A unified shear strength equation for slabs and beams longitudinally reinforced with FRP or steel bars based on simplified cracked elastic properties

Abstract

At service load conditions, deflections and stresses in reinforced concrete (RC) slabs and beams are typically calculated using the properties of the transformed cracked cross section. For more than eighty years, the depth of compression zone factor k in cracked singly-reinforced rectangular RC sections has been calculated using k=2ρn+ρn2-ρn where ρ is the reinforcement ratio and (n) is the modular ratio. It is shown that in slabs and beams with a wide range of concrete compressive strength fc′ that are reinforced longitudinally with steel or FRP bars, ρn ranges from 0.005 to 0.16. Within this range, k can be accurately calculated using k=0.95ρn0.43 or simply k=ρn0.44. The error between the former equation and the exact equation ranges from –2% to + 3 %. This equation is used to show that the shear strength of concrete vc in the ACI–440.11-22 code for GFRP–reinforced members depends effectively on ρ, fc′ and d, just like the strength of steel–reinforced members in the ACI–318-19 code equation. It is proposed to apply to the factor Er/Esteel1/3 where Er and Esteel are the moduli of elasticity of the FRP and the steel bars respectively to the ACI–318 equation to make it suitable also for FRP–reinforced members. This unifies the equation for the two types of reinforcement and provides a better correlation with the experimental results from 233 FRP–reinforced members available in the literature than the current ACI–440.11 code equations. On the other hand, the transformed cracked moment of inertia Icr can be related solely to ρn and approximated using Icr=0.36ρn0.81bd3 where b and d are the sectional width and effective depth respectively. This equation is used to show that the cracked flexural stiffness is effectively related to the modulus of elasticity and the area of the reinforcing bars raised to the power 0.8, to d2.2 and to modulus of elasticity of the concrete raised to the power 0.2.