Residual Displacement Ratio Research Articles

The effect of superstructure properties on the re-centering capability of Friction Pendulum (PS) isolator

The re-centering capability after the seismic excitation is one of the fundamental properties of a seismic isolation system. This study examines the effects of superstructure properties on the re-centering capability of the Friction Pendulum (FP) isolator. To this end, a comprehensive parametric study is conducted using a two-degree-of-freedom (2DOF) model that accounts for the nonlinear behavior of the superstructure and isolator. This study evaluates the re-centering capability of the FP isolator by calculating the ratio of residual displacement (dres) to the maximum displacement (dmax). The effect of superstructure properties, including strength reduction ratio R, fundamental vibration period Ts, post-yield stiffness ratio αs, superstructure damping ratio ξs, and the ratio of superstructure mass to total mass of the system γm on the dres/dmax ratios is thoroughly analyzed. The findings suggest that while estimating the residual displacement of isolation systems, it is important to consider the impact of superstructure parameters. An analytical equation is proposed to predict dres/dmax ratio based on effective superstructure and isolator parameters.

Data-based equations for damage evaluation of reinforced concrete columns with square cross sections

This study aims to develop simple equations for assessing the damage of reinforced concrete (RC) columns subjected to seismic loading. A fiber element model in OpenSees, validated against experimental data, was used to generate a large dataset of 56548 data values for various RC columns. The Monte Carlo Hierarchical Adaptive Random Partitioning (MC-HARP) strategy was then applied to evaluate the sufficiency of the dataset and propose reliable equations for estimating the residual displacement ratio (RDR), damage index (DI), and residual curvature of RC columns under given displacement demands. A relationship between these parameters and the structural performance levels was also established. Residual displacement is a practical measure that can be observed in the field after an earthquake. The proposed equations facilitate the determination of the performance level of the structure after an earthquake event. The findings of this paper can also be used in designing new structures within a performance-based seismic design (PBSD) context.

Flexural performances of steel–concrete composite section of self-anchored suspension bridge: Experimental and theoretical research

This paper aims to clarify the flexural performance and the cracking load, the ultimate bearing capacity calculation theories of the steel–concrete composite section of the self-anchored suspension bridge. A refined local finite-element (FE) model of the steel–concrete composite section was built to evaluate the mechanical performance of key parts under three unfavourable load conditions. A test model of the steel–concrete composite section with a length × width × height of 7 m × 1.833 m × 1.2 m was fabricated according to the scale of 1: 3, and a four-point negative bending test was carried out. The test results show that the test model is in the elastic stage, and the residual displacement ratio is 0.7 % under the design load condition. Under the overload condition (2.3 times the design load), the experimental model is in the elastic–plastic stage, and the residual displacement ratio is 19 %. At the same time, the stress of each component still meets the requirements. Under the two conditions, the deformation angle of the composite section is small, and the stiffness transition is stable. The relative slip between the top steel plate and the concrete in the composite section is small (no more than 0.25 mm). The stress of the top steel plate and bottom steel plate of the composite section gradually decreases during the transfer from the steel stiffening beam to the concrete side. The bearing plate is the main bearer of the force transmission of the steel–concrete composite section, accounting for about 51.24 %, while the top steel plate, bottom steel plate and shear connectors share the remaining load. The steel–concrete composite section of the test model has reasonable force, smooth force transmission and good deformation coordination performance. In addition, the calculation methods of cracking load and ultimate bearing capacity of the prestressed steel–concrete composite section were proposed. The theoretically calculated value of the ultimate bearing capacity of the steel–concrete composite section is in good agreement with the FEA value, and the difference does not exceed 10 %.

Impact behavior of high-strength steel beam with circular web openings

A drop-weight impact testing machine was used to conduct an impact test on 9 specimens in order to examine the impact resistance of high-strength steel beams with circular web openings (HSSBCWO). The experimental results indicated that the dynamic response of HSSBCWO is significantly affected by the change of impact energy and opening height. A coefficient β is proposed to assess the severity of localized deformation in HSSBCWO, and the range of β values for the specimens of test are found to be approximately between 0.5 and 0.8. Within the range of impact energy designed for the experiment, HSSBCWO primarily absorbed energy through localized plastic deformation. A finite element model which can predict the dynamic response of HSSBCWO under impact load is established. The results show that the optimal ratios of opening height to beam height and spacing between openings to beam height for the HSSBCWO are determined to be 0.50 and 0.93, respectively. Furthermore, a damage evaluation system for HSSBCWO under impact load was established, utilizing the ratio of mid-span residual displacement to calculated beam span Dr / L0 as the damage assessment index, with impact energy, opening height, and spacing between opening as the main control parameters.

Development and seismic behavior of an assembly composite damping self‐centering brace

AbstractA new type of assembly composite damping self‐centering brace is proposed to simplify the configuration, reduce the manufacturing cost, and improve its energy dissipation capacity, which is assembled by several modules in parallel with different functions. A brace specimen with a total length of 1.843 m was designed and numerically analyzed. The brace exhibits stable and full flag‐shaped hysteretic responses, and dissipates adequate energy during cyclic motion, with a high bearing capacity. When the brace is unloaded, the viscous damping force disappears, and the residual displacement can be basically eliminated. The hysteretic envelope area increases with the increase in loading frequency. When the frequency is 0.5 Hz and the target displacement is 30 mm, the activation force, ultimate bearing capacity, residual displacement, energy dissipation during one cycle, and equivalent viscous damping ratio are 622.5 kN, 1018.7 kN, 9.11 mm, 41.10 kJ, and 0.214, respectively. Under random loading, the damping force provided by the viscous module ensures a fuller hysteretic loop than the existing displacement‐type energy dissipation self‐centering brace. The influences of design parameters on the hysteretic performances of the brace were studied. When the disc spring stiffness increases from 3.27 to 9.80 kN/mm, the ultimate bearing capacity, residual displacement, and equivalent viscous damping ratio change by 18%, −24%, and −15%, respectively. The viscous damping coefficient and flow index of the viscous module should be increased, and the prepressed force and stiffness of disc springs in recentering module should also be appropriately increased.

Experimental study on the flexural performance of steel-concrete composite section of self-anchored suspension bridge

In order to study the flexural performance of the steel-concrete composite section of self-anchored suspension bridge, a test model of the steel-concrete composite section was fabricated according to the scale of 1 : 3 based on the Wushui Bridge, and a four-point negative bending test was carried out. The results show that the residual displacement ratio is 0.7%, and the experimental model is in the elastic stage under the design load condition, which incidate that the component is safe under normal service condition. Under the overloading condition, the residual displacement ratio is 19%, the experimental model is in the elastic-plastic stage, but the stress of materials still meets the requirements. Under the two conditions, the deformation angle of the composite section is small and the stiffness transition is stable. The relative slip between the steel plate and concrete in the composite section is small (no more than 0.25 mm). The stress in the top steel plate and bottom steel plates of the composite section are gradually decreases during the transfer from the steel stiffening beam to the concrete side. The design of the steel-concrete composite section is reasonably loaded, and the mechanical properties are smoothly transferred.

Residual displacement ratios of highly damped SDOF systems by considering soft soil conditions

In this paper, residual displacement ratios, Cr, of highly damped single degree of freedom (SDOF) systems are studied by considering soft soil effects. For this purpose, a large number of ground motions recorded on soft soil deposits are used in the analysis. In order to calculate the residual displacement ratios, linear and nonlinear time history analysis are conducted for 79 SDOF systems with lateral strengths of R = 2, 3, 4, 5, 6, damping ratios of ξ = 0.05, 0.1, 0.2, 0.3, and post-yield stiffness ratios of α = 0, 0.03, 0.1 subjected to 426 pairs of earthquake ground motions recorded on soft soil. Therefore, a large number of data is obtained for Cr by using the results of time history analysis. After investigating the impact of lateral strengths, damping ratios, and post yield stiffness ratios on residual displacement ratios, a simplified new equation is proposed with nonlinear regression analysis. The results show that higher damping ratios cause the residual displacement ratios to reduce. Furthermore, comparison of Cr for soft soil and stiff soil conditions indicates the soil condition influence on Cr.

Study on impact resistance of precast light-weight concrete sandwich panels

To investigate the dynamic impact resistance of precast lightweight sandwich panels (PLCSP) composed of non-sintered coal ash ceramsite concrete and self-developed thermal blocking shear connectors, drop hammer impact tests with different impact energy were carried out in the central area of the exterior wythe. The dynamic response of PLCSP under different steel fibers content, wythe thickness, and the different number of shear connectors was compared and analyzed. The numerical simulation was carried out by LS-DYNA. The results show that 2% volume fraction of steel fibers enhance the stiffness of the panel by improving the overall impact toughness, while the increase in wythe thickness slows down the panel amplitude to enhance the stiffness. The exterior wythe of the PLCSP has penetrated failure, and the failure mode of the interior wythe changes from bending failure to collapse failure with the increase of the number of shear connectors. Therefore, the arrangement spacing of shear connectors can be appropriately reduced to reduce the risk of collapse failure. The ratio of maximum displacement and residual displacement of the panel under different impact energies ranged from 3.37 to 5.94, and the maximum erosion depth of the hammer was twice the displacement of the bottom of the panel, indicating that the unique sandwich structure and material of the PLCSP have good impact resistance. The research results can provide reference for the design and application of PLCSP.

Differences in responses of square steel plates exposed to blast loads generated by cubic and spherical explosives

The differences in the responses of square steel plates subjected to blast loads of cubic and spherical charge explosions were investigated in the study. A finite element model (FEM) was developed by LS-DYNA, and the FEM was validated by experiment and UFC 3-340-02. Based on the validated numerical model, the incident and reflect loads of cubic and spherical detonations were compared. In the x-z plane, the incident pressures of cubic explosives are observed to be concentrated in the x and z directions while that of spherical charges are uniform in all directions. Furthermore, cubic explosives have substantially higher normal reflect pressures than spherical charges. For square steel plates, the results indicate that cubic explosives can produce more significant deformations and plastic damage than spherical explosives. Moreover, quantification metrics was proposed to assess the differences in responses of steel plates subjected to blast loads of cubic and spherical explosives. The ratios of average maximum displacement (Dam) under cubic and spherical charge detonation were analyzed, as well as the ratio of average residual displacement (Dar). Equivalent coefficient (ζ) was defined to equate a cubic explosive to a spherical explosive. Finally, a damage number for steel plates subjected to blast loads was presented. The relationships between damage number and displacements were created. The damage number and displacements have linear correlations. Moreover, the displacement ratios of steel plates under blast loads of cubic and spherical explosives can be approximated by using the damage number. The damage number can also be used to approximately derive the equivalent coefficients.

Machine learning-aided peak and residual displacement-based design method for enhancing seismic performance of steel moment-resisting frames by installing self-centering braces

Conventional steel moment-resisting frames (SMRFs) absorb seismic energy through steel yielding behavior, leading to significant residual displacement. Although steel yielding behavior can ensure the seismic safety of SMRFs under strong earthquakes, excessive residual displacement may lead to post-earthquake demolition decisions, causing a large amount of economic loss. This paper aims to develop a peak and residual displacement-based design (PRDBD) method for controlling the peak and residual inter-story drift responses of SMRFs by installing self-centering braces. The peak and residual displacements are both set as the design targets in the proposed PRDBD method. To this end, the machine learning prediction models of inelastic and residual displacement ratios were first developed based on the median responses of single-degree-of-freedom systems under earthquakes. The detailed design steps of the proposed PRDBD method were subsequently introduced. The three- and nine-story demonstration buildings were retrofitted by using the PRDBD method with two different design targets. Static and dynamic analyses were conducted to validate the effectiveness of the proposed PRDBD method. The static analysis results indicated that the self-centering braces could efficiently enhance the SMRF’s stiffness and strength. The retrofitted SMRFs showed no strength deterioration, whereas the original SMRFs showed obvious strength deterioration at the roof drifts of 3.2% and 2.5% in the three- and nine-story buildings, respectively. The dynamic analysis results confirm that the self-centering braces can efficiently reduce the peak and residual inter-story drift responses of the existing SMRFs and the retrofitted SMRFs can achieve the peak and residual inter-story performance objectives under the considered seismic intensity. Moreover, the retrofitted SMRFs can be fully recoverable after maximum considered earthquakes by controlling the maximum residual inter-story drift lower than 0.2%.

Time-Dependent Seismic Performance of Coastal Bridges Reinforced with Hybrid FRP and Steel Bars.

To increase the durability and seismic resilience of coastal bridges, a hybrid reinforced concrete (HRC) bridge that incorporates both glass fiber-reinforced polymer (GFRP) bars and steel bars is proposed. The time-dependent seismic performance of the HRC bridge is comprehensively investigated at three levels, namely the material, bridge column and bridge structure levels. First, the decrease of tensile strength of GFRP bars over time is analyzed based on the Arrhenius theory, and corrosion initiation time and performance deterioration of steel bars are determined by Fick’s second law and an empirical formula. Second, an efficient finite element modeling method for aging HRC bridge columns is proposed. Simulation of the compression/tension behavior and the fracture failure of the GFRP bar is described. Hysteretic analysis is further conducted to investigate the time-dependent energy dissipation, ductility, residual displacement, bearing capacity and post-yield stiffness ratio. Meanwhile, comparisons of HRC bridge columns to reinforced concrete (RC) references are provided. Third, the seismic demand and damage evolution of deteriorated HRC bridge structures are investigated through dynamic time-history analysis. The results indicate that the corrosion-resistant GFRP bars contribute to improving the bearing capacity and to reducing the residual displacement of the HRC bridge. With an increase in service time, the seismic damage to the bridge column, abutment and expansion bearing increases, but the damage to fixed bearing decreases. Research results presented herein show that the HRC bridge is a promising alternative structure scheme in the marine environment.

Residual displacement demand for non-degrading bilinear SDOF oscillators with self-centering viscous-hysteretic devices

The self-centering viscous-hysteretic device (SC-VHD) is a high-performance self-centering system that was recently developed to reduce structural residual displacement and damage during severe earthquakes. This study focuses on developing a residual displacement ratio C r spectrum and a residual displacement estimation method for non‐degrading bilinear SDOF oscillators with SC-VHDs. Statistical analysis is conducted to investigate the effect of design parameters on the constant-ductility residual displacement ratio C r spectrum under near-fault pulse-like earthquake ground motions (NFPE) and far-fault earthquake ground motions (FFE). Based on statistical data, a formula for the mean C r spectrum is proposed. Subsequently, an iterative calculation method for residual displacement is presented according to the equivalent linearization method and the proposed mean C r spectrum formula. Finally, using the iterative calculation method, the residual displacement of a two-story steel moment-resisting frame (MRF) with SC-VHDs is predicted, demonstrating its application. The results indicate that the mean C r spectrum under the NFPE is significantly larger than that under the FFE. Besides, the mean constant-ductility C r spectrum decreases with increasing post-yield strength ratio, ratio of preload to yield strength, ratio of loading stiffness to elastic stiffness increases, or ratio of unloading stiffness to loading stiffness, but increases with increasing viscous damping ratio. When estimating the C r spectrum, increasing the ductility ratio and decreasing the ratio of preload to yield strength can both reduce record-to-record variability. Both the formula for the mean C r spectrum and the iterative calculation method for residual displacement are validated by comparing with statistical data. • The influence of design parameters on the residual displacement C r spectrum is investigated. • A formula for the normalized mean C r spectrum is developed. • An iterative calculation method for residual displacement is proposed. • An example application is used to validate the calculation method.

Cyclic performance of RC bridge piers retrofitted with UHPC jackets: Experimental investigation

Many existing reinforced concrete (RC) bridge piers do not have sufficient seismic-resistant capacities during the long-term service period. Seismic retrofitting of these piers will be an impending problem. In this study, quasi-static cyclic tests were conducted to investigate the efficiency of ultra-high performance concrete (UHPC) in strengthening the seismic performance of RC bridge piers. Three test specimens, among which one is a normal RC pier and the others are UHPC jackets-retrofitted RC piers, were used for the cyclic tests. Based on the experimental results, the effects of UHPC jackets on hysteresis response, strength, ductility, stiffness degradation, energy dissipation, residual displacement, and curvature distribution, are analyzed and discussed. The results show that after being retrofitted by UHPC jackets, the curvature distribution and damage modes of the RC pier are changed, and the strength and stiffness are improved. Furthermore, 50 mm-thick UHPC jackets contribute to a significant reduction of residual displacement and equivalent viscous ratio, indicating improving post-earthquake serviceability of retrofitted piers. Numerical simulation was conducted to further investigate the influence of size of UHPC jacket on the envelope curve for the RC pier. It was found that for a fixed thickness or height of UHPC jacket, the strength of retrofitted pier would not increase with the UHPC jacket volume when it reached to a certain value.

Seismic evaluation of industrial steel moment resisting frames with shape memory alloys using performance-spectra-based method

Seismic evaluation of industrial facilities may need to employ multi-performance indicators in various yielding stages towards a more delicate design methodology. This paper develops a performance-spectra-based method to evaluate the seismic performance of industrial steel moment resisting frames (MRFs) with shape memory alloys (SMAs) considering multi-performance indicators and hysteresis transition. The work commences with demonstration of the hysteresis transition of a steel MRF with SMAs from the ‘self-centring stage’ to the ‘post self-centring stage’. Then, based on the single-degree-of-freedom (SDOF) analogy and the hysteretic model considering hysteresis transition, a performance-spectra-based space for quantifying the seismic behaviour of an industrial steel MRF with SMAs including multi-performance indicators is established. In this study, the performance-spectra-based space employs three essential indicators (i.e. energy modification factor, peak acceleration ratio and residual displacement ratio). To demonstrate the application and validate the adequacy of the proposed method for seismic evaluation of industrial steel MRFs with SMAs, two prototype steel MRFs equipped with SMA connections for industrial use are designed and assessed under representative earthquake ground motion ensembles. By comparing essential structural seismic response quantities (i.e. maximum roof displacement, peak interstorey drift ratio, peak floor acceleration and interstorey residual drift ratio) predicted by the proposed method with those calculated by nonlinear-response-history analyses (NL-RHAs), the effectiveness of the proposition is confirmed.

Prestressed concrete wall system with friction devices: Seismic performance and direct displacement–based seismic design

It is convenient to use the inelastic displacement ratio spectra and residual displacement ratio spectra to predict the maximum inelastic displacement and residual displacement of building structures based on linear elastic analysis directly. The prestressed concrete wall system with friction devices (PCW-FD system) can be simulated by single-degree-of-freedom (SDOF) model with self-centering behavior. To investigate the seismic performance of PCW-FD system, the SDOF models with fully and non-fully self-centering behavior are analyzed firstly, and it is concluded that the hysteresis parameters (strength reduction coefficient, post-yield stiffness coefficient, energy dissipation coefficient, and period) have significant influence on the seismic responses (such as constant relative strength inelastic displacement ratio, constant relative strength residual displacement ratio, maximum absolute acceleration, the hysteretic energy, and site classifications) during short period, and the trends of the seismic responses are similar at different site classifications. Then a large amount of the result data is summarized, and the constant relative strength inelastic displacement ratio spectra and the constant relative strength residual displacement ratio spectra with enough precision (the correlation coefficients are 0.957 and 0.947, respectively) are established by conducting regression analysis. Finally, the direct displacement–based seismic design method is improved and verified to be suitable for PCW-FD system.

Investigation on residual displacements for SDOF systems with various initial viscous damping models

Nonlinear time history analysis (NTHA) is widely used in estimation of post-earthquake residual displacement demands. However, the choice of initial viscous damping models in NTHA has attracted insufficient attention in study of residual displacements. The more versatile viscous damping model similar to the tangent-stiffness proportional Rayleigh damping commonly used in multi-degree-of-freedom (MDOF) systems is utilized in NTHA of single-degree-of-freedom (SDOF) systems. With different combinations of mass proportional and tangent-stiffness proportional damping terms, the significance of the choice of initial damping models is examined in terms of energy dissipation and residual displacement demands. The residual displacement ratio, defined as the ratio of residual displacements to elastic spectral displacements, is computed for constant-strength SDOF systems when subjected to an ensemble of 240 earthquake ground motions. The results are statistically organized to investigate the effect of initial viscous damping models on residual displacement demands. The difference in residual displacement ratios for various damping models is found to be significant, and dependent on natural periods, relative lateral strength, post-yield stiffness, unloading stiffness degradation and initial damping ratios. The effect of these factors on residual displacement demands of SDOF systems with tangent-stiffness proportional damping, which is confirmed to be most appropriate for nonlinear analysis, is also examined. Finally, the constant-strength residual displacement response spectra are constructed for seismic assessment of structures.

Mechanical, electrical and thermal properties of graphene oxide-carbon nanotube/ ABS hybrid polymer nanocomposites

Multiwalled carbon nanotubes (MWCNTs), functionalized carbon nanotubes (FCNTs) and graphene oxide-carbon nanotube (GCNTs) hybrid Bucky paper (BP) reinforced acrylonitrile-butadiene-styrene (ABS) composites are prepared via vacuum filtration followed by hot compression molding. The nanomechanical, electrical and thermal properties of these BP reinforced ABS composites are studied. The nanoindentation hardness and elastic modulus of GCNTs-ABS hybrid composites reached to 389.98±91.79 MPa and 7669.6±1179.12 MPa respectively. Other nanomechanical parameters such as plastic index parameter, elastic recovery, the ratio of residual displacement after load removal and displacement at maximum load are also investigated. The improved nanomechanical properties are correlated with Raman spectroscopy and scanning electron microscopy (SEM). It is found that GCNTs and their composites showed the higher value of defect density. The maximum value of defect density range for GCNTs and GCNTs-ABS is (297.4 to 159.6) and (16.0 to11.6), respectively. The higher defect density of GCNTs indicates that the interfacial interaction between the ABS, which was further correlated with electrical and thermal properties. Additionally, the through-plane electrical conductivities of MWCNTs, FCNTs and GCNTs based ABS composites were 6.5±0.6, 4.5±0.7 and 6.97±1.2 S/cm respectively and thermal conductivities of MWCNTs, FCNTs and GCNTs reinforced ABS composites; 1.80, 1.70 and 1.98 W/mK respectively. These GCNTs-ABS composites with this value of thermal conductivity can be used in various applications of efficient heat dissipative materials for electronic devices.

Theoretical analysis and experimental investigation of hysteretic performance of self-centering variable friction damper braces

Traditional buildings, even with normal self-centering energy dissipation (SCED) braces, might suffer from large deformations and high mode effects under an extremely strong earthquake, leading to a concentration of inter-story drift in upper floors. To satisfy the requirements of resilience, larger post-yield stiffness and higher energy dissipation, a novel brace with pretensioned basalt fiber-reinforced polymer (BFRP) tendons and variable friction dampers (VFDs) was developed, and this brace was termed the self-centering variable friction damper (SC-VFD) brace. The variable stiffness and sliding force of the VFD system were theoretically analyzed. This was followed by quasi-static experiments on two VFD and two SC-VFD braces with different parameters. The theoretical analysis and experimental results revealed the same tendencies, demonstrating the reliability and feasibility of SC-VFD braces. The hysteretic curve of the SC-VFD braces exhibited stable and normal flag-shape behavior before the second activation, while the hysteresis curve indicated a variable flag-shape behavior with second activation, variable friction force, and larger post-yield stiffness when the brace slid at slope section. Compared with the VFD brace, the SC-VFD brace had the same energy dissipation ability but less residual displacement and lower equivalent viscous damping ratio. More combinations of disc springs in series resulted in smaller axial forces and lowers post-yielding stiffness, thereby decreasing the energy dissipation capability.

Assessment of three-dimensional RC moment-resisting frames under repeated earthquakes

Earthquakes are one of the major unavoidable natural disasters. These are characterized by the recorded ground motion data at a particular location. In any real earthquake event, shaking repeatedly occur several times within a few hours or days. In view of lack of time between the occurrences of tremors, the rehabilitation of the structure often becomes impractical, and the structural damages get accumulated. Hence, it is imperative to assess the behavior of reinforced concrete (RC) buildings under repeated earthquakes to prevent a disaster. The current practices of earthquake-resistant design utilize only a single strongest earthquake input in the form of a response spectrum, corresponding to the seismic code of practice at the site location. Hence, this investigation is primarily focused on vulnerability assessment of three-dimensional RC building frames conformed to the Indian standard codes of practice under bi-directional single and repeated ground motions. Incremental dynamic analysis is carried out to assess the inelastic behavior of the structure in terms of maximum horizontal displacement, residual displacement, and maximum inter-story drift ratio for particular spectral accelerations. The results obtained in this investigation conclude that the collapse capacity of RC building under repeated earthquakes reduces significantly compared to that of the most severe single earthquake. This emphasizes the need to account for repeated earthquake forces during the design phase in order to build a seismic resilient structure.

Prediction of residual displacement ratios for simple structures built on soft-soil sites of Mexico City

This short communication discusses the prediction of residual displacement demands through residual ratios (i.e., the ratio of residual displacement demand to the maximum displacement demand) for single-degree-of-freedom (SDOF) systems subjected to earthquake ground motions recorded at soft soil sites of Mexico City. A functional form to obtain estimates of mean residual ratios for elastoplastic and stiffness-degrading SDOF systems on this type of sites is presented and its ability to capture the empirical spectral trend of residual ratios is illustrated in this paper. It is shown that the introduced equation provides good estimates of the empirical data.