Shear Capacity Model with Variable Orientation of Concrete Stress Field for RC Beams Strengthened by FRP with Different Inclinations

Notation

a

shear span;

b_w, b_f

web widths of concrete cross section and FRP sheet on the element face in tension;

d, d_f

beam and FRP effective depths;

E_f, E_sw

steel and FRP elastic modulus;

$f_{cm}^{\mathrm{\prime }},f_{cd}^{\mathrm{\prime }}$

mean and design reduced cylinder compressive strength of concrete;

f_bd

design resistance of the adhesion between FRP and concrete;

f_ck, f_ctm

characteristic cylinder compressive and mean concrete tensile strength of concrete;

f_fe, f_fed, f_fu, f_fud

mean, and design effective and ultimate stresses of FRP;

f_sy, f_syw,

yielding stresses of longitudinal steel reinforcement and steel stirrups;

h_w

beam cross-section height;

k_v, k₁, k₂

bond-reduction coefficient and modification factors;

L_e, L_max

effective and maximum bond length;

R

reduction coefficient (ratio of effective average stress or strain in the FRP sheet to its ultimate strength or elongation);

r

reduction factor as a function of the maximum steel stirrup strain ɛ, accounting for the variation of the stirrup strain along the crack;

R₁, R₂, R₃, R₄, R₅, R₆

effectiveness coefficient based on FRP sheet fracture failures (Mode 1, 5), debonding from concrete surface (Mode 2, 6), shear crack control (Mode 3), and peeling off (Mode 4);

r_c

corner radius of the section to be wrapped;

s_f, t_f, w_f

spacing, thickness, and width of the FRP strip;

s_w

spacing of the steel stirrups;

V, V_Rd, V_n

external, resisting, and nominal shear forces;

V_ACI, V_CNR

shear strength evaluated by code models;

v, v_c, v_s, v_f

nondimensional shear strength and its contributions by concrete, steel stirrups, and FRP reinforcement;

V_c, V_s, V_f

shear strength contributions: concrete, steel stirrups, FRP reinforcement;

v_exp, v_the

experimental and theoretical nondimensional shear strengths;

V_exp, V_the

experimental and theoretical shear strengths;

z

inner lever arm;

α, β

angle of steel and FRP transverse reinforcement;

β_L

bond length coefficient;

β_w

coefficient of the FRP-to-concrete width ratio;

γ_f

partial safety factor of FRP;

Γ_Fd

design value of specific fracture energy;

ɛ_fe, ɛ_fu

effective and nominal (ultimate) FRP strains;

ɛ_fe,sd

effective strain in the direction of transverse steel reinforcement;

ɛ_syw

yield strain of steel stirrup;

θ

angle between concrete stress field and member axis (yield line inclination);

λ

normalized maximum bond length;

ρ_sl, ρ_slw

chord and web longitudinal geometrical ratios of steel reinforcement;

ρ_sw, ρ_fw

transverse geometrical ratio of steel and fiber transverse reinforcement;

σ_f,max

maximum stress along the bond length;

${\tilde{\sigma }}_{cw}$

nondimensional stress of the web concrete;

${\tilde{\sigma }}_{fw}$, ${\tilde{\sigma }}_{sw}$

nondimensional tensile stress of transverse FRP, stirrups;

ϕ

rebar diameter;

φ

angle between the FRP reinforcement direction and steel stirrups;

ψ

fictitious angle taking into account the inclination of steel stirrups and FRP reinforcement;

ψ_f

additional reduction factor;

υ

efficiency factor to take into account biaxial state of stress of web concrete; and

ω_fw, ω_sw

mechanical ratio of transverse FRP, and stirrups.

References