Ductility plays a central role in seismic analysis and design of steel buildings. A numerical investigation regarding the evaluation of energy dissipation, ductility, and ductility reduction factors for local, story, and global structural levels is conducted. Some steel buildings and strong motions, which were part of the SAC Steel Project, are used. Bending local ductility capacity (
Even though building structures undergo significant nonlinear deformations when subjected to strong earthquakes, simple elastic procedures are still used to determine the seismic demands (International Building Code (IBC) [
The reduction factor receives the name of the response modification factor (
Thus, the ductility parameter and the ductility reduction factors play a central role in the seismic design of steel buildings; for that reason, it has been an important research topic during the last recent decades. However, there are many aspects that need additional attention; some of them are addressed in this research. As it is further elaborated below, the central objective of this paper is to evaluate the ductility parameter, the ductility reduction factor (
Investigations regarding nonlinear analysis of buildings under the action of earthquakes, following different objectives, have been conducted by many researchers during the last decades. In this regard, since this dissipation of energy allows for a reduction of the elastic seismic forces, the quantification of the ductility demand, the ductility reduction factor, and the force (or seismic) reduction factor is of particular interest. There have been many studies considering concrete or steel buildings modeled as SDOF or simple systems. Introduced first in ATC [
There are also several studies regarding the evaluation
The abovementioned studies represent a significant contribution regarding the evaluation of ductility or ductility reduction factors; however, in most of them, SDOF systems or a limited level of inelastic deformation were considered. Therefore, they did not explicitly consider the energy dissipation associated with nonlinear behavior of the structural elements existing in actual systems. It has been shown [
The objectives of this research are as follows: Calculate local ductility demands (in terms of curvatures) and capacities for individual structural elements (beams and columns) as well as story and global ductility demands and capacities. The ratio of local to global ductility is also estimated. Calculate the ductility reduction factors as well as the ratio of the ductility reduction factor to ductility, for the three structural levels under consideration, and compare them with those specified in the codes. Estimate the energy demands for local, story, and global levels as well as the ratio of the ductility reduction factor to dissipated energy.
As part of the SAC Steel Project [
Isometric view of the 3-level building.
Isometric view of the 10-level building.
Beam and columns sections for Models 1 and 2.
Model | Story | Columns | Girder | |
---|---|---|---|---|
Exterior | Interior | |||
1 | 1\2 | W14X257 | W14X311 | W33X118 |
2\3 | W14X257 | W14X312 | W30X116 | |
3\ROOF | W14X257 | W14X313 | W24X68 | |
|
||||
2 | −1/1 | W14X370 | W14X500 | W36X160 |
1/2 | W14X370 | W14X500 | W36X160 | |
2/3 | W14X370 | W14X500,W14X455 | W36X160 | |
3/4 | W14X370 | W14X455 | W36X135 | |
4/5 | W14X370,W14X283 | W14X455,W14X370 | W36X135 | |
5/6 | W14X283 | W14X370 | W36X135 | |
6/7 | W14X283,W14X257 | W14X370,W14X283 | W36X135 | |
7/8 | W14X257 | W14X283 | W30X99 | |
8/9 | W14X257,W14X233 | W14X283,W14X257 | W27X84 | |
9/ROOF | W14X233 | W14X257 | W24X68 |
It is worth to mention that some results of the SAC Steel Project are given in a research report [
The frames are modeled as complex 2D MDOF systems, having three degrees of freedom per node. The Newmark constant average acceleration method, lumped mass matrix, Rayleigh damping, and large displacement effects are considered within the Ruaumoko Computer Program environment while performing the required nonlinear seismic analysis; the time increment in the analysis was 0.01 s. The panel zone was considered to be rigid. Typical input data as ground accelerations, boundary conditions, node coordinates, and elastic and inelastic section properties are given or read within the computer program. No strength degradation member, bilinear behavior with 5% of the initial stiffness in the second zone, and concentrated plasticity were assumed. The interaction axial load-bending moment is given by the yield interaction surface proposed by Chen and Atsuta [
When a structure is subjected to the action of two different strong motions, even when they are normalized with respect to the same peak ground acceleration or with respect to any other parameter, it is expected to respond differently, reflecting the influence of the frequency content of the motions and of the structural vibration modes. Thus, to get meaningful results, the models under consideration are excited by twenty strong motions in time domain whose characteristics are given in Table
Earthquake records, N-S component.
No | Place | Date | Station |
|
ED (km) |
|
PGA (cm/sec2) |
---|---|---|---|---|---|---|---|
1 | Landers, California | 28/06/1992 | Fun Valley, Reservoir 361 | 0.11 | 31 | 7.3 | 213 |
2 | Mammoth Lakes, California | 27/05/1980 | Convict Creek | 0.16 | 12 | 6.3 | 316 |
3 | Victoria | 09/06/1980 | Cerro Prieto | 0.16 | 37 | 6.1 | 613 |
4 | Parkfield, California | 28/09/2004 | Parkfield; Joaquin Canyon | 0.17 | 15 | 6.0 | 609 |
5 | Puget Sound, Washington | 29/04/1965 | Olympia Hwy Test Lab | 0.17 | 89 | 6.5 | 216 |
6 | Long Beach, California | 10/03/1933 | Utilities Bldg, Long Beach | 0.20 | 29 | 6.3 | 219 |
7 | Sierra El Mayor, Mexico | 04/04/2010 | El centro, California | 0.21 | 77 | 7.2 | 544 |
8 | Petrolia/Cape Mendocino, California | 25/04/1992 | Centerville Beach, Naval Facility | 0.21 | 22 | 7.2 | 471 |
9 | Morgan Hill | 24/04/1984 | Gilroy Array Sta #4 | 0.22 | 38 | 6.2 | 395 |
10 | Western Washington | 13/04/1949 | Olympia Hwy Test Lab | 0.22 | 39 | 7.1 | 295 |
11 | San Fernando | 09/02/1971 | Castaic-Old Ridge Route | 0.23 | 24 | 6.6 | 328 |
12 | Mammoth Lakes, California | 25/05/1980 | Long Valley Dam | 0.24 | 13 | 6.5 | 418 |
13 | El Centro | 18/05/1940 | El Centro-ImpVall Irr Dist | 0.27 | 12 | 7.0 | 350 |
14 | Loma Prieta, California | 18/10/1989 | Palo Alto | 0.29 | 47 | 6.9 | 378 |
15 | Santa Barbara, California | 13/08/1978 | UCSB Goleta FF | 0.36 | 14 | 5.1 | 361 |
16 | Coalinga, California | 02/05/1983 | Parkfield Fault Zone 14 | 0.39 | 38 | 6.2 | 269 |
17 | Imperial Valley, California | 15/10/1979 | Chihuahua | 0.40 | 19 | 6.5 | 262 |
18 | Northridge, California | 17/01/1994 | Canoga Park, Santa Susana | 0.60 | 16 | 6.7 | 602 |
19 | Offshore Northern, California | 10/01/2010 | Ferndale, California | 0.61 | 43 | 6.5 | 431 |
20 | Joshua Tree, California | 23/04/1992 | Indio, Jackson Road | 0.62 | 26 | 6.1 | 400 |
The abovementioned maximum levels of seismic intensities produce a deformation state which is very close to that of a collapse mechanism, where interstory drifts of about 5% were developed for some strong motions; therefore, they are associated with the structural and ductility capacity. This is concordant with the results of some experimental studies where it has been shown that moment-resisting steel frames may undergo interstory displacements of up to 5% (and even larger) and still be able to vibrate in a stable manner [
It is important to note that UBC-1994 in Sections 1629.1 and 1629.2 states: “Dynamic analysis procedures, when used, shall conform to the criteria established in this section. The analysis shall be based on an appropriate ground motion representation … .The ground motions representation shall, as a minimum, be one having a 10% probability of being exceeded in 50% … .” The expression “as a minimum” implies that larger intensities (and larger drifts) of the strong motions can be used. In addition, according to the particular objectives stated in our paper, we need a deformation state close to that of a collapse mechanism which is developed for drifts of about 5%, for some strong motions.
In addition to the seismic loading, the following gravity loads [
The discussions made in Sections
Theoretically, ductility capacity should be reached when a collapse mechanism develops in the structure. To obtain this, it needs to be guaranteed that plastic moments are reached at positions of maximum moments before failure due to instability, namely, local buckling or lateral torsional buckling, in a member or in a connection occurs. For the case of steel buildings, local ductility (
Thus, as soon as any of the joints of a given member yields for the first time, the corresponding curvature is identified as
Story ductility is defined in terms of lateral drifts. The ductility of a story (
Regarding the Δy parameter in Equation (
As for the ductility parameter, the ductility reduction factors are estimated for different structural levels. This parameter, in general, can be expressed as
The story ductility reduction factor (
The dissipated energy is also calculated for local, story, and global levels. The normalized dissipated energy at a given joint (
For all the beams of a story, the average normalized dissipated energy per joint ((
The normalized dissipated energy by bending at a column
For all the columns of a story, it is obtained as
Finally, the normalized dissipated energy for the whole frame (
All the parameters in Equation (
The bending local ductility parameter (Equation (
For a given story, the
Mean values of
Parameter | Type of member | ST | NS direction | EW direction | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
|
| |||||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | |||
|
Beams | 1 | 2.61 | 4.27 | 5.99 | 8.23 | 10.22 | 2.19 | 3.56 | 4.56 | 6.33 | 7.81 |
2 | 2.48 | 4.60 | 7.17 | 9.57 | 11.86 | 2.26 | 4.15 | 5.87 | 7.69 | 9.32 | ||
3 | 2.01 | 4.31 | 6.71 | 8.94 | 11.20 | 1.99 | 3.73 | 5.59 | 7.67 | 9.30 | ||
Columns | 1 | 1.06 | 1.32 | 1.95 | 2.65 | 3.70 | 1.01 | 1.25 | 1.72 | 2.22 | 2.87 | |
2 | 1.00 | 1.00 | 1.09 | 1.25 | 1.41 | 1.00 | 1.01 | 1.03 | 1.21 | 1.29 | ||
3 | 1.00 | 1.00 | 1.06 | 1.12 | 1.23 | 1.00 | 1.00 | 1.02 | 1.13 | 1.26 | ||
|
||||||||||||
|
Beams | 1 | 1.68 | 2.29 | 2.89 | 3.45 | 4.06 | 1.54 | 2.12 | 2.68 | 3.22 | 3.91 |
2 | 1.66 | 2.25 | 2.79 | 3.28 | 3.86 | 1.55 | 2.11 | 2.65 | 3.16 | 3.81 | ||
3 | 1.58 | 2.11 | 2.60 | 3.06 | 3.60 | 1.50 | 2.01 | 2.50 | 2.97 | 3.54 | ||
Columns | 1 | 1.52 | 1.87 | 2.25 | 2.60 | 2.98 | 1.35 | 1.66 | 2.00 | 2.35 | 2.79 | |
2 | 1.50 | 1.85 | 2.19 | 2.51 | 2.86 | 1.37 | 1.71 | 2.01 | 2.32 | 2.71 | ||
3 | 1.55 | 1.95 | 2.28 | 2.59 | 2.95 | 1.43 | 1.81 | 2.17 | 2.51 | 2.92 | ||
|
||||||||||||
|
Beams | 1 | 0.64 | 0.54 | 0.48 | 0.42 | 0.40 | 0.70 | 0.60 | 0.59 | 0.51 | 0.50 |
2 | 0.67 | 0.49 | 0.39 | 0.34 | 0.33 | 0.69 | 0.51 | 0.45 | 0.41 | 0.41 | ||
3 | 0.79 | 0.49 | 0.39 | 0.34 | 0.32 | 0.75 | 0.54 | 0.45 | 0.39 | 0.38 | ||
Columns | 1 | 1.43 | 1.42 | 1.15 | 0.98 | 0.81 | 1.34 | 1.33 | 1.16 | 1.06 | 0.97 | |
2 | 1.50 | 1.85 | 2.01 | 2.01 | 2.03 | 1.37 | 1.69 | 1.95 | 1.92 | 2.10 | ||
3 | 1.55 | 1.95 | 2.15 | 2.31 | 2.40 | 1.43 | 1.81 | 2.13 | 2.22 | 2.32 |
10-level building. Mean values of
The results for the 10-level building resemble those of the 3-level building in the sense that the mean values of
It is worth to mention that moderate yielding occurred for seismic intensities of 0.4 g and 0.2 g for the 3- and 10-level structural models, respectively; the corresponding seismic intensities for significant deformation, as stated in Section
Similar to the
Statistics for
Parameter | ST | NS direction | EW direction | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
| ||||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | ||
|
1 | 1.61 | 1.99 | 2.81 | 3.44 | 3.95 | 1.48 | 1.90 | 2.80 | 2.89 | 4.28 |
2 | 1.70 | 2.20 | 2.86 | 3.31 | 3.99 | 1.65 | 2.15 | 2.58 | 3.09 | 3.54 | |
3 | 1.53 | 1.93 | 2.69 | 2.99 | 3.90 | 1.52 | 2.10 | 2.72 | 3.19 | 3.84 | |
|
|||||||||||
R |
1 | 1.46 | 1.90 | 2.37 | 2.76 | 3.10 | 1.33 | 1.73 | 2.11 | 2.51 | 2.89 |
2 | 1.44 | 1.86 | 2.30 | 2.68 | 3.05 | 1.36 | 1.84 | 2.25 | 2.68 | 3.06 | |
3 | 1.45 | 1.84 | 2.17 | 2.47 | 2.76 | 1.36 | 1.75 | 2.08 | 2.36 | 2.67 | |
|
|||||||||||
|
1 | 0.91 | 0.95 | 0.84 | 0.80 | 0.78 | 0.90 | 0.91 | 0.75 | 0.87 | 0.68 |
2 | 0.85 | 0.85 | 0.80 | 0.81 | 0.76 | 0.82 | 0.86 | 0.87 | 0.87 | 0.86 | |
3 | 0.95 | 0.95 | 0.81 | 0.83 | 0.71 | 0.89 | 0.83 | 0.76 | 0.74 | 0.70 |
10-level building. Mean values of
The global ductility values (
Mean of
Parameter | 3-level | 10-level | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NS direction | EW direction | NS direction | EW direction | |||||||||||||
|
|
|
| |||||||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.2 | 0.4 | 0.6 | 0.2 | 0.4 | 0.6 | |
|
1.61 | 2.04 | 2.79 | 3.25 | 3.95 | 1.55 | 2.05 | 2.70 | 3.05 | 3.89 | 1.86 | 2.92 | 4.11 | 1.66 | 2.67 | 3.60 |
Q | 0.68 | 0.46 | 0.42 | 0.36 | 0.36 | 0.72 | 0.54 | 0.51 | 0.42 | 0.44 | 0.67 | 0.43 | 0.37 | 0.72 | 0.46 | 0.35 |
|
1.35 | 1.77 | 2.15 | 2.52 | 2.87 | 1.45 | 1.87 | 2.28 | 2.64 | 2.97 | 1.47 | 2.28 | 2.97 | 1.49 | 2.34 | 3.09 |
|
0.84 | 0.87 | 0.77 | 0.78 | 0.73 | 0.94 | 0.91 | 0.84 | 0.87 | 0.76 | 0.79 | 0.78 | 0.72 | 0.90 | 0.88 | 0.86 |
As stated above, the estimation of ductility capacity is commonly based on experimental studies of individual members. For this reason, it is suggested [
The local, story, and global ductility reduction factors, as given by Equations (
It can be seen that, for beams of the 3-level building, the mean values of
The story ductility reduction factor (
The mean values of global ductility reduction factors (R
The ratios of the ductility reduction factor to ductility, for local (
The values of
The results for
10-level building,
As for the ductility demand and ductility reduction factor parameters, the dissipated energy is calculated for three levels. The mean values of (
Mean values of (
Type of member | ST | NS direction | EW direction | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
|
| ||||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | ||
Beams ( |
1 | 0.053 | 0.147 | 0.256 | 0.379 | 0.512 | 0.045 | 0.114 | 0.188 | 0.274 | 0.376 |
2 | 0.022 | 0.072 | 0.140 | 0.216 | 0.298 | 0.018 | 0.055 | 0.109 | 0.170 | 0.242 | |
3 | 0.008 | 0.033 | 0.069 | 0.113 | 0.161 | 0.007 | 0.025 | 0.057 | 0.096 | 0.141 | |
|
|||||||||||
Columns |
1 | 0.003 | 0.010 | 0.025 | 0.046 | 0.076 | 0.003 | 0.007 | 0.017 | 0.031 | 0.050 |
2 | 0.001 | 0.003 | 0.004 | 0.007 | 0.013 | 0.001 | 0.003 | 0.004 | 0.006 | 0.010 | |
3 | 0.001 | 0.001 | 0.002 | 0.004 | 0.007 | 0.001 | 0.001 | 0.002 | 0.004 | 0.007 |
Mean values of
ST | NS direction | EW direction | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
|
| |||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | |
1 | 0.020 | 0.056 | 0.102 | 0.157 | 0.221 | 0.017 | 0.043 | 0.074 | 0.112 | 0.158 |
2 | 0.008 | 0.026 | 0.050 | 0.077 | 0.108 | 0.007 | 0.020 | 0.039 | 0.061 | 0.087 |
3 | 0.003 | 0.012 | 0.024 | 0.040 | 0.058 | 0.003 | 0.009 | 0.020 | 0.035 | 0.052 |
Mean values of
Parameter | Member | 3-level | 10-level | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NS direction | EW direction | NS direction | EW direction | ||||||||||||||
|
|
|
| ||||||||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.2 | 0.4 | 0.6 | 0.2 | 0.4 | 0.6 | ||
|
|
0.011 | 0.031 | 0.059 | 0.091 | 0.129 | 0.009 | 0.024 | 0.044 | 0.069 | 0.099 | 0.026 | 0.111 | 0.218 | 0.017 | 0.083 | 0.171 |
|
0.028 | 0.084 | 0.155 | 0.236 | 0.324 | 0.023 | 0.065 | 0.118 | 0.18 | 0.253 | 0.076 | 0.314 | 0.383 | 0.047 | 0.241 | 0.475 | |
|
48.2 | 21.1 | 13.9 | 10.7 | 8.9 | 63.0 | 28.8 | 19.3 | 14.7 | 11.7 | 19.3 | 7.3 | 7.8 | 31.7 | 9.7 | 6.5 |
Mean values of local normalized energy demands, 10-level building: (a) beams NS; (b) beams EW; (c) columns NS; (d) columns EW.
Mean values of story normalized energy demands, 10-level building: (a) NS direction; (b) EW direction.
Results for the 3-level building indicate that (
It is clear that the reduction of the response from the elastic to the inelastic case due to yielding of the material, which is quantified by the ductility reduction factor, is produced by the dissipated energy. In this regard, it may be of interest to study the ratio of the ductility reduction factor to dissipated energy. In this section of the paper, the ratio of the story ductility reduction factor to story normalized dissipated energy (
The results for
ST | NS direction | EW direction | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
|
| |||||||||
0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | |
1 | 27.5 | 12.9 | 9.3 | 7.3 | 6.1 | 29.6 | 15.2 | 11.2 | 9.2 | 7.7 |
2 | 65.5 | 25.8 | 16.4 | 12.4 | 10.2 | 75.6 | 33.5 | 20.6 | 15.8 | 12.6 |
3 | 181.3 | 55.8 | 31.4 | 21.9 | 17.1 | 194.3 | 70.0 | 36.5 | 24.6 | 18.9 |
It is observed from Table
Results in Table
The ductility parameter plays a central role in seismic analysis and design of steel buildings. However, it is still used in an indirect way. A numerical investigation regarding the evaluation of local ( Bending local ductility capacity ( The mean values of The maximum values of The
Data used to support the findings of this study are included within the article, and data supporting this study are from previously reported studies, which have been cited.
Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsors.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research presented in this paper was financially supported by La Universidad Autónoma de Sinaloa (UAS) under grant PROFAPI-2015/235.