Dynamic response of a beam on a frequencyindependent damped elastic foundation to moving load

Transcription

1 60 Dynamic response of a beam on a frequencyindependent damped elastic foundation to moving load Seong-Min Kim and Jose M. Roesset Abstract: The dynamic displacement response of an infinitely long beam on an elastic foundation with frequencyindependent linear hysteretic damping subjected to a constant amplitude or a harmonic moving load was investigated. The advance velocity was assumed to be constant. Formulations were developed in the transformed field domain using (i) a Fourier transform in moving space for moving loads of constant amplitude, (ii) a double Fourier transform in time and moving space for moving loads of arbitrary amplitude variation or to include the transient due to the initial application of the load for moving harmonic loads, and (iii) a Fourier transform in moving space for the steady-state response to moving harmonic loads. The effects of velocity, damping, loaded length, and load frequency on the deflected shape and the maximum displacement were investigated. The critical (resonant) velocities and frequencies were obtained by analyses, and expressions to find them were suggested. Key words: beam on elastic foundation, damping, Fourier transform, frequency, harmonic load, moving load, transformed field, velocity. Résumé : Une poutre infiniment longue, reposant sur une fondation élastique avec un amortissement hystérétique linéaire indépendant de la fréquence, est soumise à une charge mobile d amplitude constante ou harmonique, et sa réponse dynamique en déplacement a été étudiée. La vitesse de marche a été supposée constante. Des formulations ont été développées dans le champ correspondant à la transformée du domaine en utilisant (i) une transformée de Fourier dans l espace de déplacement pour les charges mobiles d amplitude constante, (ii) une double transformée de Fourier dans le temps et dans l espace de déplacement pour des charges mobiles d amplitude variant arbitrairement ou bien pour inclure le transitoire dû à l application initiale de la charge pour les charges en mouvement harmonique, et (iii) une transformée de Fourier dans l espace de déplacement pour la réponse en régime établi à des charges en mouvement harmonique. Les effets de la vitesse, de l amortissement, de la longueur de chargement, et de la fréquence du chargement sur la forme déviée et le déplacement maximum ont été étudiés. Les vitesses et fréquences critiques (résonantes) ont été obtenues par des analyses et des expressions permettant de les calculer sont suggérées. Mots clés : poutre sur fondation élastique, amortissement, transformée de Fourier, fréquence, chargement harmonique, charge mobile, transformée du domaine, vitesse. [Traduit par la Rédaction] Kim and Roesset 67 Introduction Received 17 September 00. Revision accepted December 00. Published on the NRC Research Press Web site at on May 003. S.-M. Kim. 1 Center for Transportation Research, The University of Texas at Austin, Suite 00, 308 Red River, Austin, TX 78705, U.S.A. J.M. Roesset. Department of Civil Engineering, Texas A&M University, College Station, TX 7783, U.S.A. Written discussion of this article is welcomed and will be received by the Editor until 31 August Corresponding author ( seong-min@mail.utexas.edu). A number of studies have been conducted to find the dynamic response of layered media as models of pavement systems to moving loads (Barros and Luco 199; Zaghloul and White 1993). Most of the studies have been conducted for moving loads of constant amplitude. The moving loads created by vehicles, however, will have variations in amplitude with time due to many factors such as the roughness of the pavement surface and the mechanical systems of the vehicles. In addition, some nondestructive testing devices, such as the rolling dynamic deflectometer (RDD) (Bay et al. 1995, 1996; Bay 1997; Kim et al. 1999), apply a steady-state harmonic force while continuously moving. To analyze the response of layered media to the loads imposed by the moving vehicles, we conducted a study considering both constant and harmonic moving loads. Solutions were obtained both for simplified models, such as a beam on an elastic foundation and a plate on an elastic foundation, and for a layered half-space (Kim and Roesset 1997, 1998). Our study of a beam on an elastic foundation is presented in this paper. The beam on an elastic foundation is also used often for a model of a railroad system (Kerr and Shenton 1986; Kerr 000). The main objective of this paper is to discuss the dynamic response of a beam on an elastic foundation to a moving load with a constant advance velocity and with either constant or harmonic amplitude variations. Frequencyindependent linear hysteretic damping was considered for Can. J. Civ. Eng. 30: (003) doi: /L NRC Canada

2 Kim and Roesset 61 the elastic foundation. Because moving loads in practice will have a finite area over which they are distributed and the point load represents only an idealization (extreme case), a distributed load was considered instead of a point load. The geometry and material properties were assumed to be linearly elastic, and the beam was assumed to extend to infinity. Formulations were developed with transformed field domain analyses using (i) a Fourier transform in moving space for moving loads of constant amplitude, (ii) a double Fourier transform in time and moving space for moving loads of arbitrary amplitude variation or to include the transient due to the initial application of the load for moving harmonic loads, and (iii) a Fourier transform in moving space for the steady-state response to moving harmonic loads. Studies were conducted to investigate the effects of parameters such as load velocity, internal damping, loaded length, and load frequency on the displacements, and expressions were developed for the critical (resonant) velocities and frequencies. Formulations in transformed field domain The dynamic displacement response of a beam of infinite extent on an elastic foundation to a moving load whose amplitude changes with time can be obtained using a double Fourier transform in time and moving space. The governing differential equation for a system without damping can be written in fixed Cartesian coordinates {x, y} at time t as yx (,) t [1] EI m yx (,) + t + ky(,) x t q(,) x t x t where E is Young s modulus of elasticity, I is the moment of inertia, m is the mass of the beam per unit length, k is the stiffness of the foundation per unit length, and q(x, t) isthe external load per unit length. If the load q(x, t) is moving in the positive x direction with a constant velocity V, a moving coordinate η can be defined by x Vt. Equation [1] can then be rewritten as y( η, t) [] EI η y( η, t) y( η, t) y( η, t) + m V + V t t η η + ky( η, t) q( η, t) If ξ and Ω are assumed to be the transformed fields of η (moving space) and t (time), respectively; and if y(η, t) and q(η, t) are written in the form Y(ξ, Ω) e iωt e iξη and i t Qxi (, Ω) e Ω e iξη, where i 1; the transformed displacements Y(ξ, Ω) can be obtained by [3] Y(, ξ Ω) Q(, ξ Ω) EI ξ + k m( Ω V ξ) where the transformed load Q(ξ, Ω) is obtained using the double Fourier transform i ξη i Ω e e t d d [] Q(, ξ Ω) q(,) η t η t Lastly, the dynamic displacement response can be obtained using the double inverse Fourier transform [5] y( η, t) 1 π Q(, ξ Ω) EI ξ + k m( Ω V ξ) iξη iω t e e dξ dω In practice, eqs. [3] [5] are solved using the fast Fourier transform (FFT), which is a discrete transform. To successfully perform the FFT in the time and frequency domains, the system should have some damping. This requirement can be dropped when using the exponential window method (Kausel and Roesset 199; Kim and Roesset 1997). If the moving load has a harmonic variation of amplitude e iω t, where Ω is the load frequency, and only the steady-state response is of interest, the displacement response in eq. [5] can be rewritten as [6] y( η, t) with 1 π Q(,) ξ t EI ξ + k m( Ω V ξ) iωt iξη [7] Q(,) ξ t e q() η e dη e i ξη dξ If the response to the force sin Ωt (the imaginary component of e iω t ) is considered, the imaginary component of eq. [6] should be used. If a moving load of constant amplitude (Ω 0) is considered, eqs. [6] and [7] can be expressed as 1 Q() ξ i ξη [8] y( η) π ξ EI ξ + k mv ξ e d with ξη [9] Q() ξ q() η e dη Because η is a point on the moving axis, y(η, t) represents the response at a moving point with time. The response at a fixed point can simply be determined by the relation η x Vt, where x is the abscissa of the fixed point. If the model shown in Fig. 1 is used and the variation in load amplitude with time is qf(t), the transformed force Q defined in eq. [] is given by η + d/ [10] Q(, ξ Ω) q f() t e e dη dt 0 η d/ 0 iξη iωt sin( d/ ) ξ iξη q 0 iω t ft () ξ e e dt where d is the loaded length and η 0 is the coordinate of the center of the load. If viscous damping is assumed, c[ y( x,) t / t] and c[ y(η, t)/ t V y(η, t)/ η] should be added in eqs. [1] and [], respectively, where c is the viscous damping constant. If 003 NRC Canada

3 6 Can. J. Civ. Eng. Vol. 30, 003 Fig. 1. Beam on an elastic foundation. Fig.. Deflected shapes for various velocities (V) without damping. frequency-independent linear hysteretic damping (or material damping) is considered, an expression idk can be used for the damping term, where D is the damping ratio (Foinquinos and Roesset 1995). The transformed displacements Y(ξ, Ω) in eq. [3] can then be written as [11] Y( ξ, Ω) Q(, ξ Ω) EIξ + k( 1+ id) m( Ω Vξ) + ic( Ω Vξ) Fig. 3. Effect of damping on deflected shape (V 95.5 m/s). It is noted that the sign of the linear hysteretic damping term needs to be consistent with that of the viscous damping term. It is generally accepted that most of the dissipation of energy in soils takes place through internal friction (hysteretic damping) rather than through viscous behavior. Hysteretic damping produces an energy loss per cycle that is frequency independent. Since the elastic foundation in this study represents soil deposits, frequency-independent linear hysteretic damping has been considered in this study for the foundation. Response to a moving load of constant amplitude The displacement response caused by a moving single load of constant amplitude is investigated first using the values EI.3 kn m, k 68.9 MPa, m 8. kg/m, q 70 kn/m, and d 0.15 m, where the negative sign of q means that the load direction is opposite to the y direction shown in Fig. 1. Damping of a linear hysteretic nature is considered for the elastic foundation. Effects of velocity and damping If the beam is infinitely long and the velocity is constant, the deflected shapes under the load are the same at any instant. This means that the deflected shape is moving with the load. Figure shows the deflected shapes along the x axis when there is no damping in the system. The 0 distance in Fig. represents the location of the center of the load. The deflected shape is symmetric with respect to the vertical axis, which passes through the center of the load. As the velocity increases, the maximum displacement increases, more pronounced fluctuations occur, and the deflected region is more widely spread. The effect of damping on the deflected shapes can be observed in Fig. 3. As the damping ratio increases, the maximum displacement decreases slightly. The deflected shapes are no longer symmetric. As the damping ratio increases, the amplitude of the peak that occurs before the load passes through the observation point (positive distance) increases while the amplitude of the peak after the load passes through the observation point (negative distance) decreases. There is a lag between the position where the maximum displacement occurs and the center of the load. This lag is referred to in the paper as distance lag. The distance lag becomes more apparent with an increase in the damping ratio. Figure shows the effect of the load velocity on the maximum displacement. As the velocity increases, the maximum displacement increases until the velocity becomes close to the critical (resonant) velocity, then decreases again. In this case, the critical velocity is about 19 m/s. For a given velocity, the maximum displacement decreases as the damping ratio increases. For velocities larger than the critical velocity, as shown in Fig. 5a, the deflected region is widely spread and the responses behind and ahead of the load are very different. The frequency of the fluctuations in front of the load is larger than that behind the load, and the amplitudes of the fluctuations in front of the load are smaller than those behind the load. As the velocity increases, the amplitudes of the fluctuations in front of the load decrease significantly. The frequency of the fluctuations in front of the load increases with an increase in velocity while the corresponding frequency behind the load decreases. The distance lags are also clearly observed. As the damping ratio increases, as shown in Fig. 5b for a velocity of 15. m/s, the amplitudes of the response decrease, but the periods of the fluctuations remain almost the same. Figure 6 shows the relationship between the distance lag and the velocity. As the velocity increases, the distance lag increases, and large distance lags are observed with velocities near and above the critical ve- 003 NRC Canada

4 Kim and Roesset 63 Fig.. Relationship between maximum displacement and velocity. Fig. 6. Relationship between distance lag and velocity. Fig. 5. Deflected shapes for velocities larger than the critical velocity: (a) velocity effect (D 10%), and (b) damping effect (V 15. m/s). Fig. 7. Relationship between maximum displacement and velocity for various loaded lengths (D 10%). locity. The distance lag increases as the damping ratio increases for velocities smaller than the critical velocity, but the reverse occurs for velocities larger than the critical velocity. Effect of loaded length The effect of the loaded length for the same total value of the load was investigated assuming 10% linear hysteretic damping. The relationship between the maximum displacement and the velocity was studied first for various loaded lengths. Figure 7a shows the results for relatively short loaded lengths. As the loaded length increases, the maximum displacement decreases, but the critical velocities occur at almost the same value. As the velocity increases above the critical velocity, the differences in the maximum displacements among the loaded lengths decrease. The responses for various loaded lengths would be almost the same for a very large velocity. Figure 7b shows the results for relatively large loaded lengths. Similar trends are observed, but the effect of the loaded length continues to be important for the higher velocities considered. Figure 8a shows the variation of the distance lag with velocity for different loaded lengths. The lag increases as the loaded length increases. The effect is small but clear when the velocity is lower than the critical velocity. Figure 8b shows the corresponding results for relatively large loaded 003 NRC Canada

5 6 Can. J. Civ. Eng. Vol. 30, 003 Fig. 8. Relationship between distance lag and velocity for various loaded lengths (D 10%). the maximum displacement occurs at the rear peak. Lastly, the front peak disappears, and only one peak is present under the load, as shown in Fig. 9d. Critical velocity Through a number of parametric studies, it was found that the effects of damping and the size of the loaded length on the critical velocity were negligible within the logical ranges of values of these parameters, and that the following expression for the critical velocity could be used for all cases: [1] V EIk m Figure 10 shows the variation of the critical velocity with the parameter α EIk / m, variable part of eq. [1]. The results from eq. [1] and from the analyses are identical. lengths. There are discontinuities for loaded lengths of 0.61 and 1. m. For instance, for a loaded length of 0.61 m, the distance lag increases with increasing velocity until it approaches the critical velocity. Near the critical velocity it suddenly drops to a negative value and then increases again. A second jump occurs at a higher velocity, and then the lag converges to the curve for a loaded length of 0.3 m. When the distance lag is negative, the maximum displacement occurs in front of the center of the load. As mentioned previously, the maximum displacement normally occurs behind the center of the load. Therefore, this phenomenon represents a real change in behavior. To understand the phenomenon, the deflected shapes for the velocities around the discontinuity were investigated further. Figure 9 shows the deflected shapes for a loaded length of 0.61 m and various velocities. Normally, there is just one peak under the load with a relatively short loaded length, but two peaks appear for a loaded length of 0.61 m, and the maximum displacement occurs at the rear peak as shown in Fig. 9a. As the velocity increases, however, the maximum displacement occurs at the front peak as shown in Fig. 9b. The first discontinuity in Fig. 8b is due to the change in the position where the maximum displacement occurs from the rear peak to the front peak. After the discontinuity the maximum displacement occurs at the front peak and the distance lag increases again as the velocity increases. At a relatively high velocity, the maximum displacement occurs again at the rear peak as shown in Fig. 9c. The second discontinuity in Fig. 8b is due to this reason. After the second discontinuity Response to a moving harmonic load The relationship between the frequency of the moving harmonic load and the maximum displacement was investigated. As shown in Fig. 11a, for velocities smaller than the critical velocity of a moving load of constant amplitude (19 m/s in this case), the critical frequency decreases with an increase in velocity. For velocities larger than the critical velocity of a moving load of constant amplitude (Fig. 11b), the critical frequency increases with an increase in velocity. The critical frequency f (in hertz) of a stationary harmonic load (velocity of 0) is independent of the flexural rigidity of the beam, the size of the loaded length, and the damping ratio and is defined by [13] f 1 π k m The effect of the load velocity on the maximum displacement was studied next. As shown in Fig. 1a, for load frequencies smaller than the critical frequency of a stationary harmonic load (190 Hz in this case), there are two peaks in the response curve. The first critical velocity (peak) decreases and the second increases as the load frequency increases. For load frequencies larger than the critical frequency of a stationary harmonic load (Fig. 1b), only the peak corresponding to the second critical velocity is observed. In this case, the maximum displacement decreases initially as the velocity increases, then increases at the value of the critical velocity. Figure 13 shows the variation of the critical velocities with load frequency. The critical velocity at a load frequency of 0 represents the value for a moving load of constant amplitude. Efforts were made to find expressions for the first and second critical velocities, and the following formulae are suggested: [1] Vcr1 V 1 [15] Vcr V 1 + f f f f 7/10 3 / 003 NRC Canada

6 Kim and Roesset 65 Fig. 9. Changes in deflected shapes for various velocities (d 0.61mandD 10%). Fig. 10. Comparison of critical velocities from the analysis with those from the formula (α (EIk) 1/ /(m) 1/ ). f cr can also be obtained solving eqs. [1] and [15] for f, and then 1.3 V [16] fcr f 1, V V V [17] f f cr V V 3 / 1, V V Figure 1 compares the results from eqs. [1] and [15] with those from the analyses. Very good agreement is observed for the two critical velocities. where V cr1 and V cr are the first and the second critical velocities, respectively; f is the frequency of the moving harmonic load in hertz (1 cps 1 Hz); and V and f are the critical velocity of a moving load of constant amplitude and the critical frequency of a stationary harmonic load defined by eqs. [1] and [13], respectively. The critical frequencies Summary and conclusions The dynamic displacement response of an infinitely long beam on an elastic foundation with frequency-independent linear hysteretic damping subjected to a moving load of constant amplitude and harmonic variation was investigated using formulations in the transformed field domains of time and moving space. 003 NRC Canada

7 66 Can. J. Civ. Eng. Vol. 30, 003 Fig. 11. Effect of load frequency on maximum displacement for velocities (a) smaller and (b) larger than the critical velocity of a moving load of constant amplitude (D %). Fig. 13. Relationship between critical velocity and load frequency. Fig. 1. Results from suggested formula and analysis for (a) first critical velocity, and (b) second critical velocity. Fig. 1. Effect of velocity on maximum displacement for load frequencies (a) smaller and (b) larger than the critical frequency of a stationary harmonic load (D %). For a moving load of constant amplitude, as the velocity increases, the maximum displacement increases until the velocity becomes close to the critical velocity and then decreases after the critical velocity. Very clear distance lags are observed for a relatively large velocity and some damping. The critical velocity is independent of the damping ratio and the loaded length within logical values. For relatively large loaded lengths, there are discontinuities in the relationship between the velocity and the distance lag. For a moving harmonic load, the critical frequency decreases with an increase in velocity for velocities smaller than the critical velocity of a moving load of constant amplitude, and the reverse occurs for velocities larger than this 003 NRC Canada

8 Kim and Roesset 67 critical velocity. For load frequencies smaller than the critical frequency of a stationary harmonic load, there are two critical velocities. As the load frequency increases, the first critical velocity decreases and the second increases. For load frequencies larger than the critical frequency of a stationary harmonic load, only the second critical velocity exists. Expressions for the critical velocities and frequencies were proposed, and very good agreement was observed between the results from these expressions and those from the analyses. The results in this paper correspond to a single load. The response due to multiple loads can be obtained simply by using superposition of each response or considering multiple loads simultaneously in the formulations. The loads due to moving vehicles including nondestructive testing vehicles such as the RDD, which applies a steady-state harmonic load while continuously moving, can be simplified as combinations of moving loads of constant amplitude and of harmonic variation. Therefore, the analysis techniques explained in the paper can be used to find the dynamic response of a beam on an elastic foundation due to the loads imposed by those vehicles. Acknowledgment The authors gratefully acknowledge the support of the Texas Department of Transportation under study number 0-1. References Barros, F.C.P., and Luco, J.E Moving Green s functions for a layered visco-elastic half-space. Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, Calif. Bay, J.A Development of a rolling dynamic deflectometer for continuous deflection testing of pavements. Ph.D. thesis, University of Texas at Austin, Austin, Tex. Bay, J.A., Stokoe, K.H., II, and Jackson, J.D Development and preliminary investigation of a rolling dynamic deflectometer. Transportation Research Record 173, pp Bay, J.A., Stokoe, K.H., II, and Hudson, W.R Continuous highway pavement deflection measurements using a rolling dynamic deflectometer. In Proceedings of the Conference on Nondestructive Evaluation of Bridges and Highways, Scottsdale, Ariz., 5 Dec Edited by S.B. Chase. Proceedings of SPIE Volume 96, pp Foinquinos, R., and Roesset, J.M Dynamic nondestructive testing of pavements. Geotechnical Engineering Report GR95-, University of Texas at Austin, Austin, Tex. Kausel, E., and Roesset, J.M Frequency domain analysis of undamped systems. ASCE Journal of Engineering Mechanics, 118(): Kerr, A.D On the determination of the rail support modulus k. International Journal of Solids and Structures, 37(3): Kerr, A.D., and Shenton, H.W Railroad track analyses and determination of parameters. ASCE Journal of Engineering Mechanics, 11(11): Kim, S.-M., and Roesset, J.M Dynamic response of pavement systems to moving loads. Geotechnical Engineering Report GR97-, University of Texas at Austin, Austin, Tex. Kim, S.-M., and Roesset, J.M Moving loads on a plate on elastic foundation. ASCE Journal of Engineering Mechanics, 1(9): Kim, S.-M., Roesset, J.M., and Stokoe, K.H., II Numerical simulation of rolling dynamic deflectometer tests. ASCE Journal of Transportation Engineering, 15(): Zaghloul, S.M., and White, T.D Use of a three-dimensional dynamic finite element program for analysis of flexible pavement. Transportation Research Record 1388, pp List of symbols c viscous damping coefficient d loaded length D frequency-independent linear hysteretic damping ratio E Young s modulus of elasticity of a beam f frequency of a moving harmonic load f cr critical frequency of a moving harmonic load f critical frequency of a stationary harmonic load I moment of inertia k stiffness of foundation per unit length m mass per unit length q external load per unit length Q transformed load t time V advance velocity of a load V critical velocity of a moving load of constant amplitude V cr1, V cr first and second critical velocities of a moving harmonic load x coordinate in x direction y displacement in y direction Y transformed displacement α variable part of a critical velocity formula η moving space η 0 η coordinate of the center of the load ξ transformed field of moving space Ω frequency or transformed field of time Ω load frequency 003 NRC Canada