PSI - Issue 2_A

Patrizia Bernardi et al. / Procedia Structural Integrity 2 (2016) 2780–2787 Author name / Structural Integrity Procedia 00 (2016) 000–000

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3

Fig. 1. (a) Sketch of the two limit configurations considered in the range model, respectively corresponding to the case of a tension tie block in incipient cracking condition (continuous line) and after the opening of the crack (dashed line); (b) boundary conditions at the ends of the tensile block of length 2 l t .

2.1. Kinematics, equilibrium and compatibility equations

The basic assumptions of the adopted numerical model (see also Bernardi et al. (2014)) are sketched in Figure 2. As can be seen, the model is based on the presence of the slip s and the bond stresses τ at the interface between the reinforcing bar and the surrounding concrete. The slip s is defined as the difference between the displacement of two points that were originally in contact, respectively belonging to reinforcement and concrete (that means s = u s - u c ). The corresponding bond stress τ is defined as a function of the current slip through a suitable bond-slip relation. With reference to the free body diagram of an element with infinitesimal length, the compatibility condition (Eq. 1, Fig. 2b), as well as the axial equilibrium of the reinforcing bar (Eq. 2, Fig. 2c) and of the whole cross-section (Eq. 3, Fig. 2c) can be written as: ( ) ( ) ( ) ( ) ( ) c c s s c s E x E x x x dx ds x σ σ ε ε = − = − (1) ( ) ( ) s A n dx d x s b s τ φ π σ = (2)

( )

( )

dx d x c σ

A A

dx d x s σ

n

τ φ π

( ) s

(3)

s

b

= −

= −

A

c

c

where ε c (x) and ε s (x) are the strains in concrete and in the reinforcing bar, respectively, while σ c (x) and σ s (x) are the corresponding stresses. Moreover, A c , A s and E c , E s are the area and the Young modulus of concrete and steel rebar, while n b and φ represent the number and diameter of reinforcing bars.