Theoretical Model for the Shear Strength of Prestressed Concrete Beams with FRP Tendons

Data Availability Statement

Acknowledgments

Notation

a

shear span;

b

width;

b_v,eff

shear effective width;

b_w

web width;

b₀

effective width of the diagonal strut in the beam web (deducting the diameter of ducts in post-tensioned reinforcement);

c

neutral axis depth;

c₀

neutral axis depth for the state of zero prestressing;

c_w

vertical projection of the length along the crack where the tensile stresses are extended;

d

effective depth;

d_s

effective depth of the passive reinfrocement;

d₀

effective depth of the cross section, d, but not less than 100 mm;

e

beam offset;

f_c

concrete compressive strength;

f_ct

tensile strength of concrete;

f_yw

yield strength of the transverse reinforcement;

h

overall depth of the concrete cross section;

h_f

flange thickness;

l_av

available bonded length between the critical shear crack and the free end of the beam;

l_req

anchorage length required to develop the tensile force of the prestressing reinforcement at the cracked section.;

s

stirrups spacing;

s_cr

distance of the first crack to the zero bending moment point;

u

total perimeter of prestressing reinforcement;

v_c

nondimensional shear strength contribution of concrete;

v_cc

nondimensional shear resisted in the uncracked region subjected to flexural compression forces;

v_w

nondimensional shear transferred across the web by aggregate interlock and residual stresses;

v_l

nondimensional shear resisted by the longitudinal reinforcement;

v_s

nondimensional shear resisted by the transverse reinforcement;

x

coordinate in the longitudinal direction of the beam;

y

vertical coordinate from the top fiber of the concrete element;

z

lever arm;

A_c

concrete cross-sectional area;

A_sw

area of the transverse reinforcement;

A_fw

area of the FRP transverse reinforcement;

C

compression force in the concrete compression chord;

E_cm

modulus of elasticity of concrete;

E_f

modulus of elasticity of the FRP reinforcement;

F_p

tensile force of prestressing reinforcement at the critical crack;

G_f

concrete fracture energy;

I

moment of inertia for the section;

K_ad

factor for nonslender beams;

K_c

relative neutral axis depth (c/d);

K_p

prestressing factor;

M

bending moment;

M_cr

cracking moment;

P_red

reduced prestressing force;

R_t

ratio between the principal tensile stress and the tensile strength at the critical point;

S

first moment of area of the section;

T

tensile force in the longitudinal tensile reinforcement;

V

shear force;

V_arch

shear strength due to the arch effect;

V_beam

shear strength due to the beam effect;

V_c

shear strength contribution of concrete;

V_cc

shear resisted in the uncracked region subjected to flexural compression forces;

V_c,min

minimum shear strength contribution of concrete;

V_w

shear transferred across the web by aggregate interlock and residual stresses;

V_frp

shear FRP reinforcement contribution;

V_l

shear resisted by the longitudinal reinforcement;

V_s

shear resisted by the transverse reinforcement;

V_u

shear strength;

V_u,B

shear strength for bond loss;

V_u,max

shear strength limit (diagonal compression);

V_u,U

shear strength for uncracked sections;

α

inclination angle of shear reinforcement or ties;

α_e

modular ratio (E_f/E_c);

α_l

degree of prestressing force transferred at the considered section;

β

horizontal projection of the inclined crack;

β_w

factor affecting the effective depth to calculate the bending moment due to the aggregate interlock and residual stresses when applying equilibrium;

ɛ_fw,m

mean strain of the stirrups at shear failure;

ɛ_fw,max

maximum strain of the stirrups at shear failure;

ɛ_fw,u

ultimate strain of the stirrups;

ζ

size effect;

θ

inclination angle of the strut;

ν₁

strength reduction factor for concrete cracked in shear;

ρ_fw

FRP transverse reinforcement ratio;

ρ_l

longitudinal reinforcement ratio;

ρ_p

longitudinal active reinforcement ratio;

σ₁, σ₂

principal stresses;

σ_cp

mean compressive stress due to the prestressing force;

${\sigma }_{\mathrm{cc}}^{\mathrm{\prime }}$

effective compressive stress in the concrete (considering the reduction due to compression reinforcement);

σ_fw,m

mean stress of the stirrups at shear failure;

σ_x

normal stress in the longitudinal direction;

σ_y

normal stress in the transverse direction;

τ

shear stress;

τ_max

average bond strength along the anchorage length; and

ΔVc

stirrup confinement factor.

References