Softened Membrane Torsional Model for GFRP–Reinforced Concrete Bridge Box Girders

Data Availability Statement

Acknowledgments

Notation

A_o

area enclosed by the centerline of the shear flow path; $A_o=A_c-0.5p_c{t}_d+t_d^2$;

A_l

total cross-sectional area of the longitudinal GFRP bars;

A_t

cross-sectional area of one transverse GFRP bar;

A_c

cross-sectional area enclosed by the outer perimeter of the concrete;

d

vertical distance of the spiral link;

E_c

elastic modulus of concrete;

E_ft

elastic modulus of transverse GFRP bars;

E_fl

elastic modulus of the longitudinal GFRP bars;

$f_c^{\mathrm{\prime }}$

cylinder compressive strength of concrete;

$f_{\mathrm{cr}},{\epsilon }_{\mathrm{cr}}$

cracking stress and strain of concrete;

f_fl

tensile strength of the longitudinal GFRP bars;

f_ft

tensile strength of the transverse GFRP bars;

f_ftu

ultimate tensile strength of the GFRP stirrup's straight portion;

f_ft,(bend)

tensile strength of the GFRP stirrup at the bent portion;

f_ftu,(bend)

ultimate tensile strength of the GFRP stirrup at the bent portion;

k_1c

ratio of the average compressive stress to the peak compressive stress in the concrete strut, considering the tensile stress of concrete;

k_1t

ratio of the average tensile stress to the peak tensile stress in the concrete struts;

p_c

perimeter of the outer concrete cross section;

p_h

perimeter of the centerline of stirrups;

p_o

perimeter of the centerline of shear flow; p_o = p_c − 4t_d;

Q

variable as defined in Eq. (9);

q

shear flow;

s

spacing of the GFRP web reinforcement;

T

torsional moment;

T_u

ultimate torsional moment;

T_cr

cracking torsional moment;

t

wall thickness of the hollow section;

t_d

thickness of the shear flow zone;

α₂

angle of applied principal compressive stress (two-axis) with respect to the longitudinal GFRP bars (l-axis);

β

deviation angle; $\beta =0.5{\tan }^{-1}({\gamma }_{21}/({\epsilon }_2-{\epsilon }_1))$;

${\epsilon }_o$

concrete cylinder strain corresponding to the peak cylinder strength $f_c^{\mathrm{\prime }}$;

${\epsilon }_1,{\epsilon }_2$

average biaxial strain in one direction and two directions, respectively;

${\overline{\epsilon }}_1,{\overline{\epsilon }}_2$

average uniaxial strain in the one-direction and the two-direction, respectively;

${\overline{\epsilon }}_{1s},{\overline{\epsilon }}_{2s}$

uniaxial surface strain in the one-direction and the two-direction, respectively;

${\epsilon }_l,{\epsilon }_t$

average biaxial strain in the l-direction and the t-direction of the GFRP reinforcement, respectively;

${\overline{\epsilon }}_l,{\overline{\epsilon }}_t$

average uniaxial strain in the l-direction and the t-direction of the GFRP reinforcement, respectively;

${\overline{\epsilon }}_{\mathrm{ft}}$

average uniaxial strain of the GFRP stirrups;

${\overline{\epsilon }}_{\mathrm{fl}}$

average uniaxial strain of the longitudinal GFRP bars;

${\overline{\epsilon }}_{\mathrm{spiral}}$

average uniaxial strain in the GFRP spiral direction;

${\epsilon }_{\mathrm{ftu}}$

ultimate strain of the GFRP stirrups;

${\epsilon }_{\mathrm{flu}}$

ultimate strain of the longitudinal GFRP bars;

${\epsilon }_{\mathrm{sf}}$

average strain of the GFRP bars, considering Hsu/Zhu ratios;

γ₂₁

average shear strain in the 2–1 coordinate;

γ_lt

average shear strain in the l−t coordinate of GFRP reinforcement;

${\sigma }_1^c,{\sigma }_2^c$

average normal stresses of concrete in the one-direction and two-direction, respectively;

σ_l, σ_t

applied normal stresses in the l-direction and t-direction of the GFRP reinforcement, respectively;

${\tau }_{21}^c$

average shear stress of concrete in the 2–1 coordinate;

τ_lt

applied shear stresses in the l−t coordinate of the GFRP reinforcement;

ρ_l, ρ_t

longitudinal and transverse GFRP reinforcement ratios, respectively. ρ_l = A_l/p_ot_dand ρ_t = A_t/st_d;

(ν₁₂)_GFRP

modified Hsu/Zhu ratio used in the SMMT-H-GFRP model;

θ

angle of twist per unit length;

θ_cr

cracking angle of twist per unit length;

θ_u

ultimate angle of twist per unit length;

ϕ

curvature of the concrete struts along the two-direction;

ζ

softened coefficient of concrete in compression; and

φ

inclination angle of the spiral link.

References