Width Effect in FRP–Concrete Debonding Mechanism: A New Formula

Acknowledgments

Notation

A_cord

area of a steel cord comprising five steel filaments;

b

width of the concrete prism (Fig. 1);

b_f

width of the FRP strip (Fig. 1), which is also the width of the bonded area in single-lap shear tests;

b_f,actual

actual width of the steel FRP composite obtained as the average of three measurements along the strip;

C_F

calibration factor for the fracture energy used in Neubauer and Rostasy (1997), Brosens and Van Gemert (1999), and Brosens (2001);

D_max

maximum diameter of the aggregates in the concrete mix;

E_f

Young's modulus of the bare fibers;

E_FRP

Young's modulus of the FRP composite;

$E_{f,\mathrm{SRP}}^{\mathrm{HD}}$

Young's modulus of the steel FRP composite referred to the area of the HD fiber sheet ASTM D3039 (ASTM 2008);

f_h

surface tensile strength of concrete [EN 1542 (CEN 1999)];

f_ctm

tensile strength of concrete [EN 12390-6 (CEN 2009)];

f_f,u

average tensile strength of the steel FRP composite strip;

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

cylindrical compressive strength of concrete;

$f_{f,u}^{\mathrm{A}}$

tensile strength of the steel fiber sheet provided by the manufacturer (Kerakoll 2018). A specifies the density of the sheet;

$f_{f,u}^{\mathrm{HD}}$

tensile strength of the high-density steel fiber sheet provided by the manufacturer (Kerakoll 2018);

g

global slip, that is, average of the readings of the LVDT a and b that are mounted at the beginning of the bonded area. An alternative name for g is loaded-end slip;

g₁

value of the global slip that defines the beginning of the range of values of g used to compute the average of the load, which corresponds to P_crit;

g₂

value of the global slip that defines the end of the range of values of g used to compute the average of the load, which corresponds to P_crit;

G_F

interfacial fracture energy that corresponds to the area under the τ_xy(s) curve;

${\overline{G}}_F$

average of the fracture energy for one specimen obtained using a fitting curve for the strain profile and ten DIC images within the range [g₁,g₂];

$G_F^{P_{\mathrm{crit}}}$

fracture energy back-calculated from Eq. (17), substituting P_theor with P_crit for each specimen;

k_b

symbol used for the width effect factor in Neubauer and Rostasy (1997), Brosens and Van Gemert (1999), and Brosens (2001);

k_w

symbol used for the width effect factor by the authors in this paper;

k_c

parameter used in Brosens (2001) to take into account the effect of concrete surface condition on the interfacial fracture energy;

L_eff

effective bond length that corresponds to the length of the fully established stress transfer, that is, the minimum bonded length to obtained the bond capacity P_theor;

${\overline{L}}_{\mathrm{eff}}$

average of the effective bond length for one specimen obtained using a fitting curve for the strain profile and ten DIC images within the range [g₁,g₂];

$L_{\mathrm{eff}}^{C-T}$

effective bond length proposed by Chen and Teng (2001);

$\mathcal{l}$

length of the bonded area;

P

applied load to the FRP strip in single-lap shear tests;

P_crit

plateau load, which corresponds to the bond capacity obtained experimentally from single-lap shear test as the average of the applied load P within the range [g₁,g₂];

$P_{\mathrm{crit}}^{{\epsilon }_{yy}}$

bond capacity obtained as the integral of the longitudinal strain over the width of the composite at the loaded end (see Eq. 18);

$\overline{P_{\mathrm{crit}}^{{\epsilon }_{yy}}}$

average of the bond capacity $P_{\mathrm{crit}}^{{\epsilon }_{yy}}$ considering several values of y near the loaded end and five points of the load response (Table 7);

${\overline{P}}_{\mathrm{crit}}^{\mathrm{Y},\mathrm{C},\mathrm{E}}$

average of P_crit for specimens characterized by the same width (Y), the same face to which the composite is applied (C), and the same loading rate (E) (Table 4);

P_theor

theoretical bond capacity (or load-carrying capacity) based on the Mode-II fracture approach proposed in Täljsten (1996) [see Eq. (17)];

${\overline{P}}_{\mathrm{theor}}$

theoretical bond capacity when $G_F={\overline{G}}_F$. There are three values of ${\overline{P}}_{\mathrm{theor}}$ for each specimen corresponding to the three fitting functions of the strain profile used in Carloni et al. (2017a);

P_theor,WU

symbols used for the maximum bond force (bond capacity or load-carrying capacity) in Wu and Jiang (2013);

P_u

symbols used for the maximum bond force (bond capacity or load-carrying capacity) in Chen and Teng (2001);

P*

peak load in the response of single-lap shear tests;

$P_{\mathrm{theor}}^{{\mathrm{\Gamma }}_F}$

bond capacity expressed in terms of the width factor b_w proposed by the authors and employing Eq. (17) with G_F = Γ_F;

R_cm

cubic compressive strength of concrete [EN 12390-3 (CEN 2001)];

s

interfacial slip;

t_FRP

thickness of the FRP composite;

t_FRP,actual

actual thickness of the steel FRP composite obtained as the average of three measurements along the strip;

$t_{f,\mathrm{HD}}^{\ast }$

equivalent thickness of the high density (HD) steel fiber sheet, which is equal to 0.254 mm;

$t_{f,\mathrm{UHD}}^{\ast }$

equivalent thickness of the ultrahigh density (UHD) steel fiber sheet, which is equal to 0.381 mm;

$t_{f,A}^{\ast }$

equivalent thickness of the steel fiber sheet. A specifies the density of the sheet. For example, A = HD for high density;

T_u,max

symbols used for the maximum bond force (bond capacity or load-carrying capacity) in Neubauer and Rostasy (1997) [see Eq. (1)];

w_c

backward displacement of the concrete prism measured by LVDT c;

w_d

backward displacement of the concrete prism measured by LVDT d;

${\overline{w}}_c$

average of w_c within the range [g₁,g₂];

${\overline{w}}_d$

average of w_d within the range [g₁,g₂];

x

Cartesian coordinate along the width of the composite (Fig. 1);

y

Cartesian coordinate along the direction of the fibers of the composite (Fig. 1);

β_p

symbol used for the width effect factor in Chen and Teng (2001);

β_w

symbol used for the width effect factor in Lu et al. (2005);

${\kappa }_w^{\mathrm{WU}}$

symbol used for the width effect factor in Wu and Jiang (2013);

${\kappa }_w^{\mathrm{LIN}}$

symbol used for the width effect factor in Lin et al. (2017);

ɛ_f,u

ultimate deformation of the steel fiber sheet provided by the manufacturer (Kerakoll 2018);

ɛ_yy

longitudinal strain in the composite in the direction of the fibers;

ɛ_xy

shear strain in the composite in the plane of the composite;

ɛ_max

maximum value of ɛ_yy fitting function;

Γ_F

fracture energy expressed in terms of the properties of concrete (Eq. 24); and

τ_xy

interfacial shear stress in debonding Mode-II problems that assume a fictitious zero-thickness interface.

References