Simple Analytical Models for Beams and Slabs Under Soft Impacts at Medium Speed

A simply supported reinforced concrete slender beam is modeled by 3 rigid consecutive elementary beams, the median beam being connected to the others by 2 viscous and elastoplastic spiral springs. The model can be assimilated as a non linear SDOF system convenient for the motion study of beams within flexural deformation domain, with displacements up to the height of the beam. The characteristics of the visco-elastoplastic springs are tuned so as be consistent with the beam motion before and after impact: the rigidity of the elastic domain of the springs is consistent with natural vibration frequencies which may be reduced after the impact due to subsequent damages. The motion of the beam during and after the impact is analyzed with such model: the values of the main mechanical characteristics (rigidity, plastic limit, viscous damping) may then be obtained. The impact tests performed by VTT (Finland) on one-way concrete slabs consolidate this approach and give consistent experimental values for the elastoplastic laws to be introduced in the model. With this experimental validation, the model may be used as a predictive tool for resistance and for displacements, as far as reinforced concrete beams and slabs are concerned. A thin reinforced concrete slab, simply supported along its 4 edges, is modeled by 4 to 5 rigid trapezoidal elementary slabs connected together by visco-elastoplastic spiral springs along the hinges. A non linear SDOF system is then developed to capture the behavior of such a slab within a flexural deformation domain, with displacements up to the slab thickness. The mechanism involving large shear deformations under the impact (“punching cone”) is taken into account by adding a second degree of freedom. The existing tests on reinforced concrete slabs submitted to medium velocity impacts found in literature may be used to consolidate this approach and to specify the values to be introduced in the model. The model will be used to analyze the forthcoming results (in terms of resistance and displacements) of VTT impact tests on simply supported reinforced concrete slabs. The behavior of civil works structures submitted to impacting missiles can nowadays be analyzed either with sophisticated FE calculation codes, or with analytical models. These analytical models may constitute simple but useful engineer’s tools for sensitivity analyses and for results checking of the necessary more sophisticated computation codes, in terms of resistance and in terms of displacement. They may be simply implemented on any spreadsheet software.