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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.

In the last years, the increase of blast and terroristic attacks in the vicinity of critical structures and facilities highlighted the importance of appropriate simulation and design methods for blast loaded structures. Within a building exposed to blast, columns act as key components, being asked to sustain ordinary gravity and wind loads, but also possible additional exceptional loads like earthquakes or explosions. In this paper, the dynamic behavior of steel columns under blast pressures is numerically investigated. Careful consideration is spent – based on validated Finite Element (FE) models – for the blast performance of steel columns with different cross-sectional shapes but similar mechanical properties (i.e., total area, moment of inertia and elastic section modulus). Explicit FE analyses are carried out on a wide set of configurations, including variations in (a) cross-section properties, (b) boundary conditions and (c) explosions features. In doing so, the accuracy and reliability of different FE models (i.e., based on solid, shell and beam elements) are also investigated, both in terms of global/local results and computational cost of the dynamic simulations. As key damage parameters due to blast, the maximum displacement and the residual load carrying capacity of the examined columns are examined. The collected parametric results show that the cross-sectional shape only slightly affects the global dynamic behavior of steel columns, especially in terms of residual capacity. Moreover, compared to computationally expensive full solid models, the reliability and accuracy of geometrically simplified shell and beam is discussed, with respect to the observed responses and intrinsic computational efficiency.

Increasing terrorist attacks towards ordinary or strategic buildings and soft targets represent one of the major impetus to improve existing methods of design for blast-resistant structures. When a building undergoes an extreme dynamic event such as blast or impact, local damage of its key structural components (i.e., the columns) may lead to severe failure and even collapse of the entire building. Consequently, the availability of simplified, time efficient and reliable methods of analysis can be relevant for design. In this paper, H-section steel columns subjected to blast loads are numerically investigated, so as to derive practical formulations for damage evaluation assessment. The strategy is based on parametric Finite Element (FE) models (with up to 5600 configurations), validated towards experiments and, used as an extensive data bank, for further elaboration via Gene Expression Programming. Analytical formulations are in fact proposed for calculating some relevant parameters of design, such as (a) the initial and (b) the residual axial capacity of the examined columns. The collected results show that the proposed formulations can offer a good level of accuracy and high calculation efficiency for blast loaded H-section steel columns. In addition, an expression is proposed to relate the damage index (based residual axial capacity) to the conventional displacement/rotational index. Sensitivity analyses and some calculation examples are finally presented, to further investigate the potential of the approach for design purposes.