Source: https://jmrplens.github.io/phonometry/vibration/structural/structural-damage/ # Vibration damage to structures (DIN 4150-3) Everything else in this section follows vibration because it ends up as **sound**. This page follows it because it ends up as a **crack**. A pile driver next door, a tram at the end of the street or a blast on a nearby site puts vibration into the ground, the ground puts it into a foundation, and the question the neighbours ask has nothing to do with audibility: will the building be damaged? Answering it properly means the dynamic stresses in the structure, compared against what the material and the design allow. DIN 4150-3 offers the cheap answer first, and it is the one used in practice: **guideline values** (*Anhaltswerte*) for a single measured quantity, the peak particle velocity, drawn from a large body of measurements on real buildings. Keep under them and damage of the kind the standard defines has not been observed in that body of measurements, which is a statement about the evidence rather than a promise about this building: meeting a guideline value satisfies the DIN 4150-3 criterion and certifies nothing. Exceed them and nothing follows automatically. The standard is explicit that damage does not have to occur; what has run out is the cheap answer, and the stress calculation has to be done instead.

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import vibration freq = np.linspace(1.0, 100.0, 400) fig, ax = plt.subplots() for name in vibration.BUILDING_CLASSES: ax.plot(freq, vibration.guideline_velocity(name, freq), label=name) ax.set(xlabel="Frequency [Hz]", ylabel="Peak velocity $v_i$ [mm/s]") ax.grid(True, alpha=0.3) ax.legend() plt.show() # One measurement against the guideline it is judged by, drawn on the same # curves: res = vibration.assess_building_vibration( 4.2, building_class="residential", frequency_hz=18.0 ) print(round(res.guideline_mm_s, 2), res.within_guideline) # 7.0 True res.plot() plt.show() ```

## 1. What is measured Clause 5.1 fixes the quantity before it fixes any number. At the **foundation**, the three components $v_x$, $v_y$ and $v_z$ of the particle velocity are recorded separately and the largest peak of the three, $|v_i|_\mathrm{max}$, is the value judged; the standard then writes it $v_i$ and so does this page. The transducers go in the lowest storey, on the foundation or the outside wall, and on an unbasemented building no more than 0,5 m above ground level, preferably on the side facing the excitation. In the **topmost floor plane** the quantity changes: the larger of the two horizontal components, measured in or close to the outside wall. That measurement is not a second excitation. It is the building's horizontal answer to the one at its foundation, which is why the guideline value there does not depend on frequency: the response is already at the structure's own frequencies. **How the measurement goes.** Use a vibration meter that meets DIN 45669-1, checked and calibrated at a reference frequency and amplitude for the job. At the foundation, three transducers go close together in the lowest storey, on the outside wall or its foundation, preferably on the side facing the source and, without a basement, no higher than 0,5 m above ground level, recording $z$ and two horizontal directions at right angles, one parallel to a side wall and preferably $x$ pointing at the source. On a hard surface a transducer set down loose holds only to 100 Hz vertically and 40 Hz horizontally at 3 m/s² or less. In the topmost floor plane, $x$ and $y$ go in or close to the outside wall; where relevant, on the floors expected to vibrate most, $z$ goes at about mid-span; on a pipeline, the transducers go on the pipe itself, bared only at the measuring point. Where possible, extraneous vibration is judged by comparing the readings with and without the source acting. Read the peak of each component and its frequency, and report what Annex A lists. ## 2. Short-term vibration, and why the guideline rises with frequency Short-term vibration is vibration that does not occur often enough to build up resonance in the structure: construction work, blasting, a passing train on a line that is not busy. Table 1 gives its guideline values for three building classes, and at the foundation they depend on frequency. They rise, and the reason is that the standard is really about strain. The same velocity at a lower frequency means a larger displacement, and it is the displacement across a wall that opens a crack in the plaster. So a building tolerates a fast wiggle better than a slow one, and the guideline value climbs from 1 Hz to 100 Hz. Above 100 Hz the 100 Hz value may be used. Table 1 prints the middle band as a range, "5 bis 15" for a dwelling between 10 Hz and 50 Hz, and a range does not say what 30 Hz is worth. Bild 1 says it: the corner values joined by straight lines on a **linear** frequency axis. A dwelling at 30 Hz is allowed 10 mm/s, and that is a reading of the figure, not of the table. ```python from phonometry import vibration # The corners Table 1 prints, for a dwelling. for f in (1.0, 10.0, 50.0, 100.0): print(f, vibration.guideline_velocity("residential", f)) # 1.0 5.0 / 10.0 5.0 / 50.0 15.0 / 100.0 20.0 # Inside a band, which only Bild 1 decides. print(vibration.guideline_velocity("residential", 30.0)) # 10.0 # In the topmost floor plane there is nothing to read against frequency, # and passing one anyway is refused rather than ignored. print(vibration.guideline_velocity("residential", location="top_floor")) # 15.0 ``` The three classes are the rows of Table 1: `"commercial"` for commercial and industrial buildings and buildings of like structure, `"residential"` for dwellings and buildings of like construction or use, and `"sensitive"` for buildings that fit neither and are worth preserving, a listed building being the example the standard gives. A massive engineering structure such as a reinforced-concrete abutment or a block foundation may raise the row 1 values by up to a factor of two, which is what 5.1 allows provided nothing dangerous comes of the soil mechanics. The allowance is a sentence of Clause 5 naming Table 1, so it stops at short-term vibration and the library refuses to carry it over to Table 3: ```python print(vibration.guideline_velocity("commercial", 50.0, massive_structure=True)) # 80.0 ``` ## 3. Floors, and the one number that covers them A floor or ceiling gets its own sentence rather than a table. Clause 5.2: if the vertical velocity at the point of largest vibration, in general mid-span, stays at or below 20 mm/s, no reduction in the serviceability of the floor is expected. The value is published as `FLOOR_VERTICAL_MM_S`, and for a building in row 3 of Table 1 the standard notes that even that may have to be reduced. ## 4. Buried pipelines A pipe in the ground is judged on the pipe, not on the building above it, and by what it is made of. Table 2 has three rows, and Clause 6.3 lets long-term vibration use the same table at 50 % without further evidence: ```python from phonometry import vibration print(vibration.pipeline_guideline_velocity("welded_steel")) # 100.0 print(vibration.pipeline_guideline_velocity("masonry_or_plastic")) # 50.0 print(vibration.pipeline_guideline_velocity( "masonry_or_plastic", duration="long_term")) # 25.0 ``` House connections up to 2 m from the building are judged by the building's own foundation values instead, and drainage pipes by row 3 whatever they are made of. ## 5. Long-term vibration Long-term vibration (*Dauererschütterungen*) is the opposite case: it occurs often enough for the structure to respond at its own frequencies. Table 3 answers it with one value per class in the topmost floor plane and nothing at the foundation, because a structure ringing at its own frequency is judged where it rings. The values run from a quarter to a third of the short-term ones, and the fraction is not the same for the three classes: 10 against 40, 5 against 15 and 2.5 against 8 mm/s. ```python from phonometry import vibration for name in vibration.BUILDING_CLASSES: short = vibration.guideline_velocity(name, location="top_floor") long = vibration.guideline_velocity( name, location="top_floor", duration="long_term" ) print(name, short, long) # commercial 40.0 10.0 / residential 15.0 5.0 / sensitive 8.0 2.5 ``` Asking for a long-term value at the foundation raises, because Table 3 does not have that column. ## 6. From a velocity to a stress Clause 6.2 is the bridge back to the proper calculation. For a beam or a one-way slab of full rectangular section, constant stiffness and uniform mass, vibrating in one mode, the peak bending stress follows from the peak velocity alone: $$ \hat{\sigma}_\mathrm{max} = 1{,}73 \left(E_\mathrm{dyn}\, \varrho\, \frac{G_\mathrm{ges}}{G_\mathrm{balken}}\right)^{0,5} k_n\, \hat{v}_\mathrm{max} $$ No length, no depth, no span: the dimensions cancel, which is the whole reason the formula is worth having. A velocity measured where the amplitude is largest is enough. The mode coefficient $k_n$ lies between 1 and 1,3 in the technically important cases, so it can move the answer by less than a third, and $G_\mathrm{ges}/G_\mathrm{balken}$ is 1 for a beam carrying nothing but itself. ```python from phonometry import vibration # Concrete, 30 GPa dynamic modulus, 2400 kg/m3, first mode, 10 mm/s peak. # Note the unit: the tables are in mm/s and this formula is in m/s. sigma = vibration.bending_stress( 0.010, dynamic_modulus_pa=3.0e10, density_kg_m3=2400.0 ) print(round(sigma / 1e6, 2), "MPa") # 0.15 MPa ``` Clause 6.4 adds the estimate that says whether the topmost floor plane will answer at all: for a building of about five storeys or more, the lowest horizontal natural frequency is roughly $f_i \approx 10/n$ with $n$ the number of storeys. ```python print(vibration.storey_fundamental_frequency(10)) # 1.0 Hz ``` Below that, `BuildingDamageWarning` says so: four storeys still return 2.5 Hz, because "about five" is not a line the standard drew, but the number is an extrapolation of a rule offered for taller buildings rather than the rule itself. ## See also - [Evaluating machine vibration (ISO 20816-1)](https://jmrplens.github.io/phonometry/vibration/machinery/machine-vibration-evaluation/): the same shape of question asked of a machine rather than of a building. - [Human vibration exposure (ISO 2631, ISO 5349)](https://jmrplens.github.io/phonometry/vibration/human/human-vibration/): what the same vibration does to a person. - [Transfer stiffness of resilient elements (ISO 10846)](https://jmrplens.github.io/phonometry/vibration/structural/transfer-stiffness/): the elements inserted to keep the vibration out in the first place. ## References - Deutsches Institut für Normung. (1999). *Erschütterungen im Bauwesen — Teil 3: Einwirkungen auf bauliche Anlagen* (DIN 4150-3:1999-02). The measured quantity of 5.1, the short-term guideline values of Table 1 with the curves of Bild 1, the ceiling rule of 5.2, the buried-pipeline values of Table 2, the long-term values of Table 3 and the 50 % reduction of 6.3, the bending stress of Formula (1) in 6.2 and the storey estimate of 6.4. The measuring points, directions and report of 5.4, the pipeline notes of D.1 and the worked dwelling of D.2 are prose, set out in the measurement section. - Deutsches Institut für Normung. (2005). *Messung von Schwingungsimmissionen — Teil 2: Messverfahren* (DIN 45669-2:2005-06). Where the transducers of a building measurement go and how they are set down: the foundation positions of 5.1.2, the measuring directions of 5.2, the loose-mounting limits of 5.3.2, the measurement time of 6.1, the check before use of 6.3 and the comparison with the source stopped of 7.2.2. ## Standards DIN 4150-3:1999-02, *Erschütterungen im Bauwesen — Teil 3: Einwirkungen auf bauliche Anlagen*: the guideline values for the peak particle velocity by building class, measurement location and how often the vibration occurs (Tables 1, 2 and 3, with Bild 1 fixing the reading between the printed corner values), the vertical value for floors (5.2), the doubling of row 1 for massive engineering structures (5.1), the 50 % reduction of the pipeline table for long-term vibration (6.3), the bending stress of Formula (1) in 6.2 and the storey estimate of 6.4. Prediction (DIN 4150-1), human perception in buildings (DIN 4150-2) and the instrument requirements of DIN 45669 are not implemented.