Fiber-based shakedown analysis of three-dimensional frames under multiple load combinations: Mixed finite elements and incremental-iterative solution

This work presents an efficient fiber analysis for evaluating the shakedown safety factor of three-dimensional frames under multiple load combinations. Mixed finite elements are employed for an accurate discretization. A continuation method, similar to a standard elasto-plastic analysis, is used at structural level. It evaluates a pseudo-equilibrium path made of a sequence of safe states with a converging nondecreasing load factor. Each point of the path is obtained by finding kinematic variables corresponding to self-equilibrated stresses satisfying Melan's condition for the current load factor to be safe. The stress admissible domain is defined at fiber level as a function of the load factor using the maximum and minimum effect due to all loads. An iterative state determination provides finite element stresses corresponding to assigned kinematic variables and load factor. The overall analysis differs from previous proposals for two novelty points. Firstly, a direct application of the Newton method can be employed, without any need for constrained optimization solvers. Moreover, dimension and complexity of the load domain do not affect the computational cost of the nonlinear analysis. Numerical tests show an accurate estimate of the safety factor using a small number of fibers and an efficient solution also for large buildings.