Abstract
This article presents an analytical model, based on a refinement of the rotatingangle softened truss model (RASTM) with efficient solution procedure, to predict the full response of reinforced concrete (RC) membrane elements strengthened with fiber reinforced polymers (FRP). To extend the RASTM, equations from equilibrium conditions and smeared constitutive relationships for the materials are modified in order to account for the tensile FRP reinforcement and its interactions with the other material components. In addition, an efficient algorithm is proposed for the calculation procedure to avoid using the classical trial and error technique to compute the solution points. This new algorithm provides higher numerical efficiency and stability. The reliability of the efficient RASTM FRP solution procedure is checked against some experimental data related with FRPstrengthened RC panels tested under inplane shear and found in the literature, and also with the predictions from the softened membrane model (SMMFRP) for comparison. In general, reasonably good agreement is observed between the efficient RASTM FRP procedure and the SMMFRP, and also with the experimental response of the reference test panels.
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Abbreviations
 \(A_{{\text{c}}}\) :

Area of concrete cross section
 \(A_{{{\text{Tf}}}}\) :

Area of transverse FRP reinforcement
 \(A_{{\text{L}}}\) :

Area of longitudinal steel reinforcement
 \(A_{{\text{T}}}\) :

Area of transverse steel reinforcement
 \(E_{{\text{c}}}\) :

Young’s modulus of concrete
 \(E_{{\text{f}}}\) :

Tensile Young’s modulus of FRP reinforcement
 \(E_{{\text{S}}}\) :

Young’s modulus of steel reinforcement
 \(f^{\prime}_{{\text{c}}}\) :

Uniaxial compressive strength of concrete
 \(f_{{{\text{cr}}}}\) :

Tensile strength of concrete
 \(f_{{{\text{Lf}}}}\) :

Tensile stress in the longitudinal FRP reinforcement
 \(f_{{{\text{Sf}}}}\) :

Tensile stress in the FRP reinforcement
 \(f_{{{\text{Tf}}}}\) :

Tensile stress in the transverse FRP reinforcement
 \(f_{{{\text{fu}}}}\) :

Ultimate tensile strength of the FRP reinforcement
 \(f_{{\text{L}}}\) :

Tensile stress in the longitudinal steel reinforcement
 \(f_{{{\text{Ly}}}}\) :

Yielding stress of the longitudinal steel reinforcement
 \(f_{{\text{S}}}\) :

Tensile stress of steel reinforcement
 \(f_{{{\text{Sy}}}}\) :

Uniaxial yielding stress of steel reinforcement
 \(f_{{\text{T}}}\) :

Tensile stress in the transverse steel reinforcement
 \(f_{{{\text{Ty}}}}\) :

Yielding stress of the transverse steel reinforcement
 \(f^{\prime}_{{\text{y}}}\) :

Apparent yielding stress of the embedded steel rebars
 \(K_{{\text{f/s}}}\) :

Factor for FRP/steel stiffness ratio
 \(K_{{\text{w}}}\) :

Factor for FRP wrapping scheme
 \(k\) :

Index for the step of the calculation procedure
 \(k_{{\text{S}}}\) :

Shear stiffness in the cracked stage
 \(m_{{\text{L}}}\) :

Longitudinal proportionality coefficient
 \(m_{{{\text{LT}}}}\) :

Shear proportionality coefficient
 \(m_{{\text{T}}}\) :

Transverse proportionality coefficient
 \(\alpha_{2}\) :

Angle of the principal compressive stresses in the FRPstrengthened RC membrane element
 \(\alpha_{{\text{D}}}\) :

Angle between the L–T and R–D coordinate systems
 \(\Delta \varepsilon_{{\text{D}}}\) :

Path increment for \(\varepsilon_{{\text{D}}}\)
 \(\varepsilon_{0}\) :

Strain corresponding to \(f^{\prime}_{{\text{c}}}\)
 \(\varepsilon_{{{\text{cr}}}}\) :

Tensile strain corresponding to the tensile strength of concrete
 \(\varepsilon_{{{\text{cu}}}}\) :

Ultimate strain for concrete in compression
 \(\varepsilon_{{\text{D}}}\) :

Principal compressive strain
 \(\varepsilon_{{{\text{Lf}}}}\) :

Strain in the longitudinal FRP reinforcement
 \(\varepsilon_{{{\text{Sf}}}}\) :

Strain in the FRP reinforcement
 \(\varepsilon_{{{\text{su}}}}\) :

Ultimate strain for steel reinforcement
 \(\varepsilon_{{{\text{Tf}}}}\) :

Strain in the transverse FRP reinforcement
 \(\varepsilon_{{{\text{fu}}}}\) :

Ultimate strain for FRP reinforcement
 \(\varepsilon_{{\text{L}}}\) :

Longitudinal strain
 \(\varepsilon_{{{\text{Ly}}}}\) :

Yielding strain of the longitudinal steel reinforcement
 \(\varepsilon_{{\text{R}}}\) :

Principal tensile strain
 \(\varepsilon_{{\text{S}}}\) :

Strain in the steel reinforcement
 \(\varepsilon_{{\text{T}}}\) :

Transverse strain
 \(\varepsilon_{{{\text{Ty}}}}\) :

Yielding strain of the transverse steel reinforcement
 \(\varepsilon^{\prime}_{{\text{y}}}\) :

Strain corresponding to \(f^{\prime}_{{\text{y}}}\)
 \(\gamma_{{{\text{LT}}}}\) :

Shear strain
 \(\gamma_{{\text{u}}}\) :

Shear strain corresponding to \(\tau_{{\text{u}}}\)
 \(\zeta\) :

Softening coefficient
 \(\zeta_{{{\text{FRP}}}}\) :

Softening coefficient accounting for the FRP reinforcement
 \(\rho_{{\text{f}}}\) :

FRP reinforcement ratio in the principal direction of tensile stresses
 \(\rho_{{\text{L}}}\) :

Longitudinal steel reinforcement ratio
 \(\rho_{{{\text{Lf}}}}\) :

Longitudinal FRP reinforcement ratio
 \(\rho_{{\text{S}}}\) :

Steel reinforcement ratio
 \(\rho_{{{\text{Se}}}}\) :

Steel reinforcement ratio accounting for the FRP reinforcement
 \(\rho_{{\text{T}}}\) :

Transverse steel reinforcement ratio
 \(\rho_{{{\text{Tf}}}}\) :

Transverse FRP reinforcement ratio
 \(\sigma_{1}\) :

Principal tensile stress in the FRPstrengthened RC membrane element
 \(\sigma_{2}\) :

Principal compressive stress in the FRPstrengthened RC membrane element
 \(\sigma_{{\text{D}}}\) :

Principal compressive stress in the concrete membrane element
 \(\sigma_{{\text{L}}}\) :

Longitudinal normal stress in the FRPstrengthened RC membrane element
 \(\sigma_{{\text{L}}}^{{\text{c}}}\) :

Longitudinal normal stress in the concrete membrane element
 \(\sigma_{{\text{R}}}\) :

Principal tensile strain in the concrete membrane element
 \(\sigma_{{\text{T}}}\) :

Transverse normal stress in the FRPstrengthened RC membrane element
 \(\sigma_{{\text{T}}}^{{\text{c}}}\) :

Transverse normal stress in the concrete membrane element
 \(\tau_{{{\text{cr}}}}\) :

Cracking shear stress
 \(\tau_{{{\text{LT}}}}\) :

Shear stress in the FRPstrengthened RC membrane element
 \(\tau_{{{\text{LT}}}}^{{\text{c}}}\) :

Shear stress in the concrete membrane element
 \(\tau_{{\text{u}}}\) :

Shear strength stress (ultimate)
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Bernardo, L.F.A., de Carvalho Filho, B.M.V. & Horowitz, B. Predicting the behavior of FRPstrengthened RC membrane elements with efficient rotatingangle softened truss model procedure. Mater Struct 54, 42 (2021). https://doi.org/10.1617/s1152702101631y
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Keywords
 Rotatingangle softened truss model
 Efficient solution procedure
 FRPstrengthened RC membranes
 Shear behavior