A semi-analytical method deals with vertical vibrations of a massive, flexible rectangular foundation smoothly jointed to an elastic, homogeneous halfspace is presented. The method is based on the solution in frequency domain that represents a combination of the Integral Transform Method (ITM) used for the soil and the Dynamic Stiffness Method (DSM) used for the foundation. The coupling of the foundation and the soil is established using the modal superposition technique. The influence of the foundation mass and the foundation stiffness on the response of the system is analyzed. The accuracy of the results obtained by using the proposed technique is also discussed. 2857 EURODYN 2020 XI International Conference on Structural Dynamics M. Papadrakakis, M. Fragiadakis, C. Papadimitriou (eds.) Athens, Greece, 23–26 November 2020 Available online at www.easdprocedia.org EASD Procedia EURODYN (2020) 2857-2867 ISSN:2311-9020 © 2020 The Authors. Published by EASD Procedia. Peer-review under responsibility of the Organizing Committee of EURODYN 2020. doi: 10.47964/1120.9234.19268 Marko Radišić, Mira Petronijević and Gerhard Müller

In the paper, an overview of the development of the dynamic-stiffness-method-based computational model for the free vibration analysis of plates has been presented.Starting from several formulations of the so-called dynamic stiffness elements, formulated at the Institute for Numerical Analysis and Design of Structures (INP) at the Faculty of Civil Engineering, University of Belgrade in the last decade, a novel software framework FREEVIB has been developed and validated.FREEVIB is object-oriented software in Python environment, designed to predict free vibration characteristics in a wide range of possible structural problems (stepped, stiffened and folded plate structures, implying isotropic or orthotropic material formulations).The presented methodology still serves as a strong basis for further improvements through the extensive research efforts of authors, their collaborators and students.

In this paper general equation system for linear dynamic soil-structure interaction (SSI) in frequency domain is presented. The main objective of the paper is to provide and investigate a possibility to use spectral elements in SSI domain. Spectral elements reduce considerably number of unknowns and in some cases, e.g. in frame structures coupled with analytically obtained impedance functions of sub-grade, produce no modelling errors. Results for two different shallow-founded beam structures with identical foundations excited by harmonic free-field motions are presented.

Contemporary structural design implies nonlinear behavior of ductile members for design seismic action. Therefore, the application of nonlinear analysis in the aseismic design of structures is required. Nonlinear static (pushover) analysis has become a very popular tool for the seismic assessment of structural performance during a particular earthquake due to the lower computational cost and less time consuming in comparison to the nonlinear time-history analysis. The standard pushover analysis (SPA) has been extended to the modal pushover analysis (MPA) of buildings in order to consider higher modes effects. Since the higher modes usually play an important role in the seismic bridge analysis, the MPA has been adopted for the seismic assessment of bridges. In the paper, the MPA of the Nišava Bridge structure (233.2m long 7-span continuous bridge, curved in plan with R=540m and prestressed 13.75m wide bridge deck) has been performed in transverse direction, for two levels of excitation which are 2 and 3 times higher than the design level (ag=0.1g). For the horizontal component of the seismic action, elastic response spectrum, Type 2 for soil type B, according to EN1998-1 has been selected. Analyses considering different levels of excitation and different monitoring points are carried out using the SAP2000 commercial software package. The seismic demands of the structure (peak displacements of the deck in transverse direction), subjected to the monotonically increasing lateral forces have been calculated, considering five dominant transverse modes. Hinge distribution within the structure has been determined, too, for the target displacement obtained from the MPA. The overall performance of the bridge was very satisfactory. Neither local nor global failure was predicted, even under seismic actions that three times exceed the design level. The performed analysis showed that the fundamental transverse mode shape contributes to the final response significantly. The influence of higher modes is more pronounced for higher level of excitation.

The dynamic response of rigid and flexible foundations on the soil has been subject of extensive study in the past decades. A hybrid method using a combined finite element method (FEM) and boundary element method (BEM) is the most common method used for solving this problem. The objective of this paper is to present an effective frequency domain method to obtain the dynamic response of a flexible strip foundation resting on a viscoelastic halfspace. The foundation is treated with the spectral element method (SEM), while the soil is modelled using the integral transform method (ITM). Both SEM and ITM are based on the analytical solution of the Lame-equations in the frequency domain and therefore are suitable for combining. The solution is obtained in the transformed space-frequency or wavenumber-frequency domain using the Fourier transformation. The study is performed as a 2D plane-strain analysis, assuming that the foundation cross-section behaves as an Euler-Bernoulli beam and that there is no sliding between the foundation and the soil, nor discontinuities in terms of the displacement field. The vertical displacements field of the foundation is described by a set of modal functions corresponding to free vibration mode shapes of a SEM Euler-Bernoulli beam element. The coupling between the foundation and the soil is achieved using the modal soil impedance functions, which are determined by using the ITM. The displacements of the coupled foundation-soil system are solved by the modal superposition method. The accuracy of the proposed method is assessed by comparing the obtained results with the results obtained by a commercial software package SASSI2000. The comparison shows that the presented method is accurate and less costly in terms of computational effort, especially in the high frequency range. The presented method can be easily extended to provide the solution of the response of a flexible strip on a layered halfspace due to a horizontal and vertical excitation.

In this paper the dynamic interaction of two rigid massless foundations resting on the finite depth soil medium layer.The vertical response of a loaded and unloaded foundation of a layer over a bedrock is calculated by Integral transform method (ITM).This method is based on analytical solution of the Lamé's differential equations of motion.A parametric analysis of vertical vibrations as a function of soil depth is carried out using a computer program developed in MATLAB.The obtained results are presented.

In this paper the dynamic stiffness method is used for free vibration analysis of a circular cylindrical shell. The dynamic stiffness matrix is formulated on the base of the exact solution for free vibration of a circular cylindrical shell according to the Flügge thin shell theory. The matrix is frequency dependent and, besides the stiffness, includes inertia and damping effects. The derived dynamic stiffness matrix is implemented in the code developed in a Matlab program for computing natural frequencies and mode shapes of a circular cylindrical shell. Several numerical examples are carried out. The obtained results are validated against the results obtained by using the commercial finite element program Abaqus as well as the available analytical solutions from the literature.

In this paper, two shear deformable dynamic stiffness elements for the free vibration analysis of rectangular, transversely isotropic, single-and multi-layer plates having arbitrary boundary conditions are presented.Dynamic stifness matrices are developed for the Reddy's higher-order shear deformation theory (HSDT) and the Mindlin-Reissner's first-order shear deformation theory (FSDT).The dynamic stiffness matrices contain both the stiffness and mass properties of the plate and can be assembled similarly as in the conventional finite element method.The influence of faceto-core thickness ratio and face-to-core module ratio of sandwich plate, as well as the influence of the shear deformation on the free vibration characteristics of sandwich plates have been analysed.The results obtained by proposed HSDT and FSDT dynamic stiffness element are validated against the results obtained using the conventional finite element analysis (ABAQUS), as well as the results obtained by 4-node layered rectangular finite element.The proposed model allows accurate prediction of free vibration response of rectangular layered plate assemblies with arbitrary boundary conditions.

Marko Radišić Mira Petronijević UDK: 624.131.382 DOI:10.14415/konferencijaGFS 2016.056 Summary: In this paper the dynamic interaction of two rigid massless foundations resting on the finite depth soil medium layer. The vertical response of a loaded and unloaded foundation of a layer over a bedrock is calculated by Integral transform method (ITM). This method is based on analytical solution of the Lamé‘s differential equations of motion. A parametric analysis of vertical vibrations as a function of soil depth is carried out using a computer program developed in MATLAB. The obtained results are presented.

This paper deals with the free vibration analysis of isotropic plate assemblies using the dynamic stiffness method (DSM) based on the Reddy's higher-order shear deformation theory (HSDT). Using the proposed method, the isotropic rectangular plate assemblies of non-uniform thickness and material properties can be analyzed. The proposed model does not have any restrictions regarding the boundary conditions or the frequency limitations. It enables free vibration analysis of both thin and thick plates, making it advantageous in comparison with the conventional finite element method (FEM) regarding the computational cost and the accuracy of the results.Three coupled Euler-Lagrange equations of motion based on the HSDT have been transformed into two uncoupled equations of motion introducing a boundary layer function. The dynamic stiffness matrix for a completely free rectangular plate element has been derived using the superposition and the projection method.The proposed numerical model has been applied in the free vibration analysis of rectangular plate assemblies. Along with the convergence study, the results for natural frequencies have been validated against the existing data from the literature, the previous results from the authors as well as the results obtained by using the finite element software Abaqus. Excellent agreement has been obtained. Finally, a variety of new results is provided as a benchmark for future investigations. (C) 2015 Elsevier Ltd. All rights reserved.

A procedure for developing the dynamic stiffness matrix of a completely free laminated composite plate based on the first-order (FSDT) and higher-order shear deformation theory (HSDT) is presented. The proposed method allows the computational analysis of free transverse vibrations of the individual rectangular laminated composite plates, as well as the composite plate assemblies, without any restrictions regarding the boundary conditions or frequency limitations. The general solution of the governing differential equations of the HSDT and FSDT is established using the superposition method. Continuous boundary conditions are discretized by using the projection method. The dynamic stiffness matrices of plate elements are then formulated from the assembly of the four dynamic stiffness matrices (four symmetry contributions). The validation of the theory and its application are provided in the Part II of this two-part paper.

In this paper, the dynamic stiffness matrix of a completely free rectangular multi-layer plate element based on Reddy's higher-order shear deformation theory is derived. The reduction of the proposed model to the first-order shear deformation theory-based formulation is presented. Three coupled Euler-Lagrange equations of motion have been transformed into two uncoupled equations introducing a boundary layer function. The proposed model enables free transverse vibration analysis of rectangular multi-layer plates with (transversely) isotropic layers having arbitrary combinations of boundary conditions.The influence of transverse shear deformation is discussed along with the applicability of two shear deformable dynamic stiffness elements. Moreover, the influence of the boundary conditions on the free vibration characteristics of sandwich panels has been discussed. The natural frequencies obtained using different dynamic stiffness multi-layer plate elements have been validated against the solutions from the commercial software Abaqus and the previously verified numerical solutions using layered finite elements. The limitations of the model regarding the differences between material properties of the face and core layers within a sandwich plate are highlighted. The influence of face-to-core thickness ratio on natural frequencies is illustrated, while a variety of new results is provided as a benchmark for future investigations. (C) 2016 Elsevier Ltd. All rights reserved.

In this paper, a frequency dependent impedance matrix for four adjacent rigid foundations resting on the viscoelastic halfspace is presented. The frequency dependent impedance matrix is developed using the integral transform method. Non-relaxed boundary conditions, without separation or sliding, between the soil and foundation are assumed. First, the compliances for one rigid foundation on the half space are calculated and compared with the results from the literature. Than the impedance matrix of a single foundation as well as the impedance matrix of four rigid foundations are obtained. The influence between adjacent rigid surface foundations resting on the half space is presented.

Using the spectral element method (SEM), a 2-D numerical model of multi-storey frame structures has been developed. The model has been used to predict traffic-induced vibrations of two, six, and twelve-story concrete buildings measured along the route of a future metro line in Belgrade, the capital of Serbia. Vibration simulations of the six-storey frame match satisfactorily the measured dynamic response. However, there is a difference between numerical simulation and the actual measurements for the two-storey and twelve-story buildings. The results indicate a great potential of the proposed SEM to simulate ground-induced vibrations of frame buildings. However, more complex structural models should be developed to better replicate actual situations.

A simple numerical model for the dynamic analysis of 20 frames in the frequency domain, based on the Spectral Element Method (SEM), is presented in the paper. The influence of soil-structure interaction is taken into account. The dynamic stiffness of rigid foundations is determined by the Integral Transform Method (ITM). Three frames with a different number of storeys are analyzed with respect to vibrations caused by tram and road traffic. The influence of the soil-structure interaction on modal frequencies and amplitudes of vibrations is considered. The assessment of the way in which humans are affected by traffic-induced vibrations is conducted according to British Standard BS:6472.