# Influence of Multiple-Support Excitation on Seismic Response of Reinforced Concrete Arch Bridges
## 2 Prototype Bridges
### Numerical Modelling Strategies  
Two-dimensional beam finite-element models were created for all bridge prototypes, with precise geometry and appropriate spatial and time discretization of the problem, essential for calculation of high-frequency vibration modes [38].  
Influence of geometric non-linearity was investigated in this research. This effect takes into account increase of stiffness for elements subjected to a tension force and reduction of stiffness for elements subjected to a compression force (\"softening\" of structural elements), such as the arch. In the first case, numerical models for dynamic analysis were determined in the classical way, not taking into account the change in the stiffness matrix due to dead loads action. In the second case, the analysis was performed using the stiffness matrix inherited from geometric non-linear staged construction design. The results from the two cases were compared (dynamical properties, internal forces and displacements for the vertical load and for one case of multiple-support excitation (MSE)) and the differences were less than 10%. Hence, this effect was excluded in further analyses to MSE.  
Material nonlinearity of the bridge cross sections was not accounted for, as it was assumed that all load-bearing bridge elements respond in the elastic range. Arch response is mostly non-ductile due to high thrust [39]. The energy loss in elastic range was simulated by Rayleigh damping wherein the damping of the first mode and the mode for which the modal participating mass ratio exceeds 90% is assumed with the value of 5%. The Ritz vectors determined from inertia loads (mass for unit acceleration) along two directions were used instead of classic eigenvectors. This avoids the negative property of eigen-value problem where eigen-modes are generated independent of the load type, so many eigen-vectors contribute very little or nothing to total active mass of the structure. The latter means that they are orthogonal or almost orthogonal to inertia loads. Therefore, an approach to avoid this phenomenon in numerical models is the application of Ritz-vectors, determined from inertia forces acting on the structure. In other words, displacements from these loadings represent the starting vector for the iterative procedure for calculating appropriate Ritz vibration vectors. Fifty Ritz vectors were determined in all numerical models, to obtain static and dynamic modal load participating coefficients of almost 100% in translational directions. For comparison, more than double of classical eigen-vectors would have to be determined to obtain approximately the same values of static and dynamic modal load contributions. If the loads are proportional to mass of the structure, contribution coefficients are in accordance with the total contribution coefficient of modal masses. It should be noted, though, that for many Ritz vectors a very precise numerical model is mandatory, to minimize the discretization error in determining high oscillation modes (with many zero points), necessary for analysis of structures to MSE, to acceptable level. The algorithm for the generation of Ritz vectors may be found in [38].  
As bridges are founded on rock, supports were modeled accordingly as springs of large stiffness. Abutments were not considered in numerical models.  
Diaphragms placed inside the arch and those near abutments are not explicitly modelled, but their weight is taken into account. Additional dead load acting on the superstructure includes all bridge equipment and pavement layers.  
All numerical models were created and analyzed in Sap2000 software [40], while the benchmark examples of the bridge of 100 m span were analyzed in Wolfram Mathematica [41] and Seismostruct [42]. One of the benchmark examples was presented in [12].