The disposal of discarded tires is a problem of significant proportion. In the present experimental study, rubber produced from the granulation of discarded tires was used as an additive to replace certain portions of mineral aggregates in concrete. This rubberized concrete was used in making thin panels. A layer of polymer grid was used to reinforce the rubberized concrete panels. These panels were developed to study their performance in applications where the concrete could be subjected to flexure. Buildings constructed in areas with extreme wind pressures resulting from hurricanes or tornadoes are examples of structures that require concrete that can handle considerable deformation without failing catastrophically. Three different panel thicknesses, three different water-cement ratios, and three different rubber contents were the parameters evaluated in this study. All panels were loaded in bending with two equal loads applied at two equal distances from the supports. Test results showed that the flexural resistance of the panel increases with an increase in the thickness of the section, and with a reduction in the water-cement ratio of the concrete. The panels behaved in a ductile manner and there were no signs of brittle failure. Considerable deformation was measured during load application where loaded panels fractured but remained intact relying on the elongating polymer reinforcement. In addition to the lightweight properties, it was concluded that rubber concrete and polymer grid could be used as effective tools to impart ductility to the concrete and to control the mode and nature of the brittle failure of conventional concrete.