Study On The Effects Of Adding Diagonal Springs In A Rigid Body Spring Model With Quadrilateral Elements

== Abstract ==

An improved Rigid Body-Spring model (RBSM) for in-plane analyses is here proposed. The approach has been developed starting from the model presented by Casolo [4], according to which a continuum plate is discretized in an assembly of rigid quadrilateral elements connected by elasticplastic springs. In the proposed model, not only the elements that share one edge are connected, as it usually happens in RBSMs, but also the elements that share only one node, that are the elements on the diagonals. These elements are connected through an axial spring fixed to their centers of gravity. The spring stiffnesses have been obtained through the equivalence of the stored strain energy for a plane stress state. These, for an isotropic material, results to be function of the Young's modulus (E) and of the Poisson's ratio (). Thanks to the addition of the diagonal springs, this quadrilateral RBSM is capable of modeling isotropic materials with a Poisson's ratio different from zero. The effectiveness of the proposed computational method was proved comparing the elastic solution with a Finite element code for an uniaxial tensile test. About the failure response, on one hand, the addition of the diagonal springs introduces a new direction in which the failure condition is verified, reducing the intrinsic anisotropy of the failure response, typical of the discrete approaches [5, 14]. On the other hand, the new springs increase the complexity of the model and the spring constitutive behavior is no longer directly related to the material uni-axial behavior. In this work, the spring post elastic response was defined considering the mode I fracture energy of the material. In the end, the model was applied to the study of a concrete notched beam in a three point bending test. The results have been compared with the experimental one find in the literature. The analyses were executed with a dynamic explicit solver and different tensile spring behaviors were compared to highlight the importance of considering a bi-linear softening law for modeling the failure response of quasi-brittle materials like concrete.

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