Dynamic Analysis Of Framed Structures Research Articles

A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures

This paper presents a new C1-type beam spectral element method, which aims to ensure the continuity of the deflection and its first derivative at the junction of the adjacent spectral elements. The proposed method is based on the Euler-Bernoulli beam theory. The main novelty of the method is that two different sets of interpolation functions are adopted for the axial displacement and deflection in order to meet the C1 continuous requirement. By removing the rotational degrees of freedom of the internal nodes, a lumped mass matrix can be formed in this spectral element method. Static and dynamic analysis of various beams and frame structures are performed to examine the validity of the proposed spectral beam element model. The numerical results show that the proposed method is accurate and effective. A single element for each component of the structure is able to provide good results.

Time History Analysis of Frame Structure Systems by State-space Representation Method

This paper develops the state-space representation (SSR) in the field of seismic analysis of the building structures. Dynamic analysis of multi-degree-of-freedom structures involves the solution of second-order linear differential equations which they represent the equation of motion of the structure. In this paper, a SSR was formulated to replace differential equation with two coupled first-order linear differential equations. The objectives of this study are as follows: (i) To implement the SSR as a powerful tool in dynamic analysis of frame structures and (ii) to conduct a linear time history analysis for large structures subjected to ground acceleration and the seismic responses of the building were studied as well. The analysis was based on the assumption that the system is elastic linear time-invariant system and material nonlinearity is not considered. The 1940 El-Centro earthquake time history record has been used in the study. There are many effective traditional methods which can be used for carrying out linear dynamic analysis of the structures, however, this paper introduces a state-space model as an alternative approach to perform this analysis. The advantage of this method, it works properly with MATLAB software, gives explicit result for time-invariant systems, applied to multi-input and multi-output control systems, solve the equation of motion for complicated dynamic problems.

Adaptive numerical method algorithms for nonlinear viscous and bilinear oil damper models subjected to dynamic loading

Adaptive numerical method algorithms are presented for the numerical simulation of the hysteretic behaviour of nonlinear viscous and bilinear oil dampers within a finite element program for nonlinear dynamic analysis of frame structures under earthquake excitations. The adaptive algorithms are applicable for computing high-precision solutions for nonlinear viscous and bilinear oil dampers with valve relief that are typically represented mathematically with a nonlinear Maxwell model. The algorithms presented possess excellent convergence characteristics for viscous dampers with a wide range of velocity exponents and axial stiffness properties. The algorithms are implemented in an open source finite element software, and their applicability and computational efficiency is demonstrated through a number of validation examples with data that involve component experimentation as well as the utilization of full-scale shake table tests of a 5-story steel building equipped with nonlinear viscous and bilinear oil dampers.

Hysteretic beam element with degrading smooth models

This work presents the development of a hysteretic beam element in the context of the finite element method, that is suitable for the inelastic dynamic analysis of framed structures. The formulation proposed is able to capture the main characteristics of hysteresis in structural systems and mainly accounts for stiffness degradation, strength deterioration and pinching phenomena, as well as for non-symmetrical yielding that often characterizes their behavior. The proposed formulation is based on the decoupling of deformations into elastic and hysteretic parts by considering additional hysteretic degrees of freedom, i.e., the hysteretic curvatures and hysteretic axial deformations. The direct stiffness method is employed to establish global matrices and determine the mass and viscous damping, as well as the elastic stiffness and the hysteretic matrix of the structure that corresponds to the newly added hysteretic degrees of freedom. All the governing equations of the structure, namely the linear global equations of motion and the nonlinear evolution equations at elemental level that account for degradations and pinching, are solved simultaneously. This is accomplished by converting the system of equations into state space form and implementing a variable-order solver based on numerical differentiation formulas (NDFs) to determine the solution. Furthermore, hysteretic loops and degradation phenomena are easily controlled by modifying the model parameters at the element level enabling simulations of a more realistic response. Numerical results are presented and compared against experimental results and other finite element codes to validate the proposed formulation and verify its ability to simulate complex hysteretic behavior exhibiting cyclic degradations.

An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia

A computationally efficient beam finite element is presented for the static and dynamic analysis of frame structures with any number and location of concentrated damages, whose macroscopic effects are simulated with a set of longitudinal, rotational and transversal elastic springs at the position of each singularity. The proposed mathematical model exploits positive Dirac’s deltas in the corresponding flexibility functions of the beam elements, and allows also considering shear deformations and rotatory inertia. Such contributions may have a huge impact on the higher modes of vibration, as confirmed by the numerical examples.

A novel beam finite element with singularities for the dynamic analysis of discontinuous frames

In this work, a model of the stepped Timoshenko beam in presence of deflection and rotation discontinuities along the span is presented. The proposed model relies on the adoption of Heaviside’s and Dirac’s delta distributions to model abrupt and concentrated, both flexural and shear, stiffness discontinuities of the beam that lead to exact closed-form solutions of the elastic response in presence of static loads. Based on the latter solutions, a novel beam element for the analysis of frame structures with an arbitrary distribution of singularities is here proposed. In particular, the presented closed-form solutions are exploited to formulate the displacement shape functions of the beam element and the relevant explicit form of the stiffness matrix. The proposed beam element is adopted for a finite element discretization of discontinuous framed structures. In particular, by means of the introduction of a mass matrix consistent with the adopted shape functions, the presented model allows also the dynamic analysis of framed structures in presence of deflection and rotation discontinuities and abrupt variations of the cross-section. The presented formulation can also be easily employed to conduct a dynamic analysis of damaged frame structures in which the distributed and concentrated damage distributions are modelled by means of equivalent discontinuities. As an example, a simple portal frame, under different damage scenarios, has been analysed and the results in terms of frequency and vibration modes have been compared with exact results to show the accuracy of the presented discontinuous beam element.

Dynamic analysis of frame structures with free viscoelastic layers: New closed-form solutions of eigenvalues and a viscous approach

Materials of viscoelastic nature are of great importance for damping and vibration control in the civil engineering, automotive and aircraft fields. The frequency dependence of their mechanical properties results in integral–differential equations of motion in the time domain along with a nonlinear eigenvalue problem in the frequency domain. The main challenge of this paper is to develop closed-form expressions for the eigenvalues of framed general structures with bonded unconstrained viscoelastic layers, whose constitutive relations are based on the fractional derivative. The developed expressions allow us to obtain an equivalent viscous model which dynamic matrices explicitly depend on the viscoelastic damping parameters. Then, the time response problem solution is reduced to a simpler system of second order linear differential equations decoupled in the modal space of the undamped problem. The proposed methodology is validated by two numerical examples, a cantilever bean and a frame. Very good agreement is found between the proposed and exact (from other iterative methods) eigenvalues and transfer functions for a wide range of viscoelastic materials.

Small and Large Displacement Dynamic Analysis of Frame Structures Based on Hysteretic Beam Elements

In this work, a beam element is proposed for the nonlinear dynamic analysis of frame structures. The classical Euler-Bernoulli formulation for the elastic beam is extended by implicitly defining new hysteretic degrees of freedom, subjected to evolution equations of the Bouc-Wen type with kinematic hardening. A linear interpolation field is employed for these new degrees of freedom, which are regarded as hysteretic curvatures and hysteretic axial deformations. By means of the principle of virtual work, an elastoplastic hysteretic stiffness relation is derived, which together with the hysteretic evolution equations fully describes the behavior of the element. The elemental stiffness equations are assembled to form a system of linear global equations of motion that also depend on the introduced hysteretic variables. The solution is obtained by simultaneously solving the entire set of governing equations, namely the linear global equations of motion with constant coefficient matrices, and the nonlinear local ...

Extremely large displacement dynamic analysis of elastic–plastic plane frames

AbstractThis paper presents a numerical analysis of large displacement responses of elastic–plastic plane frames under static and dynamic loads, by applying the vector‐form intrinsic finite element (VFIFE or V‐5) method. The VFIFE method defines the structure into a number of mass points, and applies Newton's second law and the internal force equilibrium to describe the motions of each mass point. By tracing the motions of all the mass points, it can analyze the large geometrical and material nonlinear changes during the motion of the structure without using the geometrical stiffness matrix and iterative procedures. Three different numerical examples are presented to demonstrate both the capability and the accuracy of the VFIFE method in a nonlinear dynamic analysis of frame structures with extremely large displacement. Copyright © 2011 John Wiley & Sons, Ltd.

The Use of the Dynamic Magnification Factor in the Dynamic Analysis of Framed Structures

This paper examined the use of the dynamic magnification factor in the analysis of framed structures. It is a method of practice in dynamic analysis of structures to magnify static response by a dynamic magnification factor in order to obtain the equivalent dynamic response. This method has been applied to the dynamic analysis of many structures including bridges and some country’s codes of practice made specifications in respect of the dynamic magnification factor for the analysis and design of various types of structures subjected to dynamic excitation. The suitability of this method to the dynamic analysis of frames was investigated in this paper by carrying out static and dynamic analysis of four frames using the flexible frame model and the stiffness formulation. Dynamic responses were first obtained by direct analysis as solutions to the set of equations governing the motions of the frames and secondly by the magnification of the static responses using the dynamic magnification factors. By comparing the results obtained in both methods it was inferred that the practice of magnifying static responses to obtain their dynamic equivalents in frame analysis gives correct results only in the case of deflections and not in stresses. Finally, this practice should de discouraged or limited only to the case of deflections in the dynamic analysis of framed structures.

Three-Dimensional Indirect Boundary Element Method Formulation for Dynamic Analysis of Frames Buried in Semiinfinite Elastic Media

This paper presents a formulation of the indirect boundary element method based on the principle of virtual work for the dynamic analysis of frame structures buried in semi-infinite elastic media. The present formulation, which falls in the category of symmetric Galerkin boundary element methods, leads to symmetric stiffness matrices for the continuum that may be defined in terms of conventional structural analysis variables (i.e., generalized displacements and lumped forces). It is shown that, in the context of the present formulation, rotation degrees of freedom may readily be introduced in the interpolation scheme with very little additional computational effort. The consistency of the present formulation with well-established results is assessed by comparing the predictions for the static and dynamic stiffness of single piles with other results from the literature. Finally, the dynamic stiffness of a single buried frame under vertical and horizontal loading is studied. The analysis shows that the stiffness of the full frame may not always be accurately estimated by means of results for single piles, even when dynamic interaction factors are used.

Optimization and Dynamic Analysis of Frame Structures with Friction Energy Dissipation Device

An analytical solution for a single-damped-storey frame structure system including a friction energy dissipation device under sine wave excitation is presented with an example. A new mathematical model for optimizing some key system parameters in earthquakes is proposed. The model minimizes the sum of brace stiffness that complies with the Chinese seismic code for storey-drift limitations for three given earthquake records. Therefore, the seismic requirements for similar structures can be satisfied with fewer friction energy-dissipation devices. Using the Visual C++ language, a computer program, GAOFF, has been developed to optimize the parameters of a frame structure with friction energy dissipation devices based on three earthquake records using a genetic algorithm. An example of a six-storey RC frame is also presented to demonstrate the effectiveness of the proposed optimal design procedure.

Response of open-plane frames on isolated footings to an excitation characterized by a white noise

There are occasions, such as multi-storey buildings founded on soft soil, when it becomes necessary to consider the effects of deformability of the foundation. These effects are generally referred to as soil–structure interaction. While the dynamic analysis of framed structures can be made on the basis that the superstructure possesses continuously distributed properties, the analysis is complex and practicable only in the case of very simple structures. More conveniently, a finite element approach can be adopted in which framed structures are discretized into segments and the displacements of the interconnecting nodes constitute the generalized co-ordinates (or dynamic degrees of freedom) of the structure. Kinematic constraints are commonly adopted to reduce the degrees of freedom in order to save computational efforts without significant loss of accuracy.

Applications of a reduction method for reanalysis to nonlinear dynamic analysis of framed structures

This paper is concerned with the nonlinear dynamic analysis of framed structures using a reduction method recently proposed by the authors. The reduction method is originally devised for structural static reanalysis and has been applied in optimal design of structures to speed up the design process. For nonlinear dynamic analysis of framed structures, the incremental or iterative equations of motion can be transformed into an algebraic system of equations if appropriate integration methods such as Newmark's method are used to integrate the equations of motion. The resulting algebraic system, referred to as the effective system in this paper, changes during the simulation for a nonlinear dynamic problem. Therefore, from the point of view of solving systems of equations, a nonlinear dynamic problem is very similar to an optimal design problem in that the system of equations changes for both types of problems. Hence, any reanalysis technique can be readily applied to carry out a nonlinear dynamic analysis of structures. As demonstrated from the presented numerical examples, the response obtained by the adopted reduction method is as accurate as that obtained by the Cholesky method, and as estimated from the operation counts involved in the method, it is more efficient than the Cholesky method when the half-band width is greater than about 50.

Dynamic analysis of framed structures with statistical uncertainties

Dynamic analysis of framed structures with statistical uncertainties

Dynamic analysis of framed structures with statistical uncertainties

The forced harmonic vibration analysis of portal frames consisting of viscously damped beams with spatial stochastic variation of mass and stiffness properties is considered. The analysis is based on the assembly of element stochastic dynamic stiffness matrices. The solution involves inversion of the global dynamic stiffness matrix, which, in this case, turns out to be a complex-valued symmetric random matrix. Three alternative approximate procedures, namely, random eigenfunction expansion method, complex Neumann expansion method and combined analytical and simulation method are used to invert the matrix. The performance of these approximate procedures is evaluated using Monte Carlo simulation results. Copyright © 1999 John Wiley & Sons, Ltd.

A kinematic model for dynamic analysis of space frames

This paper presents a kinematic model for dynamic analysis of framed structures. The accuracy of the proposed method is demonstrated by analyzing the dynamic responses of two space frames of varying stiffness properties.

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.

Nonlinear static and dynamic analysis of framed structures

The present study is concerned with methods of nonlinear static and dynamic analysis of structures, particularly for application to geometrically nonlinear space frames. The displacement formulation of the finite element method is adopted. Large displacements are accounted for by using a material (Lagrangian) description of motion from a fixed reference frame, assuming small strains and moderately large rotations. Curved beam elements are considered by introduction of initial deflections in the element stiffness relations. Higher order axial interpolation polynomials are included in order to obtain an appropriate coupling between axial forces and bending for the frame members. Efficient methods for instability and postbuckling problems have been a main point for the study of nonlinear static behaviour. This also includes evaluation of nonlinear numerical solution methods, some of which being applicable also in dynamic analysis. The efficiency of these methods are discussed in connection with the nonlinear dynamic response analyses presented. For the integration of the incremental equations of motion, a method based on transformation to generalized coordinates on modal basis is proposed and studied as an alternative to direct integration. Numerical examples are presented for both static and dynamic analyses, and it has been considered important to show the applicability of the nonlinear analysis methods to practical structural problems. The advantages of post-processing and extensive use of computer graphics in presenting results for nonlinear space structure analyses are demonstrated.

Vibrations of frame structures according to the Timoshenko theory

The dynamic analysis of frame structures by means of the general dynamic slope-deflection equations including the effect of rotary inertia and of transverse-shear deformation is presented. A portal frame subjected to free and forced vibrations is used to illustrate the application of the proposed method and to show the rotary inertia and shear effects.