 Article
 Open Access
Application of the Direct Displacement Based Design Methodology for Different Types of RC Structural Systems
 Saleh Malekpour^{1} and
 Farhad Dashti^{2}Email author
https://doi.org/10.1007/s4006901300432
© The Author(s) 2013
 Received: 6 August 2012
 Accepted: 4 April 2013
 Published: 22 May 2013
Abstract
This study investigates the direct displacement based design (DDBD) approach for different types of reinforced concrete structural systems including single momentresisting, dual wallframe and dual steelbraced systems. In this methodology, the displacement profile is calculated and the equivalent single degree of freedom system is then modeled considering the damping characteristics of each member. Having calculated the effective period and secant stiffness of the structure, the base shear is obtained, based on which the design process can be carried out. For each system three frames are designed using DDBD approach. The frames are then analyzed using nonlinear timehistory analysis with 7 earthquake accelerograms and the damage index is investigated through lateral drift profile of the models. Results of the analyses and comparison of the nonlinear timehistory analysis results indicate efficiency of the DDBD approach for different reinforced concrete structural systems.
Keywords
 direct displacement based design
 RC frame systems
 timehistory analysis
 accelerograms
1 Introduction
In recent years, there has been a great tendency toward performancebased seismic design of structures. In this connection, various methods have been developed among which Capacity Spectrum Method (Freeman 1998), the N2 Method (Fajfar 2000), and Direct DisplacementBased Design can be enumerated. A relatively new performancebased seismic design procedure called the direct displacementbased design (DDBD) proposed by Priestley and Kowalsky (2000) has recently received notable acceptance among researchers. It seems that the methods could be a rational alternative to traditional erroneous forcebased seismic design of structures. The method defines the design performance level of the structure in terms of displacement limits. Therefore, displacement is the key parameter of the design method.
This paper investigates the DDBD of different types of reinforced concrete structural systems including single momentresisting, dual wallframe and dual steelbraced systems. There is a need for a design methodology that is applicable to dual system structures, because the dynamic behaviour of dual systems is considerably different from pure frame or wall or steel braced structures for which many design recommendations already exist. Such differences in dynamic behaviour are attributed principally to the interaction that takes place between different structural systems, which is not well accounted for in current design practice.
A further motivation for this study comes from the fact that combination of different structural systems results in an efficient earthquakeresisting system, and considering structural and aesthetic points of view combination of the structural systems presents considerable advantage over structures formed purely out of frames or walls or bracing systems.
The displacement based design of RC structures has been addressed increasingly in recent years, and the displacement based design of multiple degree of freedom RC structures has been the main philosophy of these approaches. Gulkan and Sozen (1974) investigated the nonlinear behavior of RC structures under dynamic loads, and presented equivalent equations for damping of single degree of freedom structures. Shibata and Sozen (1976) presented the substitute structure methodology for RC structures, and intended to devise a displacement based design method. Moehle (1992) suggested the general outline of a seismic resistant design approach based on the interstory relative displacement calculation using displacement response spectrum. Although the initial step of design using this approach is also calculation of stiffness, elastic time period and different strengths of the structure, it is quite different from the traditional methods as includes direct control of displacements instead of indirect control using ductility coefficients. The initial step of the approach recommended by Kowalsky et al. (1995) for single degree of freedom structures is determination of the maximum target displacement which can be obtained based on the ductility capacity being proportional to details of the members. Assuming an acceptable value for yield displacement, the designer converts the maximum displacement to the demand displacement ductility, and using a series of displacement response spectrums with different damping values (due to ductility values), calculates the effective period of the single degree of freedom structure in the maximum displacement. The ultimate results of the calculated yield strength based on the maximum displacement and secant stiffness is corresponding to the effective period. Calvi and Kingsley (1995), and Calvi and Pavese (1995) generalized this approach for the structures with multiple degrees of freedom. It should be noted that the ultimate results of all these approaches are the demand strength values based on which sections and dimensions of the members shall be computed.

Development of an expression for the displaced shape of framewall structures at maximum response, to enable equivalent SDOF characteristics to be established.

Development of an expression for the equivalent SDOF system ductility or equivalent viscous damping that takes into account the framewall interaction.
Sullivan et al. (2005) proposed that the design displacement profile be set as a function of the moment profile in the walls, using proportions of strength assigned at the start of the design procedure. There is experimental evidence that supports the validity of this approach as reported by Sullivan et al. (2004). Another recommendation made by Sullivan et al. (2005) was that the equivalent SDOF system viscous damping could be obtained by factoring the individual frame and wall components by the proportions of overturning they resist. The challenge in this paper is therefore to finalize the design procedure proposed by Sullivan et al. (2005) and to verify its accuracy through examination of a range of case study structures.
Belleri (2009) suggested the performance based design approach for RC precast buildings in 2009. Sullivan et al. (2009) investigated a DDBD code in 2009. Garcia et al. (2010) investigated the DDBD approach for steel frameRC wall dual systems in 2009. Pennucci et al. (2009) studied the DBD for RC precast walldamper systems.
RC buildings with steel bracings are the new structural systems addressed in rehabilitation of RC structures in recent years and researches are extensively investigating on application of steel bracing systems in such buildings. The research conducted by Higashi et al. (1981) on application of CBF and EBF bracing systems in rehabilitation of RC frames and the studies of Badox and Jirsa (1990) on nonlinear behavior modeling of bracing systems in RC frames can be cited as examples. Maheri and Sabahi (1997) suggested the direct connection of the internal bracing to the RC frame. Tasnimi and Masoomi (1999) experimentally investigated the direct use of steel bracings in RC frames. For this purpose, the manufactured frames were subjected to static gravity loads and cyclic lateral loads. The results indicated that adding bracing to an RC frame, depending on the utilized details, considerably increases the equivalent stiffness of the frame and leads to notable change in its behavior. Ghaffarzadeh and Maheri (2006a, b) showed that different directly connected internal bracing systems can be used effectively in retrofitting of the existing concrete frames as well as shear resisting elements for the construction of new RC structures. Having conducted two cyclic experiments on an RC momentresisting frame and an RC frame with steel bracings, Youssef et al. (2007) came to the conclusion that the braced frame has more ductility and can resist greater lateral load. Maheri and Ghaffarzadeh (2008) investigated the amount of the interaction force between the RC frame and the steel bracing analytically and experimentally using the experiments conducted on RC moment resisting frames and RC frames with steel bracings. Malekpour et al. (2012) developed steps of the Displacement Based Design method for RC frames with steel bracings.
In this study, considering the ever increasing development of the DDBD and its use in RC buildings, the DDBD methodology is investigated for three lateral load resisting systems of reinforced concrete structures, i.e., RC Frame, RC WallFrame and SteelBraced RC Frame Systems. For this purpose, the displacement profile is calculated and the equivalent single degree of freedom system is then modeled considering the damping characteristics of each member. Having calculated the effective period and secant stiffness of the structure, the base shear is obtained, based on which the design process can be carried out. For each system three frames are designed using DDBD approach. The frames are then analyzed using nonlinear timehistory analysis with 7 earthquake accelerograms and the damage index is investigated through lateral drift profile of the models. Results of the analyses and comparison of the nonlinear timehistory analysis results indicate efficiency of the DDBD approach for different reinforced concrete structural systems located in nearfield regions.
2 Description of the Design Procedure
Assuming a design ductility, which is choosed by the designer based on ultimate design displacement and yield displacement of the real structure as μ_{∆} = ∆_{D}/∆_{y}, or simply based on code requirements, and using existing ductilitydamping diagrams, a design damping is selected for the equivalent SDOF system (Fig. 1c). Then, a design period is obtained using Fig. 1d and according to the specified design displacement and the specified equivalent viscous damping.
2.1 Assignment of Strength Proportions for Dual Systems
2.2 Yield Deformation and Design Displacement Profile of Frames, Bracings and Walls
The design displacement profile is developed using the various values obtained as the following for each of the systems, together with the design story drift using Eqs. (6) to (13).
In Eq. (7), ε_{ y }, l_{ b }, h_{ b } and h_{ m } are yield strain of longitudinal bars, bay length, beam height and effective height of the structures (which can be assumed to be 0.7 h_{ n } for framed structures), respectively.
2.3 Equivalent SDOF Characteristics
2.4 Determine the Design Base Shear and Member Strengths
2.5 Equivalent Viscous Damping
One of the most important parameters required for DDBD approach is equivalent viscous damping. Thus, in this section the methodology used for calculation of this parameter is given for the structural systems investigated in this study.
Equivalent viscous damping is a function of ductility and the effective period (Malekpour et al. 2012; Priestley and Grant 2005; Blandon 2004).
When beams of equal strength are used along the height of the structure, the ductility obtained from Eq. (4) for each story can be averaged to give the frame displacement ductility demand. The proposed procedure determines the equivalent damping in such a way that when this damping is applied to a SDOF system with a definite effective period which is obtained based on the secant stiffness in the maximum displacement response, response of this SDOF system becomes consistent with the nonlinear timehistory analysis response. Finally, the objective is to propose an equation that calculates the equivalent damping factor for DDBD. In order to obtain the equivalent damping, the following method which is based on Blandon`s method (Blandon 2004) is used.
The process is repeated for effective periods from 0.5 to 4 s each 0.5 s, for 5 ductility levels from 2 to 6. Six different hysteretic curves are used and all the cases are analyzed for six records.
Step 2: The equivalent damping factor ξ is estimated. For the first iteration this was based on Jacobsen’s approach (Jacobsen 1930) according to the hysteretic loop considered. However, after the results of the first iteration were obtained, the equivalent damping was changed in the next iterations to improve the substitute structure/timehistory agreement. The significant assumption of this step would be definition of the initial viscous damping factor ξ_{ 0 } which is assumed equal to zero to prevent its effect.
Step 6: Timehistory analysis is run for each of the records and the maximum displacements are obtained.
Step 7: The displacements obtained from Step 6 are compared with that from Step 4.
Step 8: If the displacements are similar (within a tolerance of 3 %), the damping factor is not changed and the process is repeated from Step 1 with the next T_{ eff } and μ; otherwise, the damping factor is modified and the process is repeated from Step 2.
Note: ${\mathit{\zeta}}_{\mathit{frame}}$ is calculated using Eq. (24)
Constants of Ramberg–Osgood hysteresis model.
Constant  Value 

a  150 
b  0.45 
c  1 
d  4 
Note: ${\mathit{\zeta}}_{\mathit{frame}}$ is calculated using Eq. (24).
The effective period obtained from Fig. 2 is compared with the initial period. If this period does not agree with the initial period, replaces the initial period and the process is repeated. When the effective period agrees with the initial period, it will be the design period and will be used for obtaining the design base shear.
2.6 Design Flowcharts
In this section the DDBD methodology employed for the above mentioned systems is clarified via the flowcharts shown in Figs. 3, 4, and 5. For each system three 4, 8 and 12story models are designed based on the corresponding chart.
2.6.1 RC Frame Systems
DDBD parameters calculated for RC frame models.
4 storey  8 storey  12 storey  

Drift limit θ_{ d }  0.025  0.025  0.025 
Effective mass m_{ e } (kg)  192,660  375,680  567,920 
Effective height H_{ e } (mm)  9,105  17,650  26,320 
Design displacement Δ_{ d } (mm)  240  404  525 
Equivalent damping ${\mathit{\xi}}_{\mathit{eq}}$  12.58  12.64  12.2 
Effective period T (s)  1.4  2.1  3 
Base shear (kN)  914  1,282  1,332 
2.6.2 RC WallFrame Systems
DDBD parameters calculated for RC wallframe models.
4 storey  8 storey  12 storey  

Drift limit θ_{ d }  0.025  0.025  0.025 
Effectivemass m_{ e } (kg)  795,167  1,575,298  2,298,357 
Effective height H_{ e } (mm)  10,876  20,602  30,450 
Design displacement Δ_{ d } (mm)  256  475  663 
Equivalent damping ${\mathit{\xi}}_{\mathit{eq}}$  12  11.25  10.14 
Effective period T (s)  1.79  2.76  3.50 
Base shear (kN)  2,504  3,875  4,907 
2.6.3 RC Steel Braced Frame Systems
DDBD parameters calculated for steel braced RC frame models.
4 storey  8 storey  12 storey  

Drift limit θ_{ d }  0.025  0.025  0.025 
Effective mass m_{ e } (kg)  229,836  462,550  694,873 
Effective height H_{ e } (mm)  9,670  18,390  27,140 
Design displacement Δ_{ d } (mm)  230  414  587 
Equivalent damping ${\mathit{\xi}}_{\mathit{eq}}$  13.62  13.64  13.49 
Effective period T (s)  1.70  2.65  3.30 
Base shear (kN)  650  992  1,299 
3 Structural Models
Three 4story, 8story and 12story buildings with three different structural configurations (RC Frame, RC WallFrame and Steel Braced RC Frame Systems) are designed based on the DDBD approaches mentioned in each section and according to the following considerations.
Final design results of RC steel braced frame models.
4 storey  8 storey  12 storey  

Brace section  2UNP120  2UNP160  2UNP200 
Beam depth × width (mm)  400 × 300  450 × 300  500 × 400 
Int. column depth × width (mm)  400 × 400  500 × 500  550 × 550 
Ext. column depth × width (mm)  400 × 400  450 × 450  500 × 500 
Inter storey height (mm)  3,200  3,200  3,200 
Final design results of RC wallframe models.
4 storey  8 storey  12 storey  

Wall length (mm)  8,000  8,000  8,000 
Wall thickness (mm)  200  300  350 
Beam depth × width (mm)  650 × 400  750 × 450  750 × 450 
Int. column depth × width (mm)  650 × 650  750 × 650  750 × 650 
Ext. column depth × width (mm)  550 × 550  650 × 650  650 × 650 
Inter storey height (mm)  3,200  3,200  3,200 
Final design results of RC frame models.
4 storey  8 storey  12 storey  

Beam depth × width (mm)  500 × 450  500 × 450  550 × 500 
Int. column depth × width (mm)  580 × 580  600 × 600  625 × 625 
Ext. column depth × width (mm)  580 × 580  600 × 600  625 × 625 
Inter storey height (mm)  3,200  3,200  3,200 
4 Performance Verification
The model for a FEMA concrete beam is similar to that for a steel beam. However, rotation hinges are used, instead of curvature hinges. This is because FEMA356 gives the properties for concrete beams and columns in terms of plastic end rotations rather than as multiples of the yield rotation.
Most of the inelastic components in PERFORM3D have the same form for the F–D relationship. This is a trilinear relationship with optional strength loss, as shown in Fig. 10. For further description about the key points of the F–D relationship please refer to reference FEMA (2002). The widely used hysteretic models of Ramberg–Osgood (from Blandon 2004) and Takeda (from Blandon 2004) were implemented for simulating the reinforcing steel and concrete behavior, respectively (Figs. 11, 12).
Characteristics of the selected records.
Peak acceleration (g)  Minimum distance from faulting (km)  Soil type (USGS)  Magnitude (Ms)  Years  Title 

0.267  2.6  C  7.8  1999  Kocali 
0.09  2  B  7.3  1999  Duzge 
0.4966  4.38  C  6.69  1992  Erzincan 
0.519  4  C  7.62  1979  Imperial valley 
0.6357  8.34  C  6.9  1995  Kobe 
1.22  2.2  B  6.6  1971  Sanfernando 
0.852  –  C  –  1970  Tabas 
Figure 20 represents the interstory drift profiles of the RC frame models under the selected pulsetype records. In these figures, the design interstory drift profile is also displayed. Referring to these diagrams, the method performs quite satisfactorily. Maximum interstory drifts in all RC frame models fall under the specified design profile. The overall profile shapes are similar to those expected for rigid frames. The shape of the profiles for the tall RC frame model (12 story frame) are very similar to natural higher mode shapes of these structures derived from Eigenvalue analysis of frames, implying that higher mode effects are important for tall frames.
Figure 21 indicates interstory drift profile of the RC wallframe models. Displacement responses of the models are in close agreement with each other, and interstory drifts of the all the models subjected to the records are below 2.5 %. All the lateral displacement responses exhibited some decrease in increasing displacement profile in upper stories which can be attributed to the fact that frames have dominant response in upper stories in comparison with structural walls.
Figure 22 indicates interstory drift profile of the steel braced RC frame models. The interstory drifts of the 4 and 8story models subjected to the records are below 2.5 %, but interstory drift of the 12story model subjected to the records exceeded the design interstory drift in Stories 1–4, which can be attributed to the bracing buckling in lower stories. The interstory drifts of the 4 and 8story models subjected to the records are below 2.5 %, except for Imperial Valley Record in both models and Erzincan and Sanfernando Record in the 8story model. However, interstory drift of the 12story model subjected to the records exceeded the design interstory drift in Stories 1–4 demonstrating the fact that this design method was not successful for the 12story model.
5 Concluding Remarks
The present study focuses on seismic behavior of nearfault RC structures design with a new performancebased design tool called the DDBD.
For this purpose, seismic response of RC frame systems in addition to dual RC wallframe and steel braced RC frame systems designed using DDBD are investigated.
Performance verification studies show that the method can be regarded as an appropriate alternative to current forcebased seismic design of structures. The method, performed quite satisfactorily in term of maximum interstory drift, even for tall models. Some deviations, especially in tall models, from design values are mainly due to the complex and highly varying nature of frequency content of nearfault records. Another important finding of the study is that, the DDBD methodology is able to design structures with quite controlled residual behavior, an interesting subject which needs further studies.
Declarations
This article is published under license to BioMed Central Ltd. Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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