Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil-Structure Interaction Effects Using Capacity Curves

The analysis of buildings under seismic motion with a flexible base must consider two principal aspects of structural displacement: the first, being structural deformation, the second, rigid body behavior. This effect produces a modification of the inelastic behavior of structures. In addition, a consideration of the flexible base may change the distribution of internal forces along the structure that could generate variations in the ductility demands on different structural elements. This article summarizes the results of previous studies on variations in the inelastic behavior of steel and reinforced concrete structures, taking dynamic soil structure interaction into consideration. The response of buildings with a flexible base is compared and contrasted with those with rigid bases. The inelastic behavior of buildings is set out in terms of ductility capacity and demands. Pushover analysis is used to establish inelastic capacity parameters by comparing the capacity curves of buildings with rigid (fixed) and flexible bases. Some comments and general guidelines are made about how base flexibility influences the inelastic behavior of structures.


Resumen
El análisis de las estructuras bajo acciones sísmicas con base flexible debe considerar dos componentes principales del movimiento: uno asociado con la deformación propia de la estructura y el otro relacionado con el movimiento de cuerpo rígido.Adicionalmente, la consideración de la base flexible puede modificar la distribución y magnitud de las fuerzas internas dentro de la estructura y puede, incluso, modificar las demandas de ductilidad en distintos elementos estructurales.En este trabajo se resumen los resultados de estudios previos respecto a la variación del comportamiento inelástico de marcos de acero y de concreto considerando los efectos de interacción dinámica sueloestructura.Se compara la respuesta de las estructuras con base flexible y con base rígida.El comportamiento inelástico se define con base en curvas de capacidad calculadas con análisis estáticos no lineales.Se calculan las demandas de ductilidad y la capacidad de deformación inelástica de las estructuras por medio de las curvas de capacidad.Además, se incluyen algunos comentarios y conclusiones generales acerca de la influencia de la base flexible en el comportamiento inelástico de las estructuras.

Introduction
The inelastic behavior of structures has become a fundamental aspect of seismic design.It is essential to be able to predict the inelastic deformations of structural elements under seismic motion.Since demands can only be computed by a nonlinear time history analysis of the fully designed structure in conjunction with a specific seismic motion, alternative methods must be used for design procedures.The response spectrum method is by far the most commonly used approach to computing force and deformation demands.For inelastic analysis, the use of the inelastic response spectrum of uniform ductility demand allows us to establish the yield strength required by the structure to control global ductility demand.With this approach, it is possible to choose the maximum ductility demand desired on the structure and then establish the required yield strength.Once the yield strength and maximum ductility demand have been defined, the designer must ensure that the structure will be capable of resisting these demands.The structure's strength and ductility capacity should be greater than demand.Building codes around the world use the well-known approach of yield strength reduction factors (R μ ).
First, the designer chooses the maximum ductility demand desired on the structure ( μ) and then the associated R μ is computed.The relationship between μ and R μ depends on the structure's dynamic properties and the characteristics of the input motion. 1 With R μ values, the acceleration demands computed from the elastic response spectrum are reduced, so the required structural strength is defined.Most of the ideas and hypotheses for this method were developed for systems whose supports are fixed.In addition, response spectrums are built with the maximum responses of Single Degree of Freedom systems (sdof).The ratio between R" and μ is computed with sdof systems as well.Under these conditions, all structural responses are represented by a single displacement.The entire displacement of the system is associated with structural deformation, so ductility is defined as the ratio between maximum and yielding displacements.
However, in some cases, the stiffness of the soil foundation system is not enough to constitute a fixed base, so a relative displacement is produced between the foundation and the surrounding soil.Soil structure system displacement includes two principal elements, one the result of structural deformation (u) and other of rigid body behavior (u 0 and θ ) as shown in figure 1.The interaction between the soil and the foundation can modify the structure's dynamic properties, excitation characteristics and soil behavior.Those modifications which arise from the joint performance of the soil and the foundation are defined as Dynamic Soil-Structure Interaction (dssi).
In general, dssi is calculated through the modification of the structural period (lengthening) and the damping produced by system flexibilization. 2The structure will therefore be subjected to a modified spectral acceleration demand.Several building codes 3 use the base shear variation associated with the spectral acceleration shift to compute changes in the remaining response quantities (e.g.displacements, element forces, etc).Nevertheless, the presence of rigid body displacement elements modifies the relationship between μ and R μ .
Since the response spectrum method is based on the response of sdof systems, the approach for flexible base structures using the response spectrum must represent the soil structure system with an 2 John P. Wolf, Dynamic Soil-Structure Interaction (Prentice Hall, 1985).

Luciano Roberto Fernández Sola | Dossier
Equivalent Single Degree of Freedom system (esdof).Previous studies had characterized the modifications introduced by the flexible base using the concept of an equivalent system with a single degree of freedom (esdof). 4They established the equivalent properties of a single degree of freedom system (fundamental period, damping ratio and ductility) that may reproduce the inelastic response of a multi-degree system with a flexible base.This approach is the one used in several building codes that account for the dssi. 5Equivalent ductility ( ) is defined as the ratio of the maximum (ũ u ) and yield displacement (ũ y ) of the esdof (equation 1).
This ductility can be computed as a function of the fundamental period and the ductility of the system with a fixed base (T and μ) and the equivalent period (T ) with equation 2. This assumes that the esdof behaves as a perfectly elastoplastic system without considering post-yielding stiffness.In redundant systems, where many elements contribute to lateral stiffness, capacity curves show a progressive yield, which must be modeled as a bilinear system with post-yield stiffness.The effect of displacement elements due to rigid body behavior on the inelastic branch will be smaller, but not null, as for an elastoplastic model. 6quivalent ductility always yields smaller values than with a fixed base.This does not mean that a structure with a flexible base has a reduced inelastic capacity, as commonly misunderstood.The ductility factor must be corrected due to the modification of the relationship between the yield strength reduction factor (R μ ) and the ductility factor (μ ) produced by base flexibility.If elastic forces are reduced by the fixed base yield strength reduction factor without any correction, ductility demands on a structure with a flexible base may be increased.

Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil Structure Interaction Effects Using Capacity Curves
The use of an esdof is very useful and yields good results in many cases.Since just one degree of freedom is used, this procedure implicitly considers the modifications introduced by base flexibility in all structural responses throughout the structure to be linearly equivalent.However, studies have shown that, in some cases, the representation of a flexible base system with multiple degrees of freedom with an ESDOF may not yield good results. 8quivalent ductility can be computed from the capacity curve of structures with a flexible base. 9This article summarizes the results of prior studies conducted by the author and others on the inelastic behavior of buildings with fixed and flexible bases in order to show the main changes in equivalent ductility and the influence of post-yield stiffness.The capacity curves of buildings with fixed and flexible bases are presented.These curves are computed in two different ways.First, the total displacement of the soil structure system is considered (u), including rigid body elements (u 0 and Θ).This set of results is used to compute equivalent ductility.Capacity curves are then computed using only the displacement associated with structural deformation (u) to determine if the structure's inelastic parameters are modified by base flexibility due to P-Δ effects.Results for braced steel frame buildings and reinforced concrete frame buildings are presented.

The influence of post yield stiffness on equivalent ductility
The relationship between R μ and μ is determined by the percentage of the total displacement produced by rigid body behavior for yield and maximum displacement.Luciano Roberto Fernández Sola | Dossier related to u, the ratio between displacement produced by rigid body elements (u 0 and Θ) and u is different for yield and maximum displacement.In order to take this effect into consideration, Fernández Sola and Huerta Écatl have proposed the following procedure: 11 using equation 1, the maximum displacement of esdof system ductility (ũ u ) is expressed as equation 3. Expressing in terms of rigid body displacement and structural deformation yields (equation 4) Equation 4 can be expressed in terms of the deformation of the structure (u u and u y )as equation 5: To compute the ductility of the structure ( μ), only the displacement produced by structural deformation should be considered.Consequently, the relationship between ESDOF system ductility with a flexible base ( ) and ductility in the structure (μ) can be defined as equation 6:

Equivalent ductility from capacity curves
Equivalent ductility ( ) can be computed from the capacity curves of the soil-structure system using total displacement as the displacement of the ESDOF system (u).Results for three different types of structures are presented.
First, the results reported by Fernández Sola, Tapia Hernández and Dávalos Chávez 13 correspond to the inelastic behavior of individual braced steel frames with one and two braced bays and unbraced frames.All frames are part of the same building.Three different buildings of 8, 12 and 16 storeys are analyzed.All buildings are built on a soft clay layer with a shear wave velocity of Vs=65 m/s.Two types of foundations are used for each building, one that consists of a mat foundation and the other of frictional piles. 14 Áviles and Pérez Rocha, "Use of Global Ductility for Design of Structure Foundation System." 13 Fernández Sola and others, "Respuesta inelástica de marcas de acero con interacción inercial suelo-estructura." 14 See: Fernández Sola and others, "Respuesta inelástica de marcos de acero con interacción inercial suelo-estructura."Capacity curves built with total displacement (u) are shown in figure 2 for the unbraced frames of the 8 and 16 storey buildings.As expected, frames with dssi effects are more flexible.This means that ũy and ũu are larger than for the fb.For the 8 storey building, the pile foundation is stiffer than the mat foundation.The opposite happens for the 16 storey building.Yield and maximum base share are very similar for the frames with fb and dssi, so for these cases, base flexibility does not influence overall resistance.
Values for are shown in table 1.These values are computed using bilinear primary capacity curves ( cc ).Yield displacement is defined at the intersection of elastic and inelastic branches.Fernández Sola, 15 report that the overstrength factors for these frames are 1.07 and 1.10 for the 8 and 16 storey frames respectively.This means that both frames behave almost as perfect elastoplastic systems.From the results seen in table 1, it can be seen that, as the base becomes more flexible (mat foundation for the 8 storey frame and pile foundation for the 16 storey frame), values get smaller.Since the frames presented behave almost as elastoplastic systems with very low post-yield stiffness, the values computed from capacity curves are very similar to the values computed using equation 2 ( eq2 ).

Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil Structure Interaction Effects Using Capacity Curves
In order to compare the ductility of the structure (μ), capacity curves are computed using the displacement associated with the deformation of the structure (u).These curves are shown in figure 3. Capacity curves are very similar for the three booth frames.This indicates that the inelastic behavior of the structural system is basically unchanged by base flexibility, that is, if the strength reduction factor is defined by , the ductility of the structural system is the same for all cases.This is the reason for using values to compute R μ .There are some very small differences in frame resistance.Frames with a flexible base experience a reduction of maximum base shear values.In addition, a dropdown in terms of the elastic stiffness of frames with DSSI is observed.This can be associated with the amplification of P-Δ effects produced by base flexibility.16 Fernandez Sola and others, "Respuesta inelástica de marcas de acero con interacción inercial suelo-estructura." In order to study the potential strength and stiffness reduction produced by the amplification of P-Δ effects, a parametric analysis was performed.The capacity curves of the 16-storey frame with a pile foundation were computed using extremely reduced values of soil foundation system stiffness.These values are unreal and are used to amplify P-Δ effects.
Figure 4 shows the capacity curves computed with u using 100%, 20% and 5% soil pile foundation stiffness.Similar results for the other cases can be found in Fernadez Sola, Tapia Hernández and Dávalos Chávez 16 Drastic base stiffness reduction produces a reduction in elastic stiffness and yield base shear.It is worth remembering that these capacity curves show only the deformation of the structure without rigid body Luciano Roberto Fernández Sola | Dossier elements.These effects are produced by the increase of P-Δ effects due to the amplification of the relative displacement between the ends of the columns produced by base rotation.These effects have been examined in single steel columns in previous studies. 17In order to show the influence of overstrength on the inelastic behavior of systems with a flexible base, the results presented by Huerta Écatl and Fernández Sola 18 are shown.In this study, a 10 storey building with reinforced concrete (RC) frames and a mat foundation is used.Three different soil stiffness values are considered (Vs=70, 100 and 250 m/s).Again, capacity curves with total displacement (ũ) and with structural deformation (u) are computed (figure 5).In this study, results are obtained for the whole building and not for individual frames.
For the steel frames presented previously, as the soil becomes more flexible, ũ u and ũ y values become higher.The variation in the relationship between ũ u and ũ y produces changes to .When the deformation of the structure is analyzed, capacity curves for all soil conditions Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil Structure Interaction Effects Using Capacity Curves 19 Details on the role of each displacement element can be found in Huerta Écatl, "Evaluación de la interacción dinámica suelo-estructura en el comportamiento inelástico de un edificio de concreto reforzado." 20 See: Huerta Écatl, "Evaluación de la interacción dinámica suelo-estructura en el comportamiento inelástico de un edificio de concreto reforzado."As for the steel frames, table 2 shows that, as the soil becomes more flexible, values are lower.In addition, it can be seen that μ̃ values computed using an elastoplastic model ( eq.2 ) yield lower values than those computed directly from capacity curves ( CC ) due to the post-yield stiffness of the structure.When is computed taking into consideration the overstrength of the structure ( eq.6 ), values are similar to those computed directly from the capacity curves.From these results, it is clear that elastoplastic models can overestimate changes in ductility due to base flexibility. 21ernández Torres 22 studied the inelastic behavior of steel buildings with braced frames at different heights (4, 7 and 10-storey) with both fixed and flexible bases.Mat foundations on a soft soil of V s =80 m/s were used.Capacity curves considering total displacement (ũ ) are shown in figure 6.This analysis is performed for the whole building and not for individual frames. 23In these cases, the higher the structure, the larger the dssi effects.Similar effects to those from the previous results can be observed.These structures develop large overstrength factors.μ values computed from capacity curves ( 67.9 ), equation 2 ( 55) and equation 6 ( 67.-) are shown in table 3. 23 Fernández Sola and others, "Respuesta inelástica de marcas de acero con interacción inercial suelo-estructura".
As for all previous cases, equivalent ductility ( ) is lower than structural ductility ( ).Since these buildings exhibit significant post-yield stiffness, equivalent ductility computed with the elastoplastic model ( eq.2 ) yields lower values than those computed directly from capacity curves ( CC ).Equation 6predicts equivalent ductility better since it explicitly takes post-yield stiffness into account.

Final remarks
This article has presented a summary of studies of the inelastic behavior of flexible base systems and explored the use of equivalent ductility ( ) for the inelastic design of structures with dynamic soil structure interaction effects (dssi) with the response spectrum method.The relationship between the yield strength reduction factor (Rμ) and structural ductility (μ) is modified by dssi.Since the response spectrum method is based on the response of single degree of freedom systems, the use of is necessary in order to keep μ demands within design values.
Post-yield stiffness plays an important role in variations.Procedures included in building codes are based on a perfect elastoplastic equivalent system.values computed with this procedure tend to be lower than those computed for systems with post-yield stiffness.
Equivalent ductility can be computed directly from the systems' capacity curves.Results for individual steel frames, reinforced concrete (RC) buildings and steel buildings with braced frames are shown.Capacity curves are computed with two sets of results: one using the complete system displacement, including rigid body displacements, and one using only the deformation of the structure.μ values are computed with the first set of results and μ values are computed with the second set of results.
It has been confirmed that the value is smaller than the μ value for all cases with dssi effects.Elastic stiffness is always reduced by base flexibility.Yield and maximum base shears and overstrength factors are very similar.On the other hand, capacity curves that only take structural deformation into consideration are almost entirely unmodified by dssi Luciano Roberto Fernández Sola lrfs@azc.uam.mxA civil engineer with a Bachelor's degree from the National Polytechnic Institute (ipn) Higher School of Engineering and Architecture (2005) and a Master's and Doctorate in Engineering (2007 and 2011, respectively)  with a specialization in structural engineering from the National Autonomous University of Mexico (unam), graduating with an honorable mention on both of the latter occasions.
A specialist in the seismic behavior of structures and foundations and in dynamic soil-structure interactions.He has been an author and coauthor of a variety of publications, including popularization articles, papers presented at national and international conferences, articles in indexed journals and research reports.He has also participated in interinstitutional projects and interuniversity networks and served as a consultant for the private sector.
A He has also served on the working group that is giving comments on the Complementary Technical Regulations for Seismic Design for Mexico City's 2017 construction regulations.
Actualmente es miembro del comité de Interacción Suelo-Estructura de la Sociedad Mexicana de Ingeniería Geotécnica, secretario de la mesa directiva de la Sociedad Mexicana de Ingeniería Estructural, miembro de la comisión de inteligencia competitiva de la Alianza Fii-DEM, presidente del comité de educación continua y estudiantes de la Sociedad Mexicana de Ingeniería Estructural y coordinador de la Licenciatura en Ingeniería Civil de la uam-Azcapotzalco.

Figure 1 :
Figure 1: Displacement elements of a structure with a flexible base.

3
See: asce 7, "Minimum Design Loads for Buildings and Other Structures, " asce Standard asce/sei 7-10, American Society of Civil Engineers, 2010; nbcc, "National Building Code of Canada, " National Research Council of Canada, Ottawa, 2015; nzs 3101-1, "New Zealand Standard Code of Practice for General Structural Design and Design Loadings for Buildings, " Standards Association of New Zealand, Wellington, 2006; and mcbc, "Reglamento de construcciones para el Distrito Federal, " Gaceta Oficial del Departamento del Distrito Federal, Mexico, 2004. 7

Figure 2 :
Figure 2: capacity curves with total displacement (u) for unbraced frames on a fixed base (FB), mat foundation (Mat) and pile foundation (Piles) for a) 8-storey and b) 16-storey buildings.

Figure 3 :
Figure 3: capacity curves with structural deformation (u) for unbraced frames on a fixed base (fb), mat foundation (Mat) and pile foundation (Piles) of a) 8 storey and b) 16 storey buildings.

Figure 5 :
Figure 5: capacity curves for RC buildings on different soil types (Vs=70, 100 and 250 m/s) and a fixed base with a) structural deformation (u) and b) total displacement (ũ) (figures 5).
Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil Structure Interaction Effects Using Capacity Curves Evaluation of the Equivalent Ductility of Buildings with Dynamic Soil Structure Interaction Effects Using Capacity CurvesFor elastoplastic systems, rigid body displacements u u rb and u y rb are equal.Substituting u rb u = u rb y = u rb and using u rb = ũ y -u y , after some algebra, proves that equation 6 yields equation 2. However, a system's post-yield stiffness leads to larger values of , as shown byAvilés and  Pérez Rocha. 12 Equation Equation Equation • segunda época • año 9 • núm.18 • México • unam • diciembre 2018 • 57-72

Table 2 -
Equivalent ductility ( ) computed using capacity curves ( 55 ), equation 2 ( 67.-) and equation 6 ( 67.9 ). are nearly identical.In this case, differences among capacity curves for fixed and flexible bases computed with structural deformation (u) are smaller than those for steel frames.This is expected, since P-Δ values are expected to be less important on RC frames.On the other hand, these systems develop greater overstrength factors than steel frames, around 1.25.19Asmentionedabove, overstrength influences the values of .valuescomputed using capacity curves ( 55 ), equation 2 ( 67.-) and equation 6 ( 67.9 ) are shown in table 2.20 research professor at the Structural Engineering Department of the Metropolitan Autonomous University's Azcapotzalco campus, where he has taught undergraduate and graduate classes since 2011.He has been a member of the National Research System at the candidate level from 2013 to 2017 and a professor with a "sought-after" rating in the Teaching Development Program (prodep) since 2013.He served as the coordinator for the Mexican College of Civil Engineers' operations center during the evaluation of the damage caused by the September 19, 2007 earthquake in Mexico City.He is currently a member of the Mexican Geotechnical Engineering Society's Soil-Structure Interaction Committee, secretary of the Board of Directors of the Mexican Structural Engineering Society, member of the Competitive Intelligence Committee of the Alliance for the Promotion of Infrastructure Research for the Development of Mexico (Alianza FiiDEM), president of the Continuing Education Committee of the Mexican Structural Engineering Society and coordinator of the Bachelor's in Civil Engineering program at the uam-Azcapotzalco.