A brief description of these tools is detailed as follows:. In , Dumanogluid and Soyluk used the random vibration method, which is based on relating statistical values of the exciting forces with the corresponding internal forces arising as an excitation response. This method suggests a set of mutually stationary movements, finally generating three displacement components in the structural response: dynamic, pseudo-statical and covariance components.

The latter represents the statistics part of the problem, but due to practical issues, it is disregarded due to its low contribution to the total response. In addition, Soyluk compared three analysis methods based on the random vibration theory: the spectral analysis, the power spectral density function, based on the response spectrum, and the response spectrum method. The three methods used the cross-spectral density function.

The main difference between the methods was the way the maximum response was obtained. In Figure 3 shows the response of an arched bridge and a cable-stayed bridge analyzed through the three methods. The first two methods showed certain similarity, while the third method produces greater displacements at the span of both bridges. It is important to mention that the random vibration method suggests a set of mutually stationary movements, which implies a great disadvantage since the random nature of seisms produces energy processes that vary in accordance with time and space.

Another disadvantage is that in the engineering practice this method is not frequently applied since the typical practice is to determine input seismic forces through chronological analyses or spectral analyses. The method consists in generating seismograms for each of the supports by using the coherence function, which contains the wave passage effects, loss of correlation and local site effect.

The coherence is characterized by the triple product shown in equation 5 Zhang et al. However, Kassawara and Sandell propose an acceptable model based on the analysis of 12 seismographic arrays, which is recommended for any site condition, seism size and spacing between stations, except for abrupt topographical conditions. For the generation of accelerograms including asynchronous seismic excitation there are methods such as that used by Ghobarah et al.

However, the temporary nonstationarity does not guarantee the spectral nonstationarity of the movement and this last characteristic must be taken into account for the analysis of hysteretic structures according to Konakli and Kiureghian The spectral nonstationarity can be attributed through an evolving function of potential spectral density.

The disadvantage is that we still do not have a general method or studies that confirm whether kinematic in the movement is performed when an evolving function of potential spectral density is used. The response spectrum method used in the asynchronous analysis is based on the random vibration approach. It has the advantage of implicitly introducing a response spectrum to the structure, which is practical from the viewpoint of the designer Liang and Shou-lei and Cacciola and Deodatis In addition, the response spectrum obtained inherently includes the nonstationarity.

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The great disadvantage is that the method only uses modal superposition and it is limited to the linear analysis. The simplest method to model the asynchronous seismic excitation only taking into account the wave passage effect is the modification of an actual seismogram. There are seismic stations worldwide that are continuously monitoring and storing information of significant seismic events. This material could be implemented in the asynchronous analysis taking into account the wave passage effect. The method extends to nonlinear analysis according to Ghobarah et al.

Apart from that, the Eurocode 8 EC8 European Committee for Standardization proposes considering the asynchronism only if: i there are geological discontinuities, near faults or abrupt topographic characteristics; ii the length of the bridge exceeds m. The second consideration has been called into question through studies such as that carried out by A. However, proposals based on the research of the EC8 have been made in order to include lower limits with regard to the total length of the bridge, depending on the type of soil it is supported Sextos and Kappos The EC8 proposes three asynchronous analysis methods: the first method has to do with the description of the movement in the supports as a component of a random, homogeneous field that is stationary in time; the second method is about a simplified random model, and the third method is a pure kinematic model, which is based on developing a set of relative static displacements Valdebenito and Aparicio These methods are not very reliable because from the viewpoint of quantities of material there is no difference between designing under asynchronous seismic excitation and designing under uniform seismic excitation since the response does not vary substantially, while more elaborate methods do generate significant differences.

In addition, the methods of the EC8 are unable to identify fault points and do not allow working with high vibration modes, which characterize the asynchronous seismic excitation. Also, according to Sextos and Kappos , they are not applicable to curved bridges. In , Nuti and Vanzi carried out a study in order to establish design criteria for bridges under asynchronous seismic excitation in order to update the Italian code for bridges.

The analysis method used by Nuti was based on the fundamental principles of the random vibration theory and the structural elements were idealized elastic and linear, which generates a disadvantage when an inelastic and nonlinear analysis is required. In addition, the method was created in order to apply it to structures with two supports, and even though it can be extended to multiple supports, there is no correlation between supports and the site effect is not taken into account.

In that study, Nuti and Vanzi analyzed a bridge with only one span of 32 m in total length, supported in soft soil. They found out that in the asynchronous case the differential displacements in the abutments exceeded in 98 mm the 14 mm proposed in the Eurocode and the Italian Code of Civil Protection, thus emphasizing the importance of including the asynchronous analysis even in short-length bridges and the need to update the design codes. In general, Fernandez et al. The results of the works performed by researchers interested in comparing the classic analysis with the asynchronous analysis for some structural types of bridges are detailed as follows.

Alvarez et al. On the other hand, Kaiming et al. Units: mm Kaiming et al. For the first study of box girder bridges under asynchronous seismic excitation, Konakli and Kiureghian used four bridges with irregularities both in level and height see Table 1. The authors found out that in more flexible bridges such as the Penstock Bridge and the South Ingram Slough Bridge, the pier drifts increase significantly when the wave passage and a strong loss of correlation between the signals are taken into account, but the most critical scenario is the combination of the three asynchronous patterns, that is to say, wave passage, loss of correlation and local site effect.

Apart from that, the authors classify the synchronous analysis as conservative in the case of box girder bridges with low fundamental periods such as the Auburn Ravine Bridge and the Big Rock Wash Bridge. Mehanny et al. The analysis was developed in the Opensees software and 20 seismic records from the Pacific Earthquake Engineering Center database were used.

The authors determined that in the longitudinal direction, the continuous deck works as a rigid diaphragm that minimizes the wave passage effect, making the uniform seismic excitation the most conservative for the seismic design. However, in the transverse direction, the authors recommend taking into account the wave passage effect whose severity depends on the frequency content of the seism, being more critical for high-frequency ranges in order not to underestimate the probability of structural failure.

With this, it was determined that in the case of soft soils, the seismic sensitivity and seismic vulnerability under asynchronous seismic excitation are greater than in stable soils with an excess of the annual frequency of up to 7. According to Wang et al. In a study carried out by Sextos and Kappos , where 27 types of multiple-span bridges with different span lengths were analyzed, the authors found out that in bridges with a total length greater than m the results of the asynchronous analysis predominate over the classic analysis.

In , Price and Eberhard proposed a method to determine in advance whether an asynchronous analysis in bridges should be carried out, based on the participation constant Cp see equation If the participation constant tends to infinity, the dynamic component of displacement predominates, that is to say, the asynchronism is irrelevant. Otherwise, the asynchronous analysis should be taken into account.

Although the method works in the models proposed by Price and Eberhard, , the behavior of this typology could not be generalized. Vapp is the apparent wave velocity in the rocky environment, and. The loss of correlation is the pattern that produces a higher increase in internal forces and displacements according to Saxena et al. However, the authors recommend taking into account the three asynchronous patterns separately and in combination. Mezouer et al. If Tp tends to 1. Additionally, Kleoniki et al. The geological profile was the key variable in the models see Figure 9 , thus determining the influence in the nonlinear dynamic response of multiple-span bridges subjected to asynchronous seismic excitation.

Moreover, the authors propose taking into account in the asynchronous analysis factors such as the topography, the geological characteristics of the superposed layers where the structure is supported, and every discontinuity in the soil that produces changes in the frequency content of the wave in the surface due to the direct influence in the bridge response. Burdette and Elnashai , Price and Eberhard ; Wang et al.

This type of displacement could be disregarded in the design phase if an asynchronous seismic excitation or a uniform seismic excitation is considered. Sextos et al. The results of this analysis allowed them to conclude that the attack angle horizontal plane plays a secondary role in the asynchronous analysis. This conclusion was obtained after analyzing two bridges with three spans, each span of 50 m and piers of approximately 55 m in height; the difference laid in the type of support between the girder and the pier, considered as two support types: M1 elastomeric support and M2 monolithic support.

Feng and Kim and Saxena et al. Through nonlinear analyses, the authors identified an increase in the ductility demand for the columns, in comparison with the classic analysis. The study of Feng and Kim is the first study proposing fragility curves under asynchronous seismic excitation conditions, which provide useful information to be taken into account in the update of design codes.

According to Feng and Kim , the probability of failures in the structure could increase up to 2. Different authors have been interested in analyzing the effects of the asynchronous seismic excitation in the Jindo metal cable-stayed bridge, which was based on a variable soil and has a central span of m and two lateral spans of 70 m each.

Soyluk and Avanoglu found the characteristics of the Jindo Bridge interesting to carry out an asynchronous analysis, taking into account the soil structure interaction, by adding the three asynchronous patterns separately and in combination. The patters that affect more, if the soil-structure interaction is taken into account, is the local site effect, increasing the demand on the deck and the towers see Figure 12 , where F, M and S represent the stable, average and soft soils, respectively on each of the four supports of the bridge.

On the other hand, Valdebenito and Aparicio and Soyluk and Dumanoglu also studied the Jindo Bridge, and based on the results they determined that for cable-stayed bridges with big spans, the pattern that affects the response more is the wave passage effect since when increasing the apparent wave velocity, the temporary lag between seismic forces applied to the supports of the bridge increases.

This directly affects the structural response see Figure 13 and Figure Karakostas et al. From the results they determined that: the asynchronous seismic excitation is beneficial for the bending moments in the piers and for the displacements in the central span of the deck. With regard to the bending moments outside the plane and the displacements in the top of the piers, the asynchronous moment is clearly critical and the increase of the displacements vary in accordance with the change in the amplitude of the Fourier spectrum, that is to say, it depends on the peak acceleration values contained in the range of frequencies of high modes.

According to Abdel et al. Since these modes are mainly asymmetrical, the implementation of control systems under asynchronous seismic excitation is difficult, thus reducing the effectiveness of the energy dissipation devices, being these active, semi-active or passive.

### Critical Excitation Methods in Earthquake Engineering

Harichandran et al. In this analysis, the authors found out that the major influence component is the dynamic component. However, the pseudo-static component and the covariance component contribute significantly to the total displacement at the center of the main span. Therefore, in the case of the asynchronous seismic excitation, the response is critical at the center of the main span, but in the rest of the structure, the response is underestimated in relation to the uniform seismic excitation.

He calls this resistant design robust.

Practical application and base can be written as extension of critical excitation methods have then been made extensively. The ratio of where u is the displacement of the mass relative to the critical response to the corresponding response to the ground. Iyengar and Advances in Structural Engineering Vol. They expressed an allowable set of 2. Various Measures of Criticality excitation accelerations as a linear combination of In the initial stage, various quantities have been chosen recorded ground motions of which the site characteristics as objective functions to be maximized in the critical and other earthquake occurrence properties are similar.

Ahmadi considered another They selected several response quantities as the measure critical excitation problem including the response accel- for criticality and compared the response to the subcritical eration as the objective function. He showed that a rectan- excitation with those to recorded earthquake ground gular wave is the critical one and recommended to add motions as the basis functions.

They showed that the another constraint in order to make the solution more conservatism of the subcritical excitations can be realistic and acceptable. Westermo introduced the following input Abdelrahman et al. He also set a constraint on the time integral of squared Pirasteh et al. He used a variational approach and technique in one of the subcritical excitation problems.

His solution may not be a sites to create the candidate accelerograms, then used complete one because the response velocity is a function optimization and approximation techniques to find the of the excitation to be obtained. He stated that the derived most critical accelerogram. The most critical accelero- critical input acceleration includes the solution by gram was defined as the one which satisfies the constraint Drenick As for more general critical excitation peaks, Fourier spectra, intensities, growth rates and methods for earthquake input energy, see later sections.

The damage was The damage caused by input motions may be another defined as the cumulative inelastic energy dissipation or measure of criticality. The corresponding problems have the sum of interstory drifts. However, freedom MDOF models. They utilized a variational for example, the record at SCT1 during Mexico formulation and selected a quantity in terms of multiple Michoacan Earthquake and that at Kobe University responses as the objective function to be maximized in during Hyogoken-nanbu Earthquake indicate the context of critical excitation problems.

They showed clearly that ground motions unpredictable from only the that the relation among the critical displacement, velocity past knowledge can occur and inclusion of such ground and acceleration responses is similar to that among the motions is important in effective application of the displacement, velocity and acceleration response spectra. Iyengar and Manohar , , Iyengar , Srinivasan et al. To overcome stochastic problems. Srinivasan et al. They used a Manohar and Sarkar and Sarkar and Manohar stationary model of ground acceleration in the paper , set a bound on the total average energy and Iyengar and Manohar and utilized a nonstationary solved linear or nonlinear programming problems.

The function shot noise model for input motions. Manohar and Sarkar c t is a deterministic envelope function and w t is a and Sarkar and Manohar , further set stochastic function representing a stationary random a bound on the average rate of zero crossings in order to Gaussian process with zero mean. In function of w t. The auto- where f is an objective function, e. Each function is a possible realization of an uncertain event. A local energy-bound convex model, an a given envelope function c t.

Iyengar and Manohar integral energy-bound convex model, an envelope-bound expressed the square root of the PSD function convex model, a Fourier-envelope convex model and a of the excitation in terms of linear combination of response-spectrum-envelope convex model are some orthonormal functions and determined their coefficients examples Ben-Haim et al.

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## Critical excitation methods in earthquake engineering - Semantic Scholar

One of the advantages Advances in Structural Engineering Vol. He considered both deterministic and random inputs. Drenick and Park provided some interesting Maximization with respect to PSD function comments on the paper due to Iyengar However, he did not mention the applicability of the concept to actual problems Time and his concept is restricted to deterministic equivalent linearization. He restricted the class of critical excitations to periodic ones. He presented several Time interesting points inherent in the problems for nonlinear systems. Philippacopoulos , Philippacopoulos and Wang utilized a deterministic equivalent linearization technique in critical excitation problems of nonlinear SDOF hysteretic systems.

They derived several critical Objective function f t inelastic response spectra and compared them with subjected to the motion f t inelastic response spectra for recorded motions. This situation causes much difficulties of convex models is the possibility of prediction of the in finding a critical excitation for elastic-plastic structures.

The shape of the Furthermore, unlike the other methods, such as the critical PSD function has been restricted to a rectangular subcritical excitation, and stochastic excitation, another function attaining its upper bound in a certain frequency advantage of the convex model approach is that it can range.

The combination of probabilistic approaches varied in finding the critical PSD function. The critical and convex model approaches appears to be promising excitations have been obtained for two examples and Drenick a. It is not the objective of this review to compared with the corresponding recorded earthquake provide the detail of the convex models. Readers should ground motions.

This fact corre- sponds well with the result by Westermo Duffing oscillator. However, practical application of the technique is not shown sufficiently. For example, it does The critical excitation problem includes the double not appear clear for what excitation the equivalent maximization procedure with respect to time and the stiffnesses and damping coefficients should be defined. The key for finding the critical Takewaki e extended the critical excitation envelope function is the order exchange in the double method for elastic-plastic SDOF models to elastic-plastic maximization procedure.

An upper bound of the mean- MDOF models on compliant ground by employing a square drift can also be derived by the use of the Cauchy- statistical equivalent linearization method for MDOF Schwarz inequality. It can be shown that the proposed models as an approximate response simulator of the technique is systematic and the upper bound can bound the original elastic-plastic hysteretic model. The shape of the critical PSD function is further envelope function of the critical excitation can be a assumed to be a rectangular function attaining its upper function similar to an increasing exponential function in bound in a certain frequency range.

Different from the the probabilistic problem. The sum of standard deviations of story ductilities along the height 9. It is to regard the Much work has been accumulated on the topics of central frequency of the rectangular PSD function as a earthquake input energy Housner , Akiyama Most of the research have been done based on the The simulation results by time-history response analysis approach in the time domain. On the contrary, it has been for elastic-plastic models revealed that the proposed shown by Takewaki , e that the frequency- critical excitation method is reliable in the models for domain approach is essential for the development of which the validity of the statistical equivalent linearization critical excitation methods for the earthquake input method is guaranteed.

It has also been demonstrated that energy. It has been demonstrated Takewaki , the critical response representation in terms of non- d that the input energy expression can be of a exceedance probabilities can be an appropriate candidate compact form via the frequency integration of the for expressing the criticality of recorded ground motions. Magnetic Memory. Denny D. Understanding Voltammetry. Richard G Compton.

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