Issue 58

R.N. da Cunha et alii, Frattura ed Integrità Strutturale, 58 (2021) 21-32; DOI: 10.3221/IGF-ESIS.58.02

where M i and M j are bending moments on the nodes i and j of the element, respectively, N i is the axial force on the node i of the element, α b is the orientation of the frame element and L b its length; β b , χ b and R b geometrically define the circular arch element (Fig 2). Note that the second member of (3) is the sum of the internal forces of each finite element.

Figure 2: Generalised stresses and geometric definition of (a) arch and (b) frame finite elements.

The kinematic relations for a finite element are given by:              b b Φ B U U

(7)

being [ B ( U )] b the same transformation matrix previously presented in (5-6) considering large displacements and { Φ } the matrix of generalised deformations, which is conjugated to { M } and written as:         T i j i b Φ (8) where φ i and φ j are flexural rotations due to the bending moments M i and M j , respectively, and δ i is the axial deformation related to N i . If small displacements and small deformations are assumed, (7) can be rewritten as follows:        0 b b Φ B U (9) being [ B 0 ] the transformation matrix for the initial condition. The elastic relations of the element are obtained by assuming the deformation equivalence hypothesis, by applying the Castigliano’s Theorem [26-27, 29] and considering the inelastic phenomena are concentrated at hinges located at the edges of the element (Fig 3):             p b b b Φ Φ F D M (10) where { Φ p } b is the matrix of generalised plastic deformations, that accounts for reinforcement yielding, and [ F ( D )] b is the damaged flexibility matrix, both written as:        0 T p p p i j b Φ (11)

1

 

        

0

0

0

11 F F

F

12

13

d F

1

 

i

1



 

0

0 F F 22

0

   b 

(12)

 

F D

12

23



d

1

   

j

0

0

0

F

F

F

13

23

33

24