Predicting Strength of FRP-Reinforced Concrete Deep Beams with Stirrups Using the Indeterminate Strut–Tie Method

Data Availability Statement

Acknowledgments

Notation

A1–A4

abbreviations for approaches to compute ζ;

A_bar,f

area of one flexural rebar;

A_bar,v

area of one stirrup leg;

a

length of the shear span;

a/d

shear span-to-depth ratio;

b

beam width;

c

distance from extreme compression fiber to the neutral axis;

d

effective depth;

d_T

distance between the centroid of the rebars and the extreme tensile fiber;

$E_{\mathrm{c}}^s$

modulus of elasticity of concrete based on CSA A23.3:19 (CSA 2019);

$E_{\mathrm{c}}^t$

initial tangential modulus of concrete;

E_f

elastic modulus of flexural reinforcements;

$E_n^a$

elastic modulus of each strut calculated in the nth cycle, (a − 1)th subcycle and used in the nth cycle, ath subcycle during the IST analysis;

E_v

elastic modulus of shear reinforcements;

f_c

concrete compressive stress;

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

concrete compressive strength (maximum f_c);

f_cu

limited strength of concrete struts;

f_fu

ultimate tensile strength of the flexural reinforcements;

f_vu,bent

ultimate tensile strength of the bent portion of the stirrups;

g

parameter used in Approach A2;

h1, h2, h3

abbreviations for methods to compute h_C;

h

beam height;

h_C

depth of the nodes on the compression side;

h_T

depth of the nodes on the tension side;

jd

lever arm between the resultant compressive and tensile forces, equal to d − h_C/2;

k

1.0 for ${\epsilon }_c/\zeta {\epsilon }_0\le 1$, curve control parameter for the concrete compressive stress–strain model by Thorenfeldt et al. (1987);

k_size

size effect parameter from Nehdi et al. (2008), used to compute ζ in Approach A2;

l_C

horizontal size of the nodes on the compression side;

l_T

horizontal size of the nodes on the tension side;

n

$0.8+f_c^{\mathrm{\prime }}(\mathrm{MPa})/17$ (if not softened), curve control parameter for the concrete compressive stress–strain model by Thorenfeldt et al. (1987);

n_E

$E_f/E_{\mathrm{c}}^s$;

P

applied load;

P_n

load applied to the nth cycle during the IST analysis;

P_inc.

load increased in each cycle during the IST analysis;

P_Predict

predicted strengths;

P_test

tested strengths;

P_predict/P_test

ratio between predicted and tested strengths;

s

stirrups spacings;

S1, S2

abbreviations for methods to soften concrete compression stress–strain models;

S-1 to S-13

labels for struts;

${\mathit{S}}_n^a$

stiffness matrix obtained in the nth cycle, (a − 1)th subcycle and used in the nth cycle, ath subcycle during the IST analysis;

ST1, ST2, and ST3

abbreviations for Strut-and-Tie models used in this work;

T-1 to T-6

labels for ties;

w_sC

width of the strut on the compression side;

w_sT

width of the strut on the tension side;

x

variable representing the distance from any point to the neutral axis used in Method h3;

β_s

parameter from Nehdi et al. (2008), used to compute ζ in Approach A2;

γ_c

concrete density (kg/m³);

${\epsilon }_0$

concrete compressive strain corresponding to $f_c^{\mathrm{\prime }}$;

${\epsilon }_1$

principal tensile strain;

${\epsilon }_c$

concrete compressive strain;

${\epsilon }_F$

tensile strain in the tie bar located closest to the tension face of the beam and inclined at θ_strut to the strut;

${\epsilon }_f$

strain in the flexural tie of the tie sets inside the projection of the specific strut to compute $({\epsilon }_f+{\epsilon }_v{)}_{\max }$;

${\epsilon }_{f\mathrm{\_}\max }$

maximum tensile strain in the flexural FRP ties inside the projection of the strut;

${\epsilon }_s$

compressive strain in the strut, in negative;

${\epsilon }_{T\mathrm{\#}}$

strain in the tie with label T-#;

${\epsilon }_{\mathrm{Top}}$

strain of the concrete extreme compressive fiber;

${\epsilon }_v$

strain in the vertical tie of the tie sets inside the projection of the specific strut to compute $({\epsilon }_f+{\epsilon }_v{)}_{\max }$;

ζ

softening factor;

θ_strut

incline of the strut, measured as the smallest angle between the strut and the adjoining ties; and

ρ_f

flexural reinforcement ratio.

References