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On-site observations show that several damages have occurred in CRTS II slab-type ballastless track system, such as crack and interface debonding between different structural layers, which will trigger the occurrence of some failure modes of CRTS II track slab. As the damages are the prior for specific failure modes, time-dependent reliability of CRTS II track slab should be conducted considering the uncertainties associated with both the failure mode and the damage, which is defined as the conditional time-dependent reliability in this study. To appropriately express the causal relationship among the failure modes and prior damages, a conditional limit state function (LSF) model is proposed with explicit expressions provided. With the aid of the proposed model, conditional time-dependent reliability of CRTS II track slab for two individual failure modes are investigated, which are the steel yielding failure mode under longitudinal cracking, and the vertical instability failure mode under interface debonding. Furthermore, conditional time-dependent system reliability of the track slab has also been analyzed considering four safety failure modes. The advantage of the proposed method is that prior damages have been accounted in establishing the failure criteria and incorporated in the formulation of time-dependent reliability, which provides a general formwork for structural reliability involving the causal relationship among the failure modes and damages.

In this paper, a total of 70 experimental flexural tests of corroded post-tensioned prestressed concrete (PC) beams were gathered from the published literatures. Using the experimental data, a comprehensive databank of flexural capacity of the corroded PC beams was established. Meanwhile, four existing empirical models for calculating the residual flexural strength of the corroded PC beams were also gleaned and contrasted with each other based on the databank. It turns out that three of the four existing empirical models underestimate the flexural capacity. To deeply analyze the factors affecting the flexural capacity, numerous factors were discussed, e.g., the corrosion ratio, beam width, effective height of the beam section, concrete compressive strength, and the ratio of strength between the tensile reinforcement (including prestressed and non-prestressed reinforcement) and concrete. It is found that the corroded post-tensioned PC beams’ residual flexural capacity is affected most by the corrosion ratio. In the end, a better empirical model is proposed for computing the residual flexural capacity of corroded PC beams, which provides accurate and effective predictions in a wide range of corrosion ratio for engineering practice.

In this study, a total of 177 flexural experimental tests of corroded reinforced concrete (CRC) beams were collected from the published literature. The database of flexural capacity of CRC beam was established by using unified and standardized experimental data. Through this database, the effects of various parameters on the flexural capacity of CRC beams were discussed, including beam width, the effective height of beam section, ratio of strength between longitudinal reinforcement and concrete, concrete compressive strength, and longitudinal reinforcement corrosion ratio. The results indicate that the corrosion of longitudinal reinforcement has the greatest effect on the residual flexural capacity of CRC beams, while other parameters have much less effect. In addition, six available empirical models for calculating the residual flexural strength of CRC beams were also collected and compared with each other based on the established database. It indicates that though five of six existing empirical models underestimate the flexural capacity of CRC beams, there is one model overestimating the flexural capacity. Finally, a newly developed empirical model is proposed to provide accurate and effective predictions in a large range of corrosion ratio for safety assessment of flexural failure of CRC beams confirmed by the comparisons.

The existing theoretical models for the analysis of concrete-filled steel tube (CFST) columns are mainly limited to circular specimens. This paper is aimed at developing a unified theoretic...

Because of the vulnerability of roof sheathing panel to wind load, light-frame wood low-rise buildings often suffer from serious destructions in typhoon hazards. Efficient assessment of the...

Reinforcement corrosion exhibits an adverse effect on the shear strength of reinforced concrete structures. In order to investigate the effects of chloride-induced corrosion of reinforcing steel on the shear behavior and failure pattern of reinforced concrete beams, a total of 24 reinforced concrete beams with different concrete strength grades and arrangements of stirrups were fabricated, among which 22 beams were subjected to accelerated corrosion to achieve different degrees of reinforcement corrosion. The failure pattern, crack propagation, load–displacement response, and ultimate strength of these beams were investigated under a standard four-point loading test in this study. Extensive comparative analysis was conducted to investigate the effects of the concrete strength, shear span-to-depth ratio, and stirrup type on the shear behavior of the corroded reinforced concrete beams. The results show that increasing the stirrup yielding strength is more effective in improving the shear strength of corroded reinforced concrete beams than that of concrete compressive strength. In terms of three types of stirrups, the shear strength of the beams with deformed HRB-335 is least sensitive to stirrup corrosion, followed by the beams with smooth HPB-235 and the beams with deformed HRB-400. The effect of the different stirrups on the shear strength depends on the corrosion degree of stirrup and shear span-to-depth ratio of the beam. The predicted results of shear strength of corroded reinforced concrete beams by a proposed analytical model are well consistent with the experimental results.

A total of 158 experimental tests of shear behavior of corroded reinforced concrete (CRC) beams under action of concentrated load, published in the literature were collected and compiled into a shear strength database. This database was firstly used to discuss important parameters which affect the shear strength of CRC beams. The results show that the effect of stirrups’ corrosion on shear capacity of CRC beams is greater than that of longitudinal reinforcement corrosion. The shear span-to-depth ratio is also an important factor on shear strength of CRC beams. Total 9 available empirical models for predicting the residual shear strength are evaluated and compared based on the test database. It is found that eight of the nine models underestimate the shear strength of CRC beams while the other model gives the overestimated results. It is in this regard that a new empirical model for predicting the residual shear strength of CRC beams is proposed, in which a reduction coefficient is incorporated with the consideration of the effect of stirrups’ corrosion as well as shear span-to-depth ratio. The comparison studies demonstrate that the new proposal can provide an effective and accurate prediction of the shear capacity of CRC beams with a wide range of reinforcement corrosion damages.

The light-frame low-rise buildings widely used for residential and industrial purposes are very vulnerable to high winds. Postdisaster surveys indicated that envelope components such as win...

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosis > 3) and “hardening” responses (with narrower tails; for example, kurtosis

The corrosion rate of reinforcing steel is an important factor to determine the corrosion propagation of reinforced concrete structures in the chloride-laden environments. Since the corrosion rate of reinforcing steel is affected by several coupled parameters, the efficient prediction of which remains challenging. In this study, a total of 156 experimental data on corrosion rate from the literature were collected and compared. Seven empirical models for predicting the corrosion rate were reviewed and investigated using the collected experimental data. Based on the investigations, a new empirical model is proposed for predicting the corrosion rate in corrosion-affected reinforced concrete structures considering parameters including concrete resistivity, temperature, relative humidity, corrosion duration and concrete chloride content. The comparison between the experimental data and those predicted using the new empirical model demonstrates that the new model gives a good prediction of the corrosion rate. Furthermore, the uncertainty and probability characteristics of these empirical models are also investigated. It is found that the probability distributions of the model errors can be described as lognormal, normal, Weibull or Gumbel distributions. As a result, the new empirical model can provide an efficient prediction of the corrosion rate of reinforcing steel, and the model error analysis results can be utilized for reliability-based service life prediction of reinforced concrete structures under chloride-laden environments.

So far, the loading path effect on the compressive strength of confined concrete has been poorly investigated. To clarify the compressive strength of confined concrete in concrete-filled steel tube (CFT) columns, experimental tests with a total of 18 specimens are conducted to investigate the loading paths of confined concrete in concrete loaded CFT stub columns. The parameters that are varied in the experiment include the steel strength, the unconfined concrete strength and the D/t ratio. Two evaluation indices, namely the laterally dominant index and the effect index, are proposed to characterize the loading path and its effect on the compressive strength of confined concrete, respectively. A detailed parametric study is conducted on the loading paths and the corresponding effects on compressive strength. The results suggest that the loading paths of confined concrete in CFT column are affected by the column parameters. Depending on the loading paths by which the concretes are confined, the loading path effects on the compressive strength of the confined concrete in CFT columns could be totally different. Based on the test results, a compressive strength model incorporating the loading path effect is developed and good performance was found in comparison with the experimental results.

This paper proposes a new method for seismic fragility analysis of nuclear power plants (NPPs) considering the uncertainty sources of both ground motions and structural parameters. Based on the conventional seismic fragility model that only contains the uncertainty of ground motions, seismic fragility analysis conditioned on a specific seismic intensity is probabilistically reformulated by further incorporating the uncertainty of structural parameters. To improve the efficiency of the analysis, a point estimation-assisted incremental dynamic analysis was developed for evaluating the seismic fragility model including the uncertainty of structural parameter. The accuracy and efficiency of the proposed method were demonstrated by a single-degree-of-freedom system, where Monte Carlo simulation method is used for comparison. The seismic fragility based on the finite element model of an AP1000 NPP was analyzed using the proposed method. The results reveal that the safety of NPP structure would be overestimated without considering the uncertainty of structural parameter and the influence of the uncertainty of structural parameters on the seismic fragility of NPP presents an upward trend as the damage level increases. Furthermore, the sensitivity analysis of different structural parameters reveals that elastic module may have a dominant influence.

In practical engineering, the probability distributions of some random variables are often unknown, and the only available information about these may be their statistical moments. To conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions, an explicit fourth-moment standardization function has been proposed, and a single expression of its inverse transformation, that is, normal transformation, with limitations of inputting sets of the third and fourth moments (skewness and kurtosis) of random variables was derived. However, the clear definition of the complete expressions of the inverse transformation of fourth-moment standardization function under different combinations of skewness and kurtosis of random variables has not been provided yet. It is in this regard that four criteria are proposed to derive the complete inverse transformation of fourth-moment standardization function, and then the complete expressions of the inverse transformation are formulated. Through the numerical examples presented, the proposed complete expressions are found to be quite efficient for normal transformations and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability assessment.

The objectives of this paper is to investigate the mechanical behavior of steel reinforced concrete-filled square steel tubular (SRCFT) stub columns under axial loading through combined experimental and numerical studies. In total of six specimens are tested to investigate the effect of the concrete strength and steel ratio on the mechanical behavior of SRCFT stub columns. The ultimate bearing capacity, ductility, and confinement effects are discussed and clarified based on the experimental results. The inserted steel section can effectively prevent shear cracks in the core concrete from propagating quickly. The strength-weight-ratio of SRCFT stub column is larger than SCFT stub column. In addition, ABAQUS is used to establish the 3D finite element (FE) model and analyze the composite action of SRCFT stub columns under axial loading. Based on the experimental and numerical results, a simplified formula is proposed to estimate the ultimate bearing capacity of SRCFT stub columns using superposition method. The predicted results show satisfactory agreement with both experimental and FE results.

Although extensive research on steel corrosion in concrete has been undertaken it is more on longitudinal reinforcing bars than stirrups as suggested in the review of published literature. The effect of corrosion of steel stirrups on structural behaviour can be significant for shear failure mode. This paper aims to investigate the degradation of mechanical behaviour of corroded stirrups. Details of a comprehensive experiment designed to examine the mechanical behaviour of corroded stirrups in concrete structural members are presented. A micromechanical damage model for failure mechanism of corroded stirrups is proposed and a model for damage factor is derived. Based on these models, a constitutive model for corroded stirrups is developed and verified with test results. It is found in the experiment that both the strength and ductility of corroded stirrups decrease with the increase of corrosion and that the hemispherical model for the pit shape is more appropriate for the prediction of strength reduction of corroded stirrups. The constitutive model developed in the paper can be used to predict the mechanical behaviour of corroded stirrups with reasonable accuracy, paving the way for the assessment of corrosion induced shear failure of reinforced concrete structures.

This paper presents a probabilistic investigation of American and European specifications (i.e., AISC and Eurocode 4) for square concrete-filled steel tubular (CFT) stub columns. The study is based on experimental results of 100 axially loaded square CFT stub columns from the literature. By comparing experimental results for ultimate loads with code-predicted column resistances, the uncertainty of resistance models is analyzed and it is found that the modeling uncertainty parameter can be described using random variables of lognormal distribution. Reliability analyses were then performed with/without considering the modeling uncertainty parameter and the safety level of the specifications is evaluated in terms of sufficient and uniform reliability criteria. Results show that: (1) The AISC design code provided slightly conservative results of square CFT stub columns with reliability indices larger than 3.25 and the uniformness of reliability indices is no better because of the quality of the resistance model; (2) The uniformness of reliability indices for the Eurocode 4 was better than that of AISC, but the reliability indices of columns designed following the Eurocode 4 were found to be quite below the target reliability level of Eurocode 4.

In this paper, an efficient method is proposed for constructing the seismic fragility curves of bridge piers, in which the dynamical reliability assessment is developed from methods of moment. The main procedure for constructing seismic fragility curves consists of four steps. First, the non-stationary seismic acceleration process samples are simulated by the random function-spectrum representation model (RFSRM) with one elementary random variable at a given PGA value. Second, the first four moments (i.e., mean value, standard deviation, skewness and kurtosis) of extreme structural response are then evaluated from the extremum of each nonlinear time-history seismic response based on RFSRM. Third, the failure probability is then estimated by the fourth-moment reliability index, which is derived from an explicit fourth-moment transformation. Finally, the seismic fragility curves can be constructed by repeating the three steps and considering different PGA values and damage levels (i.e., slight, moderate, extensive, and complete). The efficiency and accuracy of the proposed methodology are demonstrated by a numerical example for constructing the fragility curves of a high-speed railway bridge pier in which the direct incremental dynamic analysis (IDA) method based on Monte Carlo simulations are employed for comparison.

The fibre beam element (FBE) method has been widely used to investigate the behaviors of concrete-filled steel tube (CFT) columns for its high efficiency and accuracy. The existing models were generally developed or calibrated by using specimens with relatively small size, in which the size effects were ignored or poorly incorporated. Additionally, most existing FBE models focused on CFT stub columns. In this paper, a new FBE model for nonlinear analysis of concentrically loaded stub and slender columns was proposed, in which the size effects on the effective stress-strain relationships of steel tube and core concrete were well incorporated. The proposed FBE models were then used to develop an ultimate strength model for the design of both stub and slender columns. A total of 1029 specimens assembled from 72 previous studies were used to verify the proposed models. The results suggested that the proposed FBE model performs quite well for both small- and large- size CFT stub columns and predicts the behaviors of slender columns with reasonable accuracy. Moreover, the ultimate strength model developed based on the proposed FBE models was found to make satisfying prediction of test results.

In structural reliability analysis, the input variables are often nonnormal and correlated. A procedure for efficient normal transformation, i.e., transforming dependent nonnormal random va...

Based on the change coefficient of modal flexibility curvature, an experimental research on damage recognition of cable-stayed bridge was conducted. The working conditions are introduced, as well as the design and manufacture of the model that can serve as an important guide to the similar experiments. In the design of this experiment, the optimization of cable tensions and the sensor placement optimization were discussed and the suitable methods were chosen and verified. The results verify the sensitivity of the change coefficient of modal flexibility curvature and compare the identification ability on different parts of the same damage location. In addition, how the number of modes required affects the change coefficient of modal flexibility curvature was discussed and the proper number is given.

A finite element model of an existing reinforced concrete arch bridge was established. The reliability index was investigated based on the PDF module in ANSYS software. Taking the maximum concrete stress and tendon stress as limit state, the limit state functions of main arch, upright column and deck are established respectively. There are taken as series system to study the bridge system reliability according to the structural characteristics of reinforced concrete arch bridge. The lowest reliability index superstructure is selected as the whole structure reliability index. Based on JC method and made-up MATLAB program, the reliability index of the example bridge is calculated.

This paper examines the effect of pitting corrosion of prestressing steel strands on flexural strength of prestressed concrete (PC) beams in marine environment. A reliability-based methodology for the prediction of time to structural failure after corrosion initiation was proposed and probabilities of failure are evaluated using FORM as exampled by a high-speed rail PC box beam. The methodology presented in this paper can be used as a tool for structural engineers and asset managers to assess a corrosion-affected concrete infrastructure and make decisions with regard to its maintenance and rehabilitation.

This paper presents a probabilistic investigation of design code provisions such as AISC, DL/T and Eurocode 4 for circular CFT stub columns. The study is based on experimental results of 250 axially loaded circular CFT stub columns published in the literature. By comparing experimental results for ultimate loads with code-predicted column resistance, the error of resistance models is analyzed and it is found that it can be described using a normal random variable. Based on the investigations, reliability analyses were performed for the three design codes with/without considering the model error random variables. It is found that the model error is quite relevant for column safety, especially for the case of small load ratios. Consideration of model error resulted in increased reliability of stub columns designed according to AISC code, due to conservative column resistance predicted by the code. While for design codes of DL/T or Eurocode 4, consideration of model error produces a reduction of reliability indexes. Finally, the safety level of code provisions in terms of sufficient and uniform reliability criteria is also evaluated. Results show that the AISC design code provided very conservative results and the scatter of reliability indexes is the smallest. Eurocode 4 provided suitable results in terms of the sufficient criteria. The paper also shows that the reliability indices of columns designed following DL/T are slightly below the target reliability levels of ISO 2394.

Corrosion of reinforcing and prestressing steel due to chloride contamination is one of the primary causes of deterioration of concrete structures. A review of published literatures shows that the research on the deterioration of mechanical properties of reinforcing steel is more than that on prestressing strands, even though the corrosion of prestressing strands may trigger structural collapse without warnings due to higher stress levels in the steel. This paper aims to investigate the degradation of mechanical behaviour of corroded prestressing strands. Details of a comprehensive experiment designed to examine the mechanical behaviour of corroded prestressing strands in concrete structural members are presented. A micromechanical damage model for failure mechanism of corroded prestressing strands is proposed, and a model for damage factor is derived. Based on these models, a constitutive model for corroded prestressing strands is developed and verified with test results. It is found in the paper that both the strength and ductility of corroded prestressing strands decrease with the increase of corrosion and that the hemispherical model for the pit shape is more appropriate for the prediction of strength reduction of corroded prestressing strands. The paper concludes that the constitutive model developed in the paper can be used to predict the mechanical behaviour of corroded prestressing strands accurately, paving the way for the assessment of corrosion-induced flexural failure of prestressed concrete structures.

Based on appropriate numerical constitutive model of lightweight aggregate concrete filled steel tubes (LCFST), layered method was adopted to predict the complete moment-curvature curves of LCFST composite section subjected to pure bending. A FORTRAN program for the moment-curvature curves was developed. Applying the layered method, the influences of the significant parameters, such as steel ratio, yield strength of steel and strength of lightweight aggregate concrete on the moment-curvature curves of the composite section subjected to pure bending were discussed. The practical composite flexural stiffness, ultimate moment and practical moment- curvature relationship of LCFST composite section were presented. The predicted results of both layered method and practical moment-curvature relationships of LCFST composite section are in good agreement with the test results from reference. Compared with the layered method, practical calculate method could remove the step of composite section layered and improve the calculation speed while achieving similar results.

This paper presents an investigation of ultimate strength of square CFT stub columns with compact sections. The beneficial composite action between the steel tube and the filled concrete is taken into account and a new analytical model for predicting the axial capacity of square CFT stub columns with compact sections is proposed. Experimental results of 89 axially loaded square CFT stub columns published in the literature are then used to verify the proposed strength model. Results show that the proposed strength model provides a direct, compact, and efficient representation of the ultimate strength of square CFT stub columns made with not only normal strength but also with high strength steel tubes and concrete.

In order to perfectly reflect the dynamic corrosion of reinforced concrete (RC) cover in practical engineering, an analytic model of non-uniform corrosion induced cracking was presented based on the elastic-plastic fracture mechanics theory. Comparisons with the published experimental data show that the predictions given by the present model are in good agreement with the results both for natural exposed experiments and short-time indoor tests (the best difference is about 2.7%). Also it obviously provides much better precision than those models under the assumption of uniform corrosion (the maximal improved precision is about 48%). Therefore, it is pointed out that the so-called uniform corrosion models to describe the cover cracking of RC should be adopted cautiously. Finally, the influences of thickness of local rusty layer around the reinforcing steel bar on the critical corrosion-induced crack indexes were investigated. It is found that the thickness of local rusty layer has great effect on the critical mass loss of reinforcing steel, threshold expansion pressure, and time to cover cracking. For local rusty layer thickness with a size of a = 0.5 mm, the time to cover cracking will increase by about one times when a/b (a, semi-minor axis; b, semi-major axis) changes from 0.1 to 1 mm.

In this study, a reliability-based method for predicting the initiation time of reinforced concrete bridge beams with load-induced cracks exposed to de-icing salts is presented. A practical model for predicting the diffusion coefficient of chloride ingress into load-induced cracked concrete is proposed. Probabilistic information about uncertainties related to the surface chloride content and the threshold chloride concentration has been estimated from a wide review of previous experimental or statistical studies. Probabilistic analysis to estimate the time to corrosion initiation with/without considering the effect of the load-induced cracks on the chloride ingress into concrete has been carried out. Results of the analysis demonstrate the importance of considering the effect of the load-induced cracks for correct prediction of corrosion initiation in RC bridge beams exposed to chlorides.

In this note, a number of empirical models available in the literature of the complete stress-strain curve for unconfined high-strength concrete under uniaxial compression are reviewed and investigated using the published experimental data. Based on the investigations, a new empirical model with emphasis on the softening branch is proposed to generate the complete stress-strain relationship for high-strength concrete. An application of the new empirical model to published experimental data on normal weight concretes over a wide strength range demonstrates that the present model gives a good representation of the mean behavior of the actual stress-strain response.

In this paper, a total of 250 experimental tests of axially loaded circular concrete-filled steel tube (CFT) stub columns, published in the literature was summarized. The applicability of the current design codes such as ACI, Australian Standards, AISC, AIJ, Eurocode 4, DL/T and some available empirical models proposed by various researchers for calculating the axial capacity of circular CFT stub columns was examined using these experimental data. Based on the investigations, four new empirical models for predicting the axial capacity of circular CFT stub columns are proposed. The comparisons between the experimental results and the predictions of these models show that the proposed empirical models provide a direct, compact, and efficient representation of the ultimate strength of circular CFT stub columns made with not only normal strength but also high strength steel tubes and concrete. Finally, the limiting values of the maximum effective length, the compressive strength of concrete, the yield strength of steel tubes and the diameter-to-thickness for circular CFT stub columns with respect to the present empirical models are suggested. It is expected that engineers can easily use the present empirical models to estimate the axial capacities of circular CFT stub columns for engineering designs.

In this paper, the predictions of ultimate strength of circular concrete-filled steel tube (CFT) stub columns using current design codes such as AIJ-2001, AISC-2005, DL/T-1999, and Eurocode 4-2004, are examined and compared with published experimental tests of CFT short columns made with normal- and high-strength steel and concrete. Based on the investigations, a simple formula for predicting the ultimate strength of CFT stub columns is proposed. The concrete confinement, which depends mainly on the ratio of the external diameter of the steel tube to the plate thickness, the yield stress of the steel tube, and the unconfined compressive strength of the filled concrete, was reasonably achieved in the proposed formula. From comparisons with the published experimental data for CFT short columns of normal and high strength steel and concrete, the present formula is shown to give a good representation of the ultimate strength of circular CFT columns.

In structural reliability analysis, the uncertainties related to resistance and load are generally expressed as random variables that have known cumulative distribution functions. However, in practical applications, the cumulative distribution functions of some random variables may be unknown, and the probabilistic characteristics of these variables may be expressed using only statistical moments. In the present paper, in order to conduct structural reliability analysis without the exclusion of random variables having unknown distributions, the third-order polynomial normal transformation technique using the first four central moments is investigated, and an explicit fourth-moment standardization function is proposed. Using the proposed method, the normal transformation for indepen- dent random variables with unknown cumulative distribution functions can be realized without using the Rosenblatt transformation or Nataf transformation. Through the numerical examples presented, the proposed method is found to be sufficiently accurate in its inclusion of the independent random variables which have unknown cumulative distribution functions, in structural reliability analyses with minimal additional computational effort.

It is important for obtaining the relationship between seismic energies of single degree-of-freedom (SDOF) systems and multiple degree-of-freedom (MDOF) structures in engineering. In this paper, the formula of hysteretic energy between the MDOF structures and equivalent SDOF systems is developed. Here is also presented the procedure for estimating hysteretic energy of MDOF structures subjected to severe ground motions employing the energy relation equation based on equivalent SDOF systems. Eight examples for two regular and six irregular MDOF structures show that the procedure to obtain the hysteretic energy demands of MDOF structures may be used as a simple and effective energy estimation method.