Symmetric and Asymmetric Strengthening of Two-Span RC Beams Using FRCM Systems

Data Availability Statement

Notation

A_bar

cross-sectional area of tensile steel reinforcement (mm²);

A_f

equivalent area of fabric per unit width (mm²/m);

A_st

cross-sectional area of stirrups (mm²);

b_b

cross section beam width (mm);

b_f

width of the strengthening system (mm);

c

cohesive component for normal concrete in the softening zone (MPa);

d_asc

depth of the ascending branch of concrete in the compression side (mm);

d_f

depth of the strengthening system measured from the compressive side (mm);

d_soft

depth of the concrete wedge in the softening zone (mm);

E_c

modulus of elasticity of concrete (MPa);

E_cr

cracked tensile modulus of FRCM composite (MPa);

E_f

modulus of elasticity of the fabric (MPa);

E_s

modulus of elasticity of steel;

E_sh

strain hardening of the tensile steel bars (MPa);

E_un

uncracked tensile modulus of FRCM composite (MPa);

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

concrete compressive strength (MPa);

f_frac

fracture stress of steel bars (MPa);

f_fu

ultimate tensile strength of FRCM composite (MPa);

f_y

yield stress of steel bars (MPa);

L_per

perimeter of the tensile steel bar (mm);

L_per-f

perimeter of the strengthening layer (mm);

L_soft

length of the concrete wedge (mm);

MRR

moment redistribution ratio (%);

m

parameter characterizing the concrete softening across the wedge, assumed = 0.8;

M_elastic

elastic bending moment of the sagging span (kN · m);

M_exp

experimental bending moment calculated in the sagging span (kN · m);

$M_{\exp }^{\mathrm{fail}}$

experimental bending moment at failure (kN · m);

$M_{\mathrm{pr}}^E$

predicted failure moment calculated in the east span (kN · m);

$M_{\mathrm{pr}}^W$

predicted failure moment calculated in the west span (kN · m);

P_asc

force in the ascending zone (kN);

P_rein

force in the steel bars (kN);

P_soft

force in the softening wedge of concrete (kN);

P_stren

force in the strengthening system (kN);

P_u

ultimate load-carrying capacity (kN);

$P_y^E$

yielding load in the east span (kN);

$P_y^H$

yielding load at the hogging section (kN);

$P_y^W$

yielding load in the west span (kN);

S_slide

sliding capacity of the concrete wedge (mm);

S_soft

sliding of the concrete wedge (mm);

s_sp

spacing between stirrups along the beam length (mm);

α

angle of the concrete wedge (degree);

σ_soft

concrete stress in the softening wedge zone (MPa);

β^f

strengthening ratio (%);

δ_debond

debonding slip of the strengthening system (mm);

${\delta }_D^{\mathrm{glob}}$

global ductility index;

${\delta }_D^{\mathrm{local}}$

local ductility index for each span in the asymmetric scheme;

δ_max

maximum slip in the tensile steel bars when no more shear stresses could be transferred (assumed = 15 mm);

Δ_asc

slip of the top portion of concrete in the ascending zone corresponding to concrete strain ${\epsilon }_{\mathrm{asc}}$ (mm);

${\mathrm{\Delta }}_u^W$

midspan deflection at ultimate load in the west span (mm);

${\mathrm{\Delta }}_u^E$

midspan deflection at ultimate load in the east span (mm);

${\mathrm{\Delta }}_y^W$

midspan deflection at yielding load in the west span (mm);

${\mathrm{\Delta }}_y^E$

midspan deflection at yielding load in the east span (mm);

${\mathrm{\Delta }}_y^H$

midspan deflection in the span where failure occurred corresponding to the yielding load in the hogging section (mm);

${\mathrm{\Delta }}_y^{\mathrm{last}}$

midspan deflection corresponding to the last yielding load (mm);

Δ_rein

slip of the tensile steel bars (mm);

Δ_frac

slip at fracture of the tensile steel bars (mm;

Δ_stren

slip in the strengthening system (mm);

Δ_yield

slip in the tensile steel bars at yielding (mm);

${\epsilon }_{\mathrm{asc}}$

concrete strain corresponding to the peak compressive strength, $f_c^{\mathrm{\prime }}$ (mm/mm);

${\epsilon }_{fu}$

ultimate tensile strain in the strengthening composite (mm/mm);

${\epsilon }_{\mathrm{soft}}$

concrete strain in the softening wedge zone (mm/mm);

η

enhancement ratio in the moment capacity;

λ_el and a_el

elastic parameters that characterize the load–slip relationship of steel reinforcement up to the yielding stage;

λ_sh and a_sh

parameters that determine the load–slip relationship of steel reinforcement in the strain hardening stage up to failure;

ρ_f

reinforcement ratio of the strengthening systems (%);

ρ_s

reinforcement ratio of the tensile steel bars (%);

σ_lat

parameter reflecting the lateral confinement provided by the stirrups in the softening wedge zone (MPa);

τ_max

shear capacity at the steel–concrete interface (MPa);

φ_f

aspect ratio of the strengthening system;

${\mathrm{\varnothing }}_{\exp }^{\mathrm{fail}}$

experimental curvature calculated at failure (1/mm);

${\mathrm{\varnothing }}_{\mathrm{pr}}^W$

curvature predicted at failure in the west span (1/mm); and

${\mathrm{\varnothing }}_{\mathrm{pr}}^E$

curvature predicted at failure in the east span (1/mm).

References