PSI - Issue 64

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

1560

4

( )

d

d

N z E A

d

( ) f N z p z = τ ( )

( ) z ϕ =χ

( )

(

)

(5)

s z

H d

=

−χ −

n

d

d

z

z

d

z

f

f

where ϕ ( z ) is the rotation of the cross-section at the coordinate z . The system of equations in Eq. (5) with the appropriate boundary conditions allows to determine the unknowns s ( z ), Ν ( z ), and ϕ ( z ). It should be noted that the curvature χ and the depth of neutral axis d n can be expressed as a function of known parameters and the unknown functions of the system in Eq. (5). In particular, χ and d n depend on the bending moment M 1 and axial force N of the concrete beam. Since M 1 and N can be expressed as a function of M t and N using Eq. (1) and ( ) ( ) 0 t M z M z =λ , where M 0 ( z ) is the bending moment associated with q 0 ( z ), χ and d n can be expressed as a function of λ and N : ( ) 0 M ,N χ=χ λ ( ) 0 n n d d M ,N = λ (6) Referring to Fig. 2a, the expression of functions in Eq. (6) can be obtained based on the concrete constitutive law ( ) σ ε c . If the applied load, and therefore λ , is assigned, Eq. (5) (after substitution of χ and d n expressed by Eq. (6)) allows the determination of the slip s ( z ), axial force N ( z ), and rotation ϕ ( z ) profiles along an uncracked portion of the beam once three boundary conditions for the functions s (z), N ( z ), and ϕ ( z ) are specified. If a fourth condition involving the functions s ( z ), N ( z ), and ϕ ( z ) is enforced, Eq. (5) can be employed to evaluate the load multiplier λ (and therefore the magnitude of q ( z )) that allows the extra condition to be fulfilled. Cracked cross-sections In cracked cross-sections, the curvature χ is not defined, since the rotation ϕ is discontinuous and therefore non differentiable. Assuming that the crack faces are plane and the profiles of the longitudinal displacement in the cracked cross-section are plane, the field of the relative displacement between the crack faces is identified by two parameters, namely the distance of the center of the relative rotation (CRR) from the extrados d c and the relative rotation ∆ϕ (Fig. 2b). The distance of the CRR from the extrados will be referred to as the position of the CRR herein. The parameters ∆ϕ and d c depend on the bending moment M 1 and axial force N of the concrete beam. If the bending moment M 1 is expressed as a function of M t and N and the load multiplier λ is introduced, ∆ϕ and d c can be expressed as a function of λ and N : ( ) 0 M ,N ∆ϕ=∆ϕ λ ( ) 0 c c d d M ,N = λ (7) Referring to Fig. 2b, the expression of functions in Eq. (7) can be obtained based on the relation σ cw ( w ) between concrete stress σ cw and the relative displacement w . In this work, σ cw ( w ) was derived from the concrete constitutive law ( ) σ ε c by assuming ε = c w l (Fig. 2b), where l c is a characteristic length whose influence on the result can be evaluated at the end of the procedure. In cracked cross-sections, the relations between the relative rotation and the rotations ϕ l and ϕ r and slips s l and s r at the left and right crack faces of the crack, respectively are (Fig. 2b) Eqs. (7) and (8) are employed to derive the boundary conditions at the cracked cross-section to obtain the s ( z ), N ( z ), and ϕ ( z ) profiles in the un-cracked portions of the beam with Eq. (5). It should be noted that the system in Eq. (7) does not allow to determine the axial force and the relative rotation in a cracked cross-section starting from the bending moment M t , since it contains two equations in the three unknowns N , ∆ϕ , and d c . Similarly, the system in Eq. (8) does not allow to determine the slips at and the rotations of the faces of the crack starting from ∆ϕ and w . Therefore, Eq. (5) cannot be solved individually for a generic un-cracked segment between two consecutive cracks, as the boundary conditions at the two cracked cross-sections depend on the position of the remaining cracks previously formed along the beam. The model was applied to a simply supported beam strengthened at the intrados. The beam and the load are assumed symmetric with respect to the midspan cross-section, where the origin of the z -axis is located (Fig. 3). Once the r l ∆ϕ=ϕ −ϕ ( ) c l r H d s s ∆ϕ − = − (8)