4.1 Assumptions for push-over analyses

In order to perform push-over analyses, the software SAP2000^® (Computers and Structures, Inc., Walnut Creek, CA, USA) was used. The soil-structure interaction was simulated by a fixed constraint at the bottom of the column at the ground floor. According to §7.2.6 of Italian building code N.T.C. 2008 the floors were considered to be rigid, because a solid RC plate with thickness of 4 cm was considered, simulated by imposing a “diaphragm” restraint. A “Master node”, placed at the geometrical centroid of each floor was defined and coincident with the centroid of the masses. The 3D model is characterized, from a dynamic point of view, by three degrees of freedom at each level: two translations of the centroid, in two orthogonal directions, and the rotation around the vertical axis passing through the centroid of the floor.

4.2 Seismic forces

The Italian building code NTC2008 adopts a performance-based approach at limit states for the design of new structures and the verification of existing ones. The seismic action on buildings is evaluated from a “basic seismic risk”, which is the basic element to know in order to estimate the seismic actions. Seismic actions are evaluated according to the reference period (RP), which is obtained by multiplying the nominal life (NL), for a coefficient of use (CU), which is a function of the class of use [NTC2008 – §2.4.3]. Once the RP is known, the seismic design actions are defined starting from “basic seismic hazard” in terms of ordinates of the elastic spectrum, in terms of acceleration, S_a(T), and the probability of exceedance in the reference period, PVR. The Italian building code, NTC2008, provides four limit states: (i) operational limit state (LSO), (ii) limit state of damage (LSD), (iii) limit state of life safeguarding (LSV) and limit state of collapse (LSC). The first and the second are serviceability limit states (SLS), while the third and the fourth are ultimate limit states (ULS). Once the reference period of the seismic actions is known, the spectral shapes, for each place, are defined in relation to the maximum ground acceleration, a_g, maximum value of the amplification factor of the acceleration spectrum, F₀ and the initial period of horizontal acceleration of the spectrum, where velocity is constant, T^*_C. Starting from parameters a_g, F₀, T^*_C seismic action is defined according to the return period, T, of seismic action and to latitude and longitude. However, the spectrum of elastic response (of the horizontal component of the seismic forces) is defined by equations (1)–(4):

where T_B, T_C and T_D are threshold periods of the spectrum of acceleration, defined by Italian code NTC2008, S_a(T) is the elastic spectrum of acceleration, S is a coefficient related to the subsoil condition and to topography of the site, η is a coefficient that considers the viscous damping.

Table 1 shows the main seismic parameters used to perform finite element method (FEM) simulations. Figure 3 shows the elastic spectrum of LSD and LSV, evaluated by considering a conventional damping of 5%, by using equations (1)–(4) and the values reported in Table 1. Seismic masses of the structure need to be defined in order to perform FEM analyses. Seismic masses have been evaluated according to NTC2008.

Figure 3:

Elastic spectrum of limit state of damage (LSD) and limit state of life safeguarding (LSV).

4.3 Assumptions for FEM simulations and definition of plastic hinges

The model of the plastic hinges is based on the model proposed by NTC2008. Figure 4 shows the nonlinear relationship between the bending moment and rotations characterizing the plastic hinges, where the value of the ultimate moment, M_u, is related to the definition of the ultimate curvature, ϕ_u, usually considered at 20% load drop after the maximum bending moment, M_MAX. Generally, it is possible to consider four phases: (i) elastic phase with no cracks [(0,0); (ϕ_cr, M_cr)], (ii) elastic phase with presence of cracks [(ϕ_cr, M_cr); (ϕ_y, M_y)], (iii) inelastic phase post-yielding [(ϕ_y, M_y); (ϕ_MAX, M_MAX)], and inelastic phase post-peak [(ϕ_MAX, M_MAX); (ϕ_u, M_u)], where ϕ_cr and M_cr are, respectively, curvature and bending moment at cracking, ϕ_y and M_y are yielding curvature and yielding moment, ϕ_MAX and M_MAX are curvature and bending moment at peak and ϕ_u and M_u represent the ultimate values of curvature and bending moment.

Figure 4:

Plastic hinges used for simulations.

According to NTC2008, the regression formulations to calculate the value of the maximum yielding rotation and the value of the ultimate yielding rotation (related to their bending moment) were used to define the link between the bending moment and the rotation for plastic hinges. A simplified model to define the link between the bending moment and the rotation was used; particularly it does not take in account the softening branch after the maximum value of the bending moment, while the crack rotation, θ_cr, and the yielding rotation, θ_y, were evaluated according to equations (5) and (6), respectively.

where θ_cr is the rotation at cracking, ϕ_cr is the curvature at cracking, L_v is the shear length, θ_y is rotation at yielding, ϕ_y is curvature at yielding, h is the height of the cross section of the element, d_b is the diameter of the longitudinal bar, f_y is the yielding stress of steel, and f_c is the compressive strength of concrete.

Assuming that the elements between the two hinges are rigid with concentrated plasticity, the nonlinear relationship has been defined with reference to the chord rotation, considering the software calculation model, the value of elastic rotation due to flexural deformation needs to be subtracted from the value of the calculated elastic rotation. Thus, four plastic hinges (two of them at the top and two of them at the bottom) were associated with each column, while one plastic hinge was associated with each side of the beam. The yield rotation was calculated according to Panagiotakos and Fardis (2001).

Reduction of steel bar cross sections and cracking and spalling of concrete cover were considered, while parameters such as the loss of bond between steel and concrete and the loss of ductility of corroded steel have been neglected because these issues are still under debate.

4.4 Corrosion levels and configurations

Three different corrosion levels and configurations have been considered in order to perform FEM simulations of the structure. Particularly, levels refer to damage which occurred to structural elements, while configurations refer to their position in the structure. Corrosion levels were evaluated according to Bossio et al. (2012), in terms of reduction of the bar radius, x, considering the relationship between the concrete cover cracking and the volume of oxide inducing cracks. The concrete cover spalling was considered to occur when a 0.5 mm crack appears on a concrete surface. The first level of a generalized corrosion due to carbonatation provides a reduction of bar radius such as to induce the concrete cover cracking, but it continues to be considered part of the concrete section (reduction of radius of stirrups equal to 1 mm and a lower level of corrosion of longitudinal bars). The second level provides the spalling of the concrete cover and the reduction of the radius of the stirrups is assumed to be equal to 4 mm while the reduction of the diameter of the longitudinal bars is equal to 2 mm; furthermore, it is considered that the corner-positioned columns corrode on two sides, while central-positioned columns corrode on one side, as well as the beams. The third level of corrosion considers a reduction of the bar equal to 50% of the diameter, the spalling of the concrete cover and a residual diameter of stirrups equal to 1 mm. Corrosion levels lead to us to consider different geometry for beams and columns sections, both in terms of cross section of reinforcements and concrete; in particular, considering the second level of corrosion, the eccentricity of the vertical load on the columns should be considered, due to the unsymmetrical spalling of the concrete cover. Moreover, the concrete geometry is different. The concrete cross section, at the first level of corrosion, is (30×30) cm²; considering the second level of corrosion, concrete cross section reduces to (27×30) cm² for central-positioned columns and (27×27) cm² for corner-positioned columns; when corrosion occurs on all sides, the concrete cross section is equal to (24×24) cm². Table 2 shows the values of the diameter of the bars, d_b, and stirrups, d_st, considered, for uncorroded structures and for the three levels of corrosion, the degradation level of the concrete cover, c_c, and a list of corroded/uncorroded elements referring to different configurations of corrosion. Approximately, by considering an exposure class defined as “XC4” according to the UNI EN 206:2006 with a carbon dioxide diffusion coefficient k equal to 6 mm/year^0.5 and an average corrosion rate of 70 μm/year, it needs about 26, 40 and 75 years to reach the first, the second and the third levels of corrosion, respectively.

For each corrosion level, three different configurations of corrosion were also considered. Particularly, the first configuration considers that the corrosion of one façade of the building and its structural elements are corroded along the external side only both for longitudinal bars and external legs of the stirrups, (orange colored in Figure 5), the second configuration considers the corrosion acting on all façades of the building and all external sides of structural elements are corroded both for longitudinal bars and external legs of the stirrups, while the third configuration considers the corrosion acting on all structural elements of the building along all the sides both for longitudinal bars and all legs of the stirrups. Figure 5 show the corrosion configurations.

Figure 5:

Corrosion configurations.

The behavior of the corroded columns of first corrosion configuration (Figure 5A) is not axi-symmetric, particularly referring to the plastic hinges’ formation. Considering the second configuration (Figure 5B), corner-positioned columns named “P1”, “P3”, “P4” and “P6”, are corroded along two sides, while columns named “P2” and “P5” are corroded along the external side. The behavior of the plastic hinges of the third configuration (Figure 5C) is axi-symmetric along the two main directions such that the columns and beams corroded uniformly along all sides. The push-over analyses were performed in both the X-direction and Y-direction (both positive and negative versus). Shear capacity was preliminary evaluated and related to bar reduction, x. Figure 6, shows the shear capacity related to the bar reduction for each cross-section condition.

Figure 6:

Corrosion levels and correlation between shear capacity and bar reduction.

5.1 Push-over analyses: uncorroded and corroded structures

The uncorroded structure does not show any structural problem and as a result, the structure has a capacity in terms of PGA of 4.015 m/s² with a return period of 968 years considering the X-direction, while, considering the Y-direction, it is able to resist a PGA of 3.760 m/s² with a return period of 759 years. Figure 7 shows the bilinear curve and the capacity in the acceleration-displacement response spectrum (ADRS) space (X-direction).

Figure 7:

Bi-linear curve and ADRS capacity of an uncorroded structure.

Table 3 reports the value of ductility, μ_c, the value of capacity in terms of PGA and the return period both in absolute terms and percentage variation with respect to the uncorroded structure, according to considered levels and configurations of corrosion referring to the X-direction and Y-direction both in case of positive or negative versus. Moving between corrosion levels (first, second and third levels) values of ductility, PGA and the return period decrease as the corrosion level increases. Considering the first configuration it is interesting to notice that the values of available ductility are different if considering the X+⁺ direction or X⁻ direction depending on compression or traction acting on the corroded sides of the member.

Similarly, PGA values or the return period are different because of the asymmetry of concrete and steel cross sections due to corrosion that, in this configuration, is considered as acting on one side/facade of the structure. The maximum loss of ductility in the X-direction (~ −35%) is obtained considering the third level of corrosion and the second configuration, while considering Y-direction ductility is in a range of −2.83% (second level and first configuration of corrosion) to +7.19% (third level and second configuration of corrosion).

Referring to the PGA, as expected, values decrease as corrosion increases and so does the return period. Also, in these cases there are some differences. In terms of PGA, considering the X-direction, the values are in the range of −1.59% to −93.31%, while referring to Y-direction the range is −0.72% to −24.17%. In terms of the return period, the range is 968–232 years as maximum and minimum values in the X-direction and 759–389 as extreme values in the Y-direction.

The third considered configuration of corrosion is a very severe condition. In fact, except for first level of corrosion, it is not possible to reach a ductile failure because of the occurrence of the brittle shear failure of beams.

In general, the corrosion reduces the steel reinforcement cross section, hence the strength of the plastic hinges reduces, but the ductility increases; however, in the global response, this effect could be modified by the layout of the formation of plastic hinges, involving at different degrees the plastic hinges of the corroded elements. Furthermore, the plastic hinges represent the more desired flexural failure mode, however, the corrosion impacts first on the external stirrups, hence the shear capacity reduces earlier, and the premature combined shear failure could limit the global ductility of the structural response.

5.2 Push-over analyses performed considering the first level of corrosion

Considering the first corrosion configuration, the global behavior of the structure is not symmetric considering the seismic forces acting both along the “+” direction or “−” direction and the X and Y directions. Anyway, the global behavior of the structure is very similar to the previous case; in fact, the seismic capacity and failure mode are comparable, as shown in Figure 8.

Figure 8:

Comparison of bi-linear curves and ADRS capacity of an uncorroded structure and a structure corroded at the first level of corrosion and the first configuration.

Considering the second corrosion configuration, even if the structure is considered corroded along all sides, its global behavior is comparable to the previous cases. Both bi-linear curve and ADRS capacity are almost overlapping with respect to the previous cases. Considering both the first and the second corrosion configurations, the failure mode is a ductile failure occurring at the bottom of columns P2 and P4 (exhaustion of ductility – SLV). Considering the third configuration, it is not possible to obtain a ductile failure, but a brittle shear failure (V_r/V_s<1, where V_r is the shear capacity and V_s is the shear force, bold values in Tables 4–6) occurs at T1-4-5 and T2-4-5 beams according to Biskinis et al. (2004), while they are safe when considering Italian building code NTC2008.

5.3 Push-over analyses performed considering the second level of corrosion

Considering the second level of corrosion (in first configuration), corrosion effects start to be more visible in terms of the global seismic behavior of the structure. The ADRS curve is not overlapping to the ADSR curve of the uncorroded structure, but they are almost similar to each other as shown in Figure 9. Another important aspect is the not symmetric response considering the X⁺ or X⁻ direction of seismic forces (Table 3). Considering the second configuration, the loss of global capacity of the structure is very different with respect to the uncorroded structure (Figure 10). The third configuration presents brittle failures occurring to beams named T1-3-4, T1-4-5 and T1-5-6 due to shear according to Biskinis et al. (2004) while occurring to the beam named T1-4-5 only, according to Italian building code NTC2008 (Table 5).

Figure 9:

Comparison of bi-linear curves and ADRS capacity of an uncorroded structure and a structure corroded at the second level of corrosion and the first configuration.

Figure 10:

Comparison of bi-linear curves and ADRS capacity of an uncorroded structure and a structure corroded at the second level of corrosion and the second configuration.

5.4 Push-over analyses performed considering the third level of corrosion

The third level of corrosion is a severe condition for the structure as the first configuration; in fact, the ADRS curve is very different with respect to the previous cases (Figure 11). Considering the second configuration, brittle shear failure occurs for the beams named T1-3-4 and T1-5-6 (Figure 12). The third configuration presents brittle failure considering both NTC2008 and Biskinis et al. (2004). Particularly, considering Biskinis et al. (2004), no structural element is safe (i.e. V_r/V_s<1) as shown in Table 6.

Figure 11:

Comparison of bi-linear curves and ADRS capacity of an uncorroded structure and a structure corroded at the third level of corrosion and the first configuration.

Figure 12:

Comparison of bi-linear curves and ADRS capacity of an uncorroded structure and a structure corroded at the third level of corrosion and the second configuration.

5.5 Comparison of results

Push-over analyses showed the seismic behavior of corroded reinforced concrete structures. Even if some aspects can be improved, the results obtained can represent a good starting point and a good methodology to model corroded structures. Figure 13 shows the results in terms of the return period, T_r, considering both the X-direction and Y-direction (positive verse) related to levels of corrosion.

Figure 13:

The return period, T_r, according to levels and configurations of corrosion in the case of a seismic force acting along: (A) X-direction (positive verse) and (B) Y-direction (positive verse).

The values of the return period, T_r, in the X-direction are higher than the values in Y-direction, even if they decrease rapidly as corrosion levels increase; on the other hand, the values in the Y-direction show a lower decrease rate and present less variation than the X-direction in the same configuration.