Modeling considerations in seismic assessment of RC bridges using state-of-practice structural analysis software tools

Abstract

The increasing awareness of the general society toward the seismic safety of structures has led to more restrictive performance requirements hence, many times, to the need of using new and more accurate methods of analysis of structures. Among these, nonlinear static procedures are becoming, evermore, the preferred choice of the majority of design codes, as an alternative to complete nonlinear time-history analysis for seismic design and assessment of structures. The many available software tools should therefore be evaluated and well understood, in order to be easily and soundly employed by the practitioners. The study presented herein intends to contribute to this need by providing further insight with respect to the use of commonly employed structural analysis software tools in nonlinear analysis of bridge structures. A comparison between different nonlinear modeling assumptions is presented, together with the comparison with real experimental results. Furthermore, alternative adaptive pushover procedures are proposed and applied to a case study bridge, based on a generic plastic hinge model. The adopted structural analysis program proved to be accurate, yielding reliable estimates, both in terms of local plastic hinge behavior and global structural behavior.

Keywords

nonlinear analysis / pushover / RC bridges / structural modelling software

Introduction

The existing codes and guidelines for seismic design of structures currently provide practitioners with a high variety of analysis methods, from simple equivalent static procedures to more complex nonlinear time-history analyses. Traditionally, it is common practice to adopt the so-called force-based methods, which employ simple elastic analysis and take into account the inelastic response of the structure only through the application of force reduction factors. Comprehensive reviews of research on the applicability of these elastic methods and on the definition of the force reduction factors can be found in several studies [¹–⁵], as described in further detail in the work by Costa et al. [⁶], wherein elastic methods are presented as an interesting and expeditious way to predict the elastic capacity of structures, as well as to identify the first structural members to yield. However, these methods clearly fail to predict failure mechanisms and to account for the redistribution of forces that take place as the yielding occurs hence one can assume that force-based methods tend to work quite well to primarily carry out a life safety analysis of the structure but they cannot provide an accurate damage limitation analysis.

Consequently, in the recent years, there has been a significant shift in the seismic design philosophy toward the so-called performance-based methods. As a structure undergoes significant inelastic deformation during a strong earthquake, its dynamic characteristics tend to change with time, which renders fundamental the use of nonlinear analytic procedures to account for these features when investigating the performance of such structures. Nevertheless, in spite of the general agreement on the fact that nonlinear procedures are a better tool to assess existing structures, linear elastic methods are still used, and many times preferred, due to its relative simplicity, by practitioners [⁷]. In fact, in order to assess the degree of acquaintance of the civil engineering community with nonlinear seismic assessment practices, a survey was conducted by Araújo and Castro [⁸] among Portuguese practitioners, which revealed that almost half of the engineers (42%) are not familiarized with nonlinear analysis procedures and only 20% of those who are have already applied them.

Alongside with this change in the seismic design philosophy of modern codes, current structural analysis software tools are continuously enabling new technically-sound features that allow for nonlinear analysis and modeling of structures, making them increasingly attractive to practitioners and to the scientific community [⁹–¹²]. The user-friendly interfaces and the variety of nonlinear analysis and modeling solutions proposed by current software tools may motivate practitioners to blindly use them. As such, these should be assessed and scrutinized with the aim of filling the gap between practitioners and the research community and to promote the use of such important tools and critical thinking among the former.

A critical issue regarding the seismic safety assessment of structures is related to the biaxial modeling of its members. Most of the research that has been conducted up to date, focusing on the seismic assessment of structures, systematically assumes simplified planar structural systems and neglects the biaxial nonlinear behavior of members. Nevertheless, existing structures are oftentimes irregular in plan, requiring the use of numerical models that are able to conveniently capture such monotonic and cyclic biaxial behavior [¹³,¹⁴]. Fiber models are typically recognized to be more accurate, with respect to lumped plasticity models, in reproducing the bidirectional behavior of members, also because they do not require specific expertise on the practitioner’s side. Nevertheless, Rodrigues et al. [¹⁵] demonstrated that the former may fail in representing some behavioral aspects, such as the energy dissipation evolution. On the other hand, currently available and commonly employed software tools usually enable the use of simplified concentrated plasticity models that incorporate the interaction of moments in both longitudinal and transverse directions to reproduce the biaxial behavior of the member. However, despite its evident computational advantages and efficiency when compared to fiber models, research works that focus on its assessment and validation are, to the authors’ knowledge, still lacking in literature. As such, this paper aims to provide a thorough revision of the various modeling solutions commonly adopted by most of the existing software tools, which will be of great value in demystifying the impact of different modeling assumptions and in selecting more adequate models.

Moreover, the generalization of nonlinear methods of analysis in the new generation of codes has placed nonlinear static analysis as a preferable alternative to nonlinear time-history analysis. However, the difficulties of traditional nonlinear static analysis in accurately estimating the seismic response of irregular and semi-irregular structures, such as bridges, have been widely recognized [¹⁶–²⁰], therefore, with a view to overcome such difficulties, this paper presents a methodology for conducting adaptive pushover analysis, with and without the inclusion of the higher mode effects, using the concentrated plasticity backbone model, which can be extended to any software tool.

Focusing on the specific goal of this study, SAP2000 [²¹] has been selected as software tool due to its widespread use among practitioners. Furthermore, it features the same variety of nonlinear modeling solutions enabled by other well-known available software tools, which should yield similar results.

Nonlinear structural analysis

Currently employed methods

Among the existing nonlinear analysis methods, nonlinear time-history analysis (THA) is commonly recognized in the scientific community as the most reliable method to assess the seismic behavior of structures, particularly the irregular ones. Nevertheless, it typically requires high computational onus and is time demanding, which will become particularly important when structures with a large number of elements are being considered. As a result, the nonlinear static (pushover) analysis has quickly gained popularity as an interesting alternative, reasonably simple and efficient.

Pushover analysis (PA) consists in a modern variation of the classic collapse analysis [²²,²³], which enables e.g., the determination of the ultimate lateral strength of a structure or to characterize its local deformations by applying a monotonically increasing lateral load pattern. Notwithstanding the fact that this method produces unique and important information on the structural inelastic response, its limitations [²⁴] are particularly related to the impossibility of taking into account the influence of higher modes and the changes in the dynamic characteristics of the structure due to its progressive stiffness degradation throughout the loading process.

To overcome such limitations, several attempts to improve the traditional PA have been proposed, with varying degrees of complexity and success, as described by Elnashai [²⁵]. A first proposal, presented by Paret et al. [²⁶], was the Multi-Mode Pushover procedure (MMP), which consisted in the inclusion of the higher mode effects by performing different pushover analyses with different horizontal load patterns proportional to various mode shapes of the structure. A similar procedure was later on proposed by Chopra and Goel [²⁷] designated as Modal Pushover Analysis (MPA), whereby a series of independent pushover analyses are carried out, considering different load patterns proportional to the various modal shapes, followed by a combination of the response quantities using an appropriate combination rule (CQC or SRSS). In spite of yielding more accurate estimates for the case of irregular structures, when compared to traditional PA, neither MPA nor MMP take into account the progressive stiffness degradation of the structure. To fill such gap, the so-called Adaptive Pushover Analysis based methods (APA) have been recently proposed, first by Bracci et al. [²⁸] and later on by Elnashai [²⁵] and Requena and Ayala [²⁹]. These APA methods serve as the basis to the Modal Adaptive Pushover Analysis method (MAPA) [³⁰], which consists in a single-run pushover analysis whereby the load pattern is updated after the formation of every new plastic hinge. In each step of the analysis the dynamic properties of the structure are recalculated using its actual stiffness and the force distribution pattern obtained through SRSS combination of the new modal shapes. This adaptive pushover based method takes thus into account the progressive structural stiffness degradation, the change of modal characteristics, the period elongation of the structure and the influence of higher modes. MAPA can also be applied taking into account the weight based on the spectral ordinate at the period of vibration of the structure (MAPA-RS), as foreseen by Antoniou and Pinho [³¹]. Additional enhanced adaptive pushover algorithms, displacement-based, have been proposed very recently and successfully used to estimate the nonlinear response of buildings [³²,³³] and, specifically, bridges [¹⁶,¹⁷,³¹,³⁴] although this sort of analysis, due to its advanced features, is not widely implemented in current software tools.

Selected nonlinear structural analysis software tool

Among the currently available general purpose structural analysis programmes, SAP2000 [²¹] has been selected for this study, for its significant popularity among practitioners and the scientific community, having been widely used in several scientific publications [⁹–¹²,³⁵], particularly regarding the use of traditional nonlinear pushover analysis. Moreover, it features capabilities for linear and nonlinear, static and dynamic analysis of 2D and 3D structures, with particular emphasis on dynamic and earthquake loading and provides a user-friendly interface to conduct general nonlinear analysis. SAP2000 also enables the definition of new load cases from another, already defined, load case, as well as the calculation of the modal properties of the structure by setting a different structural stiffness. This latter feature could be used with a view to test the structural analysis software tool in carrying out adaptive pushover analyses.

The present study intends therefore (i) to provide a comparison between the use of lumped and distributed plasticity models in the application of a common structural analysis program to general nonlinear PA and THA of bridge structures and (ii) to study the extension of the use of such tool to carry out alternative adaptive pushover–based methodologies. To accomplish these goals, a preliminary review of different nonlinear modeling approaches, for both uniaxial and biaxial bending, will be conducted and numerical estimates will be validated with real experimental results.

Nonlinear Modeling of RC Columns

The nonlinear response of RC frame structures under earthquake excitation is usually concentrated at certain parts of the structure, often corresponding to points of maximum internal forces, such as beam-column element ends. The early approaches to model such nonlinear behavior consisted of zero length plastic hinges in the form of nonlinear springs at the member ends [³⁶], which are widely recognized and used, particularly in the case of bridge structures, where the nonlinearities tend to concentrate in the base of its piers. Furthermore, the lumped plasticity idealization of a cantilever generally serves as the base for deformation capacity estimates given by many codes and guidelines, such as Eurocode 8 – Part 2 [³⁷], ATC40 [³⁸], ASCE41-13 [³⁹] or CALTRANS [⁴⁰]. The selected nonlinear analysis tool adopts such concentrated plasticity model, which requires the definition of the moment vs plastic-rotation relation of each plastic hinge in advance, yielding several modeling approaches, presented and evaluated next.

Nonlinear modeling approaches

The definition of the lumped plasticity behavior of beam-column elements may be carried out through two distinct ways:

1) By setting the moment vs plastic-rotation (M-q) relationship itself, through zero-length uniaxial (M_x ; M_y) or biaxial (M_xM_y) hinges including axial load interaction (PM_x ; PM_y; PM_xM_y), being x and y the main axes of the member cross-section, or through nonlinear Plastic Wen or Multi-Linear Plastic links [⁴¹];

2) By setting the sectional moment vs plastic-curvature (M-F) relationship and a plastic hinge length, proposed in several literatures [⁴²] and codes [³⁷] that will convert the plastic-curvatures into plastic-rotations.

With regard to the M-F relationship definition, and depending on the characteristics of the structure and on the type of the analysis, the following modeling solutions can be adopted using the selected software tool.

Multilinear uniaxial M_x or M_y hinges

The M-F relationship may be defined automatically, based on the recommendations in CALTRANS or ASCE41-13, or manually, by defining a five-point backbone curve, in which the first point is the zero load point; the second and the third points are the yielding and the ultimate capacity points, respectively; whereas the fourth and the fifth points correspond to the degraded capacity and failure. Given that the nonlinear behavior of the element is characterized independently in both transverse and longitudinal directions, this type of models can only be used in 2D analysis. On the other hand, as this uniaxial hinge model accounts for the cyclic behavior of the element, by selecting Isotropic, Kinematic, Takeda or Pivot [⁴³] models of hysteretic behavior, notwithstanding some numerical instability, it can be adopted in the case of time-history 2D analysis.

Multilinear interaction axial force vs uniaxial bending (PM_x; PM_y) and axial force vs biaxial bending (PM_xM_y) hinges

The definition of the interaction hinge models is similar to that of uniaxial hinge models, except for the consideration of coupled behavior in both orthogonal bending directions, in the case of interaction biaxial bending (M_xM_y) hinges, and for the interaction between axial force and bending moment, in the case of PM_x or PM_y hinges. To do so, it is necessary to previously define the M_xM_y or PM_x or PM_y interaction diagrams of the element cross-section. An additional important aspect regarding the M_xM_y hinges is related to the number of M-F curves that should be defined depending on the type of cross-section. In the case of circular symmetry of the column cross section only one curve is required, while the use of a minimum of three curves is recommended for asymmetrical configurations (transverse, longitudinal and 45° directions). The main advantage of the PM_xM_y hinges is that they can be used in the case of 3D pushover analysis however, similarly to the PM_x or PM_y hinges, they do not allow the selection of any type of hysteretic model hence should not be used in the case of nonlinear time-history analysis.

Fiber axial force vs biaxial bending (PM_xM_y) hinges

Fiber hinge models compute the M-F relation in any bending direction for varying levels of axial force by assigning particular material stress-strain relationships to individual discretized fibers of the cross section. In the case of reinforcement steel, the stress-strain relationships can be manually or automatically defined, assuming, in the second case, a Park and Paulay [⁴⁴] material law with Isotropic, Kinematic or Takeda hysteretic behavior. In the case of confined and unconfined concrete, the stress-strain relationships are defined, also manually or automatically, assuming a Mander et al. [⁴⁵] material law with the same hysteretic behavior. Typically, fiber models are more robust and stable for nonlinear analysis and can be adopted in any situation (2D/3D pushover or dynamic analysis) [¹⁴].

Assessment of different nonlinear modeling approaches

The assessment of the various nonlinear modeling approaches presented before is based on the comparative analysis with experimental results obtained in a large experimental campaign promoted by the Laboratory of Earthquake and Structural Engineering (LESE) of the Faculty of Engineering of the University of Porto for the study of RC columns under horizontal cyclic loads [⁴⁶,⁴⁷]. Two rectangular RC columns, with the section properties and reinforcement steel detailed in Fig. 1(a), were tested under uniaxial loading, in both directions, under biaxial loading, as represented in Fig. 1(c).

The confined and unconfined stress-strain relationships were defined considering an average compressive ultimate strength of 24.39 MPa [⁴⁶,⁴⁷] and the Mander et al. [⁴⁵] model, with the confinement effectiveness factor a been defined according to EC8-1 [⁴⁸] (Fig. 1(e)). The concrete was assumed to have zero strength in tension, as recommended by Johnson et al. [⁹]. In the case of the reinforcement steel, the stress-strain relationship was calibrated from experimental data [⁴⁶,⁴⁷], as displayed in Fig. 1(f), considering a yielding strength of 430 MPa, an ultimate strength of 551 MPa and an ultimate strain of 23%. The experimental tests were performed with constant axial loading of 300 kN.

For what concerns the structural model, a lumped plasticity hinge model has been adopted, whereby the moment-rotation, or curvature, relationship was defined at a single point along the element. If an M-F relationship is adopted, it will be lumped at the middle of the plastic hinge length, as represented in Fig. 1(b). The plastic hinge lengths observed experimentally [⁴⁶,⁴⁷] were adopted in the numerical models (0.12m in the square column, for both directions, and 0.28 and 0.16 m in the stronger and weaker directions of the rectangular column, respectively). The choice for one specific plastic hinge length was also based on the studies by Monteiro et al. [⁴⁹] and Monteiro [⁵⁰], who concluded that using different plastic hinge lengths in lumped plasticity bridge models, using the same software tool employed herein, did not yield relevant differences. Three distinct plastic hinge models where thus considered: (i) fiber PM_xM_y hinge model, with cross-section refinement of 200 fibers (Fig. 1(d)); (ii) uniaxial hinge model; and (iii) interaction M_xM_y hinge model.

According to Fig. 2(a), which establishes a comparison between the experimental results and the fiber hinge numerical results with different steel hysteretic laws for the uniaxial cyclic test N13 of the N13-N16 square RC column, a very good prediction of the nonlinear behavior of RC columns could be obtained in this case. However, this type of models failed to correctly predict the strength degradation of the column for higher displacements values, at which the nonlinear response is mainly governed by the steel behavior. As the response of the columns at this stage is very sensitive to the adopted value of the steel ultimate strain and the steel stress-stain relationship, changes in these two parameters will considerably affect the columns behavior.

Regarding the hysteretic behavior of the steel, two models were considered: 1) multilinear kinematic, which is based on the kinematic hardening behavior, more commonly used in metals; 2) Takeda, which is very similar to the kinematic model, but uses a degrading hysteresis loop based on the well-known Takeda model [⁴³]. The difference between the Takeda and the Kinematic models is that, in the former, when crossing the horizontal axis during unloading, the curve follows a secant path to the backbone curve for the opposite loading direction. Although some researchers [⁴³] recommend the use of kinematic hysteretic laws for the curves that define the reinforcement, the Takeda model also yielded fair estimates (Fig. 2(a)). Both confined and unconfined concrete were defined using Takeda relationship, despite the fact that it will actually not have influence in the response, given that zero strength in tension was assumed.

Figures 2(b) and 2(c) depict the comparison corresponding to the results of the biaxial cyclic test N14 of the N13-N16 square RC column in both x- and y-direction of the cross-section, respectively. Again, it can be concluded that the numerical model provides a good prediction of the biaxial response of square columns through the adoption of fiber hinge models. However, in spite of the very good results in the uniaxial analysis of the rectangular N09-N12 column in both x and y directions (Figs. 2(d) and 2(e)), the numerical model considerably underestimated the nonlinear response of the rectangular column in the biaxial N11 test. This underestimation may be most likely associated with the incapacity of the steel model available in SAP2000 to capture the exact coupled isotropic and kinematic hardening [⁵¹] of the reinforcement steel, as well as the well-known Bauschinger effect. The development of such cyclic hardening phenomena, caused by the asymmetric biaxial loading seems to explain, not only, the increase in the bending capacity of the member, but also the attainment of the onset of strength deterioration at lower rotation values in the x-direction. In fact, most models made available in current commercial software tools seem to miss the incorporation of more reliable and advanced strength and stiffness deterioration models [⁵²]. Furthermore, the adoption of Takeda hysteretic models for steel in both square and rectangular columns appears to overestimate the pinching effect and to underestimate energy dissipation. It should be noted that advanced models that incorporate bond-slip effects and reinforcement bar buckling [⁵³,⁵⁴] are generally missing in current commercial software tools, which can also contribute to the deviation between the numerical and experimental results observed in Fig. 2.

In alternative to the fiber hinge model, two multilinear biaxial and uniaxial hinge models were also analyzed, for which results are presented in Fig. 3. Given that the uniaxial models in SAP2000 allow the use of nonlinear time-history analysis, but not 3D analysis, only the uniaxial N13 test was considered in the evaluation of this type of hinges. On the other hand, as the coupled hinge models can only be used for static analysis, only 3D pushover analysis was considered in the evaluation of this type of hinges. Figures 3(a) and 3(b) therefore depict the results obtained through the use of a multilinear M_x uniaxial hinge, from which one can observe that this type of models can predict with significant accuracy the uniaxial nonlinear behavior of columns. It can also be concluded that the Takeda hysteretic law is the one leading to better results. This type of hinge model, which requires minor computational onus, represents an interesting alternative to more complex fiber hinge models, however, it can exhibit converge problems, as discussed by Aviram et al. [⁴¹], particularly when a degradation softening is taken into account in the backbone curve (Fig. 3(a)). As such, in the case study herein considered (Section 3), no degradation slope was considered. Finally, Figs. 3(c) and 3(d) present, respectively, the M_xM_y interaction diagram of the N13-N16 column cross-section, essential to the coupled hinge model definition, and the comparison of the results obtained from pushover analysis in two different incidence angles (45° and 26.5°) using both coupled and fiber hinge models.

The models agree quite well along cracking conditions and up to very close to the yielding point for both directions, individually (x, y) or together (x-y), and both incidence angles. On the other hand, as expected, the fiber hinge models are able to capture the stiffness degradation around the yielding phase of the curve, which, due to its linear nature, the couple model is not able to represent. The same difference in performance is observed for what concerns the softening branch, which is not successfully represented by the coupled hinge model. Whereas the fiber hinge model is more accurate in terms of strength degradation, the coupled hinge one is more conservative for what concerns maximum attained strength. All-in-all, the multilinear model led to good results, with the significant advantage of enabling the completion of the analyses in reduced time.

Case study – nonlinear analysis of a short and irregular PREC8 viaduct

Structural modeling, dynamic characteristics of the analyzed bridge and seismic input

To scrutinize the performance of the selected structural analysis software tool, when carrying out nonlinear pushover and time-history analysis, the well-known short and irregular P213 PREC8 bridge [¹⁶–¹⁸,⁵⁵], whose model is illustrated in Fig. 4, was selected as case study.

The bridge was modeled using the software package SAP2000 and two lumped plasticity models were adopted at the base of the columns to represent the local ductility cantilever model idealized in Eurocode 8: (i) a fiber PM_xM_y hinge model; (ii) and a multilinear uniaxial M_x hinge model, as shown in Figs. 4(b) and 4(c). It is noted that a different five-point definition of the M-F backbone curve was considered in the multilinear hinge of this study, with respect to the one previously presented, so as to better discretise the nonlinear behavior of the element. The stress-strain model for the longitudinal reinforcement of the piers was defined using the Park and Paulay [⁴⁴] model bilinear relationship with kinematic hysteretic behavior whereas the confined and unconfined concrete were characterized using the Mander et al. [⁴⁵] models with Takeda hysteretic behavior. The plastic hinge length was determined according to the Annex E of Eurocode 8 [³⁷]. The remaining elements of the structure were taken as linear-elastic, the masses were lumped at the top of the columns and mid-spans (characterized by three translational and three rotational inertia values defined with respect to the global reference system) and P-∆ effects were directly included in the numerical analysis. A uniform elastic viscous Rayleigh damping ratio of 5% was assumed, as proposed in Eurocode 8 for RC structures. Finally, the abutment restraints of the bridges were defined by a pinned support in the left abutment and a roller support in the right abutment. Given that the studied structure is regular in plan and taking into account the boundary conditions and the high stiffness of the bridge in the longitudinal direction, only one analysis in the transverse direction of the bridge was deemed necessary. Further information on the properties of the PREC8 bridges may be found in Guedes [⁵⁵].

Regarding the dynamic characteristics of the analyzed bridge (Figs. 5(a), 5(b) and 5(c)), which were obtained through a simple eigenvalue and eigenvector analysis, it can be seen from the modal mass participation ratios depicted in Fig. 5(c) that the dynamic response of the bridge is mainly governed by the two first modes of vibration, which present elastic periods and corresponding modal mass participation ratios of 0.43s and 26% and 0.31s and 67%, respectively.

Figures 5(b) and 5(c) depict the evolution of the periods of vibration and the modal mass participation ratios of the first three modes of the structure along with its degradation level, considering different behavior stages at the piers (Fig. 4(c)). It is observed that the bridge displays a highly irregular dynamic behavior, particularly due the high mass participation ratio of the 2nd mode, which will have an important influence in the elastic response of the structure. Moreover, the dynamic properties of the bridge tend to significantly change with its degradation, increasing the importance of its 1st mode in the response. The nonlinear static (PA) and time-history analyses (THA) were conducted considering the Portuguese EC8 elastic spectrum (type I, epicenter offshore) and a corresponding set of compatible artificial records, as illustrated in Fig. 6. More information regarding the selection of the suite of ground motion records may be found in Araújo et al. [⁵⁶].

Nonlinear conventional pushover (PA) and time-history analysis (THA) of the bridge

According to Eurocode 8 the pushover analysis of bridges must be performed by pushing the entire structure with two distinct load distribution patterns: 1) constant along the deck; and 2) proportional to the 1st mode shape. Figure 7(a) depicts the obtained capacity curves for each load pattern using two types of lumped plasticity models. An additional well-known structural analysis software tool (http://www.seismosoft.com), which adopts a distributed plasticity model, has also been used and depicted for validation purposes. Consistently, similar modeling assumptions to those previous described in Section 4.1 have been adopted using SeismoStruct.

It is observed that all the nonlinear modeling solutions lead to similar results, which confirms the selected program as a reliable tool for pushover analysis, whether fiber or multilinear lumped plasticity models are employed. For what concerns time-history analysis, which accounts for strength degradation of the different elements of the bridge, as well as the influence of all modes and the characteristics of the dynamic response, several time-integration methods are available, such as the Newmark or Hilber-Hughes-Taylor (HHT) algorithms. According to Araújo et al. [¹⁴], the latter proved to be more beneficial under high input ground motions, as it could reduce the high short-duration peaks in the solution therefore it will be employed in this case. A commonly employed uniform equivalent viscous damping value of 5% was assumed for all modes of vibration, for the sake of simplicity, through the use of the Rayleigh damping coefficients. The moment-rotation diagrams obtained through THA are presented in Fig. 7(b), in which an important difference between the fiber models and the lumped plasticity multilinear model can be observed. Such divergence may be due in part to the inherent convergence issues of the multilinear model, which many times lead to erroneously high results. Another reason is definitely related to the estimation of the hysteretic energy dissipation parameters (stiffness and strength degradation or pinching), which require further analysis for proper calibration.

Finally, Figs. 7(c) and 7(d) display the PA response and the maximum THA response over time of the bridge for the adopted seismic action, for two demand levels. In tandem with what previous studies have shown [¹⁴,⁵⁷,⁵⁸], the conventional pushover predictions are less accurate, when applied to irregular bridges, hence advanced pushover analysis should be taken into account.

Improvement of nonlinear static analysis of RC bridges

Adaptive pushover analysis using conventional software

As the monotonic loading of a certain distribution of forces increases, the elements of the structure tend to change its behavior, leading to the stiffness/strength degradation of the structure and, consequently, to a change in its dynamic properties (as illustrated in Figs. 5(b) and 5(c)). Adaptive pushover analysis (APA) enables the consideration of such phenomena by updating the modal forces of the structure every time a plastic hinge is formed or a change in the behavior of any structural element occurs. Having in mind the application to bridge structures, wherein the plastic hinges form at the bottom of the piers, and considering the five-point multilinear hinge model presented in Fig. 4(c), the following steps, also illustrated in the flowchart in Fig. 8, are proposed to extend the use of the employed software to adaptive pushover analysis:

1) Compute the natural periods and modes for linearly elastic vibration of the structure. In the case of reinforced concrete bridges, the elastic periods should correspond to the effective stiffness of the members [⁵⁹];

2) Perform a conventional pushover analysis for a force distribution proportional to a certain mode shape, obtained in 1, and define the corresponding capacity curve. The pushover analysis must be carried out until a plastic hinge forms or a change in the nonlinear behavior of any pier occurs, according to the five-point linear model of Fig. 4(c), when the analysis will be stopped. As an example, Figs. 9(c) and 9(d) represent, respectively, the formation of the first hinge, which occurred at the base of the central pier, and the step when all the piers reached the plastic state and the bridge mechanism was formed;

3) Once completed the pushover analysis in step. 2, the natural periods and modes of the structure are recalculated and a new pushover analysis is carried out, using the previous one as starting point, by setting a force distribution pattern proportional to the new shape of the mode selected in the previous step;

4) Repeat the procedure as many times as necessary to achieve a state of collapse in at least one plastic hinge of the bridge piers under study, as exemplified in Fig. 9(a) for a semi-regular bridge configuration (P123).

APA does not account for the influence of higher modes. To do so, an alternative modal adaptive pushover analysis (MAPA) should be adopted, by changing steps 2 and 3 of the presented APA procedure (Fig. 9(b)). Hence, contrarily to APA, in the steps 2 and 3 of the MAPA procedure, the force distribution pattern, F, must be obtained through the combination of the most important mode shapes of the structure (those whose masses contribute to at least 90% of the total mass of the structure) using an appropriate modal combination rule, e.g., SRSS, according to Eq. (1).

(1)F=\sqrt{{\displaystyle \sum {{\displaystyle F}}_i^{\mathrm{2}}}}.

F_i are the inertial forces related to each mode of vibration, given by Eq. (2).

(2){{\displaystyle F}}_i={{\displaystyle m}}_i{{\displaystyle \mathit{\Phi }}}_i.

m_i the mass in the i-th node of the structure and F_i the modal shape of the i-th mode of vibration. If the spectral amplification is to be taken into account, F_i is given by Eq. (3).

(3){{\displaystyle F}}_i={{\displaystyle m}}_i{{\displaystyle \mathit{\Phi }}}_i{{\displaystyle L}}_i^{*}{{\displaystyle S}}_{a,i},

L_i^* is the earthquake excitation factor and S_a,i is the spectral acceleration.

Evaluation of the proposed adaptive pushover procedure

The numerical results obtained by carrying out the proposed adaptive pushover procedure, with the selected software tool are present in Fig. 10, in which a and b display the evolution of the 1st mode shape proportional load pattern and the mode with higher modal mass participation factor proportional load pattern, respectively. On the one hand it is observed that the 1st mode shape tends to significantly change during the APA, distributing the inertial forces along the bridge length as the plastic hinge formation takes place on a uniform fashion. On the other hand, if the mode with higher modal mass participation ratio is considered in the force distribution definition for the various steps of the analysis, a very distinct structural response should be expected for lower displacement levels, wherein the response is governed by the 2nd mode, and for higher displacement levels, wherein the response is mostly governed by the 1st mode, as Figs. 5(b) and 5(c) also denote. Regarding MAPA (Figs. 10(c) and 10(d)), it can be concluded that the adoption of a combined load pattern will considerably change the force distribution along the bridge length, which may lead to significantly different structural responses. Secondly, the influence of the spectral amplification, defined by Eq. (3), could also be noted, which renders the load pattern different from a shape that is constant along the deck, produced by MAPA (Fig. 10(c)), and closer to the 1st mode load pattern (Fig. 10(a)). The response of the bridge for the adopted seismic action, for two demand levels, is presented in Figs. 10(e) and 10(f). According to what was expected, MAPA leads to considerably better results, particularly when the spectral amplification is taken into account, whereas the conventional pushover-based procedure underestimated, in the case of the PA 1st mode, or overestimated, in the case of the PA uniform, the bridge response.

MAPA provided thus accurate predictions with respect to time-history analysis, even if it requires a non-negligible number of steps for a static analysis procedure. However, its superiority comes from the fact that all such steps are straightforward and of easier comprehension and application for users who may instead not be at ease with the concept and the parameters required to carry out nonlinear time-history analysis.

Finally, for the sake of completeness, Fig. 11 illustrates the comparison of the capacity curves obtained considering the various pushover analysis versions that were tested, together with IDA analysis results. In addition to the case study bridge P213 (Fig. 11(a)), a second, semi-regular configuration, P123 (Fig. 11(b)) has also been considered. Good results were obtained, in terms of comparison with time-history analysis estimated response, with the application of the adaptive pushover analysis using SAP2000, for both configurations [¹³,⁵⁹].

It is also important to note, from the observation of the results in Fig. 11, how the 1st mode based approaches (PA 1st Mode and APA 1st Mode) fail to accurately estimate the response of the bridge P213, being far from the IDA envelope, which instead is better matched by MAPA and MAPA RS. This result was expected, given that the configuration P213 is very irregular, with an extremely short central pier, which renders the first mode less significant. On the other hand, for the semi-regular bridge, P123, the MAPA RS provides similar predictions to APA 1st Mode (and even PA 1st Mode), which confirms how for a more regular bridge, the 1st mode is a fair predictor of the deformed shape.

Conclusions

The use of a commonly employed structural analysis software tool, popular among practitioners, in seismic assessment of bridge structures was scrutinized in this work. Various nonlinear modeling approaches enabled by the program were presented and assessed for both uniaxial and biaxial bending. For what concerns different nonlinear static analysis methodologies, this study assessed the accuracy of the conventional pushover analysis foreseen by the program and addressed the possibility of extending its capabilities to the performance of adaptive pushover analysis, as a valid alternative approach when irregular structures are being analyzed. The following conclusions were drawn:

1) The selected nonlinear structural analysis tool revealed itself accurate, in terms of comparison of bridge response numerical predictions with experimental results. In terms of pushover analysis results, the results also matched well the ones obtained with an alternative well-known fiber-based finite element modeling computer program. The different modeling options for both uniaxial and biaxial bending were easy to implement and intuitive enough from a regular practitioner’s use viewpoint;

2) Despite the significant requirements of time demand and computational onus, the fiber hinge models within the selected structural analysis program proved to be more accurate and of wider application hence its use is recommended for the majority of the cases. Nevertheless, multilinear coupled hinge models could also be an interesting alternative, which is faster and of easier definition. However special attention should be paid to eventual convergence instability;

3) Comparing the results of the main program (SAP2000) with the ones yielded by another software tool (SeismoStruct), a fairly good matching could be found, in terms of both Pushover and Time-History analysis. Conventional pushover analysis led to inaccurate response predictions, with the load pattern constant along the deck overestimating the response and the 1st mode proportional pattern underestimating the response;

4) Finally, the capabilities of the considered nonlinear structural analysis software tool proved to be successfully extended to the computation of adaptive pushover analysis with good accuracy. The stiffness and strength degradation of the structure and the higher mode effects were duly taken into account, particularly for the highest seismic demand level (PGA = 0.70 g). The consideration of the spectral amplification in the combined SRSS load pattern plays a non-negligible role, particularly for more regular structures, in which the dynamic response is mainly governed by the fundamental mode.

References

[1] Miranda E, Bertero V V. Evaluation of strength reduction factors. Earthquake Spectra, 1994, 10(2): 357–379

[2] CEB.Seismic design of RC structures for controlled inelastic response. Bulletin no. 236, Comité Euro-International du Béton, 1997

[3] Kappos A J. Evaluation of behaviour factors on the basis of ductility and overstrength studies. Engineering Structures, 1999, 21(9): 823–835

[4] Mwafy A M, Elnashai A S. Calibration of force reduction factors for RC buildings. Journal of Earthquake Engineering, 2002, 6(2): 239–273

[5] Maheri M R, Akbari R. Seismic behaviour factor, R, for steel X-braced and knee braced RC buildings. Engineering Structures, 2003, 25(12): 1505–1513

[6] Costa A, Romão X, Oliveira C S. A methodology for the probabilistic assessment of behaviour factors. Bulletin of Earthquake Engineering, 2010, 8(1): 47–64

[7] Paret T, Searer G, Freeman S. ASCE 31 and 41: Apocalypse Now. Structures Congress, Las Vegas, Nevada, 2011

[8] Araújo M, Castro J. Critical review of codes for seismic safety assessment of existing steel buildings. Journal of Earthquake Engineering, 2016 (in Press)

[9] Johnson N, Saiidi M, Sanders D. Nonlinear earthquake response modelling of a large-scale two-span concrete bridge. Journal of Bridge Engineering, 2009, 14(6): 460–471

[10] Inel M, Ozmen H B. Effects of plastic hinge properties in nonlinear analysis of reinforced concrete buildings. Engineering Structures, 2006, 28(11): 1494–1502

[11] Habibullah A,Pyle S.Practical three dimensional nonlinear static pushover analysis. Structures Magazine, winter, 1998

[12] Varum H, Sousa R, Delgado W, Fernandes C, Costa A, Jara J M, Jara M, Álvarez J J. Comparative structural response of two steel bridges constructed 100 years apart. Structure and Infrastructure Engineering, 2011, 7(11): 843–855

[13] Araújo M, Delgado R. Seismic safety assessment of curved bridges using pushover analysis. Papadrakakis M, Papadopoulos V, Plevris V, eds. 3rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2011), Corfu, Greece, 25–28 May, 2011

[14] Araújo M, Marques M, Delgado R. Multidirectional pushover analysis for seismic assessment of irregular-in-plan bridges. Engineering Structures, 2014, 79: 375–389

[15] Rodrigues H, Varum H, Arêde A, Costa A. Comparative efficiency analysis of different nonlinear modelling strategies to simulate the biaxial response of RC columns. Earthquake Engineering and Engineering Vibration, 2012, 11(4): 553–566

[16] Casarotti C, Monteiro R, Pinho R. Verification of spectral reduction factors for seismic assessment of bridges. Bulletin of New Zealand Society for Earthquake Engineering, 2009, 42: 111–121

[17] Pinho R, Monteiro R, Casarotti C, Delgado R. Assessment of continuous span bridges through nonlinear static procedures. Earthquake Spectra, 2009, 25(1): 143–159

[18] Kohrangi M, Bento R, Lopes M. Seismic performance of irregular bridges – comparison of different nonlinear static procedures. Structure and Infrastructure Engineering, 2015, 11(12): 1642–1650

[19] Monteiro R, Delgado R, Pinho R. Probabilistic seismic assessment of RC bridges: Part I – uncertainty models. Structures, 2016a, 5: 258–273

[20] Monteiro R, Delgado R, Pinho R. Probabilistic seismic assessment of RC bridges: Part II – nonlinear demand prediction. Structures, 2016, 5: 274–283

[21] CSI.CSI Analysis Reference Manual for SAP2000. ETABS and SAFE, Computer and Structures, Inc, California, 2010

[22] Pinho R, Casarotti C, Antoniou S. A comparison of single-run pushover analysis techniques for assessment of bridges. Earthquake Engineering & Structural Dynamics, 2007, 36(10): 1347–1362

[23] Kunnath S. Identification of modal combination for nonlinear static analysis of building structures. Computer-Aided Civil and Infrastructure Engineering, 2004, 19(4): 264–259

[24] Krawinkler H, Seneviratna G. Pros and cons of a pushover analysis of seismic performance evaluation. Engineering Structures, 1998, 20(4-6): 452–464

[25] Elnashai A S. Advanced inelastic static (pushover) analysis for earthquake applications. Structural Engineering and Mechanics, 2001, 12(1): 51–69

[26] Paret T, Sasaki K, Eilbeck D, Freeman S. Approximate inelastic procedures to identify failure mechanisms from higher mode effects. In: Proceedings of the 11th World Conference on Earthquake Engineering. Acapulco, 1996

[27] Chopra A, Goel R. A modal pushover analysis procedure for estimating seismic demands for buildings. Earthquake Engineering & Structural Dynamics, 2002, 31(3): 561–582

[28] Bracci J, Kunnath S, Reinhorn A. Seismic performance and retrofit evaluation of reinforced concrete structures. Journal of Structural Engineering, 1997, 123(1): 3–10

[29] Requena M, Ayala A G. Evaluation of a simplified method for the determination of the nonlinear seismic response of RC frames. In: Proceedings of the 12th World Conference on Earthquake Engineering. Auckland, 2000

[30] Paraskeva T S, Kappos A J, Sextos A G. Extension of modal pushover analysis to seismic assessment of bridges. Earthquake Engineering & Structural Dynamics, 2006, 35(10): 1269–1293

[31] Antoniou S, Pinho R. Development and verification of a displacement-based adaptive pushover procedure. Journal of Earthquake Engineering, 2004, 8(5): 643–661

[32] Pinho R, Marques M, Monteiro R, Casarotti C, Delgado R. Evaluation of nonlinear static procedures in the assessment of building frames. Earthquake Spectra, 2013, 29(4): 1459–1476

[33] Monteiro R, Marques M, Adhikari G, Casarotti C, Pinho R. Spectral reduction factors evaluation for seismic assessment of frame buildings. Engineering Structures, 2014, 77: 129–142

[34] Casarotti C, Pinho R. An Adaptive Capacity Spectrum Method for assessment of bridges subjected to earthquake action. Bulletin of Earthquake Engineering, 2007, 5(3): 377–390

[35] Zelaschi C, De Angelis G, Giardi F, Forcellini D, Monteiro R. Performance based earthquake engineering approach applied to bridges in a road network. Papadrakakis M, Papadopoulos V, Plevris V, eds. 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), Crete Island, Greece, 25–27 May, 2015

[36] Taucer F, Spacone E, Filippou F. A fiber beam-column element for seismic response analysis of reinforced concrete structures. Report UCB/EER-91/17, Earthquake Engineering Research Center, University of California, California, 1991

[37] CEN.EN 1998-2: 2005 Eurocode 8: Design of Structures for Earthquake Resistance – Part 2: Bridges. European Committee for Standardization, Brussels, 2005

[38] ATC.ATC-40 Seismic evaluation and retrofit of concrete buildings. Volume 1, Applied Technology Council, California, 1996

[39] ASCE. Seismic evaluation and retrofit of existing buildings (ASCE/SEI 41-13). American Society of Civil Engineers, Virginia, 2014

[40] CALTRANS.Caltrans Seismic design criteria. Version 1.6, California Department of Transportation, California, 2010

[41] Aviram A, Mackie K, Stojadinović B. Guidelines for nonlinear analysis of bridge structures in California, PEER Report 2008/03, Pacific Earthquake Engineering Research Center, University of California, California, 2008

[42] Bae S, Bayrak O. Plastic hinge length of reinforced concrete columns. ACI Structural Journal, 2008, 290–300

[43] Dowell R K, Seible F, Wilson E. Pivot hysteresis model for reinforced concrete members. ACI Structural Journal, 1998, 607–617

[44] Park R, Paulay T. Reinforced concrete structures.New York: John Wiley and Sons, 1975

[45] Mander J B, Priestley M J N, Park M. Theoretical stress-strain model for confined concrete. Journal of Structural Engineering, 1988, 114(8): 1804–1826

[46] Rodrigues H, Arêde A, Varum H, Costa A. Experimental evaluation of rectangular reinforced concrete column behaviour under biaxial cyclic loading. Earthquake Engineering & Structural Dynamics, 2013, 42(2): 239–259

[47] Rodrigues H, Arêde A, Varum H, Costa A. Damage evolution in reinforced concrete columns subjected to biaxial loading. Bulletin of Earthquake Engineering, 2013, 11(5): 1517–1540

[48] CEN.EN 1998-1: 2004 Eurocode 8: Design of Structures for Earthquake Resistance – Part 1: General rules, seismic action and rules for buildings. European Committee for Standardization, Brussels, 2004

[49] Monteiro R, Ribeiro R, Marques M, Delgado R, Costa A G. Pushover Analysis of RC Bridges Using Fibre Models or Plastic Hinges. In: Proceedings of the 14th World Conference on Earthquake Engineering. Beijing, China October12–17, 2008

[50] Monteiro R. Sampling based numerical seismic assessment of continuous span RC bridges. Engineering Structures, 2016, 118: 407–420

[51] Chaboche J L. A review of some plasticity and viscoelastic constitutive theories. International Journal of Plasticity, 2008, 24(10): 1642–1693

[52] Ibarra L F, Medina R A, Krawinkler H. Hysteretic models that incorporate strength and stiffness deterioration. Earthquake Engineering & Structural Dynamics, 2005, 34(12): 1489–1511

[53] Orakcal K, Wallave J W, Conte J P. Flexural modelling of reinforced concrete walls – model attributes. ACI Structural Journal, 2004, 688–698

[54] Billah A M, Alam M S. Seismic performance of concrete columns reinforced with hybrid shape memory alloy (SMA) and fiber reinforced polymer (FRP) bars. Construction & Building Materials, 2012, 28(1): 730–742

[55] Guedes J M. Seismic behaviour of reinforced concrete bridges. Modelling, numerical analysis and experimental assessment. Dissertation for the Doctoral Degree. University of Porto, Porto, 1997

[56] Araújo M, Macedo L, Marques M, Castro J M. Code-based record selection methods for seismic performance assessment of buildings. Earthquake Engineering & Structural Dynamics, 2016, 45(1): 129–148

[57] Paraskeva T, Kappos A. Further development of a multimodal pushover analysis procedure for seismic assessment of bridges. Earthquake Engineering & Structural Dynamics, 2010, 39: 211–222

[58] Isaković T, Lazaro M, Fishinger M. Extension of pushover methods for the seismic analysis of single-column bent viaducts. Earthquake Engineering & Structural Dynamics, 2008, 37(8): 1185–1202

[59] Araújo M, Delgado R. Application of adaptive pushover methods of analysis to SAP2000, In: Proceedings of the Congress in Numerical Methods in Engineering, Coimbra, 2011

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg