Numerical Simulations and Simplified Design Approaches for Large-Rupture-Strain FRP-Strengthened Reinforced Concrete Beams under Impact

Acknowledgments

Notation

A_sc

area of the compression reinforcement;

A_st

area of the tensile reinforcement;

A_f

area of the FRP;

a

height of equivalent concrete stress block;

a_s

distance from the beam top to the center of the compression reinforcement;

b

beam cross section width;

b_c

width of the loading plate;

c

height of concrete in the compression zone;

d

effective height of the beam section;

E_D

dissipation energy consisting of deformation energy of the beam;

E_cd

dynamic elastic modulus of concrete;

E_f

elastic modulus of FRP;

E_i

inputted impact energy;

E_k

kinetic energy of the hammer;

E_L

energy loss due to hammer–beam collision;

E_P

potential energy of the hammer and beam;

E_s

static energy-dissipating capacity;

E_sc

elastic modulus of the compressive steel bar;

E_st

elastic modulus of the tensile steel bar;

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

peak stress of the unconfined concrete;

f_c1

peak stress of the confined concrete;

$f_{cd}^{\mathrm{\prime }}$

dynamic compressive strength of concrete;

f_la

actual ultimate confining stress;

f_scd

dynamic compressive stress of steel;

f_std

dynamic tensile stress of steel;

f_utd

ultimate strength of the tensile rebar;

f_ytd

dynamic yield stress of the tensile rebar;

g

acceleration of gravity;

h

height of the beam;

k

elastic depth to the cracked neutral axis;

L

beam span;

L_p

plastic hinge length;

L_n

distance between the two loading points;

m_be

equivalent mass of the beam in elastic deformation stages;

m_bp

equivalent mass of the beam in plastic deformation stages;

m_h

mass of the hammer;

M_y,f

contribution of FRP for the yield moment;

M_y,st

contribution of tensile reinforcements for the yield moment;

n

ratios of dynamic elastic moduli of the tensile reinforcement to that of concrete E_cd;

n′

ratios of dynamic elastic moduli of compressive reinforcement to that of concrete E_cd;

n_f

ratios of dynamic elastic moduli of FRP to that of concrete E_cd;

P_max

maximum impact load;

P_u

ultimate resistance;

P_y

bearing capacity at the yield point;

t_f

thickness of the FRP;

v_c

velocities of a drop hammer after collision;

v_i

initial velocity of the hammer;

w

wrapping length of transverse FRP strips;

β₁

height factor of equivalent concrete stress block;

${\dot{\epsilon }}_c$

strain rate of concrete;

ɛ_c1

strain corresponding to the peak stress of confined concrete;

ɛ_f

strain of the FRP;

${\dot{\epsilon }}_f$

FRP strain rate;

ɛ_fu

rupture strain of the longitudinal FRP laminate;

ɛ_sc

strain of the compressive steel bar;

${\dot{\epsilon }}_{sc}$

strain rate of the compressive steel;

${\dot{\epsilon }}_{st}$

strain rate of the tensile steel;

ɛ_su

ultimate strain of the tensile rebar;

ɛ_yd

dynamic yield strain of reinforcement;

ɛ_yt

yield strain of the tensile rebar;

δ

midspan deflection;

$\dot{\delta }$

deflection rate of the midspan section;

δ_max

maximum midspan deflection;

δ_u

ultimate deflection;

δ_re

residual deflection under impact;

δ_y

deflection at the yield point;

$\dot{\varphi }$

curvature rate of the midspan section;

ϕ_s

curvature corresponding to the ultimate state;

ϕ_y

curvature corresponding to the yield state;

λ

factor depending on the load distribution;

ρ

tensile reinforcement ratio;

ρ′

compression reinforcement ratio;

ρ_b

density of the beam;

ρ_f

FRP reinforcement ratio; and

σ_f

tensile stress of FRP.

References