Axial Stress–Strain Model for FRCM Confinement of Masonry Columns

Acknowledgments

Notation

A_c

net cross-sectional area of the compression member equal to bxh (mm²);

A_e

area of the effectively confined concrete (mm²);

a, b, c, d, e, f, g

constants of the polynomial equation of the simplified AOM (dimensionless);

c_b

spacing between the bars (mm);

c_bs

spacing between the bars and the border of the cross section (mm);

D

diagonal of the cross section of the column assumed equal to the square root of the sum of the squares of width and length of the cross section itself (mm);

E_f

slope of the cracked behavior of the FRCM in tensile stress state; approximately equal to Young modulus of the fabric (MPa);

E_m

Young's modulus of the unconfined column (MPa);

e

expected value in the χ² test;

f(x)

PDD of the variable x (dimensionless);

f_c

generic axial stress of the FRCM-confined column related to ɛ_c (MPa);

f_c,mat

average compressive strength of the FRCM matrix (MPa);

f_ccm,ex

experimental compressive strength of the FRCM-confined masonry column (MPa);

f_ccm,th

theoretical compressive strength of the FRCM-confined masonry column (MPa);

f_l

confining pressure (MPa);

f_l,eff

effective confining pressure (MPa);

f_m

average compressive strength of the unconfined masonry column (MPa);

f_mu

characteristic (i.e., calculated as the 5% fractile) compressive strength of the unconfined masonry column (MPa);

f_mud

design compressive strength of the unconfined masonry column calculated as f_mu divided by a safety factor equal to 2.0 in agreement with the Eurocode provisions (MPa);

g_m

mass density of the masonry (kg/m³);

h

height of the cross section of the column (mm);

k′

dimensionless confinement effectiveness coefficient;

k′′

reduction factor for the ultimate axial strain of the FRCM-confined masonry column (dimensionless);

k_b

efficiency factor computed as =(Ae/Ac)(h/b)²;

k_H

cross section shape factor (dimensionless);

k_mat

dimensionless coefficient affecting the confining pressure to achieve the FRCM matrix effect;

n_b

number of bars introduced in the masonry (dimensionless);

n_f

number of FRCM layers (dimensionless);

o

observed value in the χ² test;

R²

coefficient of determination calculated by comparing experimental and analytical axial stress–strain curves (dimensionless);

r_c

radius of curvature after the rounding of the edges (mm);

t_f

equivalent thickness of the fabric (mm²/mm);

t_mat

total thickness of mortar (mm);

w

width of the cross section of the column (mm);

x

f_ccm,ex/f_ccm,th (dimensionless);

α₁, α₂, α₃, α_4, α₅, α₆, α₇, α₈

dimensionless coefficient assumed equal to 0.5, 1, 1, 1.81, 0.7, 2.2, and 0.5 respectively;

γ_f

fabric reduction factor (dimensionless);

γ_k

compressive strength safety factor (dimensionless);

γ_m

masonry reduction factor (dimensionless);

ɛ_c

generic axial strain of the FRCM-confined column (dimensionless);

${\epsilon }_c^{\mathrm{\prime }}$

compressive strain of the unconfined concrete that corresponds to f_c′ (dimensionless);

ɛ_cc

ultimate axial strain of the FRCM-confined concrete column according to ACI-549.4R (ACI 2013) (dimensionless);

ɛ_ccm

peak axial strain of the FRCM-confined masonry column (dimensionless);

ɛ_ccmu

ultimate axial strain of the FRCM-confined masonry column (dimensionless);

ɛ_ccu

ultimate axial compressive strain of FRCM-confined concrete according to ACI-549.4R (ACI 2013) (dimensionless);

ɛ_fe

effective tensile strain level in the FRCM limited to the minimum of the design tensile strain of the FRCM itself and 0.012 according to ACI-549.4R (ACI 2013) (dimensionless);

ɛ_fu

rupture strain of FRCM in tensile stress state; approximately equal to the ultimate strain of the fabric (dimensionless);

ɛ_fud

design value of the ultimate tensile strain of the FRCM (dimensionless);

μ

mean value of the x variable (dimensionless);

ρ_f

geometrical percentage of FRCM fabric (dimensionless);

ρ_mat

geometrical percentage of FRCM matrix (dimensionless);

σ

SD related to the x variable (dimensionless): and

χ²

Pearson's chi-squared test (dimensionless).

References