Analysis of Collapse of Irregular Tall Structures Using Mixed Lagrangian Formulation

Recent extreme events, such as earthquakes, hurricanes, terrorist attacks, and fire, showed that the best designed tall structures can be vulnerable to collapse. A new approach was developed by the authors using the mixed Lagrangian formulation for analysis of tall structures. The formulation attempts to solve problems using a force-based approach in which momentum appears explicitly and can be used to deal with structures where deterioration and fracture occur before collapse. In this chapter, a time integral of functions of the response over the duration of the response is considered. The kernel of the integral consists of two functions - the Lagrangian and the dissipation function - of the response variables that describe the configuration of the structure and their rates. The Lagrangian function is energy-like and describes the conservative characteristics of the system, while the dissipation function similar to a flow potential describes the dissipative characteristics. Irregular structures are extremely sensitive to loss of supporting column, which can lead to amplified out-of-plane and torsional motions. This chapter presents the formulation using the mixed Lagrangian formulation (MLF) which allows considering the three-dimensional effects during collapse. An example, using a fifteen-story building, is presented, to illustrate the performance of the above approach. The structure is irregular and the loss of either a single or several columns on the periphery leads to an excessive sway and loss of capability to sustain the gravity loads (i.e., collapse). The failure of eccentrically located columns in an otherwise symmetric building leads to the coupled lateral-torsional collapse, well captured by the formulation presented herein.