Resonance and cancellation phenomena caused by equidistant moving loadings in a periodic structure — A pile-supported periodic viaduct

doi:10.1016/j.euromechsol.2016.03.010

European Journal of Mechanics - A/Solids

Volume 59, September–October 2016, Pages 114-123

https://doi.org/10.1016/j.euromechsol.2016.03.010 Get rights and content

Highlights

•
Resonance and cancellation phenomena in the periodic viaduct subjected to multiple equidistant moving loadings (MEML) are found for the first time.
•
The conditions for the occurrence of the resonance and cancellation phenomena in the periodic viaduct subjected to MEML are proposed.
•
Numerical simulations to demonstrate the resonance and cancellation phenomena occurring in the periodic viaduct are conducted.

Abstract

In this paper, the resonance and cancellation phenomena occurring in a periodic structure-- a pile-supported periodic viaduct subjected to multiple equidistant moving loadings (MEML) are investigated. For this purpose, the representation for the dynamic response of the periodic viaduct to a single moving loading (SML) is introduced first. In order to take account of the pile-soil-structure interaction when evaluating the response of the periodic viaduct to a SML, a wavenumber domain boundary element method (WDBEM) model for the pile-row-soil system is employed to determine the compliances of the piles. By superposing the responses of the viaduct to all SMLs constituting the MEML, the representation for the dynamic response of the periodic viaduct to the MEML is obtained. With the representation for the MEML response of the viaduct, the resonance and cancellation conditions for the viaduct are derived. For the high speed MEML, when the time interval between two neighboring moving loadings is equal to the integer or half-integer multiples of the period of one of the dominant resonance peaks of the corresponding SML response of the viaduct, resonance or cancellation occurs. The predicted resonance and cancellation phenomena are confirmed by the numerical results presented in this paper.

Introduction

Viaduct railways are now widely used in the world because they can resolve the settlement associated with the soft soil and frozen soil effectively. For example, to deal with the settlement of the soft soil, more than ninety percent of the Guangzhou-Zhuhai railway is composed of viaducts. For convenience of construction, viaducts in normal sections are usually designed such that they are composed of identical spans and they can thus be simplified as periodic structures. Hence, the investigation of the dynamic response of the periodic viaduct to the moving train is important for the design and maintenance of the viaduct. As a moving train can be usually simplified as multiple moving loadings, the research about the dynamic response of the periodic viaduct to multiple moving loadings is thus pivotal for understanding the dynamic response of the periodic viaduct to a moving train.

The most important difference between the response of the bridge to a moving loading and that to multiple moving loadings is that resonance and cancellation phenomena may occur in the bridge when it is subjected to multiple moving loadings, which are crucial for the design and maintenance of the bridge. Thus, resonance and cancellation responses caused by trains moving over bridges has become a hot topic for a long time (Yang et al., 1997, Yang et al., 2004, Yau, 2001, Museros et al., 2013). In the aforementioned studies, the bridge is simplified as a beam with a single span or several spans. For the beams consisting of one or several spans, their discrete natural frequencies and modes can be obtained using various analytical and numerical methods. Hence, their resonance and cancellation conditions can be obtained accordingly by modal superposition method (Yang et al., 1997). However, for the infinite periodic viaduct considered in this contribution, there are no discrete natural frequencies as well as modes. In contrast, the free vibration of the periodic viaduct is determined by the characteristic waves of the structure satisfying specific continuous dispersion curves. Therefore, the above-mentioned method for determining the resonance and cancellation conditions for finite-span bridges is not applicable to the infinite periodic viaduct considered in this study. It is noted that in order to understand the vibration of a railroad track caused by a moving train, many researchers have analyzed the dynamic response of the infinite periodically supported beam to a moving load (Belotserkovskiy, 1996, Metrikine and Popp, 1999, Sheng et al., 2005). However, due to the aforementioned difficulty, the resonance and cancellation phenomena in the periodically supported beam caused by multiple moving loadings are not addressed in the above researches.

Field evidence has shown that pile-soil-structure interaction has a remarkable influence on the dynamic behavior of the structure, especially for the structure constructed on the soft soil (Safak, 1995, Lou et al., 2011). Hence, when investigating the dynamic response of a pile-supported periodic viaduct to moving loadings, the coupling between the soil, pile foundations and superstructure should be taken into account. To date, a few researches have been conducted to investigate the train-induced vibration in bridges with consideration of the pile-soil-structure interaction. Ju and Lin (2003) studied the resonant vibration characteristics of the 3D vehicle-bridge system when high-speed trains pass through. However, in his model, the piles and the half-space soil are simplified as the mass and spring system. Based on the semi-analytical approach, Wu and Yang, 2004, Wu and Yang, 2009 analyzed the vibration of the superstructure and ground induced by a train moving on a multi-span elevated bridge. By using the finite element method and in-situ measurements, Takemiya and Bian (2007) investigated viaduct vibration and nearby train-induced vibration with the consideration of the soil-foundation-viaduct interaction. It is worth noting that in the researches by Ju and Lin, 2003, Wu and Yang, 2004, Wu and Yang, 2009 and Takemiya and Bian (2007), only viaducts with single span or several spans are considered and moreover, the influence of the pile-soil-structure interaction on the dynamic response of the viaduct is only accounted for approximately. Therefore, it is necessary to develop a model to deal with investigate the resonance and cancellation phenomena occurring in a periodic viaduct with a full consideration of the pile-soil-structure interaction.

In this paper, the resonance and cancellation phenomena in a pile-supported periodic viaduct subjected to multiple equidistant moving loadings (MEML) are investigated. To this end, the representation of the dynamic response of the periodic viaduct to a single moving loading (SML) is introduced first. In order to take account of the pile-soil-structure interaction in the analysis of the SML response of the viaduct, a wavenumber domain boundary element method (WDBEM) model for the periodic pile row is introduced to determine the compliances of the pile row, by means of which the superstructure of the viaduct and the pile foundations are coupled. By superposing the responses of the viaduct to all SMLs constituting the MEML, the dynamic response of the periodic viaduct to the MEML is obtained. The representation of the frequency domain MEML response of the periodic viaduct indicates that when the speed and distance between the neighboring moving loadings of the MEML are chosen such that the time interval between two neighboring moving loadings is equal to the integer multiples of the period of one of the dominant resonance peaks of the corresponding SML response of the viaduct, resonance will occur. On the contrary, when the time interval between two neighboring moving loadings is equal to the half-integer multiples of the period of one of the dominant resonance peaks of the corresponding SML response, cancellation will occur. Numerical results presented in this paper confirm the predicted resonance and cancellation phenomena.

The remainder of this manuscript is organized as follows. Section 2 outlines the Fourier and sequence Fourier transforms. In Section 3, the representation for the dynamic response of the periodic viaduct to a SML is introduced. In Section 4, the representation for the dynamic response of the periodic viaduct to MEML is derived and the resonance and cancellation conditions for the periodic viaduct subjected to the MEML are proposed. Numerical results and corresponding analysis are presented in Section 5 and conclusions are summarized in Section 6.

Section snippets

Fourier and sequence Fourier transforms and their relations

As this study involves both the Fourier and sequence Fourier transforms as well as the relation between the two transforms, the definitions for the Fourier and sequence Fourier transforms are introduced in this section. Also, the relation between the two transforms is discussed.

Fourier transforms with respect to the time and the space coordinate are defined as follows (Oppenheim et al., 1983) $\begin{array}{l} \bar{f} (ω) = \int_{- \infty}^{+ \infty} f (t) e^{- i ω t} d t, f (t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \bar{f} (ω) e^{i ω t} d ω, \\ \hat{f} (k) = \int_{- \infty}^{+ \infty} f (x) e^{i k x} d x, f (x) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \hat{f} (k) e^{- i k x} d k \end{array}$ where t

Dynamic response of the pile-supported periodic viaduct to a single moving loading (SML)

As mentioned above, the response of the periodic viaduct to MEML can be determined once the response of the periodic viaduct to a SML is obtained. Hence, the response of the periodic viaduct to a SML is addressed in this section.

Resonance and cancellation conditions of a pile-supported periodic viaduct subjected to MEML

As the periodic viaduct considered here is assumed to be a plane structure which is geometrically and materially linear, the dynamic response of the periodic viaduct produced by each loading can thus be superposed to obtain the total response of the viaduct to the MEML. Suppose that multiple equidistant loadings are moving over the periodic viaduct at a constant speed v and the number of the single loadings constituting the MEML is N_L. The distance between the neighboring loadings of the MEML

Numerical results and corresponding analysis

In this section, based on the proposed model, the resonance and cancellation phenomena of a pile-supported periodic viaduct subjected to MEML will be investigated numerically. Both the frequency and time domain responses of the periodic viaduct to a SML and MEML are given. The material and geometric parameters of the piles and soil are listed in Table 1, and those of the superstructure of the viaduct are given by Table 2. Table 3 shows the stiffnesses of the beam–beam spring, left and right

Conclusions

This paper investigated the resonance and cancellation phenomena occurring in the pile-supported periodic viaduct subjected to multiple equidistant moving loadings. To this end, the representation for the dynamic response of the periodic viaduct to a SML is introduced first. In order to account for the pile-soil-structure coupling when evaluating the dynamic response of the viaduct to moving loadings, a coupled WDBEM model for the pile-row-soil system is employed to determine the compliances of

Acknowledgements

Financial support received from the National Natural Science Foundations of China (No. 11272137) is acknowledged by the authors. Also, constructive comments from the anonymous referees during the first and second reviews of our paper are highly appreciated by the authors.

References (25)

P.M. Belotserkovskiy
On the oscillations of infinite periodic beams subjected to a moving concentrated force
J. Sound. Vibr.
(1996)
H. Chebli et al.
3D periodic BE-FE model for various transportation structures interactin with soil
Comput. Geotech.
(2008)
H. Chebli et al.
Response of periodic structures due to moving loads
C. R. Mec.
(2006)
S.H. Ju et al.
Resonance characteristics of high-speed trains passing simply supported bridges
J. Sound. Vibr.
(2003)
M. Lou et al.
Structure-soil-structure interaction: literature review
Soil. Dyn. Earthq. Eng.
(2011)
A.V. Metrikine et al.
Vibration of a periodically supported beam on an elastic half-space
Eur. J. Mech. A/Solids
(1999)
P. Museros et al.
Free vibraions of simply-supported beam bridges under moving loads: maximum resonance, cancellation and resonant vertical acceleration
J. Sound. Vibr.
(2013)
X. Sheng et al.
Responses of infinite periodic structures to moving or stationary harmonic loads
J. Sound. Vibr.
(2005)
H. Takemiya et al.
Shinkansen high-speed train induced ground vibrations in view of viaduct-ground interaction
Soil. Dyn. Earthq. Eng.
(2007)
Y.S. Wu et al.
A semi-analytical approach for analyzing ground vibrations caused by trains moving over elevated bridges
Soil Dyn. Earthq. Eng.
(2004)

Y.B. Yang et al.

Vibration of simple beams due to trains moving at high speeds

Eng. Struct.

(1997)

Y.B. Yang et al.

Mechanism of resonance and cancellation for train-induced vibrations on bridges with elastic bearings

J. Sound. Vibr.

(2004)

Cited by (14)

Internal instability of thin-walled beams under harmonic moving loads
2022, Thin-Walled Structures
Citation Excerpt :
Each of the above two types of motions consists of a series of modal components, and each can be set in resonance if any of the internal modal frequencies coincides with any of the external driving frequencies imported by the moving loads, which is also known as instability of the divergence type [23,24]. Previously, there exists a number of studies for the resonance of bridges due to moving loads [25–33]. A special but very important case of resonance occurs when any of the vertical-flexural frequencies of the bridge coincides with any of the torsional–flexuralfrequencies; both are known as internal frequencies.
Frequent resonance may result in excessive vibrations and endanger the safety of bridges. Firstly, the governing equations of motion are derived for the vertical, lateral and torsional vibrations of a mono-symmetric thin-walled box beam under a harmonic moving load. Then the resonance conditions are derived by letting the denominator of the response of concern equal to zero. For the mono-symmetric beam, the vertical resonance is uncoupled, but the torsional and lateral resonances are coupled. When in vertical resonance, the vertical motion of the beam will diverge by growing to a maximum during the acting period of the moving load. Similar phenomenon exists for the torsional–flexural resonance. A specific case occurs when the above two resonance frequencies coincide, for which the critical length of the beam can be determined. The phenomenon of simultaneous resonance is a kind of internal instability featured by the fact that the beam can transform repetitively from the vertical mode to the torsional–flexural mode, and vice versa, with no additional energy input. To avoid the inherent internal instability, a beam should be designed with a length not equal or close to the critical length presented in this paper. All the aforementioned resonances have been extensively studied for cases involving both harmonic and random moving loads and validated by the finite element analysis.
A moving load amplitude spectrum for analyzing the resonance and vibration cancellation of simply supported bridges under moving loads
2022, European Journal of Mechanics, A/Solids
Citation Excerpt :
With the rapid development of the high-speed railway, the requirements for high-speed railway bridges are getting higher and higher. Therefore, the researches on high-speed railway bridges are significant, especially the resonance and cancellation vibration of high-speed railway bridges under high-speed trains (Li and Su, 1999; Yang et al., 2004; Liu et al., 2009; Lu et al., 2016). Previous work on the resonance and cancellation problem dates back to the pioneering work by Yang et al. (1997).
A novel method for analyzing the resonance and vibration cancellation of simply supported bridges in the frequency domain by using moving load amplitude spectrum (MLAS) is presented in this paper. Firstly, the Fourier transform for the motion equation of simply supported bridge under equidistant moving loads is carried out to derive the bridge's moving load spectrum and displacement response spectrum. The MLAS, which can be obtained by taking the modulus of the moving load spectrum, is related to the moving loads and the vibration modes of the bridge. Surprisingly, the MLAS can effectively reflect the displacement amplitude laws of the bridge free vibration excited by the moving loads. Then, the resonance and vibration cancellation of the simply supported bridge under equidistant moving loads are analyzed respectively based on the proposed MLAS. It is noted that the corresponding resonance speed obtained by the proposed MLAS is a supplement to the traditional resonance speed which sometimes doesn't cause the maximum displacement response of simply supported bridges. What's more, the vibration cancellation of the simply supported bridge under equidistant moving loads is divided into two types in this paper.The corresponding cancellation speeds v_c1-can and v_c2-can are determined simply by the proposed MLAS in the frequency domain. The time-domain analysis in many references only gives the cancellation speeds v_c1-can but not the cancellation speeds v_c2-can presented in this paper. Therefore, the cancellation speeds v_c1-can and v_c2-can presented in this paper can provide a more comprehensive understanding of the influence factors of vibration cancellation. Finally, the relevant theoretical researches on the proposed MLAS are validated with several examples by comparing the results of the time-domain analysis.
Investigation of axle-span ratio and moving load speed affecting bridge extreme response using a moving load amplitude spectrum method
2020, Structures
Citation Excerpt :
The research scope also extends from the vertical vibration of bridges to the lateral vibration and torsional vibration of bridges [4], even the bridge vibration caused by trainload considering the influence of various environmental factors [5–7]. Since the beginning of the new century, with the massive construction of China's high-speed railways and the upgrading of railway systems, the vibration response of bridges caused by high-speed trains has become increasingly prominent, especially about the problem of bridge resonance response caused by trainload [8,9]. For the study of bridge resonance response under the trainload, scholars have adopted different analysis methods.
In this paper, firstly the moving load amplitude spectrum is established by the Fourier transform for the two-axle moving loads, an equivalent model of the moving vehicle, acting on a simply supported beam bridge. The established spectrum is very useful for the rapid and effective analysis of the bridge free vibration response due to the excitation of moving vehicle. Then, in order to analyze the extreme response of the bridge free vibration, an efficient method is presented based on the moving load amplitude spectrum. For the moving load amplitude spectrum, the axle-span ratio (the ratio of the load wheelbase to the bridge span) and dimensionless speed are two important factors. Corresponding to the maximum value of the moving load amplitude spectrum, the relationship between the dimensionless speed and the axle-span ratio is found by the combination of theoretical derivation and numerical analysis. Further, according to the relationship, a speed formula corresponding to the maximum displacement response of bridge free vibration under moving loads is proposed, and obviously different from the resonance speeds. Finally, taking two simply supported girder bridges with different spans as examples, the results of the frequency-domain analysis are verified by a large number of the time-domain calculations.
Effect of soil properties on the dynamic response of simply-supported bridges under railway traffic through coupled boundary element-finite element analyses
2018, Engineering Structures
Citation Excerpt :
Only a few authors have investigated the dynamic response of beams or bridges and, in particular, the conditions of resonance and cancellation phenomena taking into account the wave propagation in the soil. Lu et al. [19] prove numerically the occurrence of resonance and cancellation in a periodic viaduct subject to moving loads considering pile-soil-structure interaction. Wu and Yang [20] apply a semi-analytical approach to analyse ground vibrations induced by trains moving over elevated bridges.
Railway induced vibrations on short-to-medium span simply-supported (SS) bridges is addressed in this contribution. Such structures may experience high levels of vertical acceleration at the platform, leading to adverse consequences such as a premature degradation of the ballast layer and passenger discomfort. In the present study, the evolution of the bridge dynamic response when soil-structure interaction (SSI) is taken into account is investigated. To this end a coupled three-dimensional (3D) Boundary Element-Finite Element model (BEM-FEM) formulated in the time domain is implemented to reproduce the soil and structural behaviour, respectively. First, a set of soil-bridge systems of interest is defined, covering a wide range of lengths and natural frequencies for the structures, and an interval of expectable elastic properties and damping levels for the soil. Then, different types of analyses are performed on the soil-bridge systems extracting conclusions regarding the effect of including SSI in numerical models for predicting the bridge behaviour under railway traffic. In particular natural frequencies and modal damping levels are identified, and the structure amplification after the passage of a moving load in free vibration is investigated. Conclusions regarding how resonance and cancellation conditions may be affected by soil properties are extracted. Finally, the dynamic response of a real bridge, belonging to the Spanish railway network, is evaluated under the circulation of trains that induce second and third resonances of the bridge fundamental mode. The effect of the soil flexibility, soil material damping and the bridge resonance order are evaluated. Conclusions regarding the appropriateness of the results provided by common models which do not include SSI effects are extracted.
Wave dispersion analysis of multi-story frame building structures using the periodic structure theory
2018, Soil Dynamics and Earthquake Engineering
Citation Excerpt :
Introducing the so-called ‘open’-type periodic unit, Lu and Jeng [37] conducted a numerical study to analyze the dispersion property of a viaduct possessing periodic characteristics undergoing out-of-plane vibration. In another contribution, considering the theoretical infinite periodic viaduct mode, the resonance and cancellation phenomena occurring in the pile-supported periodic viaduct subjected to multiple equidistant moving loads were investigated [38]. Proposing a second-order computational homogenization method, Bacigalupo and Gambarotta [39,40] derived the overall constitutive moduli and inertia properties, and evaluated the influences of the materials characteristic lengths on the structural response and the propagation of elastic waves in the periodic masonry structure.
Introducing the periodic structure theory in the field of solid-state-physics into the field of civil engineering, this paper conducts a comprehensive theoretical and numerical study on the wave dispersion properties of multi-story frame building structures. At first, the theoretical infinite periodic model for the multi-story frame building structure is developed. Combining the Bloch-Floquet theorem and the State-Space-Transfer-Matrix-Method (SSTMM), the dispersion equation of the infinite periodic system is solved, from which dispersion curves and group/phase velocity curves of a practical multi-story reinforced concrete frame building structure are investigated. Second, a parametric study is performed to investigate the influences of the typical material parameters (density, strength grade of concrete), geometrical parameters (span length, floor height), theoretical simplified parameters as well as the damping parameter of frame building structures on the dispersion relations. Then, harmonic analysis and time history analysis are conducted to analyze the dynamic properties of the finite periodic frame structure in the frequency domain. Numerical results show that, when the main frequency region of the external excitations (like seismic waves) falls into the passing bands, dynamic responses of the multi-story frame building structure are very large; while the main frequency region of the external excitations falls into the attenuation zones, seismic responses of the multi-story frame building structure are very small.
A dynamic model for the response of a periodic viaduct under a moving mass
2019, European Journal of Mechanics, A/Solids
Citation Excerpt :
By using the closed form solution developed in Chebli et al. (2006), Chebli et al. (2008) investigated the response of the periodic track-soil system to the high-speed train by combining the finite element method with the boundary element method. Based on the Fourier transform and sequence Fourier transform methods, Lu et al., 2015, 2016 first established a model for a periodic viaduct subjected to a moving load or a series of equidistant moving loads. It is worth stressing that the lack of a model to treat the problem of a periodic structure subjected a moving mass without a doubt restricts the further scientific design for the high speed railway viaduct.
In this study, the dynamic response of a periodic viaduct to a moving mass is investigated. In view of the periodicity of the viaduct, the mass-viaduct interaction force is expanded into a Fourier series first, each term of which represents one component of the interaction force. By using the Fourier transform method and finite element method (FEM), the frequency domain response of the periodic viaduct to each interaction force component is obtained. The time domain response of the periodic viaduct to the force component can be recovered by applying the inverse Fourier transform to the corresponding frequency domain response. With the mass-viaduct coupling condition and the time domain response of the viaduct to the force component, the Fourier coefficients of the mass-viaduct interaction force are obtained. Superposing the responses of the periodic viaduct due to all the interaction force components yields the response of the periodic viaduct to the moving mass. Numerical results show that the force-weight ratio increases with increasing mass and speed of the mass, suggesting that the inertial effect of the moving mass should be taken into account for the large mass with high speeds. Also, it is found that for the case of the large moving mass with high speeds, the mass-viaduct interaction force may become negative, indicating that the mass has a tendency to separate from the viaduct in this case.

View all citing articles on Scopus

View full text

Resonance and cancellation phenomena caused by equidistant moving loadings in a periodic structure — A pile-supported periodic viaduct

Highlights

Abstract

Introduction

Section snippets

Fourier and sequence Fourier transforms and their relations

Dynamic response of the pile-supported periodic viaduct to a single moving loading (SML)

Resonance and cancellation conditions of a pile-supported periodic viaduct subjected to MEML

Numerical results and corresponding analysis

Conclusions

Acknowledgements

J. Sound. Vibr.

Comput. Geotech.

C. R. Mec.

J. Sound. Vibr.

Soil. Dyn. Earthq. Eng.

Eur. J. Mech. A/Solids

J. Sound. Vibr.

J. Sound. Vibr.

Soil. Dyn. Earthq. Eng.

Soil Dyn. Earthq. Eng.

Eng. Struct.

J. Sound. Vibr.