# zbMATH — the first resource for mathematics

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads. (English) Zbl 07166850
Summary: Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.

##### MSC:
 74 Mechanics of deformable solids 70 Mechanics of particles and systems
Full Text:
##### References:
 [1] Timoshenko, S. P., On the forced vibrations of bridges, Lond. Edinb. Dublin Philos. Mag. J. Sci., 43, 1018-1019 (1922) [2] Fryd Ba, L., Dynamics of Railway Bridges (1996), Thomas Telford: Thomas Telford London [3] Xia, H.; Zhang, N.; Guo, W. W., Coupling Vibrations of Train-Bridge System (2014), Science Press: Science Press Beijing [4] Law, S. S.; Zhu, X. Q., Bridge dynamic responses due to road surface roughness and braking of vehicle, J. Sound Vib., 282, 3, 805-830 (2005) [6] Tabba, M. M.; Turkstra, C. J., Free vibrations of curved box girders, J. Sound Vib., 54, 4, 501-514 (1977) · Zbl 0365.73049 [7] Heins, C. P.; Sahin, M. A., Natural frequency of curved box girder bridges, J. Struct. Div. ASCE, 105, 12, 2591-2600 (1979) [8] Snyder, J. M.; Wilson, J. F., Free vibrations of continuous horizontally curved beams, J. Sound Vib., 157, 2, 345-355 (1992) · Zbl 0925.73308 [9] Genin, J.; Ting, E. C.; Vafa, Z., Curved bridge response due to a moving vehicle, J. Sound Vib., 65, 569-575 (1982) [10] Tan, C. P.; Shore, S., Response of horizontally curved bridge to moving load, J. Struct. Div. ASCE, 94, 2135-2151 (1968) [11] Wilson, J. F.; Wang, Y.; Threlfall, I., Responses of near-optimal, continuous horizontally curved beams to transit loads, J. Sound Vib., 222, 4, 565-578 (1999) · Zbl 1235.74201 [12] Yang, Y. B.; Wu, C. M.; Yau, J. D., Dynamic response of a horizontally curved beam subjected to vertical and horizontal moving loads, J. Sound Vib., 242, 3, 519-537 (2001) [13] Iwase, T.; Hirashima, K. I., Dynamic responses of multi-span curved beams with non-uniform section subjected to moving loads, Trans. JSME, 69, 12, 1723-1730 (2003) [14] Wu, J. S.; Chiang, L. K., Out-of-plane responses of a circular curved timoshenko beam due to a moving load, Int. J. Solids Struct., 40, 7425-7448 (2003) · Zbl 1061.74027 [15] Wang, Y. M., The transient dynamics of multiple accelerating/decelerating masses traveling on an initially curved beam, J. Sound Vib., 286, 1-2, 207-228 (2005) [16] Samaan, M.; Kennedy, J. B.; Sennah, K. M., Impact Factors for curved continuous composite multiple-box girder bridges, J. Bridge Eng., 12, 80-88 (2007) [17] Senthilvasan, J.; Brameld, G. H.; Thambiratnam, D. P., Bridge – vehicle interaction in curved box girder bridges, Comput. Aided Civ. Inf., 12, 3, 171-182 (1997) [18] Senthilvasan, J.; Thambiratnam, D. P.; Brameld, G. H., Dynamic response of a curved bridge under moving truck load, Eng. Struct., 24, 1283-1293 (2002) [19] Xia, H.; Guo, WW; Wu, X., Lateral dynamic interaction analysis of a train-girder-pier system, J. Sound Vib., 318, 927-942 (2008) [20] Dimitrakopoulos, E. G.; Zeng, Q., A three-dimensional dynamic analysis scheme for the interaction between trains and curved railway bridges, Comput. Struct., 149, 43-60 (2015) [21] Zeng, Q.; Yang, Y. B.; Dimitrakopoulos, E. G., Dynamic response of high speed vehicles and sustaining curved bridges under conditions of resonance, Eng. Struct., 114, 61-74 (2016) [22] Zhao, Y. Y.; Kang, H. J.; Feng, R., Advances of research on curved beams, Adv. Mech., 36, 2, 170-186 (2006) [23] Ghayesh, M. H.; Amabili, M., Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support, Mech, Mach. Theory, 67, 1-16 (2013) [24] Ghayesh, M. H.; Kazemirad, S.; Amabili, M., Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance, Mech. Mach. Theory, 52, 18-34 (2012) [25] Ghayesh, M. H.; Païdoussis, M. P., Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array, Int. J. Nonlin. Mech., 45, 5, 507-524 (2010) [26] Ghayesh, M. H.; Amabili, M., Nonlinear dynamics of axially moving viscoelastic beams over the buckled state, Comput. Struct., 112, 406-421 (2012) [27] Ghayesh, M. H., Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide, J. Sound Vib., 314, 3, 757-774 (2008) [28] Ghayesh, M. H., Stability and bifurcations of an axially moving beam with an intermediate spring support, Nonlin. Dyn., 69, 1, 193-210 (2012) [29] Ghayesh, M. H.; Païdoussis, M. P.; Amabili, M., Nonlinear dynamics of cantilevered extensible pipes conveying fluid, J. Sound Vib., 332, 24, 6405-6418 (2013) [30] Ghayesh, M. H., Subharmonic dynamics of an axially accelerating beam, Arch. Appl. Mech., 82, 9, 1169-1181 (2012) · Zbl 1293.74175 [31] Ghayesh, M. H.; Amabili, M.; Païdoussis, M. P., Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis, Nonlin. Dyn., 70, 1, 335-354 (2012) [32] Sahebkar, S. M.; Ghazavi, M. R.; Khadem, S. E., Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well, Mech. Mach. Theory, 46, 5, 743-760 (2011) · Zbl 1385.70042 [33] Fan, L. C., Bridge Engineering (2014), China Communication Press: China Communication Press Beijing [34] Vlasov, V. Z., Thin-walled Elastic Beams (1961), Office of Technical Services, U.S. Department of Commerce: Office of Technical Services, U.S. Department of Commerce Washington, 25, DC, TT-61-11400 [35] Fletcher, C. A.J., Computational Galerkin Methods (1984), Springer-Verlag: Springer-Verlag NewYork · Zbl 0533.65069 [36] Ding, H.; Chen, L. Q.; Yang, S. P., Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load, J. Sound Vib., 331, 2426-2442 (2012) [37] Yang, Y. B.; Yau, J. D.; Hsu, L. C., Vibration of simple beams due to trains moving at high speeds, Eng. Struct., 19, 936-944 (1997) [38] Taylor, A. E., L’Hospital’s rule, Amer. Math. Mon., 59, 20-24 (1952) · Zbl 0046.06202 [39] Zhai, W. M., Two simple fast integration methods for large-scale dynamic problems in engineering, Int. J. Numer. Meth. Eng., 39, 4199-4214 (1996) · Zbl 0884.73079 [40] Camara, A.; Ruiz-Teran, A. M., Multi-mode traffic-induced vibrations incomposite ladder-deck bridges under heavy moving vehicles, J. Sound Vib., 355, 264-283 (2015) [41] Abdel-Rohman, M., The influence of the higher order modes on the dynamic response of suspension bridges, J. Vib. Control, 18, 9, 1380-1405 (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.