Issue 38

P. Lonetti et alii, Frattura ed Integrità Strutturale, 38 (2016) 359-376; DOI: 10.3221/IGF-ESIS.38.46



 ] ,

1 [

0

0

2

2

2

1 N N E A N E A U U U U M M E I M E I M E I U M M E I M E I M E I U M GJ GJ                                   1 1 1 1, 1, 2, 1 1 2 0 1 2 0 0 0 2 2 0 2 2 2 0 2 2, 3, 1 3 3 3 3 3 3 3 3 3, 2, 1 2 t t X X X t t t X t t t X

X

3,

1

,

X X

(4)

1

1

,

X X

1

1

t

t

1

1, 1 X

t E A and

2 t E I and 3 t E I are the curvatures or the bending t GJ are the torsional curvature and stiffness,

2  and

1  are the axial stiffness and strain,

3  or

where

2 X and

X

axes, respectively,  and

stiffnesses with respect to the

3

1 N is the axial stress resultant,

2 M and

3 M are the bending moments with respect to the 2 X and 3 X axes,

respectively,

0 1 N ,

0 2 M and

0 3 M are the initial axial force and bending

1 M is the torsional moment. Moreover,

respectively, and

2 X and

3 X axes, respectively. The CRC tangent modulus can be expressed as [34] (Fig. 3):

moments with respect to the

E E 

0.5 N P 

1.0 for

t

l

y

   

   

(5)

N N

1

1





0.5 N P 

E

E

4

1

for

t

l

y

P

P

y

y

P is the plastic axial strength.

where E is the elastic modulus of the member, y

Figure 3 : Stiffness reduction for the plastic hinge model. Figure 4 : Full plastic surface based on AISC-LRFD. The gradual inelastic bending stiffness reduction is expressed by a dimensionless reduction parameter  which is assumed to vary according to the following parabolic functions:

  1.0 for 4 1 for       

 

 

0.5 0.5

(6)

where  is a force-state parameter which measures the magnitude of axial force and bending moment at the element end. The damage parameter  can be expressed by AISC-LRFD interaction domain (Fig. 4), which defines the cross-section plastic strength of the element as:

M

M

N M

N M

8 9

8 9

2 9

2 9

3

3

1  

2

1

2









for

P M M

P M M

y

p

y

p

2,

3,p

2,

3,p

(7)

M

M

1 N M    2

N M

2 9

2 9

3

3

1

2







for

2 P M M

P M M

y

p

y

p

2,

3,p

2,

3,p

2, p M and

3, p M are the full plastic moments of the X 2

and X 3

axes, respectively. More details on the definition of

where

Eqs.(6)-(7) can be recovered in [33].

363