טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentTabul David
SubjectImplementation of Fracture Mechanics Concepts in Dynamic
Progressive Collapse Analysis by Mixed Lagrangian
Formulation (MLF) Approach
DepartmentDepartment of Civil and Environmental Engineering
Supervisor Professor Oren Lavan
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

Prediction of the progressive collapse of large scale buildings due to extreme events, which is the purpose of this research, is a major challenge in structural engineering. The main challenge stems from the various complex and sudden phenomena that structures experience during collapse. Those often lead to the collapse of the analysis prior to the actual analysis of collapse. With theory and tools for Progressive Collapse Prediction (PCP), deficiencies in the common practice of structural design, in the context of progressive collapse, could be identified without having to wait for the next extreme event to occur.

A new approach for nonlinear dynamic structural analysis, namely the Mixed Lagrangian Formulation (MLF), has been proposed by Sivaselvan & Reinhorn (2006).  MLF was originally developed for the analysis of elastic-plastic response while considering geometric nonlinearity. This approach may enable the investigation of existence and uniqueness of solutions. It is based on time discretization of Hamilton’s principle. Hence, the computation of the response quantities in each time step reduces to the solution of an optimization problem. This weak formulation in time leads to a very stable and accurate numerical scheme that allows for large time step sizes, while allowing for sharp changes in its variables.  The MLF can potentially present a unified approach with a strong theoretical background that accounts for all stages of collapse. An approach to include fracture in MLF seems to be the missing part of this puzzle of MLF.

The work consists of the following stages:

?  Identification of the additional states required to model fracture using fracture mechanics concepts. A parameter in the macro-model that is equivalent to the crack area in fracture mechanics is chosen. Subsequently, appropriate stored energy and dissipation functions, which lead to Griffith's theory, are formulated. A Legendre transform is used to formulate their conjugate functions. Four formulations are then used to further develop the MLF. One of these leads to a more convenient optimization problem that can eventually be used.

?  Development of a numerical scheme. The resulting governing equations are discreticized using a central difference approximation. The discreticized equations are integrated to enable the solution of the optimization problem each time step of the MLF. This optimization is not convex. However, the theoretical background of fracture mechanics supplies justifications of how the local optimal solution leads to the physical solution of the problem.