Alternative approach in Partial Capacity Design (PCD) by using predicted post-elastic story shear distribution

The Capacity Design Method is an approach widely used to design earthquake resistant structures. It allows the structures to dissipate earthquake energy by forming plastic hinges through beam side sway mechanism. In the design process, the columns need to be designed stronger than the beams connected to them. Several previous studies have been conducted to propose alternative method allowing partial side sway mechanism namely the Partial Capacity Design (PCD) Method. In this method, selected columns are designed to remain elastic and the plastic hinges are allowed to occur only at the columns base. These columns are designed to resist increased forces. Despite of some successful attempts, PCD method still needs to be developed because sometimes the intended mechanism was not observed. This study proposes a new approach to improve the Partial Capacity Design (PCD) method. Symmetrical 6 and 10 story buildings with 7 bays are analyzed using seismic load for city of Surabaya. Structure behavior under non-linear static analysis is well predicted by this approach. However, under non-linear dynamic analysis, a few unexpected plastic hinges of elastic columns were observed at upper stories. But it should be noted that the earthquake used for performance analysis (maximum considered earthquake) is 50% larger than the one used for design (earthquake level corresponding to elastic design response spectrum).


Introduction
The Capacity Design Method is an approach widely used to design earthquake resistant structures where the desired failure mechanism is beam side sway mechanism as illustrated in Figure 1 (illustration in ATC 40 [1]). This method applies strong column-weak beam concept to prevent soft story failure during earthquake. In the design process, the columns need to be designed stronger than the beams connected to them. Several previous studies have been conducted to propose alternative methods based on Partial Capacity Design (PCD).
The idea of alternative method of PCD is started from previous study about gravity load dominant structure. According to Paulay and Priestley [2], in low-rise reinforced concrete frames, especially those with long-span beams, beams design are often governed by gravity load rather than seismic requirement. Application of capacity design will result excess column capacity and the structure possesses excess lateral capacity. In such case, development of plastic hinge in several columns are allowed to attain lower lateral capacity and partial side sway mechanism as illustrated in Figure 2.  Figure 1. Beam side sway mechanism [1]. Figure 2. Partial side sway mechanism [3].
Muljati et al. [3], Muljati dan Lumantarna [4,5] developed an alternative method of PCD that propose more practical design procedure than those in capacity design, where the column-beam capacity ratio is unnecessary to be checked. This method allows interior columns and beams to develop plastic hinges while maintains exterior columns to remain elastic during targeted seismic load. The targeted failure mechanism is partial side sway mechanism as shown in Figure 2. Magnification factor is used to magnify exterior column design forces to meet such condition. This method needs further research that in some case studies it failed to meet the desired failure mechanism.
This paper proposes a new approach in designing the elastic columns (exterior columns). The idea is to increase the story shear force for each story individually using the concept from previous studies. The increase of story shear force results in increase of elastic column shear force that in turn be used to determine bending moment of the column with assumption of cantilever behavior. By using this method, structure failure mechanism is expected to meet partial side sway mechanism.

Partial Capacity Design (PCD)
Muljati and Lumantarna [5] proposed an alternative to PCD method. In this method, interior columns are designed using nominal seismic load. Thus, exceeding seismic load during targeted seismic level will be sustained by exterior columns derived as : where n ex and n in are the total number of exterior and interior columns, respectively; S ex T is the shear force in the exterior column due to the target seismic load; S in N is the shear force in the interior column due to the nominal seismic load; f 1 is the overstrength factor; and V t T is the total base shear due to the targeted seismic load. The load distribution in PCD is illustrated in Figure 3.  Magnification Factor (MF) is a ratio of exterior column shear forces due to targeted and nominal seismic loads, derived as: where C T is the spectral acceleration of the target seismic load; C 500 is the spectral acceleration of a five hundred years return period earthquake (Elastic Design Response Spectrum); μ is the structural ductility; n in and n ex are the total number of interior and exterior column; R in N and R ex N are the ratio of interior and exterior columns base shear to the total base shear due to the nominal seismic load.
During targeted seismic load, structure is designed to be in nonlinear condition so that C T need to be determined in nonlinear response spectrum using structural plastic period Tp. Unfortunately, the nonlinear response spectrum is not provided in the code. Therefore, it is proposed to determine C T from elastic design response spectrum by using pseudo plastic period, Tplastic (non-linear period which corresponds to equivalent total base shear of real building with the elastic response spectrum) as illustrated in Figure 4 according to: Figure 4. Determination of spectral acceleration in PCD [5].

Development of Partial Capacity Design (PCD)
This study proposes design procedure based on previous study by Muljati and Lumantarna [5]. Beams and plastic columns (interior columns) are designed using nominal seismic load from Elastic Design Response Spectrum (EDRS) following provisions in SNI 1726:2012 [6], except the minimum columnbeam capacity ratio. Elastic columns (exterior columns) are designed by increasing column forces using additional story shear force. Load distribution in this method is illustrated in Figure 5.
(a) (b) Figure 5. Load distribution in developed partial capacity design partial capacity design due to (a) nominal earthquake (b) targeted earthquake. In design process, plastic columns are designed using nominal seismic load. During targeted seismic load, plastic columns are predicted to sustain shear force equal to design shear from nominal seismic load including their overstrength factor. Thus, shear force exceeding nominal seismic shear will be sustained by elastic columns as in Figure 5 (b). The exceeding shear force is treated as additional story shear to nominal seismic story shear. The additional story shear for design is derived as: where V dt is total additional story shear; V et N and V pt N are total shear force of elastic and plastic columns due to nominal seismic R=8; V t T and V t N are total story shear due to targeted seismic load and nominal seismic load R=8; C T and C D are spectral acceleration of the target seismic load and the design seismic load; and R is response modification coefficient (f 1 μ as in SNI-03-1726-2002 [7]). Similar to the previous study, C T is determined in elastic design response spectrum using Tplastic in Equation 3. The total additional story shear is used to calculate additional shear force of elastic columns regarding the stiffness of each column. The additional shear force then be used to calculate additional bending moment of elastic columns. During design seismic load or targeted seismic load as well, a portion of bending moment of elastic columns from connected beams resistance is predicted to be limited to the design bending moment due to nominal seismic R=8 including overstrength factor. Thus, the additional shear force of elastic columns will be addition to a portion of bending moment where the columns behave as vertical cantilevers. Elastic columns shear force and bending moment are illustrated in Figure 6.
(a) (b) Figure 6. Additional forces to elastic column in (a) shear (b) bending moment.
The notation V and M are shear force and bending moment due to nominal seismic R=8; Vd and Md are additional shear force and bending moment; Ve and Me are predicted shear force and bending moment due to nominal seismic R=1.6 (targeted seismic at design level); and Lcol is column length.  Table 1.  Both structures are designed and analyzed using seismic load for city of Surabaya at site class C as in SNI 1726:2012. Structure performance are evaluated based on story drift and failure mechanism through Nonlinear Static Procedure (NSP) and Nonlinear Dynamic Procedure (NDP) analysis as in FEMA 273 [8].

Story shear distribution to columns
Story shear distribution to the columns in 6 and 10 story buildings as in Table 2 and Table 3 are obtained from NSP analysis. Actual story shear values due to targeted seismic V t Ta of 6 story building are close to the predicted values V t T , meanwhile in 10 story building, V t Ta are lower V t T for all stories.
For both structures, actual total shear of plastic columns V pt Ta are greater than predicted, that presumed due to overstrength in structural element. The results show that structure behavior under non-linear static analysis is well predicted for both 6 and 10 story buildings.

Failure mechanism
Failure mechanism of 6 and 10 story buildings are observed by the occurrence of plastic hinges. State from NSP analysis is observed at the Performance Point and state from NDP analysis are observed at the last time step of analysis. Results for both structures are shown in Table 4 and Table 5 respectively.    Table 5. Plastic hinges in 10-story building.

Analysis
Exterior Frame Interior Frame

NDP-MCER
NSP analysis of 6 and 10 story buildings show that the failure mechanisms of both buildings meet the partial side sway mechanism, but need to be optimised. Only NSP-MCER analysis of 6 story building show the plastic hinge occurrence at all of the column base. NDP analysis show the failure mechanism that similar to those in NPS analysis. Unfortunately, plastic hinges of elastic columns occur at upper story of 6 story building in NDP-MCER analysis and at upper story of 10 story building in both NDP-EDRS and NDP-MCER analysis. This condition is presumed due to elastic column internal forces that predicted based on one way loading.

Conclusions
Structure behavior under non-linear static analysis is well predicted by this new approach. However, under non-linear dynamic analysis, a few unexpected plastic hinges of elastic columns were observed 10 at upper stories. But it should be noted that the earthquake used for performance analysis (maximum considered earthquake) is 50% larger than the one used for design (earthquake level corresponding to elastic design response spectrum). Further researches need to be conducted to observe dynamic behavior of the structure.