112
Fwij
Weiwei Guo, He Xia and Chaoyi Xia ⎧− FLyij − FRyij − N Ly ij − N Ry ij ⎫ ⎪ ⎪ z z z z y y y y + − + − + − + ( ) ( ) ( ) ( ) a F N a F N r F N r F N ⎪ ⎪ (54) L ij Ri R ij R ij L ij L ij L ij R ij R ij R ij = ⎨ L i Lxij ⎬ x x x z z y y y y + − + + + + + − + ( ) ( ) ψ [ ( ) ( )] a F N a F N M M a F N a F N L L L R R R L R w L L L R R R i ij ij i ij ij ij ij ij i ij ij i ij ij ⎪ ⎪ ⎪− FLzij − FRz ij − N Lz ij − N Rz ij ⎪ ⎩ ⎭
where, N and F are, respectively, the normal and tangent interaction forces between the jth wheel-set of the ith vehicle and the rail, with the subscripts L and R standing for the left and right wheels of the wheel-set, and the superscripts x, y, z denoting the coordinate axes in the fixed coordinate system, respectively. The relationship between the wheel and the rail in the local coordinate system is shown in Figure 5, where A and B are contact points of the left and right wheels with the corresponding rails; aLi and aRi are the horizontal projections of the distance from the ith left and right wheel-rail contact points to the centroid of the wheel-set, respectively. Let Nq denote the mode number concerned in the analysis of the finite element model of the bridge, the modal force vector of the bridge can be expressed as: Fb = [ Fb 1
Fb 2 " Fb Nq ]T
(55)
where, Fbn (n=1,2,…, Nq) is the generalized force of the nth mode, consisting of the generalized wheel-rail contact forces Fbvn and the generalized seismic forces Fbsn, transmitted by the influence matrix from the supporting parts of the bridge. They can be expressed as: Fb n = Fbs n + Fbv n
(56)
sg Fbs n = −Φ Tn M bb R bs X
(57)
where:
in which, ΦTn is the transposition of the nth mode vector for all non-supporting nodes of the bridge; Mbb the mass matrix of the corresponding nodes; and Rbs the pseudo-static displacement influence matrix of the bridge supporting nodes on the non-supporting nodes. The minus sign in the right hand side of this equation indicates that the force is in opposite direction to the ground acceleration. In practice this has little significance inasmuch as the sg , therefore, the minus sign is engineer is interested in the maximum absolute value of X omitted in the following descriptions. The horizontal and vertical seismic excitations are considered in the analysis. When multi-point excitations are considered, the following expression can be found: N b Ns
Fbs n = ∑∑ mbb i rbs ij (φnih X sgh j + φniv X sgv j ) i =1 j =1
(n = 1,2,", N q )
(58)