A number of failure criteria for reinforced concrete columns are as follows:

• In the curve of the load-deformation or moment-rotation, the strength drop (ranging from 10% to 30%) of the curve is observed. This method has considerable discrete, it is inappropriate in some cases.
• Failure of confinement corresponding to fracture of one hoop or stirrups that cause the onset of cyclic strength degradation as to progressive failure.
• Attainment of an ultimate tensile strain, ε_su of longitudinal reinforcement, it is a measure of the likelihood of reinforcing bar rupture.
• The longitudinal reinforcement buckling for reinforced concrete columns between two consecutive layers of stirrups or series stirrups. The results will lead to a fracture of the longitudinal reinforcement, and its strength will rapidly degrade.
• Attainment of an ultimate compressive strain, ε_cu of confined core concrete, it will cause crushing of concrete and loss of bearing capacity.

At first, these criteria should be used to define available capacities (failure points) as plastic rotation, (θ_f-θ_y). For reinforced concrete columns, the most promising variable can be used to quantify the limiting values describing failure, it is found to be the attainment of an ultimate compressive strain, ε_cu , of confined core concrete. This is likely to happen in the columns before reaching an ultimate compressive strain. Thus, a limiting value for ε_cu was adopted to complete available capacity followed by Paulay and Priestley in 1992 [7].

According to Paulay and Priestl y [2], the strain at peak stress, ε_cc, shown in Fig. (2), it does not represent the maximum effective strain, as high compression stresses can be maintained at several times larger strains. The effective limit strain occurs when stirrups confine steel fractures, thus ultimate compressive strain of confined core concrete, ε_cu is obtained through stirrups at the strain energy of fracture as follows:

(4)

Where, ε_c is the maximum compressive strain of unconfined concrete (generally assumed to be 0.004); ρ_v is the volumetric ratio of stirrup. For rectangular sections ρ_v=ρ_x+ρ_y, ρ_x=A_sh,x/sh_c, ρ_y=A_sh,y/sb_c, s is the stirrup spacing, A_{sh, x} and A_{sh, y} are the stirrup-sectional area parallel to the x-direction and y-direction, respectively, h_c and b_c are width and depth of confined concrete (centerline to centerline of stirrups) respectively; f_yv is the yield strength of the stirrups; ε_sm is the reinforcement strain corresponding to the maximum tensile stress. f^'_cc is the compressive strength of confined concrete. The typical maximum compressive strain ε_cu is from 0.01 to 0.06.

The compressive strength of the confined concrete is directly related to the effective confined stress P_e, the compressive strength of the concrete, stirrups strength and volumetric ratio of stirrup, was given by:

(5)

Where, k_e is the confinement effectiveness coefficient; ρ_v is the volumetric ratio of stirrup; f_yv is the yield strength of stirrups.

Fig. (2). Stress-strain model for monotonic loading of confined and unconfined concrete in compression (Paulay and Priestley, 1992).

The confinement effectiveness coefficient k_e was given by Mander in 1988 [8]:

(6)

Where, ρ_cc is the ratio of longitudinal reinforcement area to core section area; w_i is the ith clear stirrup spacing between adjacent longitudinal reinforcement; n is the number of longitudinal reinforcement; s^' is the clear stirrup spacing.

The establishment of the equation (6) is based on ordinary strength of concrete and stirrups. To increase the range of applications, Akiyama et al. [9] have conducted the test of the large size high strength rectangular concrete columns confined with high-strength stirrups. Thus, the equation (6) was modified as follows:

(7)

Where, the notation of the equation (7) is the same with equation (6).

Although the f_yv in the equation (5), the yield strength of stirrups is generally adopted. The stirrup strength corresponding to the peak stress of confined concrete does not necessarily reach the yield strength, especially when using high-strength concrete confined with stirrups. To this end, Akiyama et al. [10] proposed stirrups at peak stress f_s,c:

(8)

Where,

(9)

(10)

(11)

Where, ε_co is the strain at the peak stress of unconfined concrete; ρ_w is the volumetric ratio of stirrup; k_e is the confinement effectiveness coefficient; f^'_cis the compressive strength of unconfined concrete; k_b is the effective coefficient of unconfined concrete; f^'_co is the cylinder compressive strength of ordinary concrete (cylinder diameter of 100mm, and a height of 200mm).

Suzuki et al. [9] used the regression analysis for test data of high-strength concrete columns confined with stirrups. The relationship of the peak stress and its corresponding peak strain was given by:

(12)

(13)

Where, the definition of notation for equations (12) and (13) are the same with that of equations (5), (8) and (11); ε_co is the unconfined concrete strain corresponding to the peak stress; f^'_c is the unconfined concrete compressive strength. Thus, ultimate compressive strain of confined core concrete, ε_cu can be determined by equations (4), (5), (7), (8), (9), (10), (11), (12). After ε_cu is determined, according to a strain-compatible moment-curvature analysis, the plastic curvature capacity of the column section is defined as (Ø_f-Ø_y). Where Ø_f is the curvature corresponding to the attainment of; ε_cuØ_y is the yield curvature; the available plastic rotation capacity,θ_p=θ_f-θ_y, could be obtained by plastic curvature ability and multiplied by the length of the plastic hinge:

(14)

Where, l_pis plastic hinge length; l_p was given by Paulay and Priestly [7]:

(15)

l is the ratio of column shear span; d is the diameter of the column longitudinal reinforcement in mm; f_y is the yield strength of column longitudinal reinforcement.

The main criteria that define failure of steel and composite beams, which involves the interaction of local and lateral buckling, unloading (or strain-weakening) mechanisms, the crushing of concrete slab, separation between concrete slab and steel cross-section that is a type of loss of composite action.

Inelastic rotation capacity of steel and composite beams θ_p can reflect its ductile failure capacity. Many researchers (1969 Lukey and Adams, [11] in 1986, Kemp [12], the 1987 Roik and Kuhlmann [13]) have conducted experimental studies. These studies have shown that the main factors affecting the inelastic rotation capacity are as follows: (1) the lateral slenderness ratio of steel beam flange under the negative moment area, (2) the width to thickness ratio of compressive flange and web. Based on the above results, Kemp and Dekker 1991 [14] established a simplified model, the model accounts for the distinction between positive moment and negative moment area of composite beams.

For lateral unbraced steel beams, the effective lateral slenderness ratio was given as follows:

(16)

Where, L_i is the length of steel beam from the section of maximum moment to the point of inflection; i_c is the radius gyration about the minor axis for the part of the cross-section in compression; ε_f is, where f_yf is the yield strength of flange in MPa; K_f and K_w are empirical factors to allow for actual flange and web slenderness, respectively. They were given as follows:

(17)

(18)

(19)

Where, b and t_f are respectively the flange width and thickness of steel beams; α=h_c/d_w is the proportion of web depth in compression, d_w and t_w are the web depth and thickness, and ε_w is, where, is the yield strength of web in Mpa.

Kemp1996 [14] introduced torsion confined coefficient K_d, thus equation (16) was modified:

(20)

(21)

(22)

(23)

Where, γ=; f is the flange or web yield strength in Mpa; h_c is the composite beam web depth in compression.

By adjusting the inelastic rotation capacity of composite beam, it is obtained [15, 16]:

(24)

Assuming linear moment gradients, the elastic yield rotational capability is given as follows:

(25)

Where, M_p is the plastic moment strength of steel beams; EI is the elastic flexural stiffness of the steel beams.

The composite frames presented in this paper consisting of steel beams and high-strength concrete columns confined with high-strength stirrups. The steel beams continuously pass through the concrete columns at the beam-column joint. The typical beam-column joint is shown in (Fig. 3). Such joint test results have revealed that it has two failure modes, one for shear failure of web yield and concrete cracking or spalling, another is the bearing failure of high-pressure concrete crushing as well as up and down to the flange yield. Shear failure also exist in the bearing failure, but the bearing failure does not produce shear deformation. The bearing failure should be avoided for seismic design, which was also reflected in the ASCE design guidelines 1994.

Fig. (3). The typical beam-column joint.

At present, no models are available to predict the ultimate deformation capacity of composite joints. Therefore, the selection of suitable values for the damage parameter is based on the results of experimental work by Kanno 1993 [17] and Bugeja 1999 [18], Parra-Montesinos et al. 2000 [19], Liang 2003 [20], Fargier-Gabaldón 2005 [21], the predetermined joint rotation capacity at the point where the joint resistance drops to below 0.8 time its maximum strength. Based on linear regression of test data for eleven specimens failing in joint shear and four specimens in bearing, as shown in Fig. (4), the joint rotation capacity is given as follows:

(26)

Where, M_ps and M_vb are the nominal shear and bearing moment capacity of the joint, respectively. The expression of M_ps and M_vb follows ASCE design guidelines 1994. Regarding the equation (26) established is based on cyclic loading, under monotonic loading, it should be multiplied by 1.2 amplification factors, which is obtained as:

(27)

In section 3.1, the model of damage index and deformation ability is verified by the test data available in the literature. The columns are as follows: one specimen (V5.5-66) by Woods et al. 2007 [22]; two specimens (HHL91 and UHL61) by YAN et al. 2006 [23]; three specimens (HSC-S1-1, HSC-S2-1 and HSSC-S3-1) by Chen et al. 2009 [24]. All specimens were conducted under cyclic loading reached failure, and these specimens were computed in terms of damage index which are listed in (Table 2). The damage index is taken as 1, which agrees well with the test results.

To verify the proposed damage index and deformation capacity for steel and composite beams, the published experimental data are adopted to assess. The test data include ordinary steel beams and steel and composite beams. These tests are considered as follows: two specimens for ordinary steel beams tested by Kanno 1993 [17]. The tested results of the two specimens (OB1-1 and OBJS1-1) have shown that they occur beam failure rather than joint failure. For the steel and composite beams, four specimens are considered, respectively, the test specimen(G2) by Tagawa 1989 [25], the specimen(sy-1) by Xue et al. 2002 [26] , two specimens by lightweight 2005 [27], one is simplified beam(SCB-14), another is continuous beam(CCB-13). The computed values of damage index parameters are listed in (Table 3). From statistical measures, as shown in Table 3, the overall damage status agrees well with damage index for the specimen, the failure mode of almost all specimens is the compression buckling of steel beams causing instability, steel beams under flange and web buckling, finally the crushing of concrete slab.

In this study, the proposed damage index and the deformation capacity model in section 3.3 are verified based on the published tested results. These tests are conducted by Kanno 1995 [17], Bugeja 1999 [18], Parra-Montesinos et al. 2000 [19], Liang 2003 [20], and Fargier-Gabaldón 2005 [21]; they comprise of ten specimens, failing mainly in joint shear failure and five specimens failing in joint bearing failure. The calculation of the damage indexes are listed in Table 4, these values are computed based on the procedure presented in section 3.3. From statistical measures, it is shown in Table 4, the overall damage status agree well with that of the tested specimen. The failure modes are the crushing of concrete, concrete spalling and the yielding of stirrups and steel beam web, finally core concrete is exposed in the joint. It is worth pointing out that the prediction of failure through the proposed damage index is much better for joints with predominately shear failure mode than for joints with bearing failure mode. Inelastic rotation capacity of shear failure joints is significantly larger than that of bearing failure joints. It has revealed that shear failure joints have a better energy dissipation capacity and ductility. Therefore, the design of the joint should be designed to shear failure as much as possible.

In the case of the progressively increasing seismic demands, the evolution of the damage index can predict the damage (necessary repairs and associated economic losses) process. The composite frame tested by Baba and Nishimura (1998) [28] is to illustrate that mentioned problems. Fig. (5a) shows the relationship of hysteretic load versus deformation, and Fig. (5b) shows the relationship of damage index versus load cycle. The two figures reflect the damage index evolution process under different damage level, which can provide a basis for designing or repairing.

Fig. (5). The damage evolution for composite frame specimen: (a) hysteretic load versus deformation; (b) damage index Du versus load cycle.