do not have the actual mass participation ratios. But the phenomenon has an observable signature, and that we do have: the clustering of modes along the frequency axis.
| Measure | Value |
|---|---|
| computing… | |
The first three modes lie within a 10% band of T₁. The smallest gap is —. This indicates that the structure does not have distinct global modes that can be separated from one another.
5. Compared with a separated-mode structure
For comparison, let us put the periods of a typical separated-mode structure through the same measure:
| Structure | T₁ (s) | Modes in 10% band | Smallest gap | Clustering |
|---|---|---|---|---|
| computing… | ||||
6. The code offers no way out
That is a scope choice rather than a defect: the Turkish code is written primarily for modern buildings with rigid diaphragms. A masonry structure with timber floors is a type that already falls outside its scope.
So the engineer is back in the situation of TAR-01: there is no rule to hide behind, and the choice and the responsibility are their own.
7. But it does have a modelling provision
The code does not say what to do when 95% cannot be reached, but it does say how the diaphragm is to be modelled:
In the source analysis the floors were defined as Semi-Rigid in ETABS. That is the approach Clause 4.5.6.2 calls for, and it distributes the lateral load among the walls realistically.
There is an important distinction here: modelling the diaphragm correctly does not solve the mass participation problem. Quite the opposite — had a rigid diaphragm been assumed, the modes would have gathered and 95% would have been met easily, but then a different structure would have been solved. The correct model is the one that exposes the problem.
8. Why is the drift check satisfied?
| Storey | δ/h (X) | δ/h (Y) | Limit | Utilisation |
|---|---|---|---|---|
| computing… | ||||
The interstorey drifts are far below the limit, which is no surprise: masonry walls are stiff, and a period as short as T₁ = — confirms it. The structure is adequate in terms of stiffness.
9. What we do not know
The limits of this article should be stated plainly:
- We do not know the actual mass participation ratios. The report says 95% was not reached but does not say what was. There is a large difference between 94% and 60%, and that difference determines how much the result can be trusted.
- We do not know where the missing mass went. The roof, the intermediate floors, particular walls?
- We do not know whether any correction was applied. Was an equivalent static load added for the missing mass?
Because the report answers none of these three, this article is rated confidence B. If the figures become available it can be raised to A — and then the genuinely interesting question can be answered: by how much does the missing mass reduce the base shear?
10. What to do in practice
- Always report the mass participation ratio. "95% was met" or "it was not met; X% was reached" — write both, and the second one especially.
- Check whether the first modes are clustered. If they are, you know early that 95% will be a struggle.
- Model the diaphragm according to its behaviour. Assuming a timber floor is rigid makes 95% easier but solves the wrong structure.
- Know that adding modes has a limit. Fifty were not enough here; a hundred may not be either, because the problem is not the number of modes but the distribution of the mass.
- Consider an alternative route. The equivalent lateral force method, a missing-mass correction, or a time-history analysis — whichever is chosen must be written down with its reasoning.
- Do not mistake a passing drift check for adequacy. In masonry, shear strength governs.
11. Test yourself
- What is the physical reasoning behind the 95% rule? If part of the mass participates in no mode, what is missing?
- Why does a flexible diaphragm scatter the mass among many modes?
- What is modal clustering a signature of? Does it substitute for mass participation?
- Assuming the timber floor is rigid makes 95% easier. Why is that not a correct solution?
- Would raising the mode count from 50 to 100 solve the problem? Why?
- In this building the drift check passes but the walls are inadequate. How is that possible?
- Why is this article rated B? Which figure would have made it A?
References
- TBDY 2018 (Turkish Building Earthquake Code) — Clause 4.5.6.2 (non-rigid floors modelled with two-dimensional finite elements), Clause 4.8.1.2(a) and Eq.(4.30) (the 95% mass participation rule), Clauses 1.1.8 and 15.1.5 (scope). Read from the full official text.
- Case data: ETABS modal analysis (50 modes) of a registered primary school building; published with the permission of the data owner. The mass participation ratios are not given in the source report.
All figures in this article are produced by engine/masonry.js and separately pinned in tests/masonry.test.js (59/59). A separate test case checks that the mass participation ratios we do not have have not been invented in the data set. That the modal clustering measure discriminates is separately tested against a separated-mode example.
The correct diaphragm model does not solve the problem; it exposes it.