PSI - Issue 8

Francesco Penta et al. / Procedia Structural Integrity 8 (2018) 399–409 Francesco Penta et al./ Structural Integrity Procedia 00 (2017) 000 – 000

401 3









ˆ  j

t

b

t    b

section nodal rotations

1 / 2

and 1 / 2 

.



   

j

j

j

j

j

t i-1 y ,v

F t

v t i

t i-1

,

F

i y

i-1 ,  t

m t

m t i

,  t i

N-1 t

N t

i t

i-1 t

i-1

0 t

1 t

2 t

F t

t

t i-1

i-1 x ,u

u t i ,

F i x

i-cell

1

2

3

N-1

N

F b i y

F b

m b i v b i

,

b i-1

i-1 y ,v

,  b i

i-1 ,  b

m b

y

i-1

F i x b

F b

i-1 b

N-1 b

0 b

1 b

2 b

i b

N b

u b

b i-1

i ,

i-1 y ,v

x

O

Fig. 2. Unit cell nodes numbering with girder nodal inner forces and displacements

b j j n F F   t j

The static quantities conjugates of the previous kinematic variables are: the axial force (

the

) / 2,





t j y j y V F F   , the b j

b j t M F F l   generated by the anti-symmetric axial forces, the shear force t j j

bending moment

resultant of the nodal moments ˆ j j m m m   . The state vector s of a girder nodal cross section consists of the displacements and forces vectors d and f . Hence, the state vectors of the end sections of the i cell are 1 1 1 [ ] T T T i i i     s d f and [ ] T i T T i i  d f s , Fig. 2. They are related by the transfer matrix G : t b j j m m m   and, finally, the difference between the same moments j t b j

1 i   Gs s i

.

As shown in Stephen and Wang (2000), the force transmission modes of the unit cell are given by the unit principal vectors of the G matrix. More in particular, the axial transmission mode is the principal vector a s generated by the unit eigen-vector of G defining a rigid axial translation a u . The unit eigen-vector corresponding to a transversal rigid translation generates instead the in plane rigid rotation principal vector and this latter gives the pure bending transmission mode b s . Finally, the shear transmission mode V s is the principal vector generated by the pure bending mode b s . Also in the simplest cases, these modes must be determined numerically and for this reason ill-conditioning of G makes very problematic the practical implementation of the transfer methods. By the approach proposed in present paper the ill-conditioning problems are altogether avoided since principal vector of G are determined in closed form by operating directly on the unit cell stiffness matrix. If , T T T e e e      s d f is a unit eigen vector, the principal vector , T T T p p p      s d f of the G matrix generated by e s is such that p p e   G s s s . The displacement and force sub vectors of e s and p s are thus linked through the sub-partitions ij Ξ of the stiffness matrix by the equation:

   f

d

ll rl Ξ Ξ Ξ Ξ

  

  

  

  

p

p

lr

(1)



 

 

 f f

 d d

p e

p

e

rr

where subscript l and r are used to denote the left and right side of the unit cell and the matrices , i j l r  ) are the four sub-partitions of the cell stiffness matrix (see Appendix A ). By adding term by term the two equations in (1) and recasting the result in order to have the known terms at left hand side, the next condition for the unknown displacement vector p d is deducted: ij Ξ (with ,

(2)

e p   f Bd Ad e