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The LATIN (acronym of LArge Time INcrement) method was originally devised as a non-incremental procedure for the solution of quasi-static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time-dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. 126D. GIVOLI, R. BHARALI AND L. SLUYS 2. A 'local' problem, which consists of the nonlinear algebraic relations, and an auxiliary equation, which determines a progress direction. This problem is nonlinear but algebraic and miniature.The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems [2,4], provides an approximate solution to the original problem.The terms 'global' and 'local' are used here as in the LATIN literature. Namely, a global problem is a problem involving differential equations, whereas a local problem is a problem involving only algebraic relations. We stress this point, because in other contexts (such as absorbing boundary conditions or certain material models in plasticity that prevent mesh-dependent behavior), the word 'nonlocal' refers to an integral operator, whereas a 'local relation' means that the relation involves only functions and derivatives at a single point.The LATIN method was applied to quasi-static problems in damage mechanics [5], anisotropic plasticity [6], cyclic viscoplasticity [7,8], poroelasticity [9], and rigid plasticity (with a special modified implementation) [10]. It was also applied to sheet cutting problems [11,12], to thermomechanical problems [13], to fatigue crack propagation, in conjunction with XFEM [14], and to tensegrity media [15].Most of the papers on LATIN consider such nonlinearity that after finite element (FE) discretization leads to a single-valued load vector as a function of the displacement vector. In [16], the LATIN method was extended to deal with more general nonlinear behavior, na...

Highlights• Effective equality constraint for fracture irreversibility inequality constraint using slack variable.• The Lagrange Multiplier Method and the Penalty method are adopted to augment the energy functional.• Numerical experiments carried out on both brittle and quasi-brittle fracture problems.

ABSTRACT:In this paper, a reliability based study has been incorporated to compare the categories of Performance Based Seismic Design (PBSD) of a reinforced concrete frame which are the ductility based design and the energy based design respectively. For this purpose, a two dimensional reinforced concrete frame consisting of one bay and one storey is modelled in Opensees 2.4.0 in order to evaluate the capacity of the structure. Based on the analysis results, an equivalent bi-linear model is developed and the two different limit states are defined based on the ductility demand and overall damage index(ODI) of the structure whereby material property of concrete, steel and loads are considered as the random variables. A dynamic analysis has been carried out in Opensees 2.4.0 on the bilinear model by varying the peak ground acceleration (PGA) from 0.05g to 0.4g of the simulated results of the El-Centro earthquake to quantify its nonlinearity in damage index scale and ductility factor scale. The Damage Index (DI) has been computed from the modified Park and Ang Damage Index while the Ductility factor (DF) has been obtained from the top node displacement of the structure. It has been observed that at low PGAs, DF has higher values, as the structure does not yield . With increment in the PGAs, it is seen that, DI values are greater than DF after a certain PGA level. Also fragility curves has been developed where probability of failure has been plotted against PGA, for both ductility and energy based design for the respective durations of 20-25 and 40-50 seconds and it has been observed that at higher values of PGA, pF values for DI will be greater than that of ductility which establishes the fact that a Damage index or energy based design is better than ductility based design for higher PGA levels.