Probabilistic Seismic Performance Assessment of Classes of Buildings Using Physics-Based Simulations and Ground-Motion Prediction Equations

We discuss the applicability of two different near field representations of seismic input within the capacity-demand diagram method for seismic performance assessment of classes of buildings in urban areas. The two representations of seismic input used are: (1) synthetic accelerograms obtained from 3D wave simulations of random rupture scenarios are used to estimate seismic risk, through the Monte-Carlo approximation; and (2) random realizations of elastic spectral displacement obtained through ground-motion prediction equations. The implementation of the proposed formulation in evaluating the seismic performance of two classes of buildings in the town of Sulmona, Italy, indicates that empirical ground-motion prediction equations, combined with non-iterative methods for estimating the inelastic seismic demand, can produce results comparable to the physics-based simulations.     

Dynamic Analysis and Seismic Response of Planar Circular Arches with Variable Cross-Section

The aim of this paper is to investigate the dynamic response of planar circular arches with variable cross-section subjected to seismic ground motions. Arches have a wide range of application (e.g. bridges, roofs) thanks to their capacity to span large areas by resolving vertical actions into compressive stresses and confining tensile stresses. The full understanding of their dynamic response is a challenging technical and computational problem, especially when seismic loading is considered. For example, the assumption of axial inextensibility simplifies the differential equations but overestimates the vibration frequencies, especially those of shallow arches since axial forces are of paramount importance (as opposed to beams). In lieu of the above, our formulation incorporates the effect of axial extension, and the arches are modeled using a new generic curved beam model that includes both axial (tangential) and transverse (normal) to the arch centerline deformations, and is able to account for variable mass and stiffness properties, as well as elastic support or restraint. The resulting dynamic governing equations of the circular arch are formulated in terms of the displacements, and solved using an efficient integral equation method. Three circular arches with variable rectangular cross-section are analyzed in order to investigate their dynamic properties and seismic performance. Using both time history and modal analysis useful conclusions are drawn with regard to the contribution of each mode on the calculation of different response quantities.     

Simplification and Assessment of Modal-Based Story Damage Index for Reinforced Concrete Frames Subjected to Seismic Excitations

Different relations have been represented for the local damage index of structures to date, while the same application is defined for them as can be an indicator of relative sustained damage by the components or stories. Since different force-resisting systems subjected to the ground motions can behave differently, some well-known story damage indices are evaluated for the reinforced concrete frames with regards to their operation during nonlinear time history analysis. Two general concepts of story damage determination are selected for this purpose. SDI is a modal-based story damage index, which is calculated by the modal frequency and mode shapes. The behavior of this local index is evaluated during the seismic excitations. The results were compared with Park-Ang and modal flexibility story damage indices. Based on analytical study on seismic responses of some RC frames subjected to a suit of earthquake records a new story damage index has been developed. It has been derived from a simple global damage equation (softening index) using a normalized ratio of inelastic story shear to its drift. A procedure is recommended to use the proposed equation without any requirement to perform nonlinear dynamic analysis, which can significantly reduce the computational efforts. Distribution of the new represented SDI along the structural height shows a good agreement with damaged state of the RC frames after seismic excitations.     

Inelastic Demand Spectra of Bi-Linear Models for Capacity Spectrum Method and Response of Simple Structures under Near-Source Records

A very useful tool for the preliminary design of structures is the elastic demand spectrum that can be used in the capacity spectrum method. A pseudo-acceleration relationship has to be assumed when constructing a demand spectrum. This assumption results in large errors for long period structures with large damping ratios and the conventional demand spectra require a substitute elastic structure. In the present study, the conventional demand spectra are extended to bi-linear models. Pseudo-acceleration is still assumed but results in acceptably small errors, when a constant viscous damping coefficient for a single-degree-of-freedom (SDF) structure is calculated from the tangent stiffness and the damping ratio is set at 5% in both elastic and yield phases. For nonlinear structures, tangent stiffness dependency of damping force could be acceptable because energy absorption is primarily the result of structural nonlinear deformation. To extend the conventional demand spectra to a bi-linear model, effective period calculated from the secant stiffness has to be used. The use of effective period introduces no approximation because the peak displacement of the SDF structure is computed from nonlinear analysis in the time domain. The method presented in this study is also valid if damping coefficient proportional to initial elastic spectra is used. In this case, the pseudo-acceleration is defined as the base shear coefficient that is required to produce the peak displacement of the SDF structure in a static manner. We present demand spectra of bi-linear models for a number of near-source records from large earthquakes, and spectral ratios of two horizontal components. The effects of different types of ground motion on the response reduction factor due to inelastic deformation are investigated.     

An Explicit Integration Scheme for Solving Dynamic Problems of Solid and Porous Media

In this article, an explicit integration scheme for dynamic finite element analysis is developed and the formulations for linear elastic solid and two-phase porous media are derived. The accuracy and stability characteristics analyses show this scheme is second-order accurate, which is the same as the Central Difference Method (CDM), and possesses a broader scope of stability within the range of normal damping ratios of media. Comparing with the CDM, this explicit scheme requires no matrix factorization even if a non diagonal damping matrix is included. Therefore, the dynamic finite element equations can be integrated economically provided that the mass matrix is diagonal. To demonstrate the validity of the present scheme, three examples are provided. First, taking a single-degree-of-freedom system as an example, the results obtained by the proposed scheme are compared with the exact solutions. Second, the dynamic responses of half-space saturated porous media, subjected to a concentrated load pulse at the surface, are analyzed. Both examples show that the results obtained by the proposed scheme agree well with the analytical solutions. Finally, the dynamic responses of a plane strain plate due to a load pulse are analyzed, respectively, by the proposed procedure and the commercial code ABAQUS (both implicit module and explicit module), and the CPU costs are compared.     

Reduction Factors for Seismic Design Spectra for Structures with Viscous Energy Dampers

A simple mathematical expression is proposed to estimate spectra reduction damping factors for seismic design of systems with viscous dampers. The expression is obtained from the ratios between ordinates of uniform hazard spectra associated with two different return intervals (50 and 125 years), corresponding to sites with different types of soil within the Valley of Mexico. The expression proposed depends on the dominant period of the soil, and on both the vibration period and damping ratio of the structural system. Values of the damping factors proposed here are comparable to those recommended by different authors and seismic design building codes.     

An Analytical Study on the Seismic Vulnerability of Masonry Buildings in India

Two analytical models for unreinforced masonry (URM) buildings are proposed with the aim to simulate their seismic response and to estimate corresponding vulnerability functions. The proposed models are implemented in SAP 2000 nonlinear software to obtain capacity curve parameters for representative Indian URM buildings, based on a field survey and statistical analysis. Vulnerability functions are estimated using the obtained capacity curves. Damage Probability Matrices (DPMs) are obtained using the approximate PGA-intensity correlation relationship as per Indian seismic building code and are compared with the commonly used intensity scales and empirical damage data observed after the 2001 Bhuj earthquake.     

Equivalent Damping Ratio Equations in Support of Displacement-Based Seismic Design for Pile-Supported Wharves

This study primarily proposes new equivalent damping ratio equations based on Jacobsen’s approach for displacement-based seismic design of pile-supported wharves to account for wharf configurations and soil-pile interaction. It is found that Pivot hysteresis model and Masing rule can accurately capture nonlinear behavior of concrete and steel wharves, respectively. To verify applicability of proposed equations, analyses were conducted to three typical wharves to make a comparison of maximum displacements obtained from nonlinear time-history analyses and substitute structure method with various damping equations. The verification reveals proposed equations are better than those in practice for their higher precision in determining displacements.     

COMPLEX MODE SUPERPOSITION ALGORITHM FOR SEISMIC RESPONSES OF NON-CLASSICALLY DAMPED LINEAR MDOF SYSTEM

This paper deals with a complex mode superposition method for the seismic responses of general multiple degrees of freedom (MDOF) discrete system with complex eigenvectors and eigenvalues. A delicate general solution, completely in real value form, for calculat-ing seismic time history response of the MDOF system which cannot be uncoupled by normal modes, is deduced based on the algorithms of the complex superposition method. This solution comprises of two parts which are in relation to the Duhamel integration to sine and cosine function respectively. The related term of the Duhamel integration to sine function is actually the displacement response of the oscillator with corresponding modal frequency and the damping ratio. The other can be transferred into a combina-tion of the displacement and velocity responses of the same oscillator. In order to meet the practical needs of seismic design based on code design spectra for various kinds of structures equipped by viscous dampers, the complex complete quadratic combination (CCQC) method is deduced following similar procedures such as the well-known CQC method, in which a new modal velocity correlation coefficient, together with a new modal displacement-velocity correlation coefficient are involved besides the modal displacement correlation coefficient in normal CQC formula. The new algorithm of CCQC is not only as concise as that of the normal CQC but also has explicit physical meaning. The results obtained from complex mode superposition approaches are discussed and verified in some examples through step by step integration computation under a prescribed earth-quake motion input. From these examplary analyses, it may be pointed that the CCQC algorithm normally yields conservative outcome and that the forced mode uncoupling approach has good approximation even the discussed examplary structures are strongly non-proportional.     

Predicting the Seismic Response of Capacity-Designed Structures by Equivalent Linearization

An equivalent linearization procedure is developed for predicting the inelastic deformations and internal forces of capacity-designed structures under earthquake excitations. The procedure employs response spectrum analysis, and mainly consists of the construction of an equivalent linear system by reducing the stiffness of structural members that are expected to respond in the inelastic range. These members are well defined in structures designed with capacity principles. Maximum modal displacement demands of the equivalent linear system are determined either from the equal displacement rule, or from independent nonlinear response history analysis of SDOF systems representing inelastic modes.

Predictions obtained from the proposed equivalent linearization procedure are evaluated comparatively by using the results of nonlinear response history analysis as benchmark, linear elastic response spectrum analysis and conventional pushover analysis. The deformations and capacity controlled actions of a 12-story symmetrical plan concrete frame and a 6-story unsymmetrical plan concrete frame are obtained by each method under 96 strong ground motions. It is observed that the proposed procedure results in better accuracy in estimating the inelastic seismic displacement response parameters and capacity controlled forces than the other two approximate methods.     

Seismic Vulnerability Analysis of RC Bridges Based on Kriging Model

This paper presents a Kriging model-based method for seismic vulnerability analysis of reinforced concrete (RC) bridges. It aims at reducing the computational effect when the Monte Carlo technique is used for establishing the structural vulnerability curves. The general procedure of the proposed method is put forward firstly. In the procedure, the uncertainties existing in the structures and ground motions are both taken into account, and the uniform design (UD) technique is adopted for generating the random samples. The reliability of the proposed method is demonstrated by the vulnerability analysis of an single degree of freedom (SDOF) system using the Latin hypercube simulation (LHS) method. Vulnerability analysis of an RC bridge system is then carried out using the proposed method. The vulnerability curves of the bridge obtained by the Kriging model-based method are compared with those obtained by the LHS method. Additionally, three simulation schemes adopting different UD tables are employed to investigate the convergence and stability of the proposed method. The results show that the proposed method used for the seismic vulnerability analysis of RC bridges can reduce the computational effort and time to a large extent without much compromise on the accuracy.     

Seismic Fragility Evaluation of RC Frame Structures Retrofitted with Controlled Concrete Rocking Column and Damping Technique

Acceleration response of simple yielding structure is proportional to its own weight, but it is limited by yield strength. Thus, using rocking columns that reduces global yield strength, a limited acceleration is achieved. However, the displacement becomes large due to lower strength and higher inelasticity, but it can be controlled by adding damping. Performing fragility analyses, the seismic response of R/C frame structures with rocking columns and viscous dampers is investigated. Near field MCEER ground motions are considered. The analyses show that the story accelerations are reduced by using rocking columns, while the story displacements are controlled by using viscous dampers.     

A Method for Selecting Ground Motions that Considers Target Response Spectrum Mean and Variance as Well as Correlation Structure

This study proposes a method for selecting ground motions from a ground motion library with response spectra that match the target response spectrum mean, variance, and correlation structures. The proposed method is conceptually simple and straightforward. In this method, a desired number of ground motions are sequentially selected from first to last. The accuracy and consistency of the proposed method are verified through comparisons of the ground motions selected using the proposed method with those selected using conventional methods. This study shows that the seismic responses of the frames vary according to ground motion selection and correlation structures.     

Seismic Evaluation of Tall Buildings Using a Simplified but Accurate Analysis Procedure

A simplified analysis procedure for evaluating the nonlinear seismic responses of tall reinforced concrete (RC) buildings is examined in this study. It is called the Uncoupled Modal Response History Analysis (UMRHA) procedure. It can be viewed as an extended version of the classical modal analysis procedure, where the nonlinear response of each vibration mode is first computed, and they are later on combined into the total response of the structure. The procedure requires the knowledge of the modal hysteretic behavior, which can be obtained from a cyclic modal pushover analysis. The responses of four tall buildings in Bangkok to distant large earthquakes are computed by this procedure and compared with those obtained from the Nonlinear Response History Analysis (NLRHA) procedure. These four buildings have different heights—varying from 20 to 44 stories, different configurations of floor plan, and different arrangement of RC walls. The comparison shows that the UMRHA procedure is able to accurately compute the story shears and story overturning moments, floor accelerations, and inter-story drifts of all these tall buildings. The required computational effort is also extremely low compared to that of the NLRHA procedure. Moreover, since the UMRHA procedure computes the response of each individual vibration mode, it provides more understanding and insight into the complex nonlinear seismic responses of these tall buildings.     

Inelastic Displacement Ratios for Seismic Assessment of Structures Subjected to Forward-Directivity Near-Fault Ground Motions

This article presents results of a statistical study focused on evaluating inelastic displacement ratios (i.e., ratio of maximum inelastic displacement with respect to maximum elastic displacement demand) of degrading and non degrading single-degree-of-freedom (SDOF) systems subjected to forward-directivity near-fault ground motions. C_R spectra are computed for normalized periods of vibration with respect to the predominant period of the ground motion to provide a better ground motion characterization. This period normalization allows reducing the record-to-record variability in the estimation of C_R. An equation to obtain estimates of C_R for the seismic assessment of structures exposed to forward-directivity near-fault ground motions is proposed.     

OUT-OF-PLANE SEISMIC RESISTANCE OF MASONRY WALLS

Seismic vulnerability of unreinforced masonry buildings is studied by means of simplified out-of-plane collapse mechanisms that take into account connections with transversal walls. According to experimental evidence, the analysis assumes that failure is reached with a rigid body motion of a part of the facade that falls down. Two classes of mechanism are examined: the overturning of the facade due either to a vertical crack at the connection or a diagonal crack on the transversal wall, both defined resorting to a simple model of masonry fabric, viewed as a regular assembly of rigid blocks and elastic plastic joints with friction but no cohesion. The use of simplified mechanisms give rise to an explicit evaluation of the seismic resistance to changes in the geometry and in the masonry fabrics, that could be used by practising engineers. This formulation is developed for both static horizontal actions and ground velocity peak, in the belief that the latter probably gives a better approximation of seismic action, while also providing, by comparison with the results of static forces, an estimate of the behaviour factor for unreinforced masonry. Eventually, the analytical forecasts are compared with numerical results obtained by means of the distinct element method.     

Numerical Evaluation of Seismic Soil Pressures on Rigid Walls Fixed to the Bedrock

Seismic soil pressures developed on a 7 m rigid retaining wall fixed to the bedrock are investigated using a finite element model that engages nonlinear soil intended materials available in OpenSees. This allows incorporation of the inelastic behavior of the soil and wave propagation effects in the soil-wall system seismic response. The nonlinear response of the soil was validated using the well-stablished, frequency-domain, linear-equivalent approach. An incremental dynamic analysis was implemented to comprehensively examine the effect of soil nonlinearity and input motion on the induced seismic pressures and to evaluate current code equations/methodologies at different levels of earthquake intensity. The results show that soil nonlinearity and seismic wave amplification may play an important role in the response of the soil-wall system. Therefore, methodologies that rely only on peak ground acceleration may introduce large bias on the estimated seismic pressures in scenarios where high nonlinearity and site amplification are expected.     

Simple Models for Inelastic Seismic Analysis of Asymmetric Multistory RC Buildings

A simple stick model is presented for the inelastic seismic analysis in 3D of two-way eccentric multistory RC buildings. It has 3 DoFs per floor, point hinges at the ends of the vertical elements connecting floors, elastic story stiffness derived from the corresponding story force-interstory deformation relations of the elastic 3D structure under inverted-triangular floor loading (by torques for torsional stiffness, by horizontal forces for the lateral ones), story yield forces derived from the total resistant shear of the story vertical elements, but no coupling between lateral and torsional inelasticity. It is evaluated on the basis of comparisons of response histories of floor displacements to those from full nonlinear models in 3D of four actual buildings. Alternative locations of the story vertical element with respect to the floor mass center are examined: (a) the floor “center of twist” of the elastic 3D building under inverted-triangular floor torques; (b) the story “effective center of rigidity,” through which application of inverted triangular lateral forces does not induce twisting of floors; (c) the centroid of the secant stiffness of the story vertical members at yielding and (d) the centroid of the lateral force resistance of story vertical elements. Among alternatives (a)–(d), the floor “center of twist” provides the best agreement with floor displacement response-histories from full 3D nonlinear models. This means that the static eccentricity that matters for torsional response may be taken as that of the floor “center of twist.” The center of resistance comes up as the second-best choice.     

A Procedure for the Displacement-Based Seismic Assessment of Infilled RC Frames

The aim of this study was to propose an extension of the displacement-based assessment procedure for infilled reinforced concrete (RC) frames. Two fundamental steps of the displacement-based approach were studied: the determination of the equivalent viscous damping and the definition of the limit-state displacement profile. The proposed criteria were derived by examining the results of two different numerical investigations regarding the nonlinear seismic response of single- and multi-story infilled RC frames. Lastly, the effectiveness of the method was verified through comparisons, in terms of displacement demand, with the results of nonlinear dynamic analyses.