Vibration and Distance

Most people understand that the farther you are away from an event, the less likely you are to be significantly and directly affected by it. Indeed, distance from the source of a vibration is probably the single most important factor, after the source energy, in determining whether a vibration will have damaging effects on a home or structure (and on the human perception of the vibration). Here, I'll provide an introduction to the complex matter of distance effects on vibration intensities, with an eye toward understanding the basic factors which affect vibration movement through the ground, how vibration velocities can be mathematically estimated (as plotted at left) and the cautions that one must observe in using those estimates. Spreading of Vibration Energy As discussed generally in the CVDG's Vibration 101 chapter, ground vibration intensities (usually measured as vibration Peak Particle Velocities, PPV's) decrease as distance from the source increases (i.e. the vibration velocities depend "inversely" on distance). This is, at root, a geometric effect. The vibration decreases as a given amount of source vibration energy spreads over a sphere (or surface layer) of increasing size. But, the specific way that the vibration velocity decreases with distance is affected by many variables, some of which are listed below:
Many of these factors can be included in calculations, at least in principle, if enough information is known about the vibration source and locale. Others are so sitespecific and require so much information (e.g. embedded objects, reflection and interference effects, presence of buildings, landscaping) that they are virtually impossible to account for in the generaluse equations often employed for predicting vibration velocities. The many variables which affect vibration transmission can make accurate calculations of vibration velocities challenging, in the absence of substantial validation of the results by comparison with measurements made on the site of interest. However, calculating (and measuring) how vibration velocity changes with distance is central to an ability to estimate "safe distances" (check the CVDG Pro's "Vibration Safety chapter for more on such estimates) from construction work or to use vibration propagation equations meaningfully in any other way. Thus, such calculations are widely used, even if, sometimes, questionably so. Simple Estimates of Vibration Velocities One of the most commonly used relationships for estimating constructionrelated vibration velocities is that offered in the U.S. Federal Transit Administration's Noise and Vibration Manual:^{1} PPV_{equip} = PPV_{ref} x (25/D)^{1.5} where: PPV_{(equip)} is the calculated peak particle velocity in in/sec of the construction equipment type at the distance of interest, PPV_{(ref)} is the reference vibration level for that type of construction heavy equipment in in/sec at the reference distance of 25 feet, from Table 122 of the FTA Noise and Vibration Manual,^{1} and D is the distance from the equipment to the structure or seismograph, in feet. This equation combines all the physical details of vibration movement ("propagation") through the ground in the 1.5 exponent and all the details of the source vibration properties in the reference velocity, PPV_{(ref)}, which is related to the source energy. In this equation, the distance from the vibration source, D, is in the denominator of the fractional rightmost factor. So, the calculated vibration PPV decreases as the distance increases, just as you would expect. The FTA equation is phenomenological (i.e. it is fitted to vibration data, rather than taking into account directly in its factors most or all of the known physical variables which affect vibration propagation). Thus, there is little direct relationship between its parameters and measurable quantities affecting vibration transmission which one can relate back to physical laws and material properties. It is a considerably simplified, but often useful, approach to portraying some aspects of vibration transmission through the ground. Using the FTA Equation Immediately below is plot of calculated ground vibration velocities for vibratory compactors, according to the FTA equation, for various values of the exponent, n, and for two different values of PPV_{(ref)}. In this semilog plot, the velocities are plotted on the vertical scale with equal distances between equal values (i.e. linear), while the distance scale is logarithmic,^{7} with powers of ten (10, 100, 1000, etc.) of the distance having equal separations. Such logarithmic plots enable one to see better the relationships when large spans of numbers are to be shown. That fact can be easily seen by comparing this plot with the small alllinear version at the start of this chapter. You can view, in a new tab or window, (and print a copy from your web browser for personal use, if you wish) a fullsize version of this diagram, as well as all the other distanceground vibration velocity relationships in this CVDG chapter just by clicking on each diagram.^{8 }Calculated ground vibration velocities can be read off any of the diagrams in this chapter for any distance from 5 to 250 feet from the vibration source just by noting where the distance to the construction work^{15} intersects the appropriate curve, then reading the velocity at that intersection from the vertical scale. The velocities calculated in the diagram above are for various values of the FTA equation exponent, as might be needed for soils of differing type, using a vibratory compactor with an FTA reference velocity of 0.210 in/sec. Also shown are values calculated from more recent, revised values of the exponent and PPV_{(ref)} from a U.S. state of New Hampshire Department of Transportation (NHDOT) study.^{2,3} This diagram illustrates that the value one calculates for a vibration velocity at a specific distance is determined by the choices of PPV_{(ref)} and the exponent that one makes. If values for these parameters are chosen which do not reflect or properly account for the actual source vibration characteristics, the local soil and geology conditions, or other factors not included in the equation (e.g. wave reflection and interference), the calculated values will differ, sometimes substantially, both from each other and from measured vibration velocities. The OSM blasting standard limit for vibration midfrequencies in undamaged homes of modern construction, without plastered walls and the FTA Class III limit for timberframed homes are also shown as horizontal lines in the diagram. "Acceptable" vibration peak particle velocities for each of those two standards lie below the lines shown. Velocities exceeding those standards lie above the dotted horizontal lines. Any vibration velocity above these standards has a higher probability of causing damage than one below the standard, at least in the circumstances for which the standards are intended (see Vibration Standards for more about ground vibration standards and their use). The OSM standard is only applicable to examples where the vibration source is blasting, either in mining or, with less assurance, in construction. Construction heavy equipmentcaused vibration should be governed by the one of the FTA limits in the U.S., of which the Class III limit is shown in the diagrams in this chapter. The related Swiss machines and traffic standard is relevant and less lenient than the FTA standard (see Vibration Standards for more detailed information). None of the curves in this chapter show home or building vibration velocities, which may be substantially higher (see Resonance/Fatigue for more on this), due to varying degrees of resonant amplification of the ground vibration in the structure.^{27} Another Depiction In the next diagram below, the same calculations are represented on a "loglog" plot, in which both the calculated velocity on the vertical scale and the distance on the horizontal scale are logarithmic. This depiction is the way that such vibration velocity/distance relationships are usually shown in the scientific literature. It allows a wide range of values to be shown on both the vertical and horizontal axes. Plotting the calculations in this way has the effect of turning the curves into lines with a slope, n, of the equation exponent. As the diagram shows, there are significant differences in the calculated velocities as one changes the value of the exponent. The lower the value of the exponent, the farther the vibrations move before their ground peak particle velocities fall below standard limits. Indeed, the NHDOT studies, based on newer data and more modern road construction equipment,^{2} found significant deviations from the FTA suggested values of 1.5 for the exponent, n, and the FTA suggested PPV_{(ref)} value of 0.210 in/sec for vibratory compactor use at 25 feet. For example, NHDOT found that its own data were fit better by an exponent of 1.1 and a PPV_{(ref)} for vibratory compactors of 0.45 in/sec (c.f. FTA PPV_{(ref)}=0.21 in/sec), as shown in the diagram above.^{3,16} Both these changes have the effect of making the "safe distance" for such compactors considerably larger than the FTA equation calculation indicates, as seen in the plots in this chapter. The more uptodate NHDOT data have the effect of doubling one estimation of "safe distance" (i.e. that distance at which the calculated vibration velocity drops below the FTA Class III standard for timberframed homes) for vibratory compactors (as seen at left) from about 30 feet to over 60 feet.^{ }Note that the compactor in the photo is being used considerably closer to homes than 60 feet.^{4,24} The New Hampshire experience and parameters may not be representative of all locales. Indeed, there are some areas where vibration propagation is more favored than in New Hampshire. For example, a study for the U.S. state of Florida^{ }DOT, with 170 data points involving vibratory and static compactor use over various compaction bases,^{ }shows an exponent, n, of 0.6, with an apparent range of PPV's at or near 25 feet (the FTA reference distance) for the compaction work of from 0.05 in/sec to 0.5 in/sec, depending on compacted material.^{20} Just below is a comparison of calculated vibration velocities for vibratory compactors, using the FTA, FLDOT and NHDOT parameters in the FTA equation. The diagram shows that a calculated "safe distance" (i.e. that distance at which the vibration velocity falls below the FTA Class III limit) ranges from around 25 feet for the FTA parameters, to nearly 60 feet for the NHDOT parameters, to 100 feet for those calculated from the extensive FLDOT study results.^{21,24} We consider the New Hampshire data to be instructive, representative and reasonably conservative with respect to vibrations in most, but not all, other locations. When there is predictable potential for damage to the property of uninvolved third parties, as is almost always the case in construction damage incidence, the conservative approach is the one which should be followed. Therefore, the New Hampshire values for PPV_{(ref)} and the exponent of 1.1 in the FTA equation are currently the recommended ones to use,^{6} in the absence of specific and detailed vibration propagation data (e.g. like those from Florida), pertaining directly to the location of interest, which indicate use of different parameters. Other, modified versions of the FTA equation have been proposed and utilized for specific equipment types (e.g. pile drivers) and situations.^{ }In most of these examples, the value of the exponent n suggested is 1.1,^{6} in accord with the NHDOT study. The lower value of n results in a slower reduction of the vibration intensity with increasing distance than the FTA equation value of 1.5 would indicate. The NHDOT and FLDOT studies, among others,^{6} demonstrate that neither the 1.5 exponent nor the reference equipment velocities given in the FTA vibration manual are current or appropriate in all circumstances.^{2,6 } Other Equipment Types While vibratory compaction is the most common damage cause reported to Vibrationdamage.com, it is not the only type of construction operation with damage potential. Following is a comparison between FTA equation velocities calculated with parameters for both compactors and pile drivers, another class of equipment with significant damage potential. As the diagram shows, both pile drivers and vibratory compactors can produce ground vibrations whose velocities exceed the FTA Class III limit for timberframed homes and, close in, even the OSM midfrequency blasting limit. The NHDOT study also provides updated PPV_{(ref) } values for a broader variety of different equipment types and operations, along with the suggested value for the exponent of 1.1, for use in the FTA equation. Just below is a loglog plot of calculations with the NHDOT PPV_{(ref) } values for a wider range of construction equipment.^{22} The plot shows that some operations, particularly vibratory compaction, pile driving, use of hoe rams (hydraulic rack breakers mounted on excavators) and driving tracked vehicles over distance, can be expected to exceed one or both standard limits over at least some distance ranges. Particular attention must be paid to those operations to make certain that they are not carried out within those distance ranges and/or are appropriately mitigated to reduce damage probabilities (see Vibration 101 in the CVDG and Vibration Mitigation in the CVDG Pro for details). These calculations are only extended to 250 feet from the source, as the diagrams demonstrate that most forms of equipment vibration will be below the FTA Class III limit for homes at that distance in most locales with "normal" vibration attenuation, even when a factor of two "safety factor" is included. Conditions of low vibration attenuation (e.g. localities in Florida), interference effects in the ground vibration, or vibration frequencyrelated resonance effects in the home may act to make specific examples of equipment use dangerous for homes at distances well beyond 250 feet in some circumstances.^{27} Using Vibration Calculations It is relatively rare for vibration monitoring to be done on construction jobs (as at right), even though most public construction contracts require it as part of their "boilerplate" conditions. Thus, the typical homeowner facing construction vibration, and possible damage from it, will have no or limited access to measurements of vibration velocities to compare with the diagrams on this page. This is one of the reasons that we have recommended that homeowners do some monitoring of their own (see our Vibration Monitoring chapter for more on this point) to get an idea if felt vibrations actually have damage potential. However, in the absence of actual data, it is important for homeowners to know which operations have significant damage potential and which ones are of lower risk. Such information will help them in making knowledgeable judgments about felt vibrations and in understanding the real meaning of vibration data, when the data are available or when conclusions from monitoring or calculations are offered to homeowners. Indeed, even the more recent NHDOT data underestimate vibration velocities at distance in some areas (e.g. many parts of the U.S. state of Florida^{9,12}) which have especially low vibration attenuation (i.e. loss of vibration velocity with distance), as indicated above. Correlations also exist between soil type and the value of the equation exponent, with softer soils attenuating vibration more efficiently (producing exponents up to 1.4); a 1.0 exponent is suggested for hard rock.^{6} There are even variations within types of operations done with different pieces of the same general equipment type.^{11} Calculating Ground Vibration Velocities for Vibratory Compactors Vibratory compactors (like the examples shown at left and above)^{19} and rollers present special problems in prediction of their produced ground vibrations and, thereby, their damage potential. Different models have different drum weights, different and often variable centrifugal force limits, operatorvariable compaction amplitudes, different rates of travel, different types of vibration motions (cf. updown vibratory vs. backforth oscillatory compactors^{11,26}) and different, sometimes operatormodifiable, vibration frequencies.^{16,19} All of these parameters are known to affect the velocity and frequency of the ground vibrations produced when compactors are used.^{23} Yet, full knowledge of all of them is rarely available in reallife compaction situations. Compactors using lower frequency vibrations create vibrations in the ground which have greater range (and larger safe distances) than those compactors whose nominal vibration frequencies are higher, due to greater attenuation (loss) of the higher frequency vibrations in the ground.^{17} Low frequency vibrations also transfer vibration more efficiently to home resonant motions than higher frequency ones. The same compactor may display different dominant frequencies, depending on distance from it, as a result of interference effects between the vibrations from the two drums (see Vibration and Damage for an example and Vibration Signatures in the CVDG Pro for more on this issue). Similarly, if vibration is not turned off when the compactor changes direction, as is often the case in construction work,^{18} relatively short duration ("transient") vibrations, of considerably higher velocity than the normal, "steadystate" ones, can be generated as a result of "pileup" of vibrations.^{11,18 }The effect of the many variables which affect compactor vibrations, whose values are often unknown in a given situation, can mean that calculated ground velocity values for compactors, even when reasonably close to some measured values, may not correctly reflect true damage potential.^{19} About Construction Blasting Vibration and DistanceAlthough Vibrationdamage.com and its Facebook page emphasize construction effects of vibration from heavy equipment, blasting is sometimes done in construction contexts to remove rock or level sites on rock. Construction blasting differs substantially from mine and quarry blasting in the typical amounts of explosive used per delay and the frequency distribution of the produced vibration.^{29} Of course, it differs from heavy equipment vibration production, too, even though it is another energy source driving vibration, broadly similar to the driving energy provided by heavy equipment.
Blasting vibration distance relationships are usually depicted differently in scientific contexts than other kinds of construction vibration. The diagram below shows how blasting vibration velocities are typically plotted vs. a quantity
called "scaled distance",^{9} rather than measured absolute distance. The scaled distance, usually denoted as SD or D_{s}, is just the measured distance between the blast and the seismograph recording the vibrations, divided by the square root of the charge weight of explosive used per explosion
delay. Charge weights per delay (usually around 89 milliseconds between the multiple closelytimed detonations which comprise one "shot") are nearly always reported in documents to local regulatory authorities, as required by local and state blasting regulations, along with the measured vibrations at the seismograph locations and
their absolute distances from the blast site.
Because the vibration velocity often is directly dependent upon the square root of the explosive charge weight per delay, dividing by the square root of the charge weight can remove most or all of the charge weight dependence of the vibration, leaving only the distance effects from vibration propagation and attenuation. This division of the distance by the square root of the charge weight is usually referred to as "square root scaling" in the scientific literature, since its goal is to place a series of blasts with different amounts of explosive on the same source energy basis. Square root scaling is normally used for any vibration data resulting from blasting in drill holes with multiple charges along a more or less linelike front  perhaps the most common situation in construction and mine blasting. For point source blasting in particular, cube root scaling, i.e. dividing the distance by the cube root of the charge weight per delay, sometimes provides better correlation of the observed velocities with distance.^{30,31} When plotted with logarithmic scales^{7} on both axes, as shown above in the diagram, the equation describing the vibration intensity as a function of scaled distance produces a line, whose slope is related to the type of soil through which the vibration moves. Very soft soils or soft rock produce values of b up to about 1.8; harder soils or rock produce values of b as low as 1.0. This kind of exponential equation transformation on a loglog plot is analogous to that generated in loglog plots of heavy equipment vibration propagation shown earlier in this chapter. The diagram shown here depicts actual blasting vibration data from a construction project involving leveling and terracing of a planned housing development area on a volcanic rock site.^{32} This may be more information than most people want to know about construction blasting. But, if your home is ever damaged by it, you may well be presented with a plot like the one here. Other information about blasting vibration science is found throughout the CVDG. The CVDG Pro chapter, Calculating Vibration Amplitudes, has considerably more information about blasting vibration and its decrease with distance. How Far Away Is "Far Enough" There is no single answer to the question, "How far away must my building be from a vibration source to be considered safe from vibration damage". Buildings have different degrees of resistance to vibration damage depending on design, construction materials and age, among other factors.^{28} As discussed above, both vibration source properties (e.g. pile driving vs. compaction) and vibration movement through the ground can be very different, depending on locale and presence or absence of structures and other obstructions which can damp, reflect or modify vibration waves. Resonance effects in homes can turn an "allowable" ground vibration into a home vibration which is damaging at considerably greater distance than the direct ground vibration itself might be. Statements like, "You are too far away from the source to have damage", have meaning, at most, only in the very limited sense of relative probability of damage (see Vibration and Damage), even when correctly founded upon real scientific insight. The FTA equation, and modifications of it, are widely used to approximate "safe distances" for construction operations, as illustrated in the diagrams above. However, sometimes, that is done improperly and without benefit of current scientific understanding of the appropriate exponent and PPV_{(ref)} for use in a given construction location and operation.^{24} Anyone presented with the results of such calculations should take care to determine which vibration propagation equation was employed for the calculation and the appropriateness of the parameters used in it. These factors entirely determine the result, its meaning and its applicability in a given situation and locale. In the end, correct and judicious use of proper ground vibration standards, combined with careful and scientifically valid vibration monitoring, can provide a sense of reassurance regarding vibration damage potential. Such an approach is not the same as a "guarantee" of vibration safety, but it is far better than guessing or hoping that damage will not be done. It is overwhelmingly more desirable than dealing with actual damage after it has occurred. A Note on Historic, CulturallyImportant or Damaged Structures The OSM velocity limit for blasting and the much lower constructionbased FTA Class III limit for timberframed homes, shown in the diagrams above, are not applicable for historic structures or those which have already suffered vibration damage. Typically, among the higher limits recommended in the scientific literature for such structures is the FTA Class IV limit of 0.12 in/sec (cf. FTA Class III limit of 0.2 in/sec shown in the diagrams above). A limit of 0.08 in/sec has been advised for the thousandyearold Pueblo Bonito (seen at right) in Chaco Culture National Historic Park in New Mexico, USA.^{13} Vibration velocity limits as low as 0.05 in/sec appear in the literature for other historic structures.^{14} These lower limits have the effect of increasing safe distances, calculated using diagrams like the ones in this chapter, for historic structures (see Vibration Safety in the CVDG Pro for examples illustrating this point). Other Ways of Calculating Vibration Velocities There are other, more complicated, less phenomenological, equations which can be used to calculate predicted vibration velocities in construction and blasting environments. Some are simple adaptations of the FTA equation, whose modifications better account for those types of equipment which have variable source energies (e.g. pile drivers, vibratory compactors and pavement breakers).^{6} A few variations in vibration calculation approaches are discussed in more detail in the CVDG Pro chapter, Calculating Vibration Amplitudes.^{9} However, none of them, no matter how many physical parameters are included, take into account locationspecific variables like vibration wave interference, wave reflection from nearby building foundations, underlying rock layers, buried obstructions, and landscaping effects. Nor do they directly account for resonant amplificationrelated damage potential in homes, especially where vibration duration becomes a factor, as in construction.^{10} Applicability of Vibration Calculations I have not tried to present here a complete analysis of all the ways of calculating vibration velocities, as such an analysis is probably beyond the needs of most people. Some of these other, more complicated, approaches^{25} are discussed in the CVDG Pro's Calculating Vibration Amplitudes chapter. However, it is fair to say that all the calculation approaches that have been developed to date have limitations in scope, accuracy and applicability. Both comparison of calculated velocities with measured data and the physical limitations of the equations themselves suggest that even the more physicallyrealistic equations can be considered only as ways of approximating actual ground vibration velocities. Even when local soil conditions are explicitly measured and taken into account, these more advanced approaches to vibration velocity/distance relationships can be in error with respect to measured vibrations by a factor of two or more.^{5 } This is the reason that we have recommended throughout the CVDG (e.g. Vibration Regulation) that, when calculations are used to estimate vibration velocities, they be subjected to careful validation in the locale of interest and at least a twofold safety factor be built into any velocity standard or "safe distance" derived from them.^{31} Any vibration with a calculated PPV near the upper limit of that twofold safety factor calls for implementation of mitigation measures. Vibration velocity calculations can be very valuable in understanding how far from the vibration source damage might occur and under what conditions. But, that value is greatly increased when the equations are employed with real understanding of how they can be scientifically applied in the local vibration environment and what their limitations might be.


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