PSI - Issue 64

Francesco Focacci et al. / Procedia Structural Integrity 64 (2024) 1557–1564 Author name / Structural Integrity Procedia 00 (2019) 000–000

1559

3

where u f ( z ) and u c ( z ) are the longitudinal displacements of the fibers and matrix at the interface, respectively.

a

b

c

d

q ( z ) = λ q 0 ( z )

d G

M t +d M t

M 1 +d M 1

M 1 +d M 1

M t

M 1

M 1

H

N t +d N t

N 1 +d N 1

N +d N

N t

N 1

N

V t +d V t

V 1 +d V 1

V t +d V t

V t

V 1

V t

z

d z

τ ( s )

τ ( s )

d z

N 2 +d N 2

N 2

N +d N

N

d z

d z

Fig. 1. (a) Composite beam; (b) internal forces in the composite beam; (c) internal forces in the layers; (d) internal forces in the layers with N t =0.

Un-cracked cross-sections In the un-cracked portions of the beam, the displacement u c ( z ) is continuous and differentiable, whereas u f ( z ) is continuous and differentiable at any z . The introduction of the curvature χ = χ( z ) and the axial force N = N (z) in the derivative of Eq. (2) yields:

d

N

( )

(

)

(3)

s z

−χ − n H d

=

d

z

E A

f

f

where A f and E f are the cross-sectional area and the elastic modulus of fibers, respectively, H is the height of the concrete beam, and d n is the neutral axis depth (Fig. 2a). Eq. (3) implies that fibers are linear elastic and the concrete beam cross-sections remain plane during the loading process.

d z

a

b

d c

w

|σ cw |

d G

d n d G

M 1

M 1

H

H

χ

∆ϕ

x

x

G

G

N

N

G

ε

|σ c |

G

G

b ( ζ )

b ( ξ )

ξ

ζ

y

y

ζ

ζ

ξ

ξ

| s r |

| s l |

∆ϕ ( H - d c )

Fig. 2. (a) Un-cracked cross-section of the concrete layer; (b) cracked cross-section.

The equilibrium of an infinitesimal segment of fibers of length d z (Fig. 1c and d) yields:

d

( ) = τ f N z p z ( )

(4)

d

z

where ( ) ( ) ( ) τ = τ z s z is the shear stress at the coordinate z , where the slip is s (z), p f is the width of the fiber-matrix interfacial surface, and ( ) s τ is the interfacial cohesive material law (CML). Eqs. (3) and (4), together with the definition of the curvature, constitute the following system of differential equations: