Keywords: dynamics, analysis, bridges, railway, double-track, train loads, steel, code.

In design practice of bridges, to allow for the effect of dynamic action, the static live load is increased by a dynamic allowance factor (impact factor) given in bridge codes. The specified value of the impact factor is usually calculated using empirical formulas based on the length of the bridge. Also, the codes limit the maximum allowable deflection of the bridge, to avoid excessive deformation stresses, fatigue effects due to excessive vibration, and undesirable psychological reactions by pedestrians or passengers in vehicles and trains. The increasing of the static live load by an empirical impact factor depends only on the bridge loaded length is an approximate procedure, because there are several factors affecting the dynamic behaviour of the bridge. Among these factors are the speed of the moving vehicles or trains, their moving accelerations, the span length and the material characteristics of the considered bridges.

The present work deals with studying and understanding the dynamic behaviour of double-track railway bridges when traversed simultaneously by two moving trains. The finite element approach and the application of a mode superposition technique are used to achieve the desired dynamic analysis. A train loads according to both Egyptian and American codes are modeled as a series of concentrated moving loads and used in the analysis. Cases of simply supported steel girder bridges are tested and analyzed. Two moving trains with constant and varying speeds traversing the tested bridges simultaneously in identical and opposite directions are studied. The free vibration characteristics for each of the tested bridges are determined.

Factors affected the dynamic response of the double-track railway bridges are studied. Among these factors are the trains' speeds, the span of the bridge, the train acceleration and the damping coefficient of the bridge. The speeds considered in the analysis are 40, 80, 120 and 160 km/hr. The considered accelerations are 2, 5, 8 m/sec. The values 0.0, 0.05 and 0.10 for the damping coefficient are used in the analysis.

A comparison between the accurate values calculated for the dynamic effect factors of the investigated cases and the corresponding impact factors given by both the Egyptian and the American specifications of railway bridges has been made. Finally, useful recommendations for the designer engineer concerning the dynamic effect on railway bridges are presented.

The major conclusions of the present study can be summarized as follows:

• The dynamic effect factor for the Egyptian railway loads is lower than the American one. Also the impact empirical formula given in the Egyptian code gives lower values than those given by the American code.
• The dynamic effect factor calculated for moments and deflections for trains moving in opposite directions is the same as that of trains moving in the same direction.
• The dynamic effect factors calculated for deflections are slightly lower than those for moments for all the studied cases.
• The dynamic effect decreases with the increase of the span length. This fact is valid for both the American and the Egyptian codes.
• The actual dynamic factor, is very small for train speeds lower than 80 km/hr, for speed ranges 120 to 140 km/hr it is approximately equal to the impact factor given in codes. For higher speeds, the actual dynamic factor is considerably greater than the impact factor prescribed in both the American and Egyptian codes.
• The maximum dynamic responses are increased with the increase of trains acceleration.
• The dynamic factor is slightly decreased with the increase of the damping ratio.

1

I.I. Ishac, M.K. Swailem, "Dynamic Behaviour of Steel Simple Span Railway Bridges Traveled by Trains", Mansoura Engineering Journal, (MEJ), Faculty of Engineering, Mansoura Univ., Egypt, Vol. 25, No. 1, March, 2000.

2

E.H. Gaylord, C.N. Gaylord, "Structural Engineering Handbook", Second Edition, McGraw-Hill Book Company, 1979.

3

"Egyptian Code of Practice for Steel Constructions and Bridges", Ministerial Decree, No. 451, 1989.

4

"Egyptian Code of Practice for Loads Calculations on Structural Works and Buildings", Ministerial Decree, No. 45, 1993.

5

Y.B. Yang, B.H. Lin, "Vehicle-Bridge Interaction Analysis by Dynamic Condensation Method", J. Struct. Eng., ASCE, Vol. 121, No. 11, Nov., 1995. doi:10.1061/(ASCE)0733-9445(1995)121:11(1636)

6

T.L. Wang, D. Huang, M. Shahawy, "Dynamic Response of Multigirder Bridges", J. Struct. Eng., ASCE, Vol. 118, No. 8, August, 1992. doi:10.1061/(ASCE)0733-9445(1992)118:8(2222)

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