Issue 38

P. Lonetti et alii, Frattura ed Integrità Strutturale, 38 (2016) 359-376; DOI: 10.3221/IGF-ESIS.38.46 363   0 0 2 2 2 1 1 1 1 1, 1, 2, 3, 1 1 1 1 0 0 0 2 2 2 2 2 2 2 2 2, 3, 1 1 1 0 0 0 3 3 3 3 3 3 3 3 3, 2, 1 1 1 1 1, 1 1 [ ] , 2 , , t t X X X X t t t X X X t t t X X X t t X 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                                   (4) where t E A and 1  are the axial stiffness and strain, 2  and 3  or 2 t E I and 3 t E I are the curvatures or the bending stiffnesses with respect to the 2 X and 3 X axes, respectively,  and t GJ are the torsional curvature and stiffness, respectively, 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, and 1 M is the torsional moment. Moreover, 0 1 N , 0 2 M and 0 3 M are the initial axial force and bending moments with respect to the 2 X and 3 X axes, respectively. The CRC tangent modulus can be expressed as [34] (Fig. 3): 1 1 1.0 for 0.5 4 1 for 0.5 t l y t l y y y E E N P N N E E N P P P              (5) where E is the elastic modulus of the member, y P is the plastic axial strength. 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 0.5 4 1 for 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: 3 3 1 2 1 2 2, 3,p 2, 3,p 3 3 1 2 1 2 2, 3,p 2, 3,p 8 8 2 2 for 9 9 9 9 2 2 for 2 9 9 y p y p y p y p M M N M N M P M M P M M M M N M N M P M M P M M             (7) where 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 Eqs.(6)-(7) can be recovered in [33].