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Dynamic Amplification of Bridge-Expansion-Joints considering Roughness induced Vehicle Vibrations
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ScienceDirect Procedia Engineering 00 (2017) 000–000 Procedia Engineering 199 (2017) 2651–2656 Procedia Engineering 00 (2017) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
X X International International Conference Conference on on Structural Structural Dynamics, Dynamics, EURODYN EURODYN 2017 2017
Dynamic Dynamic Amplification Amplification of of Bridge-Expansion-Joints Bridge-Expansion-Joints considering considering Roughness Roughness induced induced Vehicle Vehicle Vibrations Vibrations a,∗ b Roland Roland Friedl, Friedl, M.Sc. M.Sc.a,∗,, Univ.-Prof. Univ.-Prof. Dr.-Ing. Dr.-Ing. Ingbert Ingbert Mangerig Mangerigb b Universit¨ at b Universit¨ at
a bulicek+ingenieure gmbh, Sonnenstraße 19, 80331 Munich, Germany a bulicek+ingenieure gmbh, Sonnenstraße 19, 80331 Munich, Germany der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 37, 85577 Neubiberg, Germany der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 37, 85577 Neubiberg, Germany
Abstract Abstract The response of bridge expansion joints to the transient impact induced by crossing vehicles always represent the dynamic reaction The response of bridge expansion joints to the transient impact induced by crossing vehicles always represent the dynamic reaction of the former to the dynamic wheel contact forces at that particular moment. Separating the dynamic components in the vertical of the former to the dynamic wheel contact forces at that particular moment. Separating the dynamic components in the vertical wheel contact forces, the dynamic amplification of the bridge expansion joints response, which is dominated by the driving velocity wheel contact forces, the dynamic amplification of the bridge expansion joints response, which is dominated by the driving velocity of the vehicle, its crossing position in lateral direction, the dynamic behavior of the expansion joint itself and most importantly of the vehicle, its crossing position in lateral direction, the dynamic behavior of the expansion joint itself and most importantly by the interaction between the vehicle tire and the expansion joint’s lamellas can be discussed. Due to the random distribution of by the interaction between the vehicle tire and the expansion joint’s lamellas can be discussed. Due to the random distribution of the roughness amplitudes a stochastic characterization of the vehicle vibrations is necessary. In addition to the classical frequency the roughness amplitudes a stochastic characterization of the vehicle vibrations is necessary. In addition to the classical frequency approach based on complex transfer matrices completely describing the dynamic behavior of the vehicles and spectral density approach based on complex transfer matrices completely describing the dynamic behavior of the vehicles and spectral density matrices characterizing the pavement roughness, the numerically efficient covariance analysis is applied. Therefore the spectral matrices characterizing the pavement roughness, the numerically efficient covariance analysis is applied. Therefore the spectral characteristics of the roughness amplitudes, which are described on the basis of multi-correlated random processes, are considered characteristics of the roughness amplitudes, which are described on the basis of multi-correlated random processes, are considered by introducing shaping filters in combination with a state space augmentation. The dynamic behavior of the considered bridge by introducing shaping filters in combination with a state space augmentation. The dynamic behavior of the considered bridge expansion joint is dominated by the properties of the discrete elastomeric spring elements. Hence a a non-proportional damping expansion joint is dominated by the properties of the discrete elastomeric spring elements. Hence a a non-proportional damping approach has to be applied to analyze the three dimensional finite element model of the structure using a modal superposition. approach has to be applied to analyze the three dimensional finite element model of the structure using a modal superposition. c 2017 The Authors. Published by Elsevier Ltd. c 2017 2017 The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN EURODYN 2017. 2017. Keywords: random vehicle vibration, dynamic amplification, interaction between vehicle, tire and expansion joint Keywords: random vehicle vibration, dynamic amplification, interaction between vehicle, tire and expansion joint
1. Introduction 1. Introduction Beside effects like wind loads, cornering, accelerating or slowing down the main source for dynamic oscillations of Beside effects like wind loads, cornering, accelerating or slowing down the main source for dynamic oscillations of driving vehicles is given by the surface roughness. Widely neglecting discrete obstacles like single pot holes or blow driving vehicles is given by the surface roughness. Widely neglecting discrete obstacles like single pot holes or blow bars with dominating step heights the influence of randomly distributed roughness amplitudes to vehicle vibrations bars with dominating step heights the influence of randomly distributed roughness amplitudes to vehicle vibrations and further to a variation in the vertical wheel contact forces is analyzed and quantified. The standard deviation of the and further to a variation in the vertical wheel contact forces is analyzed and quantified. The standard deviation of the vertical tire contact forces is used as the basis for further investigations concerning the dynamic amplification of the vertical tire contact forces is used as the basis for further investigations concerning the dynamic amplification of the expansion joints’ response. expansion joints’ response. ∗ ∗
Corresponding author. Tel.: +49-89-189-4143-88 ; fax: +49-89-189-4143-30. Corresponding author. Tel.: +49-89-189-4143-88 ; fax: +49-89-189-4143-30. E-mail address: [email protected] E-mail address: [email protected]
c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 Peer-review under responsibility of the organizing committee of EURODYN 2017. 1877-7058 2017responsibility The Authors. Published by committee Elsevier Ltd. Peer-review©under of the organizing of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.515
Roland Friedl et al. / Procedia Engineering 199 (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017) 000–000
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2. Roughness induced wheel force variations 2.1. Random surface roughness Extensive evaluation of real pavement profiles have shown [6,14] that indeed the single roughness amplitudes are distributed randomly but many profiles show strong similarities concerning their spectral characteristics. Therefore these can be approximately described on the basis of random processes respectively the parameters defining these processes. To ensure exact solutions for the vehicle oscillations, analytical approximations for the (power) spectral density functions based on rational expressions as for example 2α α2 + β2 + 4π2 n2 (1) S ww (n) = σ2 α2 + β2 − 4π2 n2 2 + 16α2 π2 n2
are preferred. Assuming normally distributed random fields a stochastically complete characterization of the vehicles excitation caused by two dimensional pavement profiles is obtained on the basis of a (nT , nT ) spectral density matrix Sww (n), where nT defines the number of vehicle tires and n the wave number. A complete description of two dimensional surfaces based on experiments is very time consuming and tedious why theoretical roughness models, which allow a simplified but consistent evaluation of Sww (n) based on a single one dimensional roughness profile, have been developed [4,5]. Because of the treatment of the vehicle vibrations in the time or corresponding frequency domain the wave number based description of the spectral density matrix hast to be transferred into a frequency based form considering the driving velocity of the vehicle. Due to the for usual pavement profiles observed declination of the spectral density with an increasing wave number the caused excitation increases along with an increasing driving velocity. Further the roughness induced vehicle vibrations reach high amplitudes in case the dominating frequency band of periodic roughness amplitudes coincide with the relevant natural frequencies of the vehicle. 2.2. Dynamic behavior of vehicles Experiments with an instrumented vehicle clearly show, that the amplification of the vertical wheel forces, which is of interest regarding the shock impact exerted to bridge expansion joints, is mainly dominated by the global oscillations of the vehicle body and the axes [2]. Theoretically investigating the global dynamic motions of vehicles, the latter can be mathematically described on the basis of a rigid body system model where all masses are assumed to be rigid and interconnected by discrete spring and damper elements. Instead of working with the well known equation of motion ¨ + Du(t) ˙ + Ku(t) = p(t) Mu(t) (2) containing the mass matrix M, the stiffness matrix K, the damping matrix D, the force vector p(t) and the vector T of the physical coordinates u = uB φBx φBy uAv φAv x uAh φAh x a more deeper insight into thy dynamic behavior of the vehicle can be obtained on the basis of a modal analysis regarding the eigen-vectors, the corresponding eigenfrequencies and most importantly the frequency response matrix −1 (3) F( f ) = − (2π f )2 M + j2π f D + 2π f K ,
which completely describes the vehicles transfer behavior. Considering the frequency response function F11 characterizing the vertical vehicle body movements in figure 1a, the corresponding frequency of 1.16 Hz can be identified very clearly. The pitch mode of the vehicle body dominated by a frequency of about 2 Hz and represented by the frequency response function F33 has also a great influence on the investigated wheel force amplifications, especially on the coupling between the oscillations of the front and rear axis. 2.3. Quantifying the variation in the vertical wheel forces A stochastic quantification of the random vehicle vibrations induced by the surface roughness can be obtained on the basis of the spectral density matrix of the systems response ¯ f )S pp ( f )FT ( f ) Suu ( f ) = F(
with
S pp ( f ) = F¯ R ( f )Szz ( f )FTR ( f )
(4)
Roland Friedl et al. / Procedia Engineering 199 (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017) 000–000
(a) Transfer functions Fii
(b) Transfer functions Fi j
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(c) Spectral density functions of vertical wheel forces
Fig. 1. Dynamic behavior of the vehicle
representing the spectral density matrix characterizing the random tire forces induced by the surface roughness [3]. Thereby the frequency response matrix textb f FR ( f ) describes the dynamic behavior of the vehicle tires and by this the transformation of a displacement based excitation to a force controlled one. To quantify the vertical wheel force variance, the correlation between the roughness amplitudes and the vehicle vibrations has to be considered. Defining the spectral density matrix of the relative displacements between the pavement surface and the vehicle axes S∆u∆u ( f ) = Suu ( f ) − Suz ( f ) − Szu ( f ) + Szz ( f )
(5)
with the complex cross spectral density matrices ¯ f )F¯ R ( f )Szz ( f ) Suz ( f ) = F(
and
Szu ( f ) = Szz ( f )FTR ( f )FT ( f ),
(6)
the spectral density matrix of the vertical wheel forces finally results in ST T ( f ) = F¯ T ( f )S∆u∆u ( f )FTT ( f ).
(7)
Up to equation 7 the presented method yields exact solutions for a stochastic characterization of the wheel forces. Comparing the exemplary plotted spectral density functions of the vertical wheel forces in figure 1c with the frequency response functions in figure 1 it becomes obvious, that both the dynamic behavior of the vehicle and the spectral characteristics of the pavement roughness have a great influence on the induced oscillations and further on the wheel force variation. The necessary integration to obtain the standard deviation of the wheel forces is executed numerically and truncated at a frequency flimit ≤ ∞ σ2Rii
+ +∞ flimit = S Ri Ri ( f )d f ≈ S Ri Ri ( f )d f. −∞
(8)
− flimit
Assuming normally distributed roughness amplitudes and linear vehicle behavior the standard deviation σRii in combination with the mean value represents a stochastically complete description of the roughness induced wheel force amplification which is further used to discuss the loads exerted on bridge expansion joints. The results derived on the basis of the described vehicle model show a good accordance with the experimental data [2]. 3. Dynamic amplification of bridge expansion joints As all modular expansion joints also the here considered swivel joist expansion joint has several steel girders, called the lamellas, running perpendicular to the driving lane which cross the varying gap between the bridge superstructure and the abutment area [11]. Beside the skew arrangement of the cross beams bearing the lamellas, the elastomer support elements, connecting the lamellas with the cross beams and the cross beams with the support structure, are the main construction elements dominating the dynamic behavior of the whole structure. As a slight modification compared to common expansion joints the lamella has been designed as a box girder with about double the width of usual lamellas. This has been done with regard to a measurement based detection of vehicle weights on the basis of an instrumented expansion joint [1].
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Roland Friedl et al. / Procedia Engineering (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017)199 000–000
3.1. Dynamic behavior and experiment To theoretically analyze the dynamic behavior of the modified expansion joint, it has been modeled on the basis of a three dimensional beam structure applying a finite element approach. The elastomer support elements are idealized by discrete three dimensional spring and damper elements considering the experimentally found stiffness and damping coefficients. Applying strain gauges on both webs of the steel girder and in addition piezo-electric force washers between the lamella and the elastomer support elements, the vertical reaction forces between the lamella and the transverse beams, which are the basis for the further discussion of the dynamic amplification, could be measured. The measured natural frequencies of the real structure correspond very well to the eigen-values of the finite element model as can be seen in figure 2c. Because of the discrete elastomer elements which moreover dominate the dynamic behavior of the whole expansion joint, a non-proportional damping approach has to be used. Regarding the measured
(a) Vertical reaction forces beteween the lamella and the cross beams
(b) experimental frequency spectrum
(c) correlation between the structural model and experimental data
Fig. 2. Experimental data
frequency spectra in combination with the eigen-vectors of the structural model it becomes obvious, that the global oscillations of the expansion joints’ lamellas are dominated by bending modes, basically characterized by vertical displacements of the box girder and torsional modes with dominating rotations about its longitudinal axis, equally. Especially the mode shapes corresponding to lower frequency ranges show dominating rotational deformation components whereas the higher frequencies are mainly represented by vertical bending modes. Another proof for the dominating character of the elastomeric support elements can be obtained regarding short time frequency spectra of the measured reaction forces. In this content it should be stated, that for this purpose only the decaying reaction forces after the contact with a single vehicle axle but never the time series representing a whole vehicle contact representing several axle excitations may be considered. The dynamic behavior in general and the decaying behavior especially can be separated into two main parts where the first part with relatively high oscillation amplitudes is dominated by torsional mode shapes with a clearly dominating frequency of about 78 Hz whereas the second part show a wider frequency band mainly correlating to vertical bending mode shapes. Because the torsional mode shapes coincide with high deformation amplitudes and therefore a relatively high strain level in the elastomer elements, these are damped out quite quickly due to the high damping capacity of the latter. In contrast the bending mode shapes show only very little deformations in the support area. Therefore the damping capacity of the elastomer elements can hardly by effective for those which can be seen by means of lower modal damping parameters (see figure 3a). To analyze this effect a non proportional damping approach is obligatory. 3.2. Dynamic amplification considering the interaction between vehicle, tire and expansion joint The dynamic amplification of the expansion joints’ response strongly depends on the crossing position of the vehicle in lateral direction, the dynamic behavior of the expansion joint itself and most importantly on the dynamic interaction between the vehicle tire and the lamellas [1]. A first estimation regarding the dynamic amplification can be reached by a comparison of the elementary frequency response functions shown in figure 3c with the frequency content of the exciting forces. Whereas the frequency response function is defined by the structure, the frequency spectrum of the exciting force highly depends on the dynamic interaction between the vehicle tire an the lamella. At this point the often stated sine half wave shaped force input function (see e.g. [12,13]) could not be verified. The carried out theoretical investigations in combination with the experimental data clearly show [1], that the nonuniform contact pressure distribution between the tire and the pavement and respectively the lamella has to be taken into
Roland Friedl et al. / Engineering Procedia Engineering (2017) 2651–2656 Roland Friedl / Procedia 00 (2017)199 000–000
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account to achieve a realistic approximation for the expansion joints oscillations. Therefor an extended tire model [1] has been invented which allows to consider arbitrary pressure distributions and which is in addition appropriate to describe the coupling behavior between the vehicle tire and the lamellas correctly. It further shows that the varying eccentricity of the resulting vertical tire force plays a dominating role especially regarding rotational deformations of the lamellas (see also figure 4). The influence of the so called non uniformity of the vertical contact pressure distribution is quantified on the basis of dynamic amplification factors φ=
wdynamic , w static
(9)
which are defined as the quotient of the dynamic deformation to the corresponding static deformation considering the same force input but completely neglecting mass inertia forces. For the pressure shape functions plotted in figure 3a the systems response is shown in figure 3b and the corresponding amplification factors in figure 3c. The amplification ●
0.07
●
●
Grad der kritischen Dämpfung [-]
0.06
● D
●
0.05 0.04 0.03 0.02 ● 0.01 0.00
● ●
●
5
● ●
●
●
●
●
●
10
15
●
●
●
●
●
20
Eigenform Nr.
(a) modal damping
(b) Elementary transfer function of the expansion joints
(c) non uniform pressure distribution
Fig. 3. Excitation of the expansion joint vehicle factor φ is plotted against a dimensionless frequency η = bT irev+b which allows a consideration of a variation in Lamella the widths of the lamella, the length of the tire contact area and the driving velocity of the vehicle. By means of the shape of the resulting vertical force and torsional moment resulting by an integration of the vertical contact pressure within the contact area shown in figure 4 it becomes evident, that the shape is everything else than a sine half wave. The dynamic amplification factor φ shows a great influence of the non uniform pressure distribution and further of the
(a) vertical force
(b) torsional moment
(c) dynamic amplification factor
Fig. 4. Input force and dynamic amplification
shape function of the exciting force as well as of the driving velocity in relation to the length of the contact area and the width of the lamella and thus the contact duration. Compared to the dominating frequencies of the expansion joint within a range of 78 Hz to 300 Hz common driving velocities especially of heavy vehicles with about 25 m/s and a Lamella corresponding contact durations of T = bT irev+b ≈ 0.02 s denote a quite low frequent excitation which is not least vehicle due to the enlarging of the lamellas width. This is also one reason for the comparatively low dynamic amplification factors below 1.20. 4. Conclusion The response of a bridge expansion joint to an excitation by a crossing vehicle, which e.g. could be measured on the basis of strain gauges or force washers, represents both, the dynamic wheel force variation induced by the vehicles
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Roland Friedl et al. / Procedia Engineering 199 (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017) 000–000
oscillations due to the pavement roughness and the dynamic amplification of the expansion joint due to the transient force excitation trough the contact between the vehicle tires and the lamella. The order of magnitude of the dynamic wheel force variation can be obtained on the basis of theoretical investigations, considering an adequate rigid body model for the vehicles calibrated to experimental data. The dynamic interaction between the vehicle tire and the lamellas turned out to be the most important influence parameter regarding the dynamic amplification of the bridge expansion joints’ response. The theoretical investigations in combination with the experimental data revealed that the non uniform contact pressure distribution has to be considered to obtain a reliable prediction of the dynamic amplification. Based on an extended tire model the influence of the non uniform contact pressure distribution to the dynamic amplification has been quantified and plotted in a dimensionless way, which allows a general estimation for a wide parameter range concerning the driving velocity, the width of the lamellas, the length of the contact area and most importantly the pressure distribution. References [1] R. Friedl, I. Mangerig: Zur messtechnischen Ermittlung von Fahrzeuggewichten unter Ber¨ucksichtigung der dynamischen Wechselwirkung zwischen Fahrzeug und Fahrbahn¨ubergangskonstruktion, 12. Fachtagung Baustatik-Baupraxis, Technische Universit¨at M¨unchen, 2014. [2] R. Friedl, I. Mangerig: Measurement based detection of vehicle weights considering the dynamic interaction between vehicles and bridge expansion joints, Engineering Mechanics Institute Conference, Stanford University 2015. [3] W. Heinrich, I. Hennig: Zufallsschwingungen mechanischer Systeme, Akademie-Verlag Berlin, 1978. [4] K.M.A. Kamash, J.D. Robson: Implications of Isotropy in Random Surfaces, Journal of Sound and Vibration, volume 54, 1977. [5] V. Bormann: Messungen von Fahrbahn-Unebenheiten paralleler Fahrspuren und Anwendung der Ergebnisse, Vehicle Syseme Dynamics, volume 7, p.65-81, 1978 [6] H. Braun, U. Gerz, P. Sulten, A. Ueckermann: Sammlung und Auswertung von Straßenunebenheitsdaten, [7] L. Ardnold: Stochastische Differentialgleichungen, R. Oldenbourg Verlag M¨unchen Wien, 1973. [8] M.C. Davis: Optimum Systems in Multi-Dimensional random Processes, Massachusetts Institute of Technology, Cambridge, 1961. [9] H. Schwarz: Mehrfachregelungen - Zweiter Band, Springer Verlag Berlin Heidelberg GmbH, Leipzig, 1971. Forschung Stra”senbeu und Straßenverkehrstechnik, volume 589, 1991. [10] R. Friedl: Zur Beschreibung fahrbahnunebenheitsinduzierter Radkraftschwankungen von Straßenfahrzeugen auf der Basis stochastischer Differentialgleichungen, Der Stahlbau, volume 84, number 10, 2015. [11] J. Braun, J. Leendertz, T. Schulze, B. Urich, B. Volk : Dynamisches Verhalten von Lamellen-Dehnfugen, Stahlbaukalender, 2009. [12] S. Marx, C. von der Haar, J. P. Liebig, J. Gr¨unberg : Bestimmung der Verkehrseinwirkung auf Br¨uckentragwerke aus Messungen an Fahrbahn¨ubergangskonstruktionen, Bautechnik, volume 90, number 8, 2013. [13] Insitu Dynamic Testing of Swivel Joist Modular Expansion Joints installed in the Bo Between Bridge, Technical Report, Arup, Brisbane, 2011 [14] A. Ueckermann: Ein geometrisch basiertes Verfahren zur Lokalisierung und Bewertung einzelner, periodischer und regelloser Unebenheiten im Straßenl¨angsprofil, Diss. RWTH Aachen, 2004.
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 00 (2017) 000–000 Procedia Engineering 199 (2017) 2651–2656 Procedia Engineering 00 (2017) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
X X International International Conference Conference on on Structural Structural Dynamics, Dynamics, EURODYN EURODYN 2017 2017
Dynamic Dynamic Amplification Amplification of of Bridge-Expansion-Joints Bridge-Expansion-Joints considering considering Roughness Roughness induced induced Vehicle Vehicle Vibrations Vibrations a,∗ b Roland Roland Friedl, Friedl, M.Sc. M.Sc.a,∗,, Univ.-Prof. Univ.-Prof. Dr.-Ing. Dr.-Ing. Ingbert Ingbert Mangerig Mangerigb b Universit¨ at b Universit¨ at
a bulicek+ingenieure gmbh, Sonnenstraße 19, 80331 Munich, Germany a bulicek+ingenieure gmbh, Sonnenstraße 19, 80331 Munich, Germany der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 37, 85577 Neubiberg, Germany der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 37, 85577 Neubiberg, Germany
Abstract Abstract The response of bridge expansion joints to the transient impact induced by crossing vehicles always represent the dynamic reaction The response of bridge expansion joints to the transient impact induced by crossing vehicles always represent the dynamic reaction of the former to the dynamic wheel contact forces at that particular moment. Separating the dynamic components in the vertical of the former to the dynamic wheel contact forces at that particular moment. Separating the dynamic components in the vertical wheel contact forces, the dynamic amplification of the bridge expansion joints response, which is dominated by the driving velocity wheel contact forces, the dynamic amplification of the bridge expansion joints response, which is dominated by the driving velocity of the vehicle, its crossing position in lateral direction, the dynamic behavior of the expansion joint itself and most importantly of the vehicle, its crossing position in lateral direction, the dynamic behavior of the expansion joint itself and most importantly by the interaction between the vehicle tire and the expansion joint’s lamellas can be discussed. Due to the random distribution of by the interaction between the vehicle tire and the expansion joint’s lamellas can be discussed. Due to the random distribution of the roughness amplitudes a stochastic characterization of the vehicle vibrations is necessary. In addition to the classical frequency the roughness amplitudes a stochastic characterization of the vehicle vibrations is necessary. In addition to the classical frequency approach based on complex transfer matrices completely describing the dynamic behavior of the vehicles and spectral density approach based on complex transfer matrices completely describing the dynamic behavior of the vehicles and spectral density matrices characterizing the pavement roughness, the numerically efficient covariance analysis is applied. Therefore the spectral matrices characterizing the pavement roughness, the numerically efficient covariance analysis is applied. Therefore the spectral characteristics of the roughness amplitudes, which are described on the basis of multi-correlated random processes, are considered characteristics of the roughness amplitudes, which are described on the basis of multi-correlated random processes, are considered by introducing shaping filters in combination with a state space augmentation. The dynamic behavior of the considered bridge by introducing shaping filters in combination with a state space augmentation. The dynamic behavior of the considered bridge expansion joint is dominated by the properties of the discrete elastomeric spring elements. Hence a a non-proportional damping expansion joint is dominated by the properties of the discrete elastomeric spring elements. Hence a a non-proportional damping approach has to be applied to analyze the three dimensional finite element model of the structure using a modal superposition. approach has to be applied to analyze the three dimensional finite element model of the structure using a modal superposition. c 2017 The Authors. Published by Elsevier Ltd. c 2017 2017 The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN EURODYN 2017. 2017. Keywords: random vehicle vibration, dynamic amplification, interaction between vehicle, tire and expansion joint Keywords: random vehicle vibration, dynamic amplification, interaction between vehicle, tire and expansion joint
1. Introduction 1. Introduction Beside effects like wind loads, cornering, accelerating or slowing down the main source for dynamic oscillations of Beside effects like wind loads, cornering, accelerating or slowing down the main source for dynamic oscillations of driving vehicles is given by the surface roughness. Widely neglecting discrete obstacles like single pot holes or blow driving vehicles is given by the surface roughness. Widely neglecting discrete obstacles like single pot holes or blow bars with dominating step heights the influence of randomly distributed roughness amplitudes to vehicle vibrations bars with dominating step heights the influence of randomly distributed roughness amplitudes to vehicle vibrations and further to a variation in the vertical wheel contact forces is analyzed and quantified. The standard deviation of the and further to a variation in the vertical wheel contact forces is analyzed and quantified. The standard deviation of the vertical tire contact forces is used as the basis for further investigations concerning the dynamic amplification of the vertical tire contact forces is used as the basis for further investigations concerning the dynamic amplification of the expansion joints’ response. expansion joints’ response. ∗ ∗
Corresponding author. Tel.: +49-89-189-4143-88 ; fax: +49-89-189-4143-30. Corresponding author. Tel.: +49-89-189-4143-88 ; fax: +49-89-189-4143-30. E-mail address: [email protected] E-mail address: [email protected]
c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 Peer-review under responsibility of the organizing committee of EURODYN 2017. 1877-7058 2017responsibility The Authors. Published by committee Elsevier Ltd. Peer-review©under of the organizing of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.515
Roland Friedl et al. / Procedia Engineering 199 (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017) 000–000
2652 2
2. Roughness induced wheel force variations 2.1. Random surface roughness Extensive evaluation of real pavement profiles have shown [6,14] that indeed the single roughness amplitudes are distributed randomly but many profiles show strong similarities concerning their spectral characteristics. Therefore these can be approximately described on the basis of random processes respectively the parameters defining these processes. To ensure exact solutions for the vehicle oscillations, analytical approximations for the (power) spectral density functions based on rational expressions as for example 2α α2 + β2 + 4π2 n2 (1) S ww (n) = σ2 α2 + β2 − 4π2 n2 2 + 16α2 π2 n2
are preferred. Assuming normally distributed random fields a stochastically complete characterization of the vehicles excitation caused by two dimensional pavement profiles is obtained on the basis of a (nT , nT ) spectral density matrix Sww (n), where nT defines the number of vehicle tires and n the wave number. A complete description of two dimensional surfaces based on experiments is very time consuming and tedious why theoretical roughness models, which allow a simplified but consistent evaluation of Sww (n) based on a single one dimensional roughness profile, have been developed [4,5]. Because of the treatment of the vehicle vibrations in the time or corresponding frequency domain the wave number based description of the spectral density matrix hast to be transferred into a frequency based form considering the driving velocity of the vehicle. Due to the for usual pavement profiles observed declination of the spectral density with an increasing wave number the caused excitation increases along with an increasing driving velocity. Further the roughness induced vehicle vibrations reach high amplitudes in case the dominating frequency band of periodic roughness amplitudes coincide with the relevant natural frequencies of the vehicle. 2.2. Dynamic behavior of vehicles Experiments with an instrumented vehicle clearly show, that the amplification of the vertical wheel forces, which is of interest regarding the shock impact exerted to bridge expansion joints, is mainly dominated by the global oscillations of the vehicle body and the axes [2]. Theoretically investigating the global dynamic motions of vehicles, the latter can be mathematically described on the basis of a rigid body system model where all masses are assumed to be rigid and interconnected by discrete spring and damper elements. Instead of working with the well known equation of motion ¨ + Du(t) ˙ + Ku(t) = p(t) Mu(t) (2) containing the mass matrix M, the stiffness matrix K, the damping matrix D, the force vector p(t) and the vector T of the physical coordinates u = uB φBx φBy uAv φAv x uAh φAh x a more deeper insight into thy dynamic behavior of the vehicle can be obtained on the basis of a modal analysis regarding the eigen-vectors, the corresponding eigenfrequencies and most importantly the frequency response matrix −1 (3) F( f ) = − (2π f )2 M + j2π f D + 2π f K ,
which completely describes the vehicles transfer behavior. Considering the frequency response function F11 characterizing the vertical vehicle body movements in figure 1a, the corresponding frequency of 1.16 Hz can be identified very clearly. The pitch mode of the vehicle body dominated by a frequency of about 2 Hz and represented by the frequency response function F33 has also a great influence on the investigated wheel force amplifications, especially on the coupling between the oscillations of the front and rear axis. 2.3. Quantifying the variation in the vertical wheel forces A stochastic quantification of the random vehicle vibrations induced by the surface roughness can be obtained on the basis of the spectral density matrix of the systems response ¯ f )S pp ( f )FT ( f ) Suu ( f ) = F(
with
S pp ( f ) = F¯ R ( f )Szz ( f )FTR ( f )
(4)
Roland Friedl et al. / Procedia Engineering 199 (2017) 2651–2656 Roland Friedl / Procedia Engineering 00 (2017) 000–000
(a) Transfer functions Fii
(b) Transfer functions Fi j
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(c) Spectral density functions of vertical wheel forces
Fig. 1. Dynamic behavior of the vehicle
representing the spectral density matrix characterizing the random tire forces induced by the surface roughness [3]. Thereby the frequency response matrix textb f FR ( f ) describes the dynamic behavior of the vehicle tires and by this the transformation of a displacement based excitation to a force controlled one. To quantify the vertical wheel force variance, the correlation between the roughness amplitudes and the vehicle vibrations has to be considered. Defining the spectral density matrix of the relative displacements between the pavement surface and the vehicle axes S∆u∆u ( f ) = Suu ( f ) − Suz ( f ) − Szu ( f ) + Szz ( f )
(5)
with the complex cross spectral density matrices ¯ f )F¯ R ( f )Szz ( f ) Suz ( f ) = F(
and
Szu ( f ) = Szz ( f )FTR ( f )FT ( f ),
(6)
the spectral density matrix of the vertical wheel forces finally results in ST T ( f ) = F¯ T ( f )S∆u∆u ( f )FTT ( f ).
(7)
Up to equation 7 the presented method yields exact solutions for a stochastic characterization of the wheel forces. Comparing the exemplary plotted spectral density functions of the vertical wheel forces in figure 1c with the frequency response functions in figure 1 it becomes obvious, that both the dynamic behavior of the vehicle and the spectral characteristics of the pavement roughness have a great influence on the induced oscillations and further on the wheel force variation. The necessary integration to obtain the standard deviation of the wheel forces is executed numerically and truncated at a frequency flimit ≤ ∞ σ2Rii
+ +∞ flimit = S Ri Ri ( f )d f ≈ S Ri Ri ( f )d f. −∞
(8)
− flimit
Assuming normally distributed roughness amplitudes and linear vehicle behavior the standard deviation σRii in combination with the mean value represents a stochastically complete description of the roughness induced wheel force amplification which is further used to discuss the loads exerted on bridge expansion joints. The results derived on the basis of the described vehicle model show a good accordance with the experimental data [2]. 3. Dynamic amplification of bridge expansion joints As all modular expansion joints also the here considered swivel joist expansion joint has several steel girders, called the lamellas, running perpendicular to the driving lane which cross the varying gap between the bridge superstructure and the abutment area [11]. Beside the skew arrangement of the cross beams bearing the lamellas, the elastomer support elements, connecting the lamellas with the cross beams and the cross beams with the support structure, are the main construction elements dominating the dynamic behavior of the whole structure. As a slight modification compared to common expansion joints the lamella has been designed as a box girder with about double the width of usual lamellas. This has been done with regard to a measurement based detection of vehicle weights on the basis of an instrumented expansion joint [1].
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3.1. Dynamic behavior and experiment To theoretically analyze the dynamic behavior of the modified expansion joint, it has been modeled on the basis of a three dimensional beam structure applying a finite element approach. The elastomer support elements are idealized by discrete three dimensional spring and damper elements considering the experimentally found stiffness and damping coefficients. Applying strain gauges on both webs of the steel girder and in addition piezo-electric force washers between the lamella and the elastomer support elements, the vertical reaction forces between the lamella and the transverse beams, which are the basis for the further discussion of the dynamic amplification, could be measured. The measured natural frequencies of the real structure correspond very well to the eigen-values of the finite element model as can be seen in figure 2c. Because of the discrete elastomer elements which moreover dominate the dynamic behavior of the whole expansion joint, a non-proportional damping approach has to be used. Regarding the measured
(a) Vertical reaction forces beteween the lamella and the cross beams
(b) experimental frequency spectrum
(c) correlation between the structural model and experimental data
Fig. 2. Experimental data
frequency spectra in combination with the eigen-vectors of the structural model it becomes obvious, that the global oscillations of the expansion joints’ lamellas are dominated by bending modes, basically characterized by vertical displacements of the box girder and torsional modes with dominating rotations about its longitudinal axis, equally. Especially the mode shapes corresponding to lower frequency ranges show dominating rotational deformation components whereas the higher frequencies are mainly represented by vertical bending modes. Another proof for the dominating character of the elastomeric support elements can be obtained regarding short time frequency spectra of the measured reaction forces. In this content it should be stated, that for this purpose only the decaying reaction forces after the contact with a single vehicle axle but never the time series representing a whole vehicle contact representing several axle excitations may be considered. The dynamic behavior in general and the decaying behavior especially can be separated into two main parts where the first part with relatively high oscillation amplitudes is dominated by torsional mode shapes with a clearly dominating frequency of about 78 Hz whereas the second part show a wider frequency band mainly correlating to vertical bending mode shapes. Because the torsional mode shapes coincide with high deformation amplitudes and therefore a relatively high strain level in the elastomer elements, these are damped out quite quickly due to the high damping capacity of the latter. In contrast the bending mode shapes show only very little deformations in the support area. Therefore the damping capacity of the elastomer elements can hardly by effective for those which can be seen by means of lower modal damping parameters (see figure 3a). To analyze this effect a non proportional damping approach is obligatory. 3.2. Dynamic amplification considering the interaction between vehicle, tire and expansion joint The dynamic amplification of the expansion joints’ response strongly depends on the crossing position of the vehicle in lateral direction, the dynamic behavior of the expansion joint itself and most importantly on the dynamic interaction between the vehicle tire and the lamellas [1]. A first estimation regarding the dynamic amplification can be reached by a comparison of the elementary frequency response functions shown in figure 3c with the frequency content of the exciting forces. Whereas the frequency response function is defined by the structure, the frequency spectrum of the exciting force highly depends on the dynamic interaction between the vehicle tire an the lamella. At this point the often stated sine half wave shaped force input function (see e.g. [12,13]) could not be verified. The carried out theoretical investigations in combination with the experimental data clearly show [1], that the nonuniform contact pressure distribution between the tire and the pavement and respectively the lamella has to be taken into
Roland Friedl et al. / Engineering Procedia Engineering (2017) 2651–2656 Roland Friedl / Procedia 00 (2017)199 000–000
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account to achieve a realistic approximation for the expansion joints oscillations. Therefor an extended tire model [1] has been invented which allows to consider arbitrary pressure distributions and which is in addition appropriate to describe the coupling behavior between the vehicle tire and the lamellas correctly. It further shows that the varying eccentricity of the resulting vertical tire force plays a dominating role especially regarding rotational deformations of the lamellas (see also figure 4). The influence of the so called non uniformity of the vertical contact pressure distribution is quantified on the basis of dynamic amplification factors φ=
wdynamic , w static
(9)
which are defined as the quotient of the dynamic deformation to the corresponding static deformation considering the same force input but completely neglecting mass inertia forces. For the pressure shape functions plotted in figure 3a the systems response is shown in figure 3b and the corresponding amplification factors in figure 3c. The amplification ●
0.07
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●
Grad der kritischen Dämpfung [-]
0.06
● D
●
0.05 0.04 0.03 0.02 ● 0.01 0.00
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5
● ●
●
●
●
●
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10
15
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20
Eigenform Nr.
(a) modal damping
(b) Elementary transfer function of the expansion joints
(c) non uniform pressure distribution
Fig. 3. Excitation of the expansion joint vehicle factor φ is plotted against a dimensionless frequency η = bT irev+b which allows a consideration of a variation in Lamella the widths of the lamella, the length of the tire contact area and the driving velocity of the vehicle. By means of the shape of the resulting vertical force and torsional moment resulting by an integration of the vertical contact pressure within the contact area shown in figure 4 it becomes evident, that the shape is everything else than a sine half wave. The dynamic amplification factor φ shows a great influence of the non uniform pressure distribution and further of the
(a) vertical force
(b) torsional moment
(c) dynamic amplification factor
Fig. 4. Input force and dynamic amplification
shape function of the exciting force as well as of the driving velocity in relation to the length of the contact area and the width of the lamella and thus the contact duration. Compared to the dominating frequencies of the expansion joint within a range of 78 Hz to 300 Hz common driving velocities especially of heavy vehicles with about 25 m/s and a Lamella corresponding contact durations of T = bT irev+b ≈ 0.02 s denote a quite low frequent excitation which is not least vehicle due to the enlarging of the lamellas width. This is also one reason for the comparatively low dynamic amplification factors below 1.20. 4. Conclusion The response of a bridge expansion joint to an excitation by a crossing vehicle, which e.g. could be measured on the basis of strain gauges or force washers, represents both, the dynamic wheel force variation induced by the vehicles
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oscillations due to the pavement roughness and the dynamic amplification of the expansion joint due to the transient force excitation trough the contact between the vehicle tires and the lamella. The order of magnitude of the dynamic wheel force variation can be obtained on the basis of theoretical investigations, considering an adequate rigid body model for the vehicles calibrated to experimental data. The dynamic interaction between the vehicle tire and the lamellas turned out to be the most important influence parameter regarding the dynamic amplification of the bridge expansion joints’ response. The theoretical investigations in combination with the experimental data revealed that the non uniform contact pressure distribution has to be considered to obtain a reliable prediction of the dynamic amplification. Based on an extended tire model the influence of the non uniform contact pressure distribution to the dynamic amplification has been quantified and plotted in a dimensionless way, which allows a general estimation for a wide parameter range concerning the driving velocity, the width of the lamellas, the length of the contact area and most importantly the pressure distribution. References [1] R. Friedl, I. Mangerig: Zur messtechnischen Ermittlung von Fahrzeuggewichten unter Ber¨ucksichtigung der dynamischen Wechselwirkung zwischen Fahrzeug und Fahrbahn¨ubergangskonstruktion, 12. Fachtagung Baustatik-Baupraxis, Technische Universit¨at M¨unchen, 2014. [2] R. Friedl, I. Mangerig: Measurement based detection of vehicle weights considering the dynamic interaction between vehicles and bridge expansion joints, Engineering Mechanics Institute Conference, Stanford University 2015. [3] W. Heinrich, I. Hennig: Zufallsschwingungen mechanischer Systeme, Akademie-Verlag Berlin, 1978. [4] K.M.A. Kamash, J.D. Robson: Implications of Isotropy in Random Surfaces, Journal of Sound and Vibration, volume 54, 1977. [5] V. Bormann: Messungen von Fahrbahn-Unebenheiten paralleler Fahrspuren und Anwendung der Ergebnisse, Vehicle Syseme Dynamics, volume 7, p.65-81, 1978 [6] H. Braun, U. Gerz, P. Sulten, A. Ueckermann: Sammlung und Auswertung von Straßenunebenheitsdaten, [7] L. Ardnold: Stochastische Differentialgleichungen, R. Oldenbourg Verlag M¨unchen Wien, 1973. [8] M.C. Davis: Optimum Systems in Multi-Dimensional random Processes, Massachusetts Institute of Technology, Cambridge, 1961. [9] H. Schwarz: Mehrfachregelungen - Zweiter Band, Springer Verlag Berlin Heidelberg GmbH, Leipzig, 1971. Forschung Stra”senbeu und Straßenverkehrstechnik, volume 589, 1991. [10] R. Friedl: Zur Beschreibung fahrbahnunebenheitsinduzierter Radkraftschwankungen von Straßenfahrzeugen auf der Basis stochastischer Differentialgleichungen, Der Stahlbau, volume 84, number 10, 2015. [11] J. Braun, J. Leendertz, T. Schulze, B. Urich, B. Volk : Dynamisches Verhalten von Lamellen-Dehnfugen, Stahlbaukalender, 2009. [12] S. Marx, C. von der Haar, J. P. Liebig, J. Gr¨unberg : Bestimmung der Verkehrseinwirkung auf Br¨uckentragwerke aus Messungen an Fahrbahn¨ubergangskonstruktionen, Bautechnik, volume 90, number 8, 2013. [13] Insitu Dynamic Testing of Swivel Joist Modular Expansion Joints installed in the Bo Between Bridge, Technical Report, Arup, Brisbane, 2011 [14] A. Ueckermann: Ein geometrisch basiertes Verfahren zur Lokalisierung und Bewertung einzelner, periodischer und regelloser Unebenheiten im Straßenl¨angsprofil, Diss. RWTH Aachen, 2004.