Equation of Motion Governing the Dynamics of Vertically Collapsing Buildings

The present paper aims at contributing to a discussion, opened by several authors, on the proper equation of motion that governs the vertical collapse of buildings. The most striking and tragic example is that of the World Trade Center Twin Towers, in New York City, about 10 years ago. This is a very complex problem and, besides dynamics, the analysis involves several areas of knowledge in mechanics, such as structural engineering, materials sciences, and thermodynamics, among others. Therefore, the goal of this work is far from claiming to deal with the problem in its completeness, leaving aside discussions about the modeling of the resistive load to collapse, for example. However, the following analysis, restricted to the study of motion, shows that the problem in question holds great similarity to the classic falling-chain problem, very much addressed in a number of different versions as the pioneering one, by von Buquoy or the one by Cayley. Following previous works, a simple single-degree-of-freedom model was readdressed and conceptually discussed. The form of Lagrange's equation, which leads to a proper equation of motion for the collapsing building, is a general and extended dissipative form, which is proper for systems with mass varying explicitly with position. The additional dissipative generalized force term, which was present in the extended form of the Lagrange equation, was shown to be derivable from a Rayleigh-like energy function. DOI: 10.1061/(ASCE)EM.1943-7889.0000453. (C) 2012 American Society of Civil Engineers.

Probabilistic failure modelling of reinforced concrete structures subjected to chloride penetration

Structural durability is an important criterion that must be evaluated for every type of structure. Concerning reinforced concrete members, chloride diffusion process is widely used to evaluate durability, especially when these structures are constructed in aggressive atmospheres. The chloride ingress triggers the corrosion of reinforcements; therefore, by modelling this phenomenon, the corrosion process can be better evaluated as well as the structural durability. The corrosion begins when a threshold level of chloride concentration is reached at the steel bars of reinforcements. Despite the robustness of several models proposed in literature, deterministic approaches fail to predict accurately the corrosion time initiation due the inherent randomness observed in this process. In this regard, structural durability can be more realistically represented using probabilistic approaches. This paper addresses the analyses of probabilistic corrosion time initiation in reinforced concrete structures exposed to chloride penetration. The chloride penetration is modelled using the Fick's diffusion law. This law simulates the chloride diffusion process considering time-dependent effects. The probability of failure is calculated using Monte Carlo simulation and the first order reliability method, with a direct coupling approach. Some examples are considered in order to study these phenomena. Moreover, a simplified method is proposed to determine optimal values for concrete cover.