Displacement-based design of Reinforced Concrete Special Moment Frame (RC-SMF) with Vertical Irregularity

Reinforced concrete Special Moment Frame (RC-SMF) with irregularity in dimension experiences large inelastic deformation under ground motion. The building code for designing structural elements generally is focused on satisfying the strength and serviceability. On the other hand, the displacement-based design (DBD) procedure is based on building performance level. This paper presents a devoted study of designing structural elements of 8 story RC-SMF with vertical irregularity using the DBD method. This method is mainly using energy-work balance concept with pre-selected yield mechanism and target drift. Moreover, a new lateral force distribution method is used in this study which accounts for inelastic dynamic response and higher mode effects. Strong column-weak beam mechanism is used to design beam and column, and beam moment capacity of each floor is calculated by equating external work to internal work approach. Furthermore, column design strength is taken as the combination of factored gravity loads and maximum expected strength of the beam. The time history analysis results show that story drift is below than the target drift and achieve the desired performance level. Moreover, the results also show lateral force distribution is very close to the lateral shear distribution which obtained from time history analysis.


Introduction
The displacement based design method is based on strong-column weak-beam concept and used a predetermined yield mechanism and target drift. The main benefit of this method is the failures first occur in the beam, not in the column. As a result, the total collapse of the building is prevented and the life safety of structure improved. Due to damage and undesired failure of the structural and non-structural member of code based reinforced concrete (RC) building under severe ground motion [1] have indicated the need to improve the design methodology for better structural performance. However, the present design codes [2,3] have special requirements for the designing of RC buildings. Recently, a displacement based design method is namely Performance Based Plastic Design (PBPD) [4] has been achieved excellent performance against earthquake loading. Since, the irregular shaped RC SMF is widely used as an earthquake resisting structure, a enhance research has been performed using this method to obtain the satisfactory performance level.
A new design procedure of structure which has one or more performance objectives to meet is referred as Performance-Based Seismic Design (PBSD) [5]. The main objective of Performance-Based Seismic Design (PBSD) is to develop design methodologies for predictable and intended seismic performance of structure under stated level of seismic hazards [5,6]. This concept becomes very popular among the design engineers and researchers who were searching a design approach where all variables that affect the performance of building like interstory drift ratio, floor acceleration, damage measure, collapse etc. will be involved [7]. After several comprehensive research, a performance-based design method using target drift and yield mechanism is carried out on the different type of steel structure like Eccentrically Braced Frame (EBF), Concentrically Braced Frame (CBF), and Special Moment Truss Frames (STMF) [8][9][10][11][12]. Furthermore, this method is also applied to the reinforced concrete structure to validate this methodology on RC structure [12,13]. In this study, an 8 story RC-SMF with vertical irregularity is designed using this method and perform the time-history analysis.
The 8 story vertically irregular shaped RC SMF is completely designed using the proposed displacementbased design where work-energy equation is used to calculate the design base shear, and a new lateral force distribution is used which include the inelastic dynamic response and higher mode effects [8]. The target response spectrum is taken from [14] for soil type E (soft soil) of Yogyakarta. After designing the same structure using the code based method, a nonlinear time history analysis is performed by using OpenSees 2.5.0 platform [15] and the resulting interstory drift and story shear distribution are compared.

Displacement based design
The performance of a structure mostly depends on ductility demand, displacement, and inter-story drift under severe earthquake. The displacement-based design considers the effect of ductility demand, displacement, interstory drift and others factor while designing components of the structure. It is also noted that this method can also be used to achieve the performance level for various hazard levels. As a result, the displacement-based design method is considered as the main function of performance-based design.

Energy balance concept
The main point of the energy-work concepts is the work need to push the structure up to a target drift is equal to the energy need of an equivalent elastic-plastic singledegree-of-freedom (EP-SDOF) system to achieve the same condition which is shown in Figs.1 and 2, and following: where Ee and Ep are elastic and plastic component respectively of total energy needed to push the structure up to the target drift. The Sv and Sa are the design pseudo-spectral velocity and pseudo-spectral acceleration respectively, T is natural period and M is total mass of the structure.

Design Base Shear
Equation (1) is further modified and get the energy modification factor  which is used to calculate the design base shear [13]: where,  R is ductility reduction factor which is obtained from The work-energy equation (1) can be modified and re-written in the following form: where Vy/W is design base shear coefficient,  is a dimensionless parameter given by following formula: where Vy is the design base shear determined by workenergy equation, W is the total seismic weight of the structure, wn and wj is top floor seismic weight and j-th floor seismic weight respectively, hn and hj is the structure height at top floor and j-th floor respectively, i  is the shear distribution factor and p  is plastic component of target drift ratio.

Lateral force distribution
The shear distribution factor and lateral force distribution is obtained from inelastic dynamic response formula [13] which is given as: where Vi and Vn is static story shear at level i and top level n respectively. Fn is lateral force at top floor n. For lateral force at i level, F, the formula is given as:

Beam design
The beam moment capacity of each floor is calculated using plastic design approach which is external work equal to internal work. The work equilibrium formula is shown below: where  is small kinematic rotation angle of yield mechanism, pb M is the beam moment strength which is sum of positive and negative moment, pc M is plastic moment of column at base and i  is the rotational term.
Equation (9) is further simplified to calculate the beam positive and negative moment, and the equations are given below: where x is the ratio of negative to positive beam moment.

Column design
In this study, the strength in column is taken as a combination of factored gravity loads and maximum expected strength of the beam. Using this concept, exterior and interior column tree is drawn which is the main function of column design. For determining axial load acting on column, the formula is given below:

Target response spectrum
The design response spectrum of all over Indonesia is available online which is managed by PPMB-ITB and PUSKIM [14]. For this study purpose, soil type E (soft soil) of Yogyakarta has been selected. The SDS(g) and SD1(g) values are 0.727 and 0.711 respectively (Fig. 5.).

Design parameters
The seismic weight of each floor (5 th to 8 th floor) is 716 kN and (1 st to 4 th floor) 1232 kn. The tributary seismic weight for all floor is 44 (kN/m). Others parameters are given in Table 2.

Material for nonlinear behaviour
The concrete cover and concrete core properties are defined by using a uniaxialMaterial ConcreteCM which is a uniaxial hysteretic constitutive model for concrete developed by Chang and Mander [18,19]. The input format for defining this concrete properties is given below: The steel rebar properties is defined by a uniaxial Giuffre-Menegotto-Pinto steel material. The input format is given below:

Section and element
In this model, fiber section is used to define the beam and column. Each section has concrete cover (in four sides) and concrete core. Rebar is placed according to design. OpenSees has a huge variety of element type for analysis. In this study, Force-Beam-Column element type is selected for our nonlinear modelling. Previously defining section is used with 5 Gauss-Lobatto integration points along the element both for beam and column.

Selection of ground motion
PEER ground motion database [20] is very useful sources for getting ground motion with respect the target response spectrum. In this study, 7 ground motions are selected for performing the time history analysis as given in Table 3.

OpenSees analysis option
The most important part in OpenSees is selecting appropriate analysis option for performing the time history analysis. The analysis option in OpenSees contains 7 parameters to run the analysis which are listed in Table 4. In this study, it is noticed that wrong selection of these parameters resulted unrealistic results or error in sections & elements, although the model definition is correct. However, the appropriate analysis option selection has given expected and accurate results.
On the other hand, in this study, the solution algorithm is needed to change from Newton to modified Newton for different ground motions.

Analysis results and discussion
In this study, nonlinear time history analysis is performed to get the structure responses and evaluate the proposed method acceptability. After getting the response of the structure under selected 7 ground motions, the maximum interstory drift and the relative story shear distribution are analysed to understand the performance the structure under real earthquake situation.

Maximum interstory drift
The maximum interstory drift shows the building performance level. The time history analysis results for all 7 selected ground motions are shown in Fig 6 with  very satisfactory drift distribution over the building height. In this graph, it is also shown that the drift ratio is reduced at 5 th floor for all ground motion. As the vertical irregularity is started after 4 th floor, the result shows that 4 th and 5 th floor drift difference is very low. However, the results of previous research works [13] for 8 story vertical regular RC SMF showed that the maximum story drift ratio at 5 th story increased as compared to above floors.

Relative story shear distribution
It is important to see the output from nonlinear time history analysis, as an inelastic response based lateral force distribution method is used in this study. Fig 7 shows the details of relative story shear distribution for all 7 ground motions. The distribution pattern of relative story shear of the proposed method is similar to the distribution pattern obtained by nonlinear time history analysis of 7 ground motions. Moreover, the result shows that the maximum shear appeared at bottom floor for all ground motions which is 6 to 7.5 times greater than the top floor shear. However, the maximum increment of story shear is appeared at 4th floor for all ground motions including DBD method. Because of the vertical irregularity start from 4 th floor the shear distribution pattern is different than the results obtained from previous research work [13] on RC structure. On the other hand, the maximum base shear obtained from 7 ground motions are raged from 650 to 700 kn which is below the design base shear (Vy).

Conclusion
The displacement based design method which is based on energy-work balance concept with pre-selected target drift has been considered in this paper. Using this method, an 8 story building is designed and extensive nonlinear time history analysis is performed by using OpenSees with 7 PEER ground motions. The main objective of this research is to prove the validity of this method for RC SMF with vertical irregularity, as some research works have already validated this method for RC SMF regular structure. The results from nonlinear time history analysis show that the achieved drift ratio is below than the pre-selected target drift ratio. In addition, the story distribution patterns is also similar to the inelastic based lateral force distribution method. The major outcome from this enhance study is that the displacement based method proves its validity for both regular and vertically irregular RC SMF structures, and the shear distribution pattern of regular and vertically irregular RC SMF is different because of the presence of structure irregularity.