Assessment of Existing Structures Using Inelastic Static Analysis

Synonyms

Nonlinear static analysis; Pushover-based analysis

Introduction

Over the last few decades, it has been found that the traditional procedures, which are based on elastic linear analysis, can only approximately estimate the typically nonlinear seismic response. Therefore, the inelastic methods of analysis have been gradually introduced into practice in order to estimate the seismic response more realistically. Contrary to the elastic linear methods which can only implicitly predict the performance, the objective of the inelastic seismic analysis procedures is to directly estimate the magnitude of inelastic deformations.

In general, the most refined and accurate inelastic method is the nonlinear response history analysis (NRHA). Nevertheless, it is only sporadically used in the design practice, since it is, for the time being, still too complex for regular use. It requires substantial experience and knowledge about the modeling of the dynamic response of structures and seismic loading. Specialized software is also needed. The results are typically quite extensive, and their interpretation is often time-consuming and too demanding for an engineer with an average knowledge about seismic engineering.

To keep the inelastic analysis relatively simple and make it more apparent for practicing engineers, different static inelastic methods have been developed. Mostly they are based on the pushover analysis. They are considered more user-friendly and relatively easy to understand. They have a great advantage in specifying the seismic input, i.e., they employ the familiar elastic response spectra instead of selecting and scaling ground motion histories (Kappos et al. 2012).

From the historical point of view, pushover analysis has been typically used as a convenient tool to estimate the nonlinear properties and capacity of individual structural components or the whole structure. It typically represents the first step of different static inelastic methods. More specifically, the static nonlinear analysis is performed using the multi-degree-of-freedom (MDOF) model of the analyzed structure. The main purpose of this analysis is to define the properties of the equivalent single-degree-of-freedom (SDOF) model, which is then further employed to estimate the maximum global displacement demand. This idea was first explored by Saiidi and Sozen (1981). Based on the estimated maximum global displacement demand, other response quantities of interest are then evaluated (see Fig. 1).

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 1

Version of the inelastic pushover-based analysis, included in EC8/3

In the majority of pushover-based methods, the pushover analysis is performed in a similar manner (see section “Pushover Analysis: Numerical Models and the Lateral Load Pattern”). However, the techniques that are used to estimate the maximum displacement demand are quite different. The pushover-based methods can be classified with respect to different parameters, e.g., based on (a) the representation of the earthquake input, (b) the type of analysis performed on the SDOF model, and (c) the way how the stiffness of the equivalent SDOF model is defined.

The seismic input is represented by the acceleration response spectrum included in the codes and the displacement spectrum or with a set of accelerograms. The type of analysis is static, dynamic, or response history. The stiffness that is used to calculate the maximum displacement demand is the equivalent pre-yielding stiffness or the equivalent secant stiffness, usually obtained based on the pushover analysis. This variety of solutions can result in a different estimation of the response. In section “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines,” those solutions (methods), which are adopted in Eurocode 8/3 (CEN 2005), ATC-40 (ATC 1996), and FEMA-356 (ASCE 2000), the recommended modifications in FEMA-440 (ATC FEMA 2005), and the displacement-based method, included in the NZSEE (2006), are presented. They were selected since they illustrate many of the basic concepts that are currently in use.

These methods are nowadays extensively and successfully used for the analyses of different types of buildings and bridges. Their popularity has increased since they are relatively simple to use but at the same time provide valuable information about the inelastic response, which is not possible to obtain with the elastic methods. They are a useful tool for understanding the general structural behavior corresponding to different seismic intensity levels. Besides other superior features, they can provide more realistic information about the force demand on brittle elements and the information on the influence of the strength degradation of individual components on the global behavior of the structure. They can identify the critical regions in the structure. This information is particularly useful when the existing structures have been evaluated and the strengthening techniques have been selected.

However, the pushover-based analysis includes certain assumptions which limit its capabilities. Basic variants of the above code methods can be characterized as single-mode nonadaptive methods, since they assume that the response is controlled by a single predominant mode, which remains unchanged after yielding occurs. This assumption limits their application, particularly when they are used for the assessment of those existing structures which are irregular in plan and/or in elevation. Consequently, the in-plan torsion and higher modes can significantly influence their response. In such cases the extended versions of the single-mode methods are needed. As an alternative, the other types of pushover methods, accounting for the influence of higher modes (multimode methods), can be used. Some issues that influence the applicability and the way of application of the single-mode methods are briefly discussed in section “Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods.” Some examples of the multimode pushover methods are briefly presented in section “Some Alternatives to Single-Mode Pushover-Based Methods.”

The assumption that the fundamental mode shape is almost invariant can also limit and complicate the application of nonadaptive methods in existing structures, where the fundamental mode can change considerably depending on the seismic intensity. This can affect the choice of the lateral load pattern in the pushover analysis and make it more complex (see section “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines”). As an alternative, adaptive pushover methods can be used. An example of these methods is briefly presented in section “Some Alternatives to Single-Mode Pushover-Based Methods.”

The methods presented in sections “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines” and “Some Alternatives to Single-Mode Pushover-Based Methods” were primarily developed for the analysis of buildings. The response of bridges, particularly in the transverse direction, can be significantly different and more complex. Therefore, some modifications of these procedures are needed. A short overview of the application of the inelastic static methods for the analysis of bridges is presented in section “Application of the Inelastic Static Methods to the Analysis of Bridges.”

In this contribution, only those inelastic static methods that are included or referred to in the codes and national guidelines are presented. Quite a long list of other inelastic static procedures currently available in the literature can be created. Some of them are general, and some are specialized for certain types of structures. Many of them are presented in different state-of-the art reports (e.g., FEMA 440 2005; CEB-FIB 2003; Kappos et al. 2012, etc.) or other specialized literature (e.g., Bhatt 2011).

Pushover Analysis: Numerical Models and the Lateral Load Pattern

Pushover analysis is the static nonlinear analysis which is typically performed as the first step of the majority of the inelastic static methods for the seismic assessment of the (existing) structures. The structure is subjected to the lateral load (representing the inertial forces), the intensity of which is gradually increased. The corresponding lateral displacement at a certain location (the reference point or control point) in the structure is recorded. Then the pushover curve, representing the relationship between the base shear of the structure and the registered displacements, is constructed (see Fig. 1c).

The pushover analysis is usually performed employing the MDOF model of the structure which is similar to the linear elastic finite-element models (see Fig. 1a). The most important difference is that the properties of some or all of the components of the model include the post-elastic strength and deformation characteristics in addition to the initial elastic properties. They are usually defined approximating the response observed in the experiments or the response defined by the theoretical analysis of individual components. The envelopes or backbone curves are approximated based on the type of the response. Typical examples are presented in Fig. 2.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 2

Idealized numerical models

Contrary to the structures designed according to modern codes, where the brittle types of failure are in general avoided, in structures which were designed before the modern principles of seismic engineering were established, different types of brittle or semi-ductile failure of their structural components may be expected. Thus for the reliable estimation of their response, the appropriate numerical models should be employed. Their properties can be defined based on different procedures, defined in the (pre)standards, guidelines, and literature (e.g., CEN 2005, FEMA-356, ATC-40, CEB-FIP 2003). It is, however, worth noting that the available models which describe quite complicated mechanisms that reduce the ductility capacity of the structural components have been less frequently investigated and are often less reliable. The properties of these models are often defined using empirical or semiempirical procedures. The results of these procedures can be considerably different.

For example, the value of the shear strength of RC components can strongly depend on the method used to estimate it. The differences between different methods depend on many parameters, e.g., the amount of the flexural and shear reinforcement, the shear span ratio, the axial force, etc. Moreover, the numerical models that can take into account the complex interaction between shear and flexural response in the nonlinear range are still under investigation and evaluation (e.g., Mergos and Kappos 2008; Fischinger et al. 2012). Similar observations can be applied to other mechanisms, which reduce the ductility capacity of the structural components and the whole structure. Therefore, it is feasible to explore and compare different available options before the numerical model, and its properties are defined. All uncertainties related to the material properties should also be properly explored.

In the pushover analysis, the MDOF model of the structure is subjected to lateral forces that are intended to simulate the inertial forces expected in the building during an earthquake. They are usually distributed according to a selected (mostly invariant) pattern. It significantly influences the pushover curve and further determines the relative magnitudes of the shear forces, the moments, and deformations within the structure; therefore, it should be carefully selected, depending on the properties of the analyzed structure and the expected response. The distribution of the lateral inertial forces will in general vary continuously during the earthquake response, depending on the gradual formation of the plastic hinges in the structure. To take into account these changes, in many codes and guidelines, it is recommended to perform an analysis, taking into consideration at least two lateral force patterns. Typically the distribution of forces proportional to the fundamental mode of vibration in the elastic range and the uniform distribution are recommended. Then the most adverse results are taken into consideration. Other solutions recommended in the codes are presented in section “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines.”

The approach described in the previous paragraph is in general appropriate for many structures, which are designed according to modern standards. However, in some older existing structures, this solution is not always suitable.

For example, in medium-rise frame buildings, which are designed according to capacity-design principles (where the columns’ flexural strength exceeds the beam flexural strength), the triangular and uniform distribution are typically applied. In existing medium-rise buildings, this is also an appropriate solution, but only if the beam-sway mechanism is the most likely to be developed. In other words, the structure does not include a so-called soft. In opposite cases, the uniform distribution may not be able to frame the range of actions that may occur during the actual dynamic response. Thus a rather complex choice of the lateral force pattern may be needed. The response of such structures in the elastic range will be approximately linear (see Fig. 3). In the nonlinear range, the displacement pattern can considerably change after the formation of the plastic hinges, as is illustrated in Fig. 3. In such cases, some (pre)standards (e.g., FEMA-356, ATC-40) recommend the use of the adaptive load pattern, which can take into account changes of the displacement response shape corresponding to the seismic intensity applied. An alternative solution is the adaptive pushover-based methods (one of which is presented in section “Some Alternatives to Single-Mode Pushover-Based Methods”). Note, however, that if either the adaptive load pattern is used or the adaptive methods are applied, the analysis procedure is more complex and more demanding.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 3

In some structures (e.g., with “soft” story), the deflection line can considerably change after yielding

Similar observations to those presented above can be applied, e.g., to the RC dual and wall buildings, with the reinforcement that does not meet the requirements of current codes (e.g., where plastic hinges can form at the elevations above the foundations).

The accuracy of different lateral load patterns in structures where the higher modes have an important influence on the response is discussed in sections “Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods” and “Application of the Inelastic Static Methods to the Analysis of Bridges”.

Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines

The static pushover analysis has no rigorous theoretical foundation. It is based on the assumption that the response of the structure (MDOF system) can be estimated using the results of the analysis of an equivalent SDOF oscillator (see Fig. 1). This means that it is assumed that the response is governed by one invariant mode of vibration. In general this is incorrect. However, the assumption is approximately fulfilled in many (regular) structures, where the influence of the higher modes is negligible and the deflection shape is almost invariable. Thus the seismic response of these MDOF systems is quite accurately estimated based on the analysis of an equivalent SDOF model. It is not the intention of this article to explain the theoretical background of the formulation of the equivalent SDOF system, since it can be found elsewhere (e.g., Krawinkler and Seneviratna 1998). Here, different applications of this basic concept, which are typically used for the assessment of the existing structures, are overviewed.

In most of the modern (pre)standards and guidelines for the assessment of existing structures, the simplest form of the pushover-based analysis is included. All methods that are included in these documents are based on the same basic concept presented above. However, these methods are not the same. The procedures which are used to define the properties of the equivalent SDOF model and those which are used to define the seismic demand of this model are different. In general two different approaches are used. The properties of the equivalent SDOF system are defined either based on the equivalent pre-yielding stiffness or based on the equivalent secant stiffness (see Fig. 4a, b).

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 4

Two different concepts, which are used to define the properties of the equivalent SDOF model

If the first approach is employed, the maximum response of the SDOF oscillator is typically defined based on the 5 % damped acceleration spectra proposed in the codes. The target displacement of the equivalent SDOF system is obtained using the equal displacement rule approximation (see Fig. 4c). Since this approximation is only suitable for the medium- and long-period structures, the displacements are corrected for short-period structures. This approach is applied in, e.g., Eurocode 8/3 (EC8/3) and implicitly in FEMA-356 (see sections “Eurocode 8/3 (CEN 2005)” and “FEMA-356”). In FEMA-356, the maximum seismic displacements, estimated based on the analysis of SDOF system, are additionally corrected to take into account different issues which are not included in the pushover-based analysis such as strength degradation, P-Δ effect, etc.

If the secant stiffness is used to define the properties of the equivalent SDOF oscillator, typically the overdamped acceleration spectra are used (see Fig. 4d). Actually the capacity spectrum method approach is followed. It is explained later in section “ATC-40” where the method included into the ATC-40 is presented.

In the NZSEE 2006, different inelastic methods are included. The one which is conceptually different from those in other standards is described in this article (see section “FEMA-440”). Contrary to the previously mentioned procedures, the maximum seismic displacement at the reference point is assumed at the beginning of the analysis taking into account the estimated capacity of the structure. It is then used to estimate the properties of an equivalent SDOF model. The dynamic properties of the SDOF model are defined based on the secant stiffness corresponding to the assumed maximum displacement. The seismic demand is estimated using the overdamped displacement spectra, derived from the 5 % damped acceleration spectra.

Since the methods included in different codes are conceptually different, they are presented in more detail in the next subsections. The FEMA-356 and ATC-40 methods were evaluated in the document FEMA-440, where their modifications are proposed. The main observations and the proposed modifications are presented in section “ATC-40.” The matters of accuracy and different issues that can influence the pushover-based analysis of buildings are discussed in section “Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods.” The estimation of the response of bridges using the inelastic pushover-based analysis is presented in section “Application of the Inelastic Static Methods to the Analysis of Bridges.”

Eurocode 8/3 (CEN 2005)

The method that is included in Eurocode 8/3 (EC8/3) was developed by Fajfar in the 1980s. Its description can be found in (Fajfar 1999). It is included in different parts of the Eurocode 8 standards. Actually the EC8/3 refers to Eurocode 8/1 (CEN 2004) – EC8/1 – and its informative annex B, where the suggested way of application of the method is presented. It is overviewed in Fig. 1.

In the first step, the pushover curve is constructed applying forces proportional to the assumed displacement shape. In EC8/3 it is required to consider at least two force patterns: (a) modal pattern – proportional to the lateral forces consistent with the lateral force distribution determined in an elastic analysis – and (b) uniform load pattern.

Based on the pushover analysis, the properties of the equivalent SDOF model are calculated. First, the pushover curve is converted to the capacity curve. Displacements and forces are divided by the transformation coefficient:

$$ \Gamma =\frac{\varSigma {m}_i{\phi}_i}{\varSigma {m}_i{\phi}_i^2} $$

(1)

m _i is the mass at the location i in the structure (e.g., mass of ith floor of the building), and ϕ _i is a corresponding component of the assumed displacement shape.

Then the equivalent pre-yielding stiffness of the equivalent SDOF model k* is defined as is shown in Fig. 5a. The elastic-perfectly plastic idealization is proposed.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 5

(a) Idealization of the capacity curve in EC8/2, (b) idealization of the pushover curve in FEMA 356

The mass of the SDOF oscillator is determined as

$$ {m}^{*}={\displaystyle \sum {m}_i}{\phi}_i $$

(2)

The period of vibration of the SDOF model is defined based on the m* and k*:

$$ {T}^{*}=2\pi \sqrt{\frac{m^{*}}{k^{*}}} $$

(3)

The target displacement d _t * (inelastic displacement of the equivalent SDOF oscillator) is defined as

$$ {d}_{et}^{*}={S}_e\left({T}^{*}\right)\cdot {\left(\frac{T^{*}}{2\pi}\right)}^2 $$

(4)

$$ {d}_t^{*}={d}_{et}^{*} $$

(5)

for medium- and long-period structures with T* ≥ T _c , where T _c is the corner period between the short- and medium-period range. S _e (T*) is the elastic acceleration in the response spectrum at the period T*.

The same relationship is used for short-period structures if their response is elastic. If the response is nonlinear, then the target displacement is defined as

$$ {d}_t^{*}=\frac{d_{et}^{*}}{q_u}\left(1+\left({q}_u-1\right)\frac{T_C}{T^{*}}\right)\ge {d}_{et}^{*} $$

(6)

where q _u is the ratio between the acceleration in the structure with unlimited elastic behavior S _e (T*) and in the structure with limited strength F _y */m*:

$$ {q}_u=\frac{S_e\left({T}^{*}\right){m}^{*}}{F_y^{*}} $$

(7)

In the next step, the seismic displacement of the structure (MDOF model) is defined multiplying the target displacement by the transformation coefficient Γ.

The static nonlinear analysis of the MDOF system is repeated up to the estimated seismic displacement in order to be able to analyze different aspects of the response (e.g., story drifts, shear forces, bending moments). The same loading pattern as in the first pushover analysis is employed.

If the target displacement d _t * is quite different from the displacement d _m * used to determine the idealized elastic-perfectly plastic force-displacement relationship, an iterative procedure may be applied.

FEMA-356

In the first step, the pushover analysis is performed. As in the EC8/3, at least two lateral force patterns need to be applied. Three possibilities are defined for the first load pattern: (a) forces proportional to the fundamental mode; (b) forces proportional to the values of coefficient C _vx , defined in the standard; and (c) proportional to the story shear distribution calculated by combining the modal responses from a response spectrum analysis of the building. The use of patterns (a) and (b) is limited to structures where more than a 75 % mass participate in the fundamental mode. The pattern (c) can be used in structures where the period of the fundamental mode exceeds 1 s.

For the second load pattern, two options are defined: (a) uniform distributions and (b) an adaptive load distribution that changes when the new plastic hinges are formed (see the discussion in section “Pushover Analysis: Numerical Models and the Lateral Load Pattern”). Different options for the adaptive load distribution are referred to in Fajfar and Fischinger (1988), Eberhard and Sozen (1993), and Bracci et al. (1997).

The pushover curve is idealized as is shown in Fig. 5b. The idealization is bilinear. Note that only the case with a positive post-yield slope is presented in Fig. 5b. In the standard, the case with the negative post-yield slope is also considered. The effective period T _e is defined based on the pre-yielding stiffness, determined in the idealized pushover curve.

The seismic displacement (the term target displacement is used in FEMA-356, but it does not have the same meaning as in the EC8/3) is defined using the so-called coefficient method as

$$ {\delta}_t={C}_0{C}_1{C}_2{C}_3\cdot {S}_a\frac{T_e^2}{4{\pi}^2}g $$

(8)

where

T _e is the effective fundamental period

S _a response spectrum acceleration at the effective fundamental period

g gravitational acceleration

C ₀ is the modification factor to relate the spectral displacement of an equivalent SDOF system to the roof displacement of the building’s MDOF system. Actually it has a similar meaning as factor Γ in EC8/3

C₁ is the modification factor to relate the expected maximum inelastic displacements to the displacements calculated for the linear elastic response. The meaning is similar to the relationship between d _t * and d _et *in EC8/3. For long- and medium-period structures, C ₁ is 1, and for short-period structures, it is

$$ {C}_1=\frac{\left(1+\frac{\left(R-1\right)\cdot {T}_s}{T_e}\right)}{R} $$

(9)

R is the ratio of elastic demand to the calculated strength capacity (a similar meaning to q _u in EC8/3).

The C ₂ modification factor represents the effect of a pinched hysteretic shape, stiffness degradation, and strength deterioration on the maximum displacement response.

The C ₃ modification factor represents increased displacements due to dynamic P-Δ effects.

ATC-40

While several similarities can be found between the EC8/3 and FEMA-356, the simplified nonlinear procedure in ATC-40 is rather different. It is based on the equivalent linearization. This is a version of the capacity spectrum method, which was first introduced by Freeman et al. (1975). The basic assumption is that the maximum displacement of the nonlinear SDOF system can be estimated from the maximum displacement of a linear elastic SDOF system that has the period and damping ratios that are larger than those of the initial values for the nonlinear system. The elastic SDOF system that is used to estimate the maximum inelastic displacements of the nonlinear system is usually referred to as the equivalent or the substitute system. The period and damping of the equivalent system are referred to as the equivalent period and equivalent damping ratio, respectively.

As in the previous two methods, the pushover analysis is performed first. In general, ATC-40 recommends the distribution of the lateral forces proportional to the fundamental mode pattern. In structures where the response considerably changes after yielding (e.g., in structures with soft stories – see the discussion in section “Pushover Analysis: Numerical Models and the Lateral Load Pattern”), the adaptive load pattern is required. In high-rise buildings or irregular buildings, the influence of the higher modes should be properly taken into account. The higher mode effects may be determined by doing higher mode pushover analyses.

The application of the capacity spectrum technique means that both the structural capacity curves and the demand response spectra are plotted in the spectral acceleration versus the spectral displacement domain and compared. Therefore in the next step, the pushover curve is converted to the capacity spectrum curve using the modal shape vectors, participation factors, and modal masses obtained from a modal analysis of the structure. The capacity spectrum curve represents the relationship between accelerations S_a and displacements S_d of the equivalent SDOF oscillator. Then the standard elastic acceleration spectrum (corresponding to 5 % damping) is converted to the ADRS format, where the spectral accelerations are presented as a function of the corresponding spectral displacements (see Fig 4d). In this way, the capacity curve and the seismic demand can be plotted on the same axes and compared.

The capacity spectrum method of equivalent linearization assumes that the equivalent damping of the system is proportional to the area enclosed by the capacity curve. The equivalent period, T_eq, is assumed to be the secant period at which the seismic ground motion demands, reduced by the equivalent damping, intersect the capacity curve (FEMA-440). Since the equivalent period and damping are both a function of the displacement, the solution to determine the maximum inelastic displacement (i.e., performance point) is iterative.

The equivalent period T _eq and effective viscous damping β _eq (which is used to reduce the seismic demand) are defined as

$$ {T}_{eq}={T}_0\sqrt{\frac{\mu }{1+\alpha \mu -\alpha }} $$

(10)

$$ {\beta}_{eq}=0,05+\kappa \frac{2\cdot \left(\mu -1\right)\cdot \left(1-\alpha \right)}{\pi \cdot \mu \cdot \left(1+\alpha \mu -\alpha \right)} $$

(11)

where

T₀ is the initial period of vibration of the nonlinear system, α is the post-yield stiffness ratio, and κ is an adjustment factor to approximately account for the changes in the hysteretic behavior in reinforced concrete structures.

The adjustment factor κ depends on the hysteretic behavior of the system. Three equivalent damping levels are defined. Type A corresponds to structures with reasonably full hysteretic loops, similar to the elastic-perfectly plastic oscillator. Type C corresponds to structures with severely degraded loops, and type B denotes the hysteretic behavior between types A and C. The value of κ and the corresponding equivalent damping is the largest for systems with a hysteretic behavior of type A. Their values decrease for degrading systems B and C. The existing buildings are in general categorized as structures of type C or B, depending on the shaking duration and hysteretic behavior of the structural components.

Based on the comparison of the capacity curve and seismic demand, the target displacement is defined. This displacement is then converted to roof displacement and other aspects of the response are defined.

The ATC-40 method is also well accepted due to the clear and useful visualization of the procedure. Note, however, that the procedure, defined in EC8/3, can also be presented in a similar manner; nevertheless, it is essentially different. The capacity curve as well as the seismic demand, defined in EC8/3, can also be converted to the ADRS format, plotted on the same axes and compared, as is presented in Fajfar (1999) and illustrated in Fig. 4c.

FEMA-440

FEMA-440 is the document where the previously described procedures (FEMA-356 and ATC-40) are evaluated. Improvements of both methods are recommended. Since the document is extended with many important observations, particularly for the existing structures, only some of the conclusions related to the methods included in FEMA-356 and ATC-40 are provided in this section. Some of them are also presented in section “Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods.”

Some Observations and Proposed Improvements of the Procedure, Included in FEMA-356

It was observed that the characteristic periods which are used to differentiate the response of short- versus medium- and long-period structures were found to be shorter than those observed from nonlinear response history analyses. This can result in an underestimation of the inelastic deformations for the periods between the characteristic period and the periods that are approximately 1.5 times the characteristic period.

The equal displacement rule approximation was found to lead to a relatively good approximation of the maximum inelastic deformations for systems with elastic-perfectly plastic behavior and periods longer than 1 s. This is not always applicable to soft soil sites and near-fault records.

The limiting values of coefficient C₁, which defines the ratio of elastic and inelastic deformations, can lead to a large underestimation of the displacements in short-period structures. Even if this limitation was not taken into account, the magnification of inelastic displacement demands with a decreasing lateral strength for short-period structures was found to be larger than that suggested by FEMA-356. The corrections of coefficient C₁ were proposed for short-period structures.

It was observed that in many cases the cyclic degradation does not increase the maximum displacements. Thus the use of the related coefficient C₂ is recommended only for structures with significant stiffness and/or strength degradation. Coefficient C₂ is corrected.

It was found that coefficient C₃, which is intended to represent an increase of displacements due to the dynamic P-Δ effects, cannot adequately take into account the possibility of dynamic instability. It was proposed that this coefficient be eliminated. Instead, it was proposed to use the NRHA for all structures where the strength is below the limit proposed in FEMA-440.

Some Observations and Proposed Improvements in the Procedure Included, in ATC-40

The accuracy of the estimated maximum displacement response in long-period structures depends on the hysteretic behavior type. In structures with hysteretic behavior type A and periods longer than about 0.7 s, the ATC-40 procedure underestimates the maximum displacements. In structures with a hysteretic behavior type B and periods longer than about 0.6 s, small underestimations or overestimations of the maximum displacements were observed. This depends on the level of lateral strength and on the site class. For structures with hysteretic behavior type C, the ATC-40 procedure leads to an overestimation of the lateral displacements regardless of the period of the structure.

In short-period structures (with periods shorter than noted in the previous paragraph), a significant overestimation of the maximum displacements were observed. The overestimation increases with decreasing strength.

It was also found that the ATC-40 assumption, where it is supposed that the inelastic deformation demands of structures with hysteretic type B will be larger than those in structures with hysteretic type A, does not agree with the results of the NRHA. According to the NRHA, these deformations were approximately the same and in some cases even slightly larger in structures with hysteretic behavior type A. The provisions of ATC-40 do not address the potential dynamic instability that can arise in systems with in-cycle strength degradation and P-D effects.

The suggested improvements of the ATC-40 procedure are presented in chapter 6 of FEMA-440. They are focused on improved estimates of the equivalent period and equivalent damping. It is concluded that generally the optimal effective period is less than the secant period and the optimal effective damping is also less than that specified in ATC-40.

More details about the accuracy, advantages, and drawbacks of both methods can be found in Appendix A of FEMA-440.

NZSEE 2006

The NZSEE 2006 addresses different methods of analysis, referred to as (a) force-based method, (b) displacement-based method, (c) consolidated force-displacement-based method, and (d) method using a nonlinear pushover analysis (the pushover analysis is performed taking into consideration the same distribution of lateral load as in FEMA-356).

Since the displacement-based method (DBM) is different from those presented in the previous sections, it is described in more detail in the next paragraphs. The basis of the method was developed by Priestley (1995) and further evaluated in, e.g., Priestley et al. (2007). The method places a direct emphasis on establishing the ultimate displacement capacity of the lateral force-resisting elements. Contrary to the methods presented in the previous sections, where the seismic displacements were calculated using acceleration spectra, in the DBM, the displacement spectra are considered.

In the first step, the flexural and shear strengths of the critical sections of the structural components, assuming that no strength degradation occurs due to the cyclic lateral loading in the post-elastic range, are estimated. In the second step, the post-elastic deformation mechanism of the structure and the probable horizontal seismic base shear capacity (V _prob ) are determined. It is recommended that the post-elastic mechanism is defined using the (a) simple lateral mechanism analysis (presented in the guidelines), (b) inelastic response history analysis, and (c) nonlinear pushover analysis. When the nonlinear pushover analysis is used to estimate the nonlinear response, it is performed in the same manner as in FEMA-356 (the same distribution of the lateral load is taken into account).

In the third step, the plastic rotation capacities of the structural members are determined. It is then eventually corrected if the shear failure occurs before the limits to the flexural plastic rotation capacity are reached. The story inelastic drift capacity is estimated from the plastic rotation capacities.

Considering the critical storey drift and the previously defined post-elastic deformation mechanisms, the overall displacement capacity U _sc and the ductility capacity μ are determined at the effective height h _eff of the substitute structure (an equivalent SDOF model of the structure). The effective secant stiffness k _eff at the maximum displacement U _sc is determined as k _eff  = V _prob /U _sc . Then the corresponding effective period of vibration T _eff is calculated as

$$ {T}_{eff}=2\pi \sqrt{\frac{M}{k_{eff}}} $$

(12)

where M is the effective mass of the substitute structure (an equivalent SDOF model of the structure). Alternatively T _eff can be estimated directly using the Rayleigh-Ritz equation.

Based on the evaluated ductility capacity, the equivalent viscous damping ξ _eff of the structure is defined. It is recommended to use the method suggested in Pekcan et al. (1999). In general the equivalent viscous damping is determined in a similar manner as in ATC-40, summing the viscous and hysteretic damping.

The structural performance factor S _p is calculated according to NZS 1170.5 (2004) and taking into consideration the detailing used in the structure. It is used to reduce the seismic design actions on a structure. The displacement demand is defined using the overdamped elastic displacement spectrum. It is defined reducing the displacement spectrum δ(T), derived from the 5 % damped elastic acceleration spectrum. The displacement spectrum δ(T) is reduced using the correction factor K _ξ :

$$ {K}_{\xi }=\sqrt{\frac{7}{2+\xi }} $$

(13)

where ξ = ξ _eff .

Thus the spectral displacement demand of the analyzed structure at height h _eff is defined as

$$ {U}_{sd}={S}_p{\left(\%NBS\right)}_t\delta \left({T}_{eff}\right){K}_{\xi } $$

(14)

where (%NBS)_t is the target percentage of the new building standard. The %NBS is essentially the assessed structural performance of the building compared with the requirements for a new building.

Then the displacement capacity U _sc and the demand U _sd are compared. If U _sc /U _sd  ≥ 1, retrofit is unnecessary to achieve (%NBS)_t and vice versa.

Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods

In this section, some issues that can influence the accuracy of the pushover-based methods are presented: the idealization of the capacity curve, the influence of the higher modes, and the influence of the in-plan torsion. In the last two subsections, the basic observations from FEMA-440 about the strength degradation and the soil structure interaction are cited.

Idealization of the Capacity Curve

In some methods presented in section “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines,” the actual capacity curve is idealized either by elastic-perfectly plastic or by a bilinear relationship (see Figs. 4 and 5). This approximation is one of the key issues, since it defines the equivalent stiffness of the equivalent SDOF model. This stiffness further influences the period of the SDOF oscillator and the estimated value of the target displacement. Thus the approximation should be carefully performed.

The approximation of the capacity curve is usually performed equating the area bounded (energy) with the actual and idealized curve (see Fig. 5). At this stage, the target displacement should be assumed, because the compared areas depend on the maximum displacement. Thus if the assumed target displacement differs considerably from the value obtained at the end of the analysis of the SDOF system, iterations are strongly recommended. Otherwise the target displacement can be poorly estimated.

In structures that do not exhibit considerable strain hardening in the nonlinear range, the elastic-perfectly plastic idealization is accurate. When the strain hardening is considerable, e.g., in some types of bridges, bilinear idealization is more appropriate (see section “Application of the Inelastic Static Methods to the Analysis of Bridges” for more details).

The Influence of Higher Modes

It was mentioned before that the basic variants of the methods described in section “Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines” can be classified as single-mode nonadaptive methods since they assume that the response is controlled by a single predominant mode which remains unchanged after yielding occurs. This assumption limits their application. For example, when they are used for the analysis of existing structures which are irregular in elevation and/or in plan (see an example in Fig. 6), certain extensions are needed in order to take into account the important influence of higher modes and the in-plan torsion. In high-rise buildings, the influence of the higher modes can be important regardless of their irregularity. In such structures, multimode pushover methods (see section “Some Alternatives to Single-Mode Pushover-Based Methods” for more details) or NRHA can be used as an alternative to single-mode pushover-based methods.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 6

An example of the structure, irregular in plan and elevation

The influence of the higher modes has been the subject of different studies. The one presented in FEMA-440 outlines the most important observations. Some of them are provided in the following paragraphs.

The nonlinear static pushover procedures appeared to be reliable for the design and evaluation of low-rise buildings. In relatively tall frame buildings, where the higher-mode response is significant, interstory drifts, story shears, and overturning moments can deviate significantly from the NRHA.

In buildings the importance of a higher-mode effect increases with the amount of inelasticity (note that in some bridges quite the opposite trend was observed – see section “Application of the Inelastic Static Methods to the Analysis of Bridges” for more details). Typical examples are, e.g., RC shear walls, where this phenomenon has been observed years ago. An explanation can be found elsewhere (e.g., Rejec et al. 2012).

Higher-mode contributions become more significant for structures with fundamental periods that fall into the constant-velocity part of the response spectrum. Forces which are developed due to the important influence of higher modes can considerably influence the failure mechanism. Thus the pushover analysis cannot always identify this mechanism.

In FEMA-440, a single first-mode distribution was found sufficient for the estimation of displacement and other quantities that were not significantly affected by higher modes. The adaptive load distribution was sometimes better and sometimes worse than the first-mode distribution. The uniform distribution is recommended in most of the codes as the second choice in order to frame the response quantities. However, in FEMA-440 it was found that it often did not fulfill this role. The uniform distribution was found to be the worst with regard to all the monitored response quantities. Thus it was not recommended as a stand-alone option.

All the codes addressed in this entry recognize that the single-mode pushover-based methods are less efficient when the higher modes are important. Thus they require these methods to be combined with the results of the linear dynamic procedures or the multimode pushover analysis is required. Specific solutions can be found in the particular standard.

The Influence of In-Plan Torsion, Two-Dimensional (2D) Versus Three-Dimensional (3D) Analysis

The in-plan torsion can importantly influence the response of the existing structures which are irregular in plan. In such structures, the 3D analysis is generally needed. The considerable influence of in-plan torsion can be expected in torsionally flexible structures, where the first mode of vibration is torsional. Substantial torsional effects can also be obtained in one direction of structures where the second mode of vibration is torsional.

It was observed that the inelastic static methods can significantly underestimate the displacements on the stiff/strong side of such buildings (see, e.g., Kreslin and Fajfar 2012). Thus the displacements on the stiff/strong side of the structure should be increased.

Most of the (pre)codes recognized this phenomenon. Thus they require an increase in the seismic displacements, defined by inelastic static methods, due to the torsional effects. All of them require a 3D analysis of the structures where the in-plan torsion is important. According to all codes, the accidental eccentricity should be taken into account in all analyses regardless of their torsional flexibility. The specific requirements related to an increase of displacements due to the torsional effects differ from standard to standard.

In general, the 3D pushover-based analysis and torsional effects are topics that require additional investigations. Thus, they are the subjects of numerous researches, reported in the literature. An overview of these researches is recently provided in Bhatt (2011).

The Strength and Stiffness Degradation

The strength degradation, including P-Δ effects, can lead to an apparent negative post-elastic stiffness in a force-deformation relationship for a structural model using nonlinear static procedures. The performance implications depend on the type of strength degradation (cyclic or in-cycle strength degradation). For structures that are affected by component strength losses, including P-Δ effects, occurring in the same cycle as yielding (in-cycle strength degradation), the negative post-elastic slope can lead to the dynamic instability of the structural model (FEMA-440 2005). For this reason, it is suggested that the pushover-based methods are only used if the strength of the structures is above a certain prescribed limit. Otherwise the use of the NRHA is recommended.

The Soil-Structure Interaction

There is a perception among many in the practicing engineering community that short, stiff buildings do not respond to seismic shaking as adversely as might be predicted analytically. There are several reasons why short-period structures may not respond as conventional analysis procedures predict. Among these are (a) radiation and material damping in supporting soils, (b) structures with basements that experience reduced levels of shaking, (c) incoherent input to buildings with relatively large plan dimensions, and (d) inaccuracies in modeling, including dumping of masses, neglecting the foundation’s flexibility, and some elements that contribute to the strength (FEMA-440 2006). In FEMA-440 procedures, it is proposed that soil-structure interaction is incorporated into the nonlinear static analyses.

Some Alternatives to Single-Mode Pushover-Based Methods

Multimode procedures are considered as an alternative approach for the analysis of structures where the single-mode methods are less accurate. There are many multimode pushover methods described in the literature. In this section, two of them are briefly presented. The multimode pushover analysis (MPA) (Chopra et al. 2004) is selected as being representative of the nonadaptive multimode pushover-based methods, and incremental response spectrum analysis (IRSA) (Aydinoğlu 2003) is selected as the representative of the adaptive multimode pushover-based methods. Both methods were addressed in FEMA-440. The MPA was evaluated as an alternative to the single-mode pushover-based methods, and the IRSA was recognized as the potential improvement of the inelastic analysis techniques that can be used to reliably address the MDOF effects.

The MPA Method

The MPA method was developed by Chopra and Goel (e.g., Chopra and Goel 2002). The analysis is performed in a similar manner as that presented in section “Eurocode 8/3 (CEN 2005).” However, the number of the analyses depends on the number of important modes, identified in the initial – elastic – state. The pushover analysis is performed separately for each important mode. The lateral load is proportional to the shape of the vibration mode. Based on each pushover analysis, the contribution of the related mode of vibration to the seismic displacements is defined. These contributions are then combined using the appropriate combination rule (e.g., SRSS). The method supposes that the modes of vibrations are invariant. Thus it can be classified as the nonadaptive method. This limits its applicability. When considerable changes of the mode shapes can be observed in the nonlinear range (an example is described in section “Pushover Analysis: Numerical Models and the Lateral Load Pattern”), the method is in general less reliable.

It was examined in FEMA-440 as an alternative to single-mode pushover-based methods. Taking into account a study of five buildings of very different properties, it was found to be more accurate than the single-mode pushover methods, but not completely reliable. Similar observations were obtained based on other studies, e.g., studies of bridges, presented in Kappos et al. (2012).

The IRSA Method

The IRSA method was developed by Aydinoğlu (2003). It is the multimode adaptive pushover method, which means that it can take into account the influence of the higher modes as well as their changes depending on the seismic intensity. The contributions of the different modes are considered in an incremental pushover analysis.

When the structure enters the nonlinear range, its dynamic properties are changed each time a new plastic hinge is developed. In regular structures, these changes are typically quite small. In irregular structures, the mode shapes as well as their contributions to the overall response can significantly change. IRSA can take into account these changes. More importantly, IRSA is also capable of taking into account the effect of modal coupling to the formation of the plastic hinges.

Different studies (e.g., Kappos et al. 2012) have confirmed that the method is quite reliable; however, it also has certain limitations, and thus it is not universal and cannot replace the NRHA in all cases, for example, in certain types of bridges.

Application of the Inelastic Static Methods to the Analysis of Bridges

With regard to their dimensions and structural systems, and in general with regard to their seismic response, bridges are quite different from buildings. Therefore the application of different pushover-based methods, which were originally developed for buildings, is not straightforward, particularly when the bridges are analyzed in the transverse direction. For the analysis of bridges in the transverse direction, the pushover methods in general should be applied in a slightly different manner. The analysis differs mainly regarding: (1) the choice of the reference point where the displacements are registered, (2) the distribution of the lateral load, and (3) the idealization of the capacity curve. These issues are briefly presented in the following subsections. In the last subsection, the applicability of the pushover methods for the analysis of bridges is discussed.

The Reference Point

In buildings, the center of mass of the roof is typically selected as the reference point. In bridges, this choice is not straightforward. The specialized standards for the design of bridges often recommend the center of mass of the deformed deck as the reference point. An alternative solution could be the top of a certain column. However, in irregular bridges, both of these solutions could be inadequate.

In general, in highly irregular bridges, such as the one presented in Fig. 7, the position of the maximum displacement can considerably vary depending on the seismic intensity. In the bridge, presented in Fig. 7, the mode shapes, their importance, and the ratios are changing depending on the seismic intensity. When the response is in the elastic range, the maximum displacement is observed close to the top of the right pier. When the yielding of the central shortest pier is reached, the station of the maximum displacement is moved toward the center of the bridge (center of mass).

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 7

An example of a highly irregular bridge

Three quite different pushover curves were obtained when each of the three columns was considered as the reference point (curves P1–P3 in Fig. 8a). Consequently, the dynamic properties of the equivalent SDOF model were also different. Since the importance of the different modes is changing considerably, depending on the seismic intensity, significantly different displacements of the structure (see curves P1–P3 in Fig. 8c) were estimated based on the pushover curves, presented in Fig. 8a. One can conclude that the pushover curve, corresponding to the column, where the maximum displacements were observed, should be used in the analysis. This is true, so far as this is the station of the maximum displacement of the superstructure, too.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 8

Pushover curves and displacement envelopes, obtained in a highly irregular viaduct, based on different reference points

The station of the maximum displacement of the superstructure in a viaduct, presented in Fig. 7, considerably changes depending on the seismic intensity. Thus, the corresponding pushover curve (see curve MD in Fig. 8b) does not coincide with any of the pushover curves constructed based on the displacements monitored at the top of a particular column. The corresponding displacements of the bridge are also significantly different from those calculated using the top of the columns as the reference points (see Fig. 8c).

Since the position of the maximum displacement is variable, it coincides with the center of mass only during stronger seismic intensities. Thus, the same conclusions as those presented in previous paragraphs can be applied for the center of mass, too.

One of the possible solutions is to consider the variable reference point when constructing the pushover curve. This means that the maximum displacement is monitored, wherever it is, in its variable position, which corresponds to a certain load level. In the case presented above, this further means that in the elastic range, the reference point is above the right pier, and after yielding of the central column, it is moved toward the center of mass. The resulting pushover curve is presented by a solid line in Fig. 8b, and the displacements of the superstructure with the bold solid line in Fig. 8c.

The Distribution of the Lateral Load

The lateral load pattern, in general, can be defined following the same basic recommendations as those for buildings: (a) distribution proportional to the fundamental mode of the bridge in the elastic range and (b) uniform distribution. Note however that the shape of the load pattern, proportional to the fundamental mode, depends on the type of the supports above the abutments as it is presented in Figs. 9 and 10. In regular bridges, pinned at the abutments, the parabolic distribution (see Fig. 9c) is also feasible.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 9

Distributions of the lateral load, appropriate for bridges that are pinned at the abutments

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 10

Distributions of the lateral load, appropriate for bridges with roller supports at the abutments

The Idealization of the Capacity Curve

The idealization of the capacity curve can significantly influence the stiffness of the equivalent SDOF model and the estimated value of the maximum seismic displacement. When this stiffness is not adequately estimated, the actual and estimated maximum displacement can be significantly different. In some methods, such as that included in the EC8/3, the capacity curve is approximated using the elastic perfectly plastic idealization. However, in viaducts which are pinned at the abutments, this idealization can be inappropriate, since an underestimated equivalent stiffness of the SDOF system and an overestimated maximum displacement (see Fig. 11) can be obtained. Namely, in bridges with pinned abutments, the capacity curve can exhibit a considerable strain hardening slope, which should be properly taken into account. This is illustrated in Fig. 11.

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 11

Idealization of the pushover curve

Applicability of Pushover-Based Methods for the Analysis of Bridges

The single-mode methods can accurately predict the response of regular bridges where the influence of higher modes is not important. This is the case where the effective mass of the predominant mode exceeds 80 % of the total mass. When the methods are nonadaptive, they can be accurately used if the mode shapes do not significantly change based on the seismic intensity.

In bridges where the superstructure is considerably stiffer than the supporting elements (piers), the influence of the higher modes is in general negligible. Typical representatives are short- and medium-span bridges (e.g., the length of a bridge is less than 500 m), which are not supported by very stiff (very short) piers (e.g., in single-column piers, the height of the columns exceeds 10 m).

Considerable changes of the mode shapes can be expected (and were observed) first of all in short bridges, where the displacements of the superstructure above the abutments are not restrained. In such bridges, the predominant mode usually changes considerably, when the damage of the side columns reduces their stiffness to such an extent that the torsional stiffness of the bridge becomes lower than its translational stiffness.

The accuracy of the single-mode pushover-based methods depends on: (a) the ratio of the superstructure stiffness and the stiffness of the bents (the length of the bridge and number and the location of the short columns along the bridge); (b) the relative strength of the columns, compared to the seismic intensity; and (c) the boundary conditions at the abutments (mostly in short bridges).

When the superstructure is stiff compared to the supporting columns (piers), it has a predominant role defining the response of the bridge. In such cases, the response is typically influenced by one predominant mode. However, if the bridge is supported by short stiff columns, they govern the response and cause the significant influence of the higher modes to the deflection line of the superstructure.

In many bridges, the accuracy of the single-mode methods increases with the seismic intensities (nonlinearity). Columns yield, their stiffness is reduced, and thus the superstructure has a more important role for the overall response. However, this is not the rule. In bridges where the torsional stiffness decreases when the columns yield, the single-mode methods cannot accurately predict the response at high intensity levels, due to the emphasized torsional rotations.

In bridges with roller supports above the abutments, considerable changes of mode shapes can occur, particularly when the side spans are relatively long. When they are supported by short stiff columns in the central part of the superstructure, considerable torsional effects can be obtained.

In long bridges, the influence of the higher modes in the majority of cases does not depend on the stiffness of the columns and their strength. In such bridges, the superstructure of the standard types becomes quite flexible, and consequently the higher modes become important, regardless of the stiffness and the strength of the columns.

When the response of the bridge is considerably influenced by the higher modes or the modes of vibration are changing considerably depending on the seismic intensity, the multimode pushover methods are needed. However, note that they also have certain limitations, which depend on their basic assumptions. For example, even quite accurate methods, such as IRSA (see section “Some Alternatives to Single-Mode Pushover Based Methods”), can fail to predict the response accurately when the bridge is supported by short stiff central columns. In such cases, the NRHA is needed.

More details and recommendations about the use of the pushover-based methods for the analysis of bridges can found in Kappos et al. (2012).

Summary

The inelastic static pushover-based methods, which are typically used for the assessment of the existing structures, are presented. Some of the basic concepts that are nowadays used are presented in the example of the four single-mode pushover-based methods included in the codes. Some parameters that influence the accuracy of these methods are presented: selection of the lateral load pattern, idealization of the capacity curve, numerical models, influence of the higher modes, and changes of the shapes of the vibration modes depending on the seismic intensity, in-plan torsion, strength degradation, and soil-structure interaction.

Two multimode pushover-based methods, which are typical examples of nonadaptive and adaptive multimode methods, are briefly overviewed. The application of the inelastic pushover-based methods to the analysis of bridges is discussed.

Cross-References

• Assessment and Strengthening of Partitions in Buildings

• Assessment of Existing Structures Using Response History Analysis

• Earthquake Response Spectra and Design Spectra

• Earthquake Return Period and Its Incorporation into Seismic Actions

• Equivalent Static Analysis of Structures Subjected to Seismic Actions

• European Structural Design Codes: Seismic Actions

• Incremental Dynamic Analysis

• Modal Analysis

• Nonlinear Dynamic Seismic Analysis

• Nonlinear Finite Element Analysis

• Numerical Modeling of Masonry Infilled Reinforced Concrete Frame Buildings

• Response Spectrum Analysis of Structures Subjected to Seismic Actions

• Retrofitting and Strengthening Masonries of Heritage Structures: Materials Used

• Retrofitting and Strengthening Measures: Liability and Quality Assurance

• Retrofitting and Strengthening of Contemporary Structures: Materials Used

• Retrofitting and Strengthening of Structures: Basic Principles of Structural Interventions

• Seismic Analysis of Concrete Bridges: Numerical Modeling

• Seismic Analysis of Masonry Buildings: Numerical Modeling

• Seismic Analysis of Steel and Composite Bridges: Numerical Modeling

• Seismic Analysis of Steel Buildings: Numerical Modeling

• Seismic Analysis of Steel–Concrete Composite Buildings: Numerical Modeling

• Strengthened Structural Members and Structures: Analytical Assessment

• Strengthening Techniques: Bridges

• Strengthening Techniques: Code-Deficient R/C Buildings

• Strengthening Techniques: Code-Deficient Steel Buildings

• Strengthening Techniques: Masonry and Heritage Structures