Effective Flexural Stiffness of Beams Reinforced with FRP Bars in Reinforced Concrete Moment Frames

Acknowledgments

Notation

A_f

area of tensile fiber-reinforced polymer (FRP) reinforcement (mm²);

a

shear span, equal to distance from center of concentrated load to center of support for beams under four-point loading (mm);

b

width of rectangular cross section (mm);

c

distance from extreme compression fiber to the neutral axis (mm);

d

distance from extreme compression fiber to the centroid of tension reinforcement (mm);

d_c

thickness of concrete cover measured from extreme tension fiber to the centroid of closest bars (mm);

E

modulus of elasticity;

E_c

modulus of elasticity of concrete (MPa);

E_f

modulus of elasticity of FRP (MPa);

EI_e

effective flexural stiffness (N · mm²);

EI_e/E_cI_g

effective flexural stiffness ratio;

E_s

modulus of elasticity of steel, E_s = 200 GPa;

E_sec

secant modulus of elasticity at specified concrete compressive strength $(f_c^{\mathrm{\prime }})$ (MPa);

E_sh

tangent modulus at the onset of strain hardening (MPa);

E_t

tangent modulus (MPa);

F

available force (N);

F_c

compression force in cross section (N);

F_peak

maximum force (N);

F_sd

equivalent force for stiffness degradation limit (N);

F_T

tension force in cross section (N);

f_c

available compressive strength of concrete (MPa);

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

specified cylinder compressive strength of concrete (MPa);

f_f

available stress in FRP;

f_fu

ultimate tensile strength of FRP (MPa);

f_s

available stress in steel (MPa);

f_su

peak ultimate stress (MPa);

f_y

specified yield strength (MPa);

h

overall height of the rectangular cross section (mm);

I

moment of inertia of section about the centroidal axis (mm⁴);

I_cr

moment of inertia of transformed cracked section (mm⁴);

I_e

effective moment of inertia (mm⁴);

I_g

gross moment of inertia of the rectangular section (mm⁴);

K

flexural stiffness of beam under four-point loading (N/mm);

k

postpeak decay factor;

l

span length (mm);

M

bending moment (N · mm);

M_a

maximum service load moment in member (N · mm);

M_cr

cracking moment (N · mm);

M_peak

maximum moment (N · mm);

M_sd

equivalent moment for the stiffness degradation limit (N · mm);

n

curve-fitting factor;

p

curve-fitting factor;

sign

function that extracts the sign of a real number;

s_max

maximum permissible center-to-center bar spacing for flexural crack control (mm);

ɛ_c

available strain in concrete;

ɛ_cu

maximum usable strain at extreme concrete compression fiber;

ɛ_f

available strain in FRP;

ɛ_fu

ultimate strain in FRP;

ɛ_s

available strain in steel;

ɛ_sh

strain at the onset of strain hardening;

ɛ_su

strain at peak stress;

ɛ_u

ultimate strain;

ɛ_y

yield strain;

ɛ₀

strain in extreme concrete compression fiber at specified concrete compressive strength $(f_c^{\mathrm{\prime }})$;

ρ

steel reinforcement ratio;

ρ′

compression reinforcement ratio;

ρ_f

FRP reinforcement ratio;

ρ_fb

FRP reinforcement ratio producing balanced strain conditions;

φ

curvature (1/mm); and

δ

midspan deflection (mm).

References