The failure of strengthened beams with fibre-reinforced polymer (FRP) materials is due to high stress concentration of FRP–concrete interface. Understanding the cause and mechanism of the debonding of the FRP plate and the prediction of the stress distribution at the concrete–FRP interface are important for more effective strengthening technique. This paper presents an analytical solution, based on Smith and Teng’s equations, for interfacial shear and normal stresses in reinforced concrete (RC) beams strengthened with a fibre reinforced polymer (FRP) plate. However, the shear stress–strain relationship is considered to be bilinear curve. The effects of the shear deformations are calculated in an RC beam, an adhesive layer, and an FRP plate. The results of parametric study are compared with those of Smith and Teng. They confirm the accuracy of the proposed approach in predicting both interfacial shear and normal stresses.