C = 0.2; w = 3.359; (0.78 ÷ 50 m). DH = 17.26 mm2; RMSH = 4.154 mm; Sk = -0.469; Ex = -0.96;. IRI80 = 0.54 mm/m; IRI120 = 0.555 mm/m; IRIcon = 0.77 mm/m; ...
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Rapid Approximate Assessment of Basic Parameters of Longitudinal Road/Runway Profiles Rychlé přibližné hodnocení základních parametrů podélných profilů vozovek a letištních drah Ing. Oldřich Kropáč, DrSc, Aeronautical Research and Test Institute (emeritus), Prague; Ing. Peter Múčka., PhD, Institute of Materials and Machine Mechanics, Slovak Academy of Sciences, Bratislava, Slovak Republic This discourse is a follow-up to the earlier paper of the authors entitled "Indicators of longitudinal unevenness of roads/runways" published in the issue 1/2006 of these Proceedings. It contains simple "thumb-rules" for estimating the RMS-value and waviness of the measured longitudinal profile. As a practical aid to this processing, an atlas of 25 typical profiles is added. Measurements on a great number of in-service roads indicate that up to 80 % of them exhibit different waviness values in long and short wavelength bands. Therefore profiles were selected at which their waviness values change in the range from 1.5 to 3.5 with increments 0.5 and all their mutual combinations are considered. To increase the homogeneity of all set parameters the numeric simulation of profiles was used. Graphical presentation is given of these profiles and of all their statistical characteristics such as the power spectral density (PSD), correlation function (CF), probability density function (PDF) and distribution function (DF). In addition, all their numerical indicators characterizing the unevenness in one-, two- and three-wavelength bands of the total effectively acting wavelength band of random longitudinal unevenness are given. Indicators for particular band systems are compared in graphs. From them, systematic departures of estimated values caused by incorrect choice of the band system may be evaluated. Examples of some profiles measured in-situ with their complete documentation are also added. Toto pojednání navazuje na dřívější článek autorů "Ukazatele podélné nerovnosti vozovek a letištních drah" publikovaný v čísle 1/2006 tohoto Zpravodaje. Obsahuje jednoduchý přibližný postup pro odhadování středně kvadratické hodnoty a vlnitosti podle naměřeného podélného profilu. Jako praktická pomůcka k tomuto postupu je připojen atlas 25 typických profilů. Přitom se vychází z poznatků získaných měřením velkého počtu vozovek v provozu, že až 80 % z nich vykazuje rozdílné hodnoty vlnitosti pro dlouhé a krátké vlnové délky. Proto byly zvoleny profily, u nichž se jejich vlnitosti mění v rozmezí od 1,5 do 3,5 s krokem 0,5 a uvažují se všechny jejich vzájemné kombinace. K zabezpečení homogenity nastavených parametrů byla použita numerická simulace profilů. Vedle těchto profilů byly graficky zobrazeny jejich hlavní statistické charakteristiky, zejména výkonová spektrální hustota (PSD), korelační funkce (CF), hustota pravděpodobnosti (PDF) a distribuční funkce (DF). Rovněž jsou uvedeny všechny číselné ukazatele pro charakterizaci nerovnosti v jedno-, dvou- a třípásmových úsecích celkového rozmezí efektivně působících vlnových délek. Ukazatele pro jednotlivé pásmové systémy jsou porovnány graficky. Z nich se zejména dají vyhodnotit systematické odchylky odhadů v důsledku nesprávného výběru pásmového systému ukazatelů. Jsou připojeny též ukázky několika profilů vozovek v provozu s jejich úplnou dokumentací.
Keywords: longitudinal road unevenness/roughness, road profile, power spectral density (PSD), unevenness index, waviness, correlation function (CF), probability density function (PDF), distribution function (DF), International Roughness Index (IRI), Three-band indicators (TBI), uncertainty, systematic departures. 1. Introduction The paper links to an earlier paper of the authors [1], but for the sake of self-contained presentation the most important relationships, numerical indicators and reference citations will be reproduced here. Power spectral density GH [m3/rad] of unevenness H(l), the co-called longitudinal profile: GH(7) = C.7-w, (1) where 7 [rad/m] is the circular spatial frequency (circular wave number), C[radw-1.m3-w] = GH(1) is the unevenness index, w [1] is the waviness. Variance DH [m3] of the unevenness H(l): DH = C.(2P)-w+1.(w-1)-1.(Lmaxw-1-Lminw-1) (2) where Lmin, Lmax are lower and upper limit wavelength values, respectively. International Roughness Index (IRI) [mm/m] is defined by
a program by Sayers [2] and its relationship to C,w is given by IRI = 2.21. C.exp( 0.356w + 0.13.($w)2, (3) where $w = w - 2. Waviness wIRI [1] computed using the Sayers' algorithm: (4) wIRI = 3 - 11.3577.log(IRI/IRI120), where IRI120 is computed according to the IRI algorithm for the travel speed 120 km/h.
2. Rough estimation of the RMS value of the longitudinal road elevations From the visualized graphical road profile a rough estimate of the RMS value may be obtained by using the following procedure: Establish by eye envelopes, upper and lower, to the given graph in such a way that it is allowed for one peak per about 100 m on each side to overcross these envelopes. In the case that the unevenness is homogeneous on the given road section,
C Z E C H A E R O S PA C E P R O C E E D I N G S both the envelopes form straight lines parallel to the x-axis. The difference between them is approximately equal to quadruple of the RMS, thus the estimate of the RMS is a forth part of the difference between envelopes. The possible non-homogeneity can manifest in different ways. The simplest one consists in a sudden jump on the border between neighboring partial sub-sections within which the courses are otherwise locally quasi-homogeneous. Thus, two different estimates of the RMS for both partial sub-sections are necessary. Another simple non-homogeneity manifests as rising or descending straight-line envelopes where their slopes are equal. This case gives an account of the non-homogeneity in the mean value reflecting the overall rising or descending trend of the track. Here again, the vertical difference between both envelopes, divided by four gives the estimate of the RMS. When the upper and lower envelopes are non-parallel and possibly non-straight, the non-homogeneity appears also in the variance and therefore, the mean RMS-value over the nonhomogeneity range can be estimated. Also, the dependence of the variance on the distance can be roughly estimated. This processing requires a higher degree of experience, but yet, the uncertainty of the estimate is high.
3. Rough estimation of the waviness The global appearance of the longitudinal profile is prevailingly influenced by the waviness in the long wavelength band, which in the greatest extent contributes to the total variance of the elevation course, see equation (2). Accordingly, the highest wavelength, which appears in the measured record, is somewhat less than the highest wavelength inherent in the profile. This is usually limited by filtering in the frame of pre-processing stage of elaborating original rough data. The contribution of shorter wavelengths manifests as dense local peaks on the mentioned long waves the amplitudes of which are greater when the overall (or in the long wavelength band) waviness is low and rather small where the overall (or in the long wavelength band) waviness is high. Obviously, when the waviness values in long and short wavelength bands are considerably different, the mentioned tendencies are more involved and the meaningful estimation is more difficult. Long-term experience is needed especially when systematically comparing preliminary approximate estimates with those obtained by fully processing given profile records. It is useful to collect data sheets of typical profiles obtained in-situ and extract from them further detailed knowledge which may contribute to improved quality of discussed preliminary estimates.
4. Examples of typical road profiles In order to facilitate the approximate estimation of longitudinal road unevenness indicators from given visualized road profile, an atlas of 25 data sheets is given in Appendix 1. All these simulated profiles are given with a common unevenness index C0 = 1.10-6 rad.m and nominal waviness values wLnom and wSnom varying from 1.5 to 3.5 with steps 0.5. On each sheet, the typical features of the longitudinal profile, the power spectral density (PSD), the correlation function (CF), the probability density function (PDF) and the distribution function (DF) are plotted, the PSD in double-logarithmic, the DF in probability,
22 and the remaining variables in linear scales. Three systems of unevenness indicators are added: One-band indicators, which are defined for the wavelength band 0.3534 to 90.9 m as recommended in the ISO 8608 [3] and embrace the following indicators: parameters of the PSD function, considered as a straight line in the log-log plot, i.e. the unevenness index C and the waviness w, see equation (1), variance DH [mm2] of the longitudinal road elevations, equation (2) and its root-mean square value (RMS)[mm], further on skewness Sk [1] and excess (kurtosis) Ex [1] as indicators of deviation from the Gaussian feature of the probability density function, the International Roughness Index IRI [mm/m] as defined by Sayers [2], IRICON = IRI converted from (C,w) according to equation (3) and wIRI computed according to equation (4). Two-band indicators defined for long wavelength band 6.28 to 90.9 m as CL and wL and for short wavelength band 0.3534 to 6.28 m as CS and wS, supplemented with their difference $w = wL - wS. Three-band indicators corresponding to the long (12.5 to 50m), medium (3.125 to 12.5m) and short (0.78125 to 3.125 m) wavelength bands according to the EN 13036-5 [4], i.e. CL,wL, CM,wM and CS,wS, respectively; also, the corresponding variances DL, DM and DS and their sum DH are added. All the presented indicators are summarized in Figs. 1a to 3j in dependence on wL and wS (as a parameter). Except for C's and D's (and RMSH), which are plotted in logarithmic scales in order to (at least approximately) linearize the relationships in question, all the remaining quantities are plotted in linear scales. For these dependences, simple deterministic relations holds (cf.[1]) which are in the figures indicated as broken lines. Deviations of the presented simulation results indicate partly statistical uncertainties due to the methodology of simulation providing independent individual realizations, partly due to the application of simplified PSD models. This last is best evident when approximating a PSD with different wL and wS values by means of a single straight-line, which has evidently its waviness value between wL and wS and simultaneously, the original C-value also changes. Thus, a short commentary on the figures presented seems to be in order. In Fig. 1a, the dependence of C on wL and wS is given. The broken line corresponds to cases with wLnom = wSnom and therefore indicates the uncertainty of simulated C's which is negligible (may be, with the except of some greater deviation for w = 3.5). For cases wL x wS the deviations from the broken line manifest the systematic departures due to the approximation of the broken-line PSD by a single straight-line. Note, however, systematic and practically at equal steps identically shifted straight-line dependencies. The same holdsfor Fig. 1b concerning the simulated waviness values. Here, the cases wLnom = wSnom lie on a slant straight-line and the remaining dependences lie on parallel straight lines with their sequence being opposite to that for C's. In Figs. 1c and 1d, the dependences of DH and RMSH respectively, are plotted showing pronounced nonlinear dependence on wL but practical independence of wS. In Figs. 1e and 1f, the departures of statistical characteristics indicating the non-Gaussianity of the probability distribution function are plotted. While the
23 variability of the skewness SkH may be judged as random, the excess ExH indicates a slight overall tendency to negative values, i.e to peakedness shapes what also manifests on the prevailing cases presented on pictures in the Appendix 1. Let us mention the known properties of statistics that with increasing their order, the uncertainties of their estimates progressively increase. In Fig. 1g, the dependence of IRI on wL and wS reflects for cases wL = wS the conversion relation (3) between IRI and (C,w). In Figs. 2a and 2b, the effect on wLnom and wSnom on CL and CS is visualized. While for CL, systematic component of deviations is evident (which increases with increasing waviness), for CS the deviations have only stochastic character. These different results may follow from the different numbers of data, which are at disposal for the evaluation of unevenness indices in both wavelength bands. i.e small in long and large in short wavelength bands, respectively. Figs. 2c to 2e demonstrate the dependences of wL, wS on wLnom, wSnom. As could be expected, wL is linearly dependent on wLnom (Fig. 2c), but independent of wSnom, the opposite holds for wS which is linearly dependent on wSnom, but independent of wLnom (Fig. 2d). These features reflect in the dependence of $w on both wLnom and wSnom in Fig. 2e. Figs. 3a to 3j relate to the three-band representation of the PSD and the feature of variances in the partial wavelength bands. Concerning CL (Fig. 3a) and CS (Fig. 3c), these dependences are similar to those demonstrated for the two-band partitioning of the whole wavelength band (see Figs. 2a, 2b, respectively). CM exhibits systematic deviations dependent both on wMnom and wSnom (Fig. 3b) with some local stochastic deviations from the otherwise linear dependences. In Figs. 3d to 3f the expressed dependences of the simulated waviness values are for wLnom (Fig. 3d) and wSnom (Fig. 3f) compatible again with those for the two-band representation (Figs. 2c, 2d, respectively). Simulated waviness wM depends both on wLnom and wSnom as for the CM, but with the opposite tendencies, i.e. to a positive deviation (both systematic and stochastic) of CM, negative deviation of wM corresponds. In Figs. 3g to 3j the dependences of variances on waviness values wLnom and wSnom are presented. Evidently, DH and DL are dependent only on wLnom, while DS is dependent only on wSnom. The variance DM is dependent both on wLnom and wSnom with the prevailing influence of wLnom. In Appendix 2, some examples are given of longitudinal road profiles measured in-situ in the United States, the road profile data being taken from [5]. The layout of individual data sheets is the same as in Appendix 1. Six examples are given with waviness values wL z wS and two examples with rather great differences between wL and wS. The estimated PSD functions are fitted with straight lines in the long and short wavelength bands. Rather great deviations of some profiles from those obtained by simulation may be observed. Besides the logical differences due to the multiform nature of real roads the relative short length plays its role, as well. Notice also different scales for the distance in Appendices 1 and 2.
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References: [1]
Kropáč, O., Múčka, P (2006): Indicators of longitudinal unevenness of roads/runways; Czech Aerospace Proceedings, 1/2006, pp34-45.
[2]
Sayers, M. W. (1995): On the calculation of IRI from longitudinal road profile, Paper No. 950842, Transport Research Board, Washington, D.C., 24 pp.
[3]
ISO 8608 (1995). Mechanical vibration — Road surface profiles — Reporting of merasured data.
[4]
prEN 13036-5 (2006): Road and airfield surface characteristics. Test methods. Determination of longitudinal unevenness indices, CEN.
[5]
Karamihas, S. (2005): Long-term pavement performance road profile data (Nov. 9, 2004). http://www.umtri.umich.edu/erd/roughness/ltpp erd.html.
Figures
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1b
Acknowledgment This work was partially supported by Grant No. 2/6161/26 of the Slovak Grant Agency for Science VEGA.
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1g Fig. 1. Dependences of one-band indicators on w Lnom, w Snom (a) C , (b) w , (c) D H, (d) RMS H, (e) Sk H, (f) Ex H, (g) IRI
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3d Fig. 2. Dependences of two-band indicators on w Lnom, w Snom (a) C L, (b) C S, (c) w L, (d) w S, (e) $ w
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26 Window type: Detrending mode: PSD fitting: PSD fitting interval: Smoothing in octave bands:
cosine digital tapering window (prEN 13 036-5) mean ISO 8608 0.011 ÷ 2.83 m-1 / 0.063 ÷ 17.77 rad/m none
Common layout of all considered cases: Heading: serial number, nominal values of C0, wL and wS a) ROAD PROFILE: Example of a typical feature of the longitudinal profile h(l) b) PSD: Power spectral density GH [m3/rad] (in log-log coordinate net) c) CF: Correlation function RH [1] d) PDF: Probability density function f(h) [cm-1] e) DF: Distribution function P [1] (in normal probability coordinate net) f) Estimates of numerical indicators obtained from selected profiles The sequence of numerical indicators is as follows: One-band indicators related to the whole wavelength band 0.3534 to 90.9 recommended in ISO 8608: 1995, i.e. C (10-6 omitted), w and further on, variance DH, RMSH, Sk and Ex of the road elevation. Indicators IRI, IRIcon and wIRI are also added. Two-band indicators assuming different waviness values wL in the wavelength band 6.25 to 90.9 m and wS in the wavelength band 0.3534 to 6.25 m giving values CL, wL, CS, wS. Three-band indicators are related to the long (12.5 to 50 m), medium (3.125 to 12.5 m) and short (0.78 to 3.125 m) wavelength bands recommended in prEN 13036-5, thus giving CL, wL, CM, wM, CS, wS. Added are corresponding variances DL, DM, DS and the total variance DH = DL+DM+DS.
3h
Effect of arbitrary value C on a) to e):
3i
a) multiply by C/C0 b) shift upwards by C/C0 c) without change d) in compliance with a) e) in compliance with a)
1. C0 = 1 s 10-6 rad·m; wL = 1.5; wS = 1.5; ROAD PROFILE
PSD
CF
PDF
DF
3j Fig. 3. Dependences of three-band indicators on w Lnom, w Snom (a) C L, (b) C M, (c) C S (d) w L, (e) w M, (f) w S, (g) D H, (h) D L, (i) D M, (j) D S
Appendix 1 Atlas of typical simulated longitudinal profiles of roads considering different waviness values in long and short wavelength bands, wL, wS, respectively, wL, wS 1.5 (0.5) 3.5, C0 = 1 s 10-6 rad·m Simulation conditions: Length of the road section Sampling interval: Sampling frequency: Total number of signal samples: Number of Fourier coefficients: Section overlapping: Number of blocks: Frequency resolution:
L = 0 ÷ 819.6 m $L = 0.1 m nsamp = 10 m-1 8196 1024 64 samples 113 by 1024 samples 0.0098
One-band ind.: C = 0.978; w = 1.488; (0.35 ÷ 90.9 m) DH = 6.642 mm2; RMSH = 2.577 mm; Sk = 0.049; Ex = -0.052; IRI80 = 2.65 mm/m; IRI120 = 1.97 mm/m; IRIcon = 2.71 mm/m; wIRI = 1.54; Two-bands ind.: CL = 1.072; wL = 1.424; CS = 0.966; wS = 1.483; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.175; wL = 1.429; CM = 0.986; wM = 1.496; CS = 0.933; wS = 1.461; (0.78 ÷ 50 m) DL = 2.995 mm2; DM = 1.392 mm2; DS = 0.694 mm2; DH = 5.08 mm2;
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2. C0 = 1 s 10-6 rad·m; wL = 1.5; wS = 2;
3. C0 = 1 s 10-6 rad·m; wL = 1.5; wS = 2.5;
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One-band ind.: C = 0.844; w = 1.925; (0.35 ÷ 90.9 m) DH = 6.244 mm2; RMSH = 2.499 mm; Sk = 0.049; Ex = -0.274; IRI80 = 2.12 mm/m; IRI120 = 1.7 mm/m; IRIcon = 2.09 mm/m; wIRI = 1.93; Two-bands ind.: CL = 1.108; wL = 1.411; CS = 1.014; wS = 2.01; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 0.872; wL = 1.66; CM = 0.949; wM = 1.763; CS = 0.999; wS = 2.002; (0.78 ÷ 50 m) DL = 3.116 mm2; DM = 1.377 mm2; DS = 0.372 mm2; DH = 4.87 mm2;
One-band ind.: C = 0.701; w = 2.337; (0.35 ÷ 90.9 m) DH = 5.652 mm2; RMSH = 2.377 mm; Sk = 0.023; Ex = -0.051; IRI80 = 1.78 mm/m; IRI120 = 1.53 mm/m; IRIcon = 1.67 mm/m; wIRI = 2.24; Two-bands ind.: CL = 1.062; wL = 1.402; CS = 1.025; wS = 2.513; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 0.988; wL = 1.513; CM = 0.858; wM = 1.979; CS = 1.018; wS = 2.51; (0.78 ÷ 50 m) DL = 2.842 mm2; DM = 1.279 mm2; DS = 0.207 mm2; DH = 4.33 mm2;
4. C0 = 1 s 10-6 rad·m; wL = 1.5; wS = 3;
5. C0 = 1 s 10-6 rad·m; wL = 1.5; wS = 3.5;
ROAD PROFILE
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One-band ind.: C = 0.585; w = 2.753; (0.35 ÷ 90.9 m) DH = 5.491 mm2; RMSH = 2.343 mm; Sk = -0.024; Ex = -0.535; IRI80 = 1.52 mm/m; IRI120 = 1.41 mm/m; IRIcon = 1.39 mm/m; wIRI = 2.63; Two-bands ind.: CL = 1.113; wL = 1.377; CS = 0.998; wS = 3; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.137; wL = 1.423; CM = 0.78; wM = 2.268; CS = 1.038; wS = 3.029; (0.78 ÷ 50 m) DL = 2.87 mm2; DM = 1.221 mm2; DS = 0.117 mm2; DH = 4.21 mm2;
One-band ind.: C = 0.499; w = 3.182; (0.35 ÷ 90.9 m) DH = 5.788 mm2; RMSH = 2.406 mm; Sk = -0.021; Ex = -0.078; IRI80 = 1.32 mm/m; IRI120 = 1.28 mm/m; IRIcon = 1.23 mm/m; wIRI = 2.88; Two-bands ind.: CL = 1.098; wL = 1.435; CS = 0.998; wS = 3.503; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 0.949; wL = 1.603; CM = 0.732; wM = 2.591; CS = 1.036; wS = 3.531; (0.78 ÷ 50 m) DL = 3.116 mm2; DM = 1.213 mm2; DS = 0.068 mm2; DH = 4.4 mm2;
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6. C0 = 1 s 10-6 rad·m; wL = 2; wS = 1.5;
7. C0 = 1 s 10-6 rad·m; wL = 2; wS = 2;
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One-band ind.: C = 1.21; w = 1.589; (0.35 ÷ 90.9 m) DH = 13.97 mm2; RMSH = 3.738 mm; Sk = -0.052; Ex = -0.211; IRI80 = 2.72 mm/m; IRI120 = 2.04 mm/m; IRIcon = 2.88 mm/m; wIRI = 1.58; Two-bands ind.: CL = 1.087; wL = 1.958; CS = 1.013; wS = 1.508; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.022; wL = 2.073; CM = 1.106; wM = 1.838; CS = 0.97; wS = 1.471; (0.78 ÷ 50 m) DL = 6.828 mm2; DM = 1.622 mm2; DS = 0.712 mm2; DH = 9.16 mm2;
One-band ind.: C = 1.004; w = 2.002; (0.35 ÷ 90.9 m) DH = 13.84 mm2; RMSH = 3.72 mm; Sk = 0.036; Ex = -0.431; IRI80 = 2.2 mm/m; IRI120 = 1.79 mm/m; IRIcon = 2.21 mm/m; wIRI = 1.99; Two-bands ind.: CL = 1.057; wL = 2.01; CS = 0.976; wS = 1.99; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.057; wL = 2.111; CM = 1.0; wM = 2.077; CS = 1.01; wS = 2.007; (0.78 ÷ 50 m) DL = 7.487 mm2; DM = 1.514 mm2; DS = 0.371 mm2; DH = 9.37 mm2;
8. C0 = 1 s 10-6 rad·m; wL = 2; wS = 2.5;
9. C0 = 1 s 10-6 rad·m; wL = 2; wS = 3;
ROAD PROFILE
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One-band ind.: C = 0.853; w = 2.427; (0.35 ÷ 90.9 m) DH = 13.23 mm2; RMSH = 3.637 mm; Sk = -0.073; Ex = -0.272; IRI80 = 1.83 mm/m; IRI120 = 1.61 mm/m; IRIcon = 1.80 mm/m; wIRI = 2.36; Two-bands ind.: CL = 1.046; wL = 1.986; CS = 1.025; wS = 2.512; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.014; wL = 2.106; CM = 0.933; wM = 2.162; CS = 1.035; wS = 2.524; (0.78 ÷ 50 m) DL = 7.117 mm2; DM = 1.428 mm2; DS = 0.206 mm2; DH = 8.75 mm2;
One-band ind.: C = 0.723; w = 2.848; (0.35 ÷ 90.9 m) DH = 12.24 mm2; RMSH = 3.498 mm; Sk = 0.551; Ex = -0.376; IRI80 = 1.58 mm/m; IRI120 = 1.5 mm/m; IRIcon = 1.53 mm/m; wIRI = 2.73; Two-bands ind.: CL = 1.083; wL = 1.928; CS = 1.058; wS = 3.024; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 0.98; wL = 2.078; CM = 0.888; wM = 2.489; CS = 1.038; wS = 3.014; (0.78 ÷ 50 m) DL = 6.588 mm2; DM = 1.44 mm2; DS = 0.119 mm2; DH = 8.15 mm2;
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10. C0 = 1 s 10-6 rad·m; wL = 2; wS = 3.5;
11. C0 = 1 s 10-6 rad·m; wL = 2.5; wS = 1.5;
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One-band ind.: C = 0.584; w = 3.252; (0.35 ÷ 90.9 m) DH = 12.2 mm2; RMSH = 3.492 mm; Sk = -0.014; Ex = 0.135; IRI80 = 1.4 mm/m; IRI120 = 1.33 mm/m; IRIcon = 1.33 mm/m; wIRI = 2.75; Two-bands ind.: CL = 1.072; wL = 1.95; CS = 0.972; wS = 3.487; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.147; wL = 1.983; CM = 0.776; wM = 2.865; CS = 0.986; wS = 3.5; (0.78 ÷ 50 m) DL = 6.667 mm2; DM = 1.365 mm2; DS = 0.067 mm2; DH = 8.1 mm2;
One-band ind.: C = 1.44; w = 1.67; (0.35 ÷ 90.9 m) DH = 35.02 mm2; RMSH = 5.918 mm; Sk = 0.1; Ex = -0.139; IRI80 = 2.71 mm/m; IRI120 = 2.08 mm/m; IRIcon = 3.03 mm/m; wIRI = 1.7; Two-bands ind.: CL = 1.06; wL = 2.53; CS = 0.982; wS = 1.494; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 0.964; wL = 2.704; CM = 1.218; wM = 2.072; CS = 1.019; wS = 1.524; (0.78 ÷ 50 m) DL = 17.49 mm2; DM = 1.879 mm2; DS = 0.697 mm2; DH = 20.1 mm2;
12. C0 = 1 s 10-6 rad·m; wL = 2.5; wS = 2;
13. C0 = 1 s 10-6 rad·m; wL = 2.5; wS = 2.5;
ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 1.227; w = 2.095; (0.35 ÷ 90.9 m) DH = 33.64 mm2; RMSH = 5.8 mm; Sk = -0.064; Ex = -0.446; IRI80 = 2.27 mm/m; IRI120 = 1.89 mm/m; IRIcon = 2.37 mm/m; wIRI = 2.09; Two-bands ind.: CL = 1.104; wL = 2.496; CS = 1.006; wS = 2.004; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.099; wL = 2.599; CM = 1.118; wM = 2.363; CS = 1.032; wS = 2.016; (0.78 ÷ 50 m) DL = 16.81 mm2; DM = 1.797 mm2; DS = 0.378 mm2; DH = 19 mm2;
One-band ind.: C = 1.021; w = 2.51; (0.35 ÷ 90.9 m) DH = 33.85 mm2; RMSH = 5.818 mm; Sk = 0.162; Ex = -0.206; IRI80 = 1.9 mm/m; IRI120 = 1.71 mm/m; IRIcon = 1.93 mm/m; wIRI = 2.5; Two-bands ind.: CL = 1.121; wL = 2.493; CS = 0.977; wS = 2.49; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.22; wL = 2.517; CM = 1.031; wM = 2.647; CS = 0.962; wS = 2.479; (0.78 ÷ 50 m) DL = 16.47 mm2; DM = 1.761 mm2; DS = 0.202 mm2; DH = 18.4 mm2;
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14. C0 = 1 s 10-6 rad·m; wL = 2.5; wS = 3;
15. C0 = 1 s 10-6 rad·m; wL = 2.5; wS = 3.5; ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 0.584; w = 2.93; (0.35 ÷ 90.9 m) DH = 31.15 mm2; RMSH = 5.581 mm; Sk = 0.042; Ex = -0.275; IRI80 = 1.66 mm/m; IRI120 = 1.61 mm/m; IRIcon = 1.64 mm/m; wIRI = 2.87; Two-bands ind.: CL = 1.118; wL = 5.454; CS = 1.002; wS = 3.004; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.159; wL = 2.525; CM = 0.926; wM = 2.885; CS = 1.024; wS = 3.012; (0.78 ÷ 50 m) DL = 15.81 mm2; DM = 1.652 mm2; DS = 0.117 mm2; DH = 17.6 mm2;
One-band ind.: C = 0.726; w = 3.354; (0.35 ÷ 90.9 m) DH = 32.48 mm2; RMSH = 5.699 mm; Sk = 0.091; Ex = -0.471; IRI80 = 1.5 mm/m; IRI120 = 1.49 mm/m; IRIcon = 1.07 mm/m; wIRI = 2.96; Two-bands ind.: CL = 1.098; wL = 2.484; CS = 1.022; wS = 3.512; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.075; wL = 2.593; CM = 0.891; wM = 3.036; CS = 0.985; wS = 3.499; (0.78 ÷ 50 m) DL = 16.42 mm2; DM = 1.647 mm2; DS = 0.067 mm2; DH = 18.1 mm2;
16. C0 = 1 s 10-6 rad·m; wL = 3; wS = 1.5;
17. C0 = 1 s 10-6 rad·m; wL = 3; wS = 2;
ROAD PROFILE
ROAD PROFILE
PSD
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PSD
CF
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DF
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DF
One-band ind.: C = 1.77; w = 1.769; (0.35 ÷ 90.9 m) DH = 93.34 mm2; RMSH = 9.661 mm; Sk = -0.056; Ex = -0.645; IRI80 = 2.85 mm/m; IRI120 = 2.26 mm/m; IRIcon = 3.61 mm/m; wIRI = 1.86; Two-bands ind.: CL = 1.103; wL = 3.053; CS = 1.007; wS = 1.51; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.365; wL = 3.03; CM = 1.342; wM = 2.333; CS = 0.98; wS = 1.491; (0.78 ÷ 50 m) DL = 42.2 mm2; DM = 2.21 mm2; DS = 0.7 mm2; DH = 45,1 mm2;
One-band ind.: C = 0.713; w = 2.846; (0.35 ÷ 90.9 m) DH = 13.04 mm2; RMSH = 3.611 mm; Sk = -0.031; Ex = -0.06; IRI80 = 1.57 mm/m; IRI120 = 1.45 mm/m; IRIcon = 1.16 mm/m; wIRI = 2.62; Two-bands ind.: CL = 1.105; wL = 1.964; CS = 0.991; wS = 2.998; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.131; wL = 2.03; CM = 0.87; wM = 2.592; CS = 0.943; wS = 2.969; (0.78 ÷ 50 m) DL = 7.072 mm2; DM = 1.445 mm2; DS = 0.113 mm2; DH = 8.63 mm2;
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LETECKÝ 18. C0 = 1 s 10-6 rad·m; wL = 3; wS = 2.5;
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19. C0 = 1 s 10-6 rad·m; wL = 3; wS = 3;
ROAD PROFILE
ROAD PROFILE
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PSD
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DF
One-band ind.: C = 1.247; w = 2.606; (0.35 ÷ 90.9 m) DH = 94.22 mm2; RMSH = 9.707 mm; Sk = 0.052; Ex = -0.565; IRI80 = 1.99 mm/m; IRI120 = 1.89 mm/m; IRIcon = 1.68 mm/m; wIRI = 2.74; Two-bands ind.: CL = 1.115; wL = 3.042; CS = 1.002; wS = 2.505; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.232; wL = 3.087; CM = 1.147; wM = 2.842; CS = 1.008; wS = 2.514; (0.78 ÷ 50 m) DL = 41.9 mm2; DM = 2.079 mm2; DS = 0.204 mm2; DH = 44.2 mm2;
One-band ind.: C = 1.039; w = 3.017; (0.35 ÷ 90.9 m) DH = 85.02 mm2; RMSH = 9.221 mm; Sk = 0.155; Ex = -0.349; IRI80 = 1.84 mm/m; IRI120 = 1.83 mm/m; IRIcon = 1.33 mm/m; wIRI = 2.96; Two-bands ind.: CL = 1.194; wL = 2.962; CS = 0.991; wS = 2.996; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.347; wL = 2.983; CM = 1.071; wM = 3.19; CS = 0.976; wS = 2.984; (0.78 ÷ 50 m) DL = 38.56 mm2; DM = 2.114 mm2; DS = 0.116 mm2; DH = 40.8 mm2;
20. C0 = 1 s 10-6 rad·m; wL = 3; wS = 3.5;
21. C0 = 1 s 10-6 rad·m; wL = 3.5; wS = 1.5;
ROAD PROFILE
ROAD PROFILE
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CF
PSD
CF
PDF
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DF
One-band ind.: C = 0.881; w = 3.438; (0.35 ÷ 90.9 m) DH = 90.57 mm2; RMSH = 9.517 mm; Sk = 0.04; Ex = -1.02; IRI80 = 1.65 mm/m; IRI120 = 1.71 mm/m; IRIcon = 1.18 mm/m; wIRI = 3.18; Two-bands ind.: CL = 1.138; wL = 3.006; CS = 1.014; wS = 3.503; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.32; wL = 3.01; CM = 0.962; wM = 3.344; CS = 1.015; wS = 3.502; (0.78 ÷ 50 m) DL = 39.7 mm2; DM = 1.963 mm2; DS = 0.068 mm2; DH = 41,7 mm2;
One-band ind.: C = 2.219; w = 1.863; (0.35 ÷ 90.9 m) DH = 272.7 mm2; RMSH = 16.52 mm; Sk = -0.041; Ex = -1.07; IRI80 = 2.97 mm/m; IRI120 = 2.54 mm/m; IRIcon = 3.7 mm/m; wIRI = 2.23; Two-bands ind.: CL = 1.289; wL = 3.521; CS = 1.013; wS = 1.502; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.72; wL = 3.446; CM = 1.585; wM = 2.737; CS = 1.025; wS = 1.514; (0.78 ÷ 50 m) DL = 106.7 mm2; DM = 2.939 mm2; DS = 0.711 mm2; DH = 110 mm2;
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22. C0 = 1 s 10-6 rad·m; wL = 3.5; wS = 2;
23. C0 = 1 s 10-6 rad·m; wL = 3.5; wS = 2.5; ROAD PROFILE
ROAD PROFILE
PSD
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CF
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DF
One-band ind.: C = 1.853; w = 2.281; (0.35 ÷ 90.9 m) DH = 296.5 mm2; RMSH = 17.22 mm; Sk = -0.024; Ex = -0.828; IRI80 = 2.53 mm/m; IRI120 = 2.37 mm/m; IRIcon = 2.45 mm/m; wIRI = 2.66; Two-bands ind.: CL = 1.255; wL = 3.555; CS = 0.999; wS = 1.997; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.467; wL = 3.569; CM = 1.451; wM = 2.984; CS = 1.063; wS = 2.046; (0.78 ÷ 50 m) DL = 114.8 mm2; DM = 2.855 mm2; DS = 0.376 mm2; DH = 118 mm2;
One-band ind.: C = 1.568; w = 2.706; (0.35 ÷ 90.9 m) DH = 277.8 mm2; RMSH = 16.67 mm; Sk = 0.044; Ex = -0.611; IRI80 = 2.23 mm/m; IRI120 = 2.26 mm/m; IRIcon = 1.8 mm/m; wIRI = 3.06; Two-bands ind.: CL = 1.239; wL = 3.543; CS = 1.037; wS = 2.517; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.682; wL = 3.457; CM = 1.33; wM = 3.163; CS = 0.983; wS = 2.487; (0.78 ÷ 50 m) DL = 107.4 mm2; DM = 2.72 mm2; DS = 0.204 mm2; DH = 110 mm2;
24. C0 = 1 s 10-6 rad·m; wL = 3.5; wS = 3;
25. C0 = 1 s 10-6 rad·m; wL = 3.5; wS = 3.5;
ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 1.322; w = 3.129; (0.35 ÷ 90.9 m) DH = 312.6 mm2; RMSH = 17.68 mm; Sk = -0.046; Ex = -0.108; IRI80 = 1.94 mm/m; IRI120 = 2.11 mm/m; IRIcon = 1.47 mm/m; wIRI = 3.42; Two-bands ind.: CL = 1.217; wL = 3.589; CS = 1.025; wS = 3.013; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.229; wL = 3.713; CM = 1.215; wM = 3.448; CS = 0.971; wS = 2.98; (0.78 ÷ 50 m) DL = 122.9 mm2; DM = 2.673 mm2; DS = 0.115 mm2; DH = 126 mm2;
One-band ind.: C = 1.088; w = 3.539; (0.35 ÷ 90.9 m) DH = 277.9 mm2; RMSH = 16.67 mm; Sk = -0.041; Ex = -0.437; IRI80 = 1.81 mm/m; IRI120 = 2.07 mm/m; IRIcon = 1.33 mm/m; wIRI = 3.67; Two-bands ind.: CL = 1.2; wL = 3.568; CS = 1.009; wS = 3.505; (0.35 ÷ 90.9 m) Three-bands ind.: CL = 1.506; wL = 3.539; CM = 1.081; wM = 3.753; CS = 0.982; wS = 3.484; (0.78 ÷ 50 m) DL = 109.1 mm2; DM = 2.604 mm2; DS = 0.068 mm2; DH = 112 mm2;
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LETECKÝ
Appendix 2 Examples of longitudinal profiles obtained from in-situ measurements (Karamihas 2005) Simulation conditions: Length of the road section: Sampling interval: Re-sampling interval: Sampling frequency: Total number of signal samples: Number of Fourier coefficients: Section overlapping: Number of blocks: Frequency resolution: Window type: Detrending mode: PSD fitting interval: Smoothing in octave bands:
L = 0 ÷ 152.4 m $L = 0.1524 m Lsamp = 0.1 m nsamp = 10 m-1 1524 512 32 samples 32 by 512 samples 0.0195 cosine digital tapering window (prEN 13 036-5) mean / linear 0.78125 ÷ 50 m none
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The sequence of numerical indicators is the same as in Appendix 1. Index of illustrations: 1. 2. 3. 4. 5. 6. 7. 8.
wL z wS z 1.5, with several potholes, distinctly non-Gaussian wL z wS z 2.0, wL z wS z 2.5, distinctly non-Gaussian wL z wS z 3.0, distinctly non-Gaussian wL z wS z 3.5, distinctly non-Gaussian wL z wS z 4.0, distinctly non-Gaussian wL >> wS: wL z 3.4, wS z 1.8, $w z 1.6, distinctly non-Gaussian wL >> wS: wL z 1.2, wS z 3.2, $w z -2.0, distinctly non-Gaussian
Common layout of all considered cases: Heading: serial number, nominal values of C0, wL and wS a) ROAD PROFILE: Example of a typical feature of the longitudinal profile h(l) b) PSD: Power spectral density GH [m3/rad] (in log-log coordinate net) c) CF: Correlation function RH [1] d) PDF: Probability density function f(h) [cm-1] e) DF: Distribution function P [1] (in normal probability coordinate net) f) Estimates of numerical indicators obtained from selected profiles
Intentionally left void
1. wL z wS z 1.5, with several potholes, distinctly non-Gaussian
2. wL z wS z 2
ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 4.507; w = 1.682; (0.78 ÷ 50 m) DH = 20.66 mm2; RMSH = 4.546 mm; Sk = -1.04; Ex = 2.65; IRI80 = 4.02 mm/m; IRI120 = 3.04 mm/m; IRIcon = 5.32 mm/m; wIRI = 1.62; Two-bands ind.: CL = 4.881; wL = 1.313; CS = 6.139; wS = 1.89; (0.78 ÷ 50 m) Three-bands ind.: CL = 4.429; wL = 1.407; CM = 4.308; wM = 1.208; CS = 8.707; wS = 2.097; (0.78 ÷ 50 m) DL = 11.18 mm2; DM = 6.339 mm2; DS = 3.147 mm2; DH = 20.7 mm2;
One-band ind.: C = 0.193; w = 2.046; (0.78 ÷ 50 m) DH = 1.751 mm2; RMSH = 1.323 mm; Sk = 0.064; Ex = -0.667; IRI80 = 1.06 mm/m; IRI120 = 0.987 mm/m; IRIcon = 0.96 mm/m; wIRI = 2.63; Two-bands ind.: CL = 0.243; wL = 1.844; CS = 0.177; wS = 1.991; (0.78 ÷ 50 m) Three-bands ind.: CL = 0.112; wL = 2.418; CM = 0.213; wM = 1.772; CS = 0.19; wS = 2.063; (0.78 ÷ 50 m) DL = 1.296 mm2; DM = 0.379 mm2; DS = 0.076 mm2; DH = 1.75 mm2;
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3. wL z wS z 2.5;
4. wL z wS z 3
ROAD PROFILE
ROAD PROFILE
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CF
PSD
CF
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DF
One-band ind.: C = 0.651; w = 2.443; (0.78 ÷ 50 m) DH = 8.956 mm2; RMSH = 2.993 mm; Sk = 0.747; Ex = 0.787; IRI80 = 1.58 mm/m; IRI120 = 1.49 mm/m; IRIcon = 1.56 mm/m; wIRI = 2.71; Two-bands ind.: CL = 0.838; wL = 2.289; CS = 0.558; wS = 2.343; (0.78 ÷ 50 m) Three-bands ind.: CL = 1.749; wL = 1.793; CM = 0.673; wM = 2.371; CS = 0.42; wS = 2.175; (0.78 ÷ 50 m) DL = 7.723 mm2; DM = 1.094 mm2; DS = 0.137 mm2; DH = 8.95 mm2;
One-band ind.: C = 0.724; w = 2.98; (0.78 ÷ 50 m) DH = 25.3 mm2; RMSH = 5.03 mm; Sk = 0.013; Ex = -1.01; IRI80 = 1.11 mm/m; IRI120 = 1.02 mm/m; IRIcon = 1.5 mm/m; wIRI = 2.56; Two-bands ind.: CL = 0.827; wL = 2.819; CS = 0.724; wS = 2.982; (0.78 ÷ 50 m) Three-bands ind.: CL = 0.279; wL = 3.573; CM = 0.791; wM = 3.45; CS = 0.798; wS = 3.035; (0.78 ÷ 50 m) DL = 23.48 mm2; DM = 1.709 mm2; DS = 0.098 mm2; DH = 25.3 mm2;
5. wL z wS z 3.5;
6. wL z wS z 4 ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 0.2; w = 3.359; (0.78 ÷ 50 m) DH = 17.26 mm2; RMSH = 4.154 mm; Sk = -0.469; Ex = -0.96; IRI80 = 0.54 mm/m; IRI120 = 0.555 mm/m; IRIcon = 0.77 mm/m; wIRI = 3.15; Two-bands ind.: CL = 0.369; wL = 2.992; CS = 0.133; wS = 3.097; (0.78 ÷ 50 m) Three-bands ind.: CL = 0.09; wL = 4.03; CM = 0.251; wM = 4.176; CS = 0.104; wS = 2.957; (0.78 ÷ 50 m) DL = 16.58 mm2; DM = 0.654 mm2; DS = 0.014 mm2; DH = 17.2 mm2;
One-band ind.: C = 0.871; w = 3.986; (0.78 ÷ 50 m) DH = 114 mm2; RMSH = 10.68 mm; Sk = 0.627; Ex = -0.73; IRI80 = 1.11 mm/m; IRI120 = 1.38 mm/m; IRIcon = 1.7 mm/m; wIRI = 4.07; Two-bands ind.: CL = 1.915; wL = 3.309; CS = 0.656; wS = 3.807; (0.78 ÷ 50 m) Three-bands ind.: CL = 1.051; wL = 3.746; CM = 1.189; wM = 4.6; CS = 0.533; wS = 3.706; (0.78 ÷ 50 m) DL = 110.3 mm2; DM = 3.614 mm2; DS = 0.033 mm2; DH = 114 mm2;
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7. wL wS: wL z 3.4, wS z 1.8, $w z 1.6, distinctly non-Gaussian;
8. wL wS: wL z 1.2, wS z 3.2, $w z 2.0, distinctly non-Gaussian;
ROAD PROFILE
ROAD PROFILE
PSD
CF
PSD
CF
PDF
DF
PDF
DF
One-band ind.: C = 0.182; w = 2.413; (0.78 ÷ 50 m) DH = 10.15 mm2; RMSH = 3.185 mm; Sk = 1.03; Ex = 1.23; IRI80 = 0.68 mm/m; IRI120 = 0.717 mm/m; IRIcon = 0.83 mm/m; wIRI = 3.25; Two-bands ind.: CL = 0.154; wL = 3.565; CS = 0.063; wS = 1.7; (0.78 ÷ 50 m) Three-bands ind.: CL = 0.276; wL = 3.123; CM = 0.149; wM = 3.551; CS = 0.035; wS = 1.329; (0.78 ÷ 50 m) DL = 9.767 mm2; DM = 0.339 mm2; DS = 0.036 mm2; DH = 10.1 mm2;
One-band ind.: C = 1.131; w = 3.018; (0.78 ÷ 50 m) DH = 11.87 mm2; RMSH = 3.445 mm; Sk = -0.393; Ex = -0.331; IRI80 = 1.99 mm/m; IRI120 = 1.91 mm/m; IRIcon = 1.87 mm/m; wIRI = 2.81; Two-bands ind.: CL = 4.379; wL = 1.18; CS = 1.469; wS = 3.211; (0.78 ÷ 50 m) Three-bands ind.: CL = 5.64; wL = 1; CM = 1.786; wM = 3.39; CS = 1.429; wS = 3.195; (0.78 ÷ 50 m) DL = 7.868 mm2; DM = 3.848 mm2; DS = 0.15 mm2; DH = 11.9 mm2;
Fatigue Characteristics of Steel Clevis with Two or Three Lugs Únavové charakteristiky ocelových závěsů s dvěma a třemi oky Zdeněk Maléř, Head of the Structure Analyses Dept., retired / Moravan, Inc. Otrokovice In this report fatigue test results of a steel clevis with two or three lugs [1] are described. On this basis generalized fatigue characteristics of attachment lugs [2] are established. Test specimens represent a common type product of this attachment type. The test specimens were produced of commonly used low-alloy steel and were heat treated to the relatively high strength at high ductility and notch sensitivity. V tomto příspěvku jsou uvedeny výsledky únavových zkoušek ”ocelových závěsných kování s dvěma a třemi oky“ [1] a na tomto základě jsou vytvořeny zobecněné únavové charakteristiky závěsných ok [2]. Zkušební vzorky představují obvyklé výrobní provedení tohoto typu závěsu. Zkušební vzorky byly vyrobeny z běžně užívané nízkolegované oceli, tepelně zpracované na vyšší pevnost při poměrně vysoké tažnosti a vrubové citlivosti. Uvedené únavové charakteristiky tohoto konstrukčního prvku byly stanoveny na základě únavových zkoušek. Jedná se tedy o hodnoty hodnověrné, které z hlediska obecné platnosti jsou ovlivněny vlastnostmi konkrétního materiálu, výrobou, způsobem zavádění zatížení, frekvencí zatěžování a dalšími vlivy. Proto jsou dále uvedeny dostupné informace, které mají popsat okolnosti za nichž bylo uvedených výsledků dosaženo.
Keywords: Built-up aircraft structure; fatigue characteristics; fatigue tests; steel clevis with two or three lugs; S/N curves; static strength.
