Approximate modal analysis of multistory symmetrical buildings with restricted inelasticity

Simplified capacity curves are presented for modal decomposition of multistory inelastic cantilever bents. These structural systems are uniform over the height and plasticity is assumed to be restricted into the beams, since significant rotation ductility factors may be attained in these members without loss of strength. The approximate method of modal decomposition of multistory inelastic bents is based on the concept that the total response may be obtained from the contributions of equivalent nonlinear single‐degree‐of‐freedom (SDOF) modal systems, in combination with the technique of modal superposition. In particular, the contribution of the first mode equivalent inelastic SDOF system is examined, since its modal contribution is of higher importance. The response of such SDOF systems basically depends on their capacity curve, which may be formulated by using the dynamic properties (frequency, effective modal mass and mode shape) of the initially elastic bent, as well as the corresponding properties of the bent when the entire set of coupling beams is assumed to be yielded. The procedure is presented for plane cantilever bents, but it can be easily extended to symmetrical structural systems composed of different types of inelastic bents. Its application is illustrated by means of a 10‐story inelastic coupled‐wall bent subject to a strong earthquake motion, equal to 1·5× El Centro N–S ground excitation, and the results compare well with those obtained from a step‐by‐step nonlinear time history analysis of the discrete member model. Copyright © 2007 John Wiley & Sons, Ltd.     

An alternative approach for assessing eccentricities in asymmetric multistory buildings. 1. Elastic systems

A simple hand method for assessing modal eccentricities of multistory structures with partial symmetry is presented. The concept of the modal center of rigidity is introduced and the determination of eccentricities of the first two modal shears is obtained by means of an equivalent one‐story eccentric system. The method is based on Southwell's formula for calculating a lower bound of the fundamental frequency and is applicable to uniform structures composed of different types of bents, such as walls, rigid frames, coupled walls and wall–frame assemblies. These structures do not belong to the special class of proportionate buildings for which dynamic properties can be accurately obtained by analyzing two simpler systems: a multistory torsionally uncoupled counterpart of the actual structure and an associated one‐story coupled system. The accuracy of the proposed method is illustrated in a parametric study that includes common types of mixed‐bent‐type multistory structures and comparisons are made with the results obtained by three‐dimensional dynamic analyses on discrete member models. Copyright © 2007 John Wiley & Sons, Ltd.     

Earthquake analysis of isotropic asymmetric multistory buildings

Isotropic multistory buildings are the ones characterized by the property: all load‐resisting planar frames have proportional lateral stiffness matrices. In the present paper it is proved that the modal analysis of an N‐story isotropic asymmetric, torsionally coupled, building (a problem of order 3N) can be separated into two independent sub‐problems: (a) a sub‐problem that corresponds to a single‐story asymmetric, torsionally coupled, building (a problem of order 3); and (b) a sub‐problem that corresponds to an N‐story, torsionally uncoupled, planar frame (a problem of order N). It is also demonstrated that the orientation of peak modal seismic forces of the building is independent of the orientation of seismic excitation, which affects only their size. The separation provides a better insight into the structural behavior of asymmetric multistory buildings under earthquake ground motion. Copyright © 2006 John Wiley & Sons, Ltd.     

Collapse behavior evaluation of asymmetric buildings subjected to bi‐directional ground motion

This paper discusses the collapse behavior of low‐rise plan‐asymmetric buildings under bi‐directional horizontal ground motions and utilizing strength and stiffness degrading nonlinear models. For this purpose, three‐dimensional three‐story and six‐story reinforced concrete frame buildings with uni‐directional mass eccentricities equal to 0% (symmetrical), 10%, 20% and 30% are subjected to nonlinear static (pushover) as well as incremental dynamic analyses using a set of far‐field two‐component ground motions and their performance are assessed on the basis of the safety margin against collapse and its probability of occurrence. Comparison of the collapse margin ratios as well as the fragility curves demonstrates significant reduction of the collapse‐level ground motion intensity with increasing eccentricity in plan. Results also indicate that current seismic design parameters including the response modification (R), overstrength (Ω) and ductility (μ) factors are not appropriate for buildings with high levels of plan eccentricity. Buildings with high values of plan eccentricity do not meet the design target life safety performance level on the basis of the calculated probability of collapse and safety margin against collapse. It appears that re‐evaluation of their design parameters is necessary. Copyright © 2015 John Wiley & Sons, Ltd.     

Modal eccentricities of asymmetric structures

Modal eccentricities of monosymmetric structures are given by a simple hand method, which is based on the concept of the continuum medium. The method is applicable to structures consisting of any combination of walls, rigid frames and coupled wall bents that are uniform over the height of the structure. These structures do not belong to the special class of proportionate buildings for which the dynamic properties can be obtained approximately from the properties of an equivalent single‐storey torsionally coupled building, in combination with those of a torsionally uncoupled multistorey structure. In the proposed method the equation of motion of each bent of the structure is approximated by that of an incompressible wall‐frame system (model) of equal flexural rigidity, but of an equivalent modal shear rigidity, which accounts for axial deformations in the vertical members of the bent. With this model any uniform structure composed of different types of bents may be analysed by standard methods and dynamic properties may be determined by simple and rapid means. Modal eccentricities are given in respect to the centre of flexural rigidities, which is easily determined in uniform structures, and as the method is based on the continuous approach, a deep insight into the structural behaviour is possible by consideration of parameter variations. Approximate expressions are also given for frequencies and modal base shears, as these properties play a key role in the response spectrum analysis. In order to illustrate the accuracy of the proposed method comparisons are made with accurate results obtained by 3D computer dynamic analyses on discrete member models. Copyright © 2006 John Wiley & Sons, Ltd.     

Evaluation of lateral‐torsional coupling in earthquake response of asymmetric multistory buildings

This paper presents a practical method for evaluating lateral‐torsional coupling in the elastic earthquake response of asymmetric multistory buildings. A transformation technique is first developed to shift the floor centers of mass of an asymmetric building to new reference positions where the sum of the squares of all floor rotations of the building due to lateral inertia loads is a minimum. By setting the locus of the floor centers of mass of the building at the new reference positions, a representative eccentricity and an effectively uncoupled system for the building can be established on the basis which an equivalent eccentric single mass system can be developed. The additional lateral translations caused by seismic torsional effects in the building can be analytically determined and expressed in terms of the representative building eccentricity and the uncoupled periods evaluated using the effectively uncoupled system. The effectiveness and practicality of the proposed method are illustrated with two 30‐story practical buildings. Copyright © 2013 John Wiley & Sons, Ltd.     

Simplified dynamic analysis of eccentric buildings with a setback. 2: the effect of stiffness irregularity

The effect of stiffness irregularity is investigated in eccentric buildings with a setback. Stiffness irregularity is assumed to be created when some of the lateral load resisting bents are curtailed at the top of the base structure. The methodology suggested in the companion paper for assessing vibration frequencies, base shears and torques is extended to include the effect of curtailed bents. Their contribution is evaluated by an indirect method after the modal stiffness of the symmetrical counterpart structure is determined and the modal contribution of the full height bents is assessed by Southwell's approach. A number of numerical examples, representing common types of building structures, are presented to illustrate the procedure and provide an insight into the response of such setback buildings. The results are compared with more accurate results obtained by three‐dimensional dynamic analyses. Copyright © 2009 John Wiley & Sons, Ltd.     

Equivalent lateral force procedure for the seismic story drift design of tall frames using a hand‐calculated approach

The basic theory of the cantilever moment distribution method and the application of this method to conduct the equivalent lateral force procedure for the seismic design of tall frames are discussed in detail. Deflection–rotation formulas are introduced in this paper to determine the relative lateral story deflections and the joint rotations of laterally loaded rigid frames. An example is presented to conduct the seismic story drift design of a multistory, multi‐bay, moment‐resisting reinforced concrete frame using the cantilever moment distribution method and the deflection–rotation formulas developed in this paper. The hand‐calculated approach presented in the example can be used as a rapid and accurate method to determine the story drift for any laterally loaded multistory, multi‐bay rigid frames that are composed of identical single‐bay symmetrical bents. The following are the advantages of using this rapid approach: (1) this approach can be carried out by using hand calculations only (without the use of computer computations); (2) the results obtained from this approach are as accurate as those derived from the traditional moment distribution and the slope deflection methods; and (3) the results obtained from this approach can be used to verify the accuracy of the results obtained from computer computations. Copyright © 2005 John Wiley & Sons, Ltd.     

Energy evaluation of inelastic structures subjected to random earthquake excitations

The modified force analogy method combined with static condensation is applied to study the seismic behaviour of inelastic structures, while energy transfer and dissipation in the structure during an earthquake hazard are investigated through energy formulation based on the equation of motion. The earthquake ground motion is modelled as a non‐stationary Gaussian random process. Rigid‐end offsets of the plastic hinges forming on the members, as well as the ability for panel zones to deform, are included in the structural analysis model. Monte Carlo simulation method is performed on a six‐story real moment‐resisting frame to determine the mean and the standard deviation of seismic energy dissipation time histories. Based on this combination of the force analogy method and energy formulation in the equation of motion with the stochastic earthquake model, the feasibility of the analysis procedure for studying the dynamics of inelastic structures is demonstrated through varying the rigidity of panel zones. Copyright © 2008 John Wiley & Sons, Ltd.     

Evaluation of deflection amplification factor in steel moment‐resisting frames

Steel moment‐resisting frames (SMRFs) are the most common type of structural systems used in steel structures. The first step of structural design for SMRFs starts with the selection of the structural sections on the basis of story drift limitation. ASCE 7 (2010) requires that the inelastic story drifts be obtained by multiplying the deflections determined by elastic analysis under design earthquake forces with a deflection amplification factor (C_d). For special moment‐resisting frames, C_d is given as 5.5 in ASCE 7 (2010). Lower C_d values will increase the overall inelastic response of the structure. On the other hand, the inelastic response of the structure is expected to be less severe when designed for higher C_d values. The performance objective is that the structure should sustain the inelastic deformation demand imposed due to design earthquake ground motions. This study aims at investigating the inelastic seismic response that low‐rise, medium‐rise and high‐rise SMRFs can experience under design earthquake ground motions and maximum considered earthquake (MCE) level ground motions and evaluating the deflection amplification factors (C_d) for SMRFs in a rational way. For this purpose, nonlinear dynamic time history and pushover analyses will be carried out on SMRFs with 4, 9 and 20 stories. The results indicate that the current practice for computing the inelastic story drifts for SMRFs is rational and the frames designed complying with the current code requirements can sustain the inelastic deformations imposed during design earthquake ground motions when seismically designed and detailed. Copyright © 2013 John Wiley & Sons, Ltd.     

Inelastic earthquake control of weld failure Part I: Limit state approach

Active structural control of inelastic response is proposed for the first time on existing buildings. The optimal linear control theory and the force analogy method are combined in state space form to calculate the response of the structure. Application of this combined method is performed to reduce the risk of weld failure in steel buildings. A six‐story moment resisting building damaged during the 1994 Northridge earthquake is used to study the sensitivity of the response and the magnitude of the structural control force to the earthquake ground motion. The limit state approach is used to design the structural control system by limiting the maximum plastic rotations in the building to an acceptable level. In the process, structural control is shown to be very effective in reducing the plastic rotations during excitations, and therefore reducing the risk of weld failure. In addition, structural control is effective in reducing the responses, which include displacement, velocity, acceleration and drift of the structure. Copyright © 1999 John Wiley & Sons, Ltd.     

A modified response spectrum analysis procedure to determine nonlinear seismic demands of high‐rise buildings with shear walls

The standard response spectrum analysis (RSA) procedure prescribed in various design codes is commonly used by practicing engineers to determine the seismic demands for structural design purpose. In this procedure, the elastic force demands of all significant vibration modes are first combined and then reduced by a response modification factor (R) to get the inelastic design demands. Recent studies, however, have shown that the response of higher vibration modes may experience much lower level of nonlinearity, and therefore, it may not be appropriate to reduce their demand contributions by the same factor. In this study, a modified RSA procedure based on equivalent linearization concept is presented. The underlying assumptions are that the nonlinear seismic demands can be approximately obtained by summing up the individual modal responses and that the responses of each vibration mode can be approximately represented by those of an equivalent linear SDF system. Using 3 high‐rise buildings with reinforced concrete shear walls (20‐, 33‐, and 44‐story high), the accuracy of this procedure is examined. The inelastic demands computed by the nonlinear response history analysis procedure are used as benchmark. The modified RSA procedure is found to provide reasonably accurate demand estimations for all case study buildings.     

Ductility factors of symmetrical structures with restricted inelasticity

A simple elasto‐plastic analysis of symmetrical buildings composed of different types of bents is presented. Plasticity in bents is assumed to be restricted to the beams, where significant curvature ductility factors may be attained without loss of strength, and the behaviour of inelastic structures may be traced from elastic to ultimate deflections on the grounds that the external load is monotonically increasing with an invariant shape throughout the loading history. The outlined procedure is based on the concept of the continuous medium and a comprehensive consideration of the structural post‐elastic behaviour of common mixed‐bent‐type buildings is possible by means of parameter variations. Therefore it is suitable for a simple pushover elasto‐plastic analysis in a parametric form. The proposed method of analysis is based on the technique of decomposing a cantilever structural system into two components: a flexural and a shear–flexure subsystem. With this decomposition, not only the axial deformations in the column members of all bents are taken into account in the calculation of deflections, but also different types of bents may be combined in symmetrical buildings. Design charts relating the overall ductility factor with the maximum rotation ductility factor at the ends of the beams are presented for the case of a triangularly distributed external load, together with the formulation that defines the post‐elastic load–deflection curve. These data may be found useful for practising engineers at the stage of a preliminary structural application. Copyright © 2007 John Wiley & Sons, Ltd.     

Forward directivity near‐fault and far‐fault ground motion effects on the behavior of reinforced concrete wall tall buildings with one and more plastic hinges

Near‐fault (NF) ground motion having forward directivity and far‐fault (FF) earthquakes can generate different responses on tall reinforced concrete cantilever walls. In this paper, the behavior of the core wall buildings were examined by performing nonlinear time history analyses on 20‐story, 30‐story and 40‐story fiber element models. The concept of one, two, three and extended plastic hinge in the core walls subjected to the NF motions having forward directivity (pulse‐like) and FF motion was studied by carrying out inelastic dynamic analysis. At the upper levels of the walls, NF pulse‐like ground motions can produce considerably larger curvature ductility, inter‐story drift and displacement demands as compared with the FF motions. A new approach was proposed to obtain the moment demand and reinforcement required to balance the curvature ductility demand along the height of a core wall. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of inelastic displacement factor of soil‐shallow‐foundation MDOF systems incorporating higher modes effect

This study attempts to investigate the higher‐mode effects on the constant‐ductility inelastic displacement factors of multi‐degree‐of‐freedom (MDOF) systems considering soil‐structure interaction. These factors were computed for 12,600 two‐dimensional superstructure models of shear buildings and their corresponding equivalent single‐degree‐of‐freedom (ESDOF) systems under 26 ground motions recorded on alluvium and soft soil. An intensive parametric study was carried out for a wide range of non‐dimensional parameters, which completely define the problem. The underlying soil is considered as a homogeneous half‐space based on the concept of cone model. The higher‐mode effects were then investigated through defining the ratio of inelastic displacement factor of MDOF system to that of the corresponding ESDOF one. The influence of soil‐structure interaction key parameters, fundamental period, ductility ratio, the number of stories, and dispersion of the results are evaluated and discussed. Results indicate that as the base becomes very flexible, unlike to the fixed‐base systems, in which the defined ratios are greater than unity, using the inelastic displacement factors of ESDOF models for MDOF ones would result in a remarkable overestimation of maximum inter‐story displacement demand. A new expression is proposed to estimate the ratio of inelastic displacement factor of MDOF soil‐structure systems to that of SDOF counterpart.     

A simple model for assessing periods of vibration and modal response quantities in symmetrical buildings

A simple mathematical model is proposed for assessing periods of vibration and mode shapes of common cantilever bents used in concrete structures, such as shear walls, coupled walls, rigid frames and wall‐frame assemblies. The bent is treated as a continuum and the proposed model is based on the technique of decomposing a cantilever bent into two complementary subsystems (a flexural and a wall‐frame bent) and on the finding that the use of Dunkerley's formula for calculating natural frequencies yields reasonable results for the first three modes of vibration. The objective in proposing this model is to consider the effect of column axial shortenings in the analysis of structural bents. With this model any cantilever bent may be approximated by a simple incompressible shear–flexure system of equal flexural rigidity, but of equivalent modal shear rigidity. This approach has the advantage that the response of different structural bents may be combined in buildings composed of these bents in any arrangement. All bents are approximated by equivalent shear–flexure models and therefore the complete structure may be analysed by a simple methodology, which has been extensively used in the past. Particularly in symmetrical buildings, frequencies may be determined by a simple formula and modal response quantities by available design charts. A quick estimate of these quantities is of particular importance at the early stages of structural design, prior to a full dynamic analysis. In order to illustrate the application of the proposed model a symmetrical building of varying height, composed of different structural bents, is analysed and comparisons are made with more accurate results obtained by 3D computer dynamic analyses. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparative study of special and ordinary braced frames

The collapse probability of ductile and non‐ductile concentrically braced frames was investigated using nonlinear dynamic response analysis. Two buildings with three and nine stories located in Boston and Los Angeles, respectively, were designed and subjected to ground motions from the areas. In Boston area, three‐story and nine‐story buildings were designed as ordinary concentrically braced frame with response modification reduction factor R equal to 3 1/4 to be considered as non‐ductile structural systems; comparatively, in Los Angeles area, three‐story and nine‐story buildings were designed as special concentrically braced frame with response modification reduction factor R equal to 6 to be considered as ductile structural systems. In order to evaluate the performance of ductile and non‐ductile concentrically braced frames in moderate and severe seismic regions, ATC‐63 would be used as reference to assess the seismic behaviors. Evaluation approach recommended by ATC‐63 was adopted, and hundreds of nonlinear dynamic analyses were performed. Through alternating the scale factors of designated ground motions, median of structural collapse intensity was presented for each structure. By observing the results of statistical performance assessment, the seismic performance of the systems was evaluated, and some observations are made based on the study. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal reduction of inelastic seismic demands in high‐rise reinforced concrete core wall buildings using energy‐dissipating devices

Recently, the issue of large inelastic seismic force demands at severe ground shakings such as maximum considered earthquake level has been highlighted in the conventionally designed high‐rise reinforced concrete core wall buildings. Uncoupled modal response history analysis was used in this study to identify the modes responsible for the large inelastic seismic force demands. The identification of dominant modes and mean elastic design spectra of seven representative ground motions for different damping ratios has led to the identification of three control measures: plastic hinges (PHs), buckling‐restrained braces (BRBs) and fluid viscous dampers (FVDs). The identified control measures were designed to suppress the dominant modes responsible for the large inelastic seismic force demands. A case‐study building was examined in detail. Comparison of the modal as well as the total responses of the case‐study building with and without the control measures shows that all the control measures were effective and able to reduce the inelastic seismic demands. A reduction of 33%, 22% and 27% in the inelastic shear demand at the base and a reduction of 60%, 22% and 26% in the inelastic moment demand at mid‐height were achieved using the PHs, BRBs and FVDs, respectively. Furthermore, a reduction of about 30–40% in the inelastic seismic deformation demands was achieved for the case of the BRBs and FVDs. The study enables us to gain insight to the complex inelastic behavior of high‐rise wall buildings with and without the control measures. Copyright © 2011 John Wiley & Sons, Ltd.     

Considering dynamic parameters of damaged structure in MPA and YPS nonlinear static procedures for asymmetric buildings

The yield point spectra and modal pushover procedures have already been used in seismic design of new buildings and vulnerability evaluation of existing structures. This paper initially identifies the similarities and differences of the two procedures. Then, the modal characteristics of damaged structure are used to modify their methodologies. The applications of the procedures in estimating displacement, interstory drift index and plastic hinge rotation responses of asymmetric buildings are investigated for three 5‐story reinforced concrete moment‐resisting frame‐building models. The results show considerable improvement in estimating the responses of those asymmetric buildings. Copyright © 2013 John Wiley & Sons, Ltd.     

Equivalent non‐linear single‐degree‐of‐freedom system of spatial asymmetric multi‐storey buildings in pushover procedure: theory and applications

In the present paper, the issue of the approximate definition of a new equivalent non‐linear single‐degree‐of‐freedom (NLSDF) system on spatial asymmetric reinforced concrete (r/c) tall multi‐storey buildings is presented. In order to achieve this goal, three different types of r/c systems are examined: the first type refers to multi‐storey planar r/c frames; the second type refers to asymmetric single‐storey r/c building; and the third type refers to asymmetric multi‐storey r/c buildings. The definition of the NLSDF system is mathematically derived, considering suitable dynamic loadings on the masses of each r/c system using simplified assumptions. The NLSDF system is very useful in the seismic design of the r/c systems, since it is widely used in all forms of various pushover analyses that have been published in the past. The use of the equivalent NLSDF system in combination with the inelastic design spectra can give an acceptable evaluation of the maximum required seismic floor displacement for a known design earthquake. The present paper concludes the total theory of definition of the optimum equivalent NLSDF system for the above three types of buildings that possess the required normality by the contemporary seismic codes in elevation. In order to illustrate the theory, three numerical examples are presented, respectively. The final numerical required displacement results by the use of the equivalent NLSDF system are verified and checked by non‐linear response history analyses. Copyright © 2008 John Wiley & Sons, Ltd.