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In the present study, finite element dynamic analysis or time history analysis of two-span beams subjected to asynchronous multi-support motions is carried out by using the moving support finite element. The elemental equation of the element is based on total displacements and is derived under the concept of the quasi-static displacement decomposition. The use of moving support element shows that the element is very simple and convenient to represent continuous beam moving, deforming and vibrating simultaneously due to support motions. The comparison between the numerical results and analytical solutions indicates that the FE result agrees with the analytical solution.

Long and slender structures are often excited dynamically through support motions rather than by applied external loadings, e.g., piers, chimneys, towers [

According to the quasi-static decomposition method, the transverse displacement of the beam subjected to support motions is composed of the quasi-static part and dynamic part. Statically determinate beams subjected to ground motions at supports are accompanied only by quasi-static displacement of rigid-body motion. Kim and Jhung [

For a beam in flexure shown in

The motion of a Rayleigh-damped Euler-Bernoulli beam with uniform cross-section is described by the following partial differential equation.

where a superimposed dot denotes a time derivative, L denotes length of the beam, and

For simplicity, we consider the dynamic response of two-span Rayleigh-damped Euler-Bernoulli beams subjected to multi-support excitation, which are shown in

where

The moving support elemental equation is given, from [

where

In Equations (9) and (10), the double prime denotes a twice spatial differentiation with respect to the element coordinates

where l is element length. In Equation (8),

where

where

Note that the underlined terms in right hand side of Equation (8) are peculiar to the moving support element and they contain the quasi-static displacement and velocity. The static components can be obtained exactly by static FE analysis, which will be considered in the next section.

According the quasi-static decomposition method, the solution can be decomposed into two parts:

where

where

and Equation (17) is subjected to the support conditions in Equations (4)-(6).

Let

The variables in Equations (18) and (19) will be called ‘quasi-static support variables’ or simply ‘support variables’ in this paper. The unknown support variables are determined by using the conventional static finite element method. The finite element equation is given by

For the hinged beam in

and the external moments at supports are

Using Equations (21) and (22), we obtain

where

Using the support variables in Equations (21), (22) and (23), we obtain the distribution of the static displacement

where

that the displacements

The two beams in

and analytical solutions will be made to check the validity of the moving support element for dynamic responses of the beams due to support motions.

The input data are as follows: D_{1} = D_{2} = 60 m, EI = 2.45 × 10^{9} N×m^{2}, and m = 2400 kg/m; α = 0.0844 s^{−}^{1} and β = 0.0141 s for the beam in ^{−1} and β = 0.0094 s for the beam in

The analytic series solutions for displacement, slope, acceleration, moment and shear force are obtained by eigenfunction expansion method with 10 modes. The numerical results such as displacement, velocity and acceleration at x = 30 m are compared with their analytical solutions in

FE dynamic analysis or time history analysis on the two-span Rayleigh-damped Bernoulli-Euler beams subjected to asynchronous support motions is carried out by using the moving support element. And the corresponding analytical solutions are obtained by using eigenfunction expansion method with 10 modes. The numerical results such as displacement, velocity, acceleration, slope, bending moment and shear force are compared

with the analytical ones to show that the moving support element describes moving, deforming and vibrating of multi-span beams subjected to support motions accurately. The numerical results agree with analytical solutions well.

Yong-WooKim,Seoung YealLee, (2015) FE Dynamic Analysis Using Moving Support Element on Multi-Span Beams Subjected to Support Motions. Modern Mechanical Engineering,05,112-121. doi: 10.4236/mme.2015.54011