By Andrew Y. T. Leung MSc, PhD, CEng, FRAeS (auth.)

**Dynamic Stiffness and Substructures** types a posh dynamic process and gives an answer to the complex dynamical challenge linked to the consequences of wind and earthquakes on buildings. because the method matrices are unavoidably frequency dependant, these are completely thought of during this booklet. The relation among the frequency matrices through the Leung's theorem is most crucial within the improvement of effective algorithms for the common modes. This new process was once constructed through the writer during the last 15 years. It bargains training engineers and researchers a large selection for structural modelling and research. considerable numerical examples allow the reader to appreciate the theory and to use the methods.

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Rx. 3. Beam Elements The beam element, as shown in Fig. 1, is the simplest bending element and one of the most widely used finite elements in structural dynamics. Beam-element matrices have been standardized and incorporated into most structural engineering computer packages. Research is still being carried out to reduce other complex 29 Beam Elements elements such as plates and shells into equivalent beams by certain energy criteria, and therefore special attention should be paid to the formulation and use of beam elements.

5) to find the dynamic stiffness matrix, one must first calculate the functions Ym(y) in Eq. 1). 7) per unit area Substituting Eq. 6) gives I N m=l mnx sin - { (mn)4 -- Ym a a - 2 (mn)2 Y';; a + Y~v Ny" --Y. Pm } =0 - -phW2 -Y. -N -x (mn)2 Y. +-Y. D m Dam D m D m Multiplication by sin(nnx/a), and integration from x = 0 to x = a, where n is a positive integer, and use of the orthogonality of sine functions, yields N } Y';; + {(mn)4 phw 2 - N Y~v. - 2 {(mn)2 ~ - 2~ ~ - ~ ; (mn)2 ~ - p} ; Ym m = 1, 2, .

5). This results in where = [(l e = [(1 h - S2p2)(X2 + b2s2]/(XI - S2p2)fJ2 - b2s 2]/fJI The remaining four constants, Cj , are determined by the boundary conditions, v(O) = VI' ~(O) = ~I' v(l) = v2 , ~(l) = ~2 With the displacement functions now given in terms of the end displacements, it is a simple matter to form the dynamic stiffness matrix through the following relationships: 1. M = EI~', the generalized stress-strain relationship for a Timoshenko beam. 2. V = M' + PV' - pI o~, the moment equilibrium of all forces acting on an element of length dx.