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Damage, Repair, Strengthening
A laboratory sequence in which every state is known: intact, damaged, repaired, strengthened. Most indicators behave as expected — but one of them moved the wrong way after repair, and that anomaly is the most instructive part of the data.
1. The experimental sequence and the data
Field data has a structural weakness for method development: the true state is never known. You do not know whether the building is damaged, where, or how much — that is precisely what you are trying to find out. So a method's success cannot be scored.
Laboratory sequences remove that weakness. Here the same specimen is measured in a series of states, and each state is known because it was created deliberately: intact, progressively damaged, repaired, then strengthened with CFRP. Mode shapes are recorded at — measurement points.
2. How damage lowers MAC
Each state is compared against the intact reference with MAC:
| State | Mode 1 | Mode 2 | Mode 3 | Note |
|---|---|---|---|---|
| computing… | ||||
The general trend is as expected: as damage increases MAC falls, and it falls further in the higher modes.
3. Why are high modes more sensitive?
A high mode has more nodal points along its length, so the curvature of its shape is greater. Damage is a local loss of stiffness, and its effect on the shape is concentrated where the curvature is high.
That tension is a recurring theme of the series. In SHM-04 the third mode could not be resolved at all because of the sensor layout. Here, in a laboratory with dense instrumentation, it can be — which is why this anomaly is visible in the first place.
4. Repair: the low modes came back, the third did not
This is not a small deviation. The repair improved the first and second modes as intended, and made the third mode worse than the damaged state. An indicator moved in the opposite direction to the physical intervention.
5. Two explanations for the anomaly
First: the repair really did change the shape. Repair is not restoration to the original. Injecting a crack with epoxy produces a region locally stiffer than the surrounding material. For low modes that is close enough to the original. For the third mode, whose curvature is concentrated exactly in that region, the local stiffness anomaly shows up as a shape change.
Second: the third mode identification is noisier. As section 3 said, high modes have a worse signal-to-noise ratio. Part of the fall may be measurement error rather than physics.
6. What did the CFRP strengthening do?
In the final state the specimen is strengthened with carbon fibre. The second mode MAC returns to — — nearly the intact value.
But note what that does not mean. A high MAC says the mode shape resembles the original; it says nothing about capacity. A strengthened member may carry considerably more load than the original while having almost the same mode shape. The converse also holds: a member whose shape has changed is not necessarily weaker.
7. COMAC: where the change concentrates
| State | Mean indicator | Four most divergent points |
|---|---|---|
| computing… | ||
8. The "worst node" is not a fixed defect
Look along the last column. The set of most divergent points changes from state to state. That is worth pausing on, because it is easy to expect the opposite: if there is a defect at one location, surely it should show up at the same node throughout?
It does not, and there are two reasons. First, each intervention changes the structure — the damaged, repaired and strengthened specimens are genuinely different structures, not three views of one. Second, the COMAC value at a point depends on the mode shapes summed over all modes, so a change in any mode redistributes the whole ranking.
9. What follows in practice
- Do not read one indicator on one mode. The whole article rests on this; the third mode alone would have given a false answer.
- Do not treat a returning MAC as proof of successful repair. Nor a falling MAC as proof of failure.
- Do not confuse MAC with capacity. MAC sees stiffness distribution, not strength.
- Report the worst node together with its comparison. It is not a fixed property of the structure.
- Interpret high-mode results with their noise. They carry more damage information and more error at the same time.
- Where you can, test methods on known states. A field data set cannot score a method; a laboratory sequence can.
10. Test yourself
- Why are high modes more sensitive to damage? Which geometric property of the mode shape is behind it?
- After repair the third mode MAC fell below its damaged value. Give two candidate explanations and say how you would separate them.
- Why does a high MAC not mean the member is strong?
- What kind of strengthening would leave the mode shape unchanged and MAC at 1?
- Why does the worst node change from state to state?
- What can a laboratory sequence show that field data cannot?
References
- MAC and COMAC definitions — standard modal analysis results.
- Data: mode shapes from a laboratory sequence of progressive damage, repair and CFRP strengthening on the same specimen, with each state known.
All figures in this article are produced by engine/shm.js and separately pinned in tests/shm.test.js (50/50). The same comparison can be repeated with your own mode shapes in the calculation tool. The anomaly discussed in section 5 is deliberately left unresolved: this data set does not permit a choice between the two explanations, and asserting one would go beyond the evidence.
MAC measures the distribution of stiffness, not its magnitude — and not capacity at all.