PSI - Issue 64

Luigi Granata et al. / Procedia Structural Integrity 64 (2024) 1073–1080 Luigi Granata, Francesco Ascione, Saverio Spadea / Structural Integrity Procedia 00 (2019) 000 – 000

1076

4

{ =− =−

(2a-b) It is important to remark that no distributed moment load is considered ( =0 ). By some algebra, it is possible to transform the previous system of six differential equations into a system of only two differential equations in function of axial force ( ) and bending moment ( ) . { 1 3 (− 1 − 1 ) 2 2 + 1 2 2 2 + 1 − (− 1 2 + 1 ) = 0 1 3 4 4 −( 1 3 + ) 2 2 + 1 2 ( 1 + 1 ) 2 2 + ( 1 + ) − 1 = 0 (3a-b) The boundary conditions are:

( = =0)0=) =0 0 ( ( = = 0⁄)2)==0 0 (= = ⁄2 ) ⁄=2) = 0 , (

1

1 (

(4a-f)

{

= 1 − 1 2 = 1 2 2 2 + 1 = 1

The expressions of all other unknown parameters before being introduced are reported in Eqs (5a-d).

(5a-d)

=−

3 3 −( 1 2

+ 1 2 ) +( 1 2 + ) 1

1 3

{

2.2. Evaluation of e A procedure to evaluate the stiffness of the tangential and radial springs is given below. Concerning the radial spring, its stiffness was determined based on a typical value of the shear modulus of elasticity of a CFRP bar (3000 MPa). The friction between the CFRP stirrup and the surrounding concrete was evaluated by considering the experimental results available in the literature [16], where tests on several types of FRP rebars were conducted to evaluate the interaction phenomena between FRP rebars and the concrete matrix. The results were expressed in terms of bond-slip laws characterised by a linear ascending branch and a non-linear softening one. The corresponding elastic stiffness was then evaluated and assumed to be equal to 4,58 N/mm 3 ( 0 ). To evaluate the tangential stiffness of the springs, the above-mentioned elastic stiffness was multiplied by the thickness of the stirrups cross-section. As the thickness is variable in function of the number of wrapped layers considered, several values of the stiffness were adopted according to the expression: = 0 ∙ 2 . It is important to remark that no friction was hypothesised between the upper side of the stirrup cross-section and concrete due to the possible presence of shear cracks and between the lower side of the stirrup cross-section and CFRP bar. 2.3. Failure criteria The stress criteria for Tsai-Hill and Tsai-Wu have been adopted in order to evaluate the failure load , . Their analytical expressions relative to the case study are reported below: