Vielma-Perez 2017a - Revision history

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← Older revision		Revision as of 11:39, 20 July 2018
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	\| style="vertical-align: top;"\|R. I. Herrera, J. C. Vielma, R. Ugel, A. Alfaro, A. H. Barbat y L. Pujades, «Seismic response and torsional eﬀects of RC structure with irregular plant and variations in diaphragms, designed with Venezuelan codes,» de ''in Earthquake resistant Engineering Structures'', B. C. y S. Hernandez, Edits., Southampton, WIT Press, 2013, pp. 85-96.		\| style="vertical-align: top;"\|R. I. Herrera, J. C. Vielma, R. Ugel, A. Alfaro, A. H. Barbat y L. Pujades, «Seismic response and torsional eﬀects of RC structure with irregular plant and variations in diaphragms, designed with Venezuelan codes,» de ''in Earthquake resistant Engineering Structures'', B. C. y S. Hernandez, Edits., Southampton, WIT Press, 2013, pp. 85-96.
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 ==References==
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 |  style="vertical-align: top;"|[1]
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 |  style="vertical-align: top;"|R. I. Herrera, J. C. Vielma, R. Ugel, A. Alfaro, A. H. Barbat y L. Pujades, «Seismic response and torsional eﬀects of RC structure with irregular plant and variations in diaphragms, designed with Venezuelan codes,» de ''in Earthquake resistant Engineering Structures'', B. C. y S. Hernandez, Edits., Southampton, WIT Press, 2013, pp. 85-96.
 |}

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 ==1. Introduction==
-Displacement ductility is an important characteristic that lead to determine whether seismic response of structures meet the early goals for which they were designed. Displacement ductility is also important because some of the most relevant seismic codes worldwide prescribe modal-spectral analysis for which the response reduction factors are estimated based on displacement ductility [1,2]. This procedure is also present in seismic codes in Latin American countries [3,4,5]. The components of response reduction factors are defined in [6] allowing to determine the response reduction factors following Eq. (1).
+Displacement ductility is an important characteristic that lead to determine whether seismic response of structures meet the early goals for which they were designed. Displacement ductility is also important because some of the most relevant seismic codes worldwide prescribe modal-spectral analysis for which the response reduction factors are estimated based on displacement ductility [1,2]. This procedure is also present in seismic codes in Latin American countries [3,4,5]. The components of response reduction factors are defined in [6] allowing to determine the response reduction factors following Eq.(1).
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 The concept used to determine displacement ductility is too simple: to stablish any relationship between the elastic and plastic behavior through specific displacements. Since Freeman et al. [1975], scientific and engineering community have accepted the pseudo-static analysis procedure with lateral forces, also called Pushover Analysis.
-The problem with the definition of displacement ductility arise when its components must be defined. First, there are some ways to determine the yield displacement ( <math display="inline">{\Delta }_{y}</math>) of the whole structure, see Fig. 1 (adapted from [10]). Fig.1a show the definition of yield displacement at the point of occurrence of the first yield in any of the reinforcement bars of the structure. Fig.1b show the criteria based on the tangent stiﬀness, defined by the elastic stiﬀness. In Fig.1c, the yield displacement is obtained from the secant stiﬀness, defined by equalizing the dissipated energy of the capacity curve and the idealized bi-linear shape. Finally, Fig.1d show the yield displacement defined by [11]. Note that according to this procedure, the secant stiﬀness corresponds to a line from the origin to a point on the capacity curve with 75% of the maximum base shear capacity. Although the convenience of whether use tangent or secant stiﬀness has not be concluded yet [12], the preference for the latter has been increasing, because it was included in the seismic assessment procedure N2 [13], adopted in Eurocode 8 [2].
+The problem with the definition of displacement ductility arise when its components must be defined. First, there are some ways to determine the yield displacement (<math display="inline">{\Delta }_{y}</math>) of the whole structure (Fig. 1, adapted from [10]). Fig.1a show the definition of yield displacement at the point of occurrence of the first yield in any of the reinforcement bars of the structure. Fig. 1b show the criteria based on the tangent stiﬀness, defined by the elastic stiﬀness. In Fig. 1c, the yield displacement is obtained from the secant stiﬀness, defined by equalizing the dissipated energy of the capacity curve and the idealized bi-linear shape. Finally, Fig. 1d show the yield displacement defined by [11]. Note that according to this procedure, the secant stiﬀness corresponds to a line from the origin to a point on the capacity curve with 75% of the maximum base shear capacity. Although the convenience of whether use tangent or secant stiﬀness has not be concluded yet [12], the preference for the latter has been increasing, because it was included in the seismic assessment procedure N2 [13], adopted in Eurocode 8 [2].
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-Determination of the ultimate displacement ( <math display="inline">{\Delta }_{u}</math>) is also controversial. One of the most popular criterion is to assume that the ultimate displacement corresponds to the point where the structure reaches the maximum base shear (Fig. 2a). Another used criterion to determine the ultimate displacement assumes that the ultimate displacement correspond to the point in which compressive strain in concrete or the fracture or buckling in transversal reinforcement in any structural member is achieved (Fig.2b and 2c). The criterion showed in Fig2d is also very popular among researchers. This criterion leads to obtain the ultimate displacement from the point in which the maximum base shear capacity drops a fixed value (usually 10 or 20%). Fig.2e summarizes ductility calculations based on the above-mentioned criteria.
+Determination of the ultimate displacement (<math display="inline">{\Delta }_{u}</math>) is also controversial. One of the most popular criterion is to assume that the ultimate displacement corresponds to the point where the structure reaches the maximum base shear (Fig. 2a). Another used criterion to determine the ultimate displacement assumes that the ultimate displacement correspond to the point in which compressive strain in concrete or the fracture or buckling in transversal reinforcement in any structural member is achieved (Figs. 2b and 2c). The criterion showed in Fig2d is also very popular among researchers. This criterion leads to obtain the ultimate displacement from the point in which the maximum base shear capacity drops a fixed value (usually 10 or 20%). Fig. 2e summarizes ductility calculations based on the above-mentioned criteria.
-Once both displacements are calculated, displacement ductility is determined according to Eq. 2.
+Once both displacements are calculated, displacement ductility is determined according to Eq.(2).
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 • Second step. Determine the ultimate displacement. To this end, it is necessary to find the base shear for which ultimate rotation capacity in the extremes of the beams and the inferior extremes of the columns of any level are achieved.
-• Third step. Obtain the yield displacement equalizing the areas under the capacity curve and the generated tri-linear idealized shape. This step is performed by means of a couple of curves obtained from the integration of the capacity curve and the tri-linear idealized shape. The point where these two curves intersect, is compared with the point for the ultimate rotation displacement, if the diﬀerence between the abscissas of both points was lower than 1%, the chose value of the abscissa for the yield point is accepted, and on the contrary, the yield point must move horizontally in order to adjust the dissipated energy. This way leads to obtain a balance of the dissipated energy of the capacity curve and the idealized bi-linear shape, see Fig. 3.
+• Third step. Obtain the yield displacement equalizing the areas under the capacity curve and the generated tri-linear idealized shape. This step is performed by means of a couple of curves obtained from the integration of the capacity curve and the tri-linear idealized shape. The point where these two curves intersect, is compared with the point for the ultimate rotation displacement, if the diﬀerence between the abscissas of both points was lower than 1%, the chose value of the abscissa for the yield point is accepted, and on the contrary, the yield point must move horizontally in order to adjust the dissipated energy. This way leads to obtain a balance of the dissipated energy of the capacity curve and the idealized bi-linear shape (Fig. 3).
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-The procedure is carried out by assigning an ultimate chord rotation to each structural element, according to the results of Eq. (2). In this way, for every increment of applied lateral load, computed chord rotations were compared versus the assigned values of ultimate rotation capacity, indicating whether an element has achieved its ultimate rotation [15], see Fig. 4. When all elements located at the ends of the beams belonging to one story and the lower ends of the columns of the immediately below story, exceeded their ultimate chord rotations, then, it is assumed that the structure is not be able to sustain additional lateral forces, so it has been reached the collapse threshold displacement. Then, the ultimate base shear corresponds to the sum of the reactions in supports opposing to lateral forces in the indicated load step.
+The procedure is carried out by assigning an ultimate chord rotation to each structural element, according to the results of Eq.(2). In this way, for every increment of applied lateral load, computed chord rotations were compared versus the assigned values of ultimate rotation capacity, indicating whether an element has achieved its ultimate rotation [15] (Fig. 4). When all elements located at the ends of the beams belonging to one story and the lower ends of the columns of the immediately below story, exceeded their ultimate chord rotations, then, it is assumed that the structure is not be able to sustain additional lateral forces, so it has been reached the collapse threshold displacement. Then, the ultimate base shear corresponds to the sum of the reactions in supports opposing to lateral forces in the indicated load step.
 ===2.2. Rotation ductility demand===
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-In this equation φy is the yield curvature, obtained from Eq. (8); <math display="inline">{d}_{bL}</math> is the diameter of longitudinal reinforcement bars, <math display="inline">{f}_{yL}</math> is the yield strength of the steel, and <math display="inline">{f}_{c}^{'}</math> is the characteristic strength of the concrete. Yield curvature can be approximated to:
+In this equation φy is the yield curvature, obtained from Eq.(8); <math display="inline">{d}_{bL}</math> is the diameter of longitudinal reinforcement bars, <math display="inline">{f}_{yL}</math> is the yield strength of the steel, and <math display="inline">{f}_{c}^{'}</math> is the characteristic strength of the concrete. Yield curvature can be approximated to:
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-The analysis, design and detailing of the buildings was performed according to the current Venezuelan seismic code [2001] for residential use. However, interstory drift check was performed using an alternative energy-based procedure [18], [19], thereby producing stiﬀer structures than the obtained according to the standard code procedure.
+The analysis, design and detailing of the buildings was performed according to the current Venezuelan seismic code [2001] for residential use. However, interstory drift check was performed using an alternative energy-based procedure [18,19], thereby producing stiﬀer structures than the obtained according to the standard code procedure.
 ===3.1. Non-linear analysis===
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 ==4. Results==
-The capacity curves obtained for each building were processed according the Park’s procedure and the new procedure reported herein, in order to find the idealized shapes in both analysis directions and to determine the values of displacement ductility ( <math display="inline">\mathrm{\mu}</math>) and structural overstrength ( <math display="inline">{\Omega }_{d}</math>). The values of both coeﬃcients are summarized in Tables
+The capacity curves obtained for each building were processed according the Park’s procedure and the new procedure reported herein, in order to find the idealized shapes in both analysis directions and to determine the values of displacement ductility ( <math display="inline">\mathrm{\mu}</math>) and structural overstrength (<math display="inline">{\Omega }_{d}</math>). The values of both coeﬃcients are summarized in Tables
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-In general, calculated values of displacement ductility using the proposed procedure, show small variations, despite on the plan irregularity of the buildings. For analysis performed in the irregular direction of the cases (x direction) resulting values are less than 6, with the exception of cases 4 and 7, in which the values are near to 7, see Fig. 7. It can be observed the uniformity of the displacement ductility values, which are slightly less than 6.
+In general, calculated values of displacement ductility using the proposed procedure, show small variations, despite on the plan irregularity of the buildings. For analysis performed in the irregular direction of the cases (x direction) resulting values are less than 6, with the exception of cases 4 and 7, in which the values are near to 7 (Fig. 7). It can be observed the uniformity of the displacement ductility values, which are slightly less than 6.
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-Values obtained using the proposed procedure are consistent with those expected for the used structural typology combined with the seismic hazard level. Further- more, the use of the new procedure may serve for the calculation of the inherent response reduction factor of the cases studied according to eq. (1). The inherent response reduction factor, also called response reduction factor supply [10], may serve as a reference value in order to evaluate the seismic design obtained through the response reduction factor prescribed by the seismic code (also called response reduction factor demand [10].
+Values obtained using the proposed procedure are consistent with those expected for the used structural typology combined with the seismic hazard level. Further- more, the use of the new procedure may serve for the calculation of the inherent response reduction factor of the cases studied according to Eq.(1). The inherent response reduction factor, also called response reduction factor supply [10], may serve as a reference value in order to evaluate the seismic design obtained through the response reduction factor prescribed by the seismic code (also called response reduction factor demand [10].
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-Both procedures used in this paper have produced values of the response reduction factor ( <math display="inline">{R}_{Supply}</math>) which are greater than the response reduction factor value prescribed by code ( <math display="inline">{R}_{Demand}</math> ), see Fig. 8. In spite of this, values calculated from the proposed procedure are consistent with the values reported in recent works, obtained using incremental dynamic analysis (IDA) in framed structures, designed in high seismic-prone regions [22,23,24,25].
+Both procedures used in this paper have produced values of the response reduction factor (<math display="inline">{R}_{Supply}</math>) which are greater than the response reduction factor value prescribed by code (<math display="inline">{R}_{Demand}</math>) (Fig. 8). In spite of this, values calculated from the proposed procedure are consistent with the values reported in recent works, obtained using incremental dynamic analysis (IDA) in framed structures, designed in high seismic-prone regions [22,23,24,25].
-Finally, Figs. 9 and 10 show a graphical representation of the values of rotation ductility demands, computed following eq. (6) for the ultimate displacement, obtained according to the proposed procedure. Distribution of rotation ductility demands is an important issue of the seismic behavior of the structure, because it synthetizes the failure mode under lateral loads. Note that the maximum values of rotation ductility demands correspond to higher ends and lower ends of columns of stories 2 and 1, respectively and both ends of fist floor beams, indicating that such kind of structures have an intermediate type of failure, according to [17].
+Finally, Figs. 9 and 10 show a graphical representation of the values of rotation ductility demands, computed following Eq.(6) for the ultimate displacement, obtained according to the proposed procedure. Distribution of rotation ductility demands is an important issue of the seismic behavior of the structure, because it synthetizes the failure mode under lateral loads. Note that the maximum values of rotation ductility demands correspond to higher ends and lower ends of columns of stories 2 and 1, respectively and both ends of fist floor beams, indicating that such kind of structures have an intermediate type of failure, according to [17].

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 [[Image:draft_Vielma-Perez_286876972-image11.png|120px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;text-align:center;"| '''Figure 5'''. Plan configurations of the archetypes studied.
+| colspan="4" style="padding:10px;text-align:center;"| '''Figure 5'''. Plan configurations of the archetypes studied.
 |}

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 [[Image:draft_Vielma-Perez_286876972-image11.png|120px]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 5'''. Plan configurations of the archetypes studied.
+| colspan="1" style="padding:10px;text-align:center;"| '''Figure 5'''. Plan configurations of the archetypes studied.
 |}
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 | colspan="1" style="padding:10px;"| '''Figure 9'''.  Rotation ductility demands in external frame of the case 1.
 |}
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 Resulting values of the displacement ductility calculated according the proposed procedure are, in general, nearly uniform in the regular direction of case studied, and have values that lightly vary according the irregular direction of analysis. These values are greater than the ductility values the designer expect the structures develop during a severe earthquake, and are also consistent with values of response reduction factors computed from most refined, and consequently time-consuming analysis, applied in recent works.

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	Displacement ductility is a parameter that characterizes the seismic response of structures. Moreover, displacement ductility can be used in order to determine whether a structural design, performed according to a specific seismic code or not, may achieve the main goal of the seismic design: to develop energy dissipation in a stable manner. Determination of displacement ductility is not an easy task, because the structural response usually does not show a clear location of the points that define yield and ultimate displacements. In this paper, some of the main procedures for ductility displacement are revised and compared, and then improvements are performed to such procedures in order to compute the displacement ductility. A new procedure is then introduced, leading to determine the ultimate displacement using the seismic collapse threshold and the yield displacement, achieving the balance of dissipated energy. The procedure has been used to calculate displacement ductility of reinforced concrete framed buildings.		Displacement ductility is a parameter that characterizes the seismic response of structures. Moreover, displacement ductility can be used in order to determine whether a structural design, performed according to a specific seismic code or not, may achieve the main goal of the seismic design: to develop energy dissipation in a stable manner. Determination of displacement ductility is not an easy task, because the structural response usually does not show a clear location of the points that define yield and ultimate displacements. In this paper, some of the main procedures for ductility displacement are revised and compared, and then improvements are performed to such procedures in order to compute the displacement ductility. A new procedure is then introduced, leading to determine the ultimate displacement using the seismic collapse threshold and the yield displacement, achieving the balance of dissipated energy. The procedure has been used to calculate displacement ductility of reinforced concrete framed buildings.
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	=1. Introduction=		=1. Introduction=

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−	~~<sup>1</sup>School of Civil Engineering, Pontificia Universidad Católica de Valparaíso~~
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−	~~<sup>2</sup>Departament of Structural Engineering, UCLA-CIMNE joint-lab, Universidad Centroccidental Lisandro Alvarado~~
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	==Abstract==		==Abstract==

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−	<sup>2</sup>Departament of Structural Engineering, UCLA-CIMNE ~~Classroom~~, Universidad Centroccidental Lisandro Alvarado	+	<sup>2</sup>Departament of Structural Engineering, UCLA-CIMNE joint-lab, Universidad Centroccidental Lisandro Alvarado