Theory of matrix structural analysis

Vibrating Spatially Periodic Structures with Simply Supported and Guided Ends

This paper introduces a powerful convenience method for automatically building largemotion models of flexible bodies of general structural form (not just beams). The starting point is the assembled standard mass matrix of a small-motion finite-element model. The goal point is a fully-coupled, all-terms-included large motion model that incorporates the same arbitrarily detailed structural geometry as the source finite-element model. An explicit straightforward procedure is provided for constructing both full and reduced-order models of flexible bodies that are able to undergo arbitrary displacement and rotation. The body translational and angular velocities are allowed to be large. Within the context of small deformations, all terms of the equations of motion are included. A standard finite-element small (absolute) motion model is used as the base model from which the equations of motion of the unrestrained body are derived. A fully consistent inertia formulation is assumed and encouraged; rotational coordinates are naturally admitted; and no low-level knowledge of the finite-element nodal shape functions is required. Lumped-mass formulations are included as a special sub-class. Automatic generic approximation is used for certain shape function products that cannot be directly extracted from the mass matrix of the small motion model. The critical strategy used to produce the approximations is called the “LISA Method” (Localized Implicit Shape-Function Approximation). The LISA Method has been demonstrated to reproduce exact results for a number of practical element types.

Practical Large-Motion Modeling of Geometrically Complex Flexible Vehicles: A Consistent-Mass Standard-FEM Based, All Terms Included Formulation

A major environmental concern related to railway traffic is vibration. A lot of re- search has been carried out to understand vibration of straight tracks, with less attention been paid to curved tracks. Modelling the dynamic behaviour of a curved railway track is important to understand the physics of generation and propagation of vibration fromtrains at non-straight sections of tracks. Modelling is also important to assess the current and any alternative track designs from an environmental point of view. In this paper a curved track is modelled and the effect of curvature is investigated. Two models have been developed and their results have been compared. In the first, the curved track is modelled using straight beam elements. In the second curved beam elements are used. For both, the Euler-Bernoulli beam theory has been adopted to describe their bending behaviour. The elements have 12 degrees of freedom accounting for displacements and rotations in the lateral, transverse and longitudinal directions. The excitation comes from an axle traversing the rails with subcritical velocity, accounting for the wheel-rail contact forces. The describedmodels are solved using the Finite Element Method. The time domain response of the versine of the curved track due to the passage of the axle is computed. A comparison is made on the efficiency of the two models for different curve radii and frequencies. The two models provide very similar results showing that the piecewise straight beam approximation represents the behaviour of the curved track accurately. Also the curved beam model used in this study shows some limitations for the specific application and therefore the straight element method is recommended

background

A comparison between the use of straight and curved beam elements for modelling curved railway tracks

In this chapter the principles of the finite element method will be presented through the application to problems of steady and nonsteady groundwater flow. The method was developed in the 1950’s, first for problems of aeronautical engineering (the construction of airplanes), mechanical engineering (nuclear reactor vessels), and civil engineering (bridges). In later years, the method was generalized to practically all areas of engineering, including groundwater flow, where the solution to field equations is required.

The Finite Element Method

Increased importance has been placed on the design and analysis of structures to account for potential terrorist attacks. Designers must account for many options that include various levels of mitigation and protection measures to ensure the safety of the building and its occupants. One critical ingredient to the analysis is an understanding of the structural performance of structures subjected to blast loading especially when considering issues of collapse. The approach presented in this paper provides an efficient alternative to dynamic analysis of a moment-framed building subject to blast induced damage. First, framing members (girders and columns) are designed to withstand typical dead and live loads. Factored design loads are calculated using a first order (load path) static analysis of the frame. Structural steel, reinforced concrete, reinforced/unreinforced masonry, and wood members are designed in accordance to the applicable building code requirements and design specifications. After structural damage due to blast loads is predicted, progressive collapse of the building is modeled as a series of events. Events include local plastic hinge formation, member failure from overstress, local buckling, lateral torsion buckling, or axial buckling, and global buckling or instability. Element section yield and failure loads are evaluated for axial, strong and weak bending and torsion using the applicable codes (striped of load and resistance factors). Stiffness at hinges is assumed to be bilinear. Global buckling is evaluated by a generalized Eigenvalue approach and instability by simple Eigenvalues. The approach uses a course finite element model of the building that includes beams, columns, walls and floor slabs. The approach can determine if progressive collapse will propagate to the entire building or if a stable condition can be achieved. An optional model that includes the debris loading of removed structural elements is also included.

Theory of matrix structural analysis

Citations

Vibrating Spatially Periodic Structures with Simply Supported and Guided Ends

Practical Large-Motion Modeling of Geometrically Complex Flexible Vehicles: A Consistent-Mass Standard-FEM Based, All Terms Included Formulation

A comparison between the use of straight and curved beam elements for modelling curved railway tracks

The Finite Element Method

A Model for Progressive Collapse of Conventional Framed Buildings

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