Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

Abstract

The solution of the motion equation for a structural system under prescribed loading and the prediction of the induced accelerations, velocities, and displacements is of special importance in structural engineering applications. In most cases, however, it is impossible to propose an exact analytical solution, as in the case of systems subjected to stochastic input motions or forces. This is also the case of non-linear systems, where numerical approaches shall be taken into account to handle the governing differential equations. The Legendre–Galerkin matrix (LGM) method, in this regard, is one of the basic approaches to solving systems of differential equations. As a spectral method, it estimates the system response as a set of polynomials. Using Legendre’s orthogonal basis and considering Galerkin’s method, this approach transforms the governing differential equation of a system into algebraic polynomials and then solves the acquired equations which eventually yield the problem solution. In this paper, the LGM method is used to solve the motion equations of single-degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems. The obtained outputs are compared with methods of exact solution (when available), or with the numerical step-by-step linear Newmark-β method. The presented results show that the LGM method offers outstanding accuracy.

Special Issue on “Buildings and Structures under Extreme Loads II”

“…Besides, critical situations for buildings and components may derive from a multitude of scenarios. First, in any case, it is required to properly describe the structural dynamic problem with reliable mathematical models [7]. Successively, the use of more complex and advanced numerical tools can provide additional important indications about the actual performance and capacity.…”

The Influence of Different Dynamic Material Constitutive Models on the Impact Performance of Circular CFST Columns

“…Impact effects include car, ship and plane collisions; the accidental fall of elevators, billboards, temporary construction facilities and other construction equipment or attachments; secondary disasters caused by earthquakes, fires, mountain torrents and mudslides; and even terrorist attacks. As a result, impact effects have the distinctive characteristics of low frequency and high hazard, and are an accidental effect that must be considered for most major civil engineering structures, crash-prone structures, crash-proof enclosures, national defense structures, and anti-explosion and anti-terror facilities [1][2][3][4][5]. In particular, accidents such as vehicle collisions, ship collisions and terrorist attacks occur frequently (see Figure 1); therefore, more attention is being drawn toward investigating the impact resistance performance of important architectural structures and components.…”

The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method

“…In recent attempts to find an optimal method for nonlinear vibrations of structures, it was discovered that the Galerkin method and its alternative form, the Rayleigh-Ritz method, can be effective for approximate analysis of nonlinear vibrations in certain circumstances [22][23][24][25]. The Legendre-Galerkin matrix (LGM) method is used to solve the vibration equations of single (SDOF) and multiple degree-of-freedom (MDOF) structural systems [26]. The nonresonance and primary resonance of microbeams with consideration of small-scale effects and nonlinear terms were obtained by the Galerkin and multiple-scale methods [27].…”