Keywords

1 Introduction

India is classified into IV zones under the category of the high seismic zone, moderate seismic zone, and low seismic zone. The zone factor for each zone corresponds to the maximum considered earthquake of that zone. The value of the zone factor is damped to get the design-based earthquake by dividing the value of zone factor by denominator 2 given in Indian Seismic code IS1893:2016 [1]. The seismic ground motion is characterised by intensity, duration which is a function of magnitude, the distance between focus and epicentre, and characteristics of the propagation of waves; the waves show impulsive and complex behaviour due to which the structure responds by introducing the inertia forces as an opposing force to horizontal seismic force which is a type of surface wave. The fundamental natural period also plays an important role in calculating the average response acceleration coefficients for different soil conditions. The short period of vibration generates large inertia forces within a structure which may cause large stiffness degradation in members so to damp the seismic forces it is feasible to elongate the period which is done by using advanced techniques like base isolation, but these techniques are not cost effective for everyone. The partial factor of safety, redundancy and ductility help in extrapolating the design base shear to ultimate elastic force; thus, in Indian code, the nonlinearity is incorporated by response reduction factor denoted by R. By following the ductile detailing guidelines, we ignore to analyse the ductility part as the response beyond elastic is not a simple extrapolation. For analysing the response of structure beyond the elastic stage, the nonlinear method or dynamic method of time history is followed by analysing the structure on finite element software which is time consuming and needs an experienced designer to analyse the results of nonlinear time-history analyses [2]. Pushover analysis is used as an alternative for time-history analysis which saves computational time and can analyse the structure as per various international guidelines. Performance-based design is a design approach that eases out the process of designing for the owner to engineer by giving the flexibility to the owner to discuss the performance goals with the designer and engineer which helps to set up multiple performance objectives which is a function of performance level which expresses the extent of damage in structure due to hazard event [3, 4]. It is necessary to introduce PBD to meet the performance requirement in emerging society [5, 6]. In this case study, we will analyse the structure by cooperating the performance design approach with user-defined functions to reflect the values based on Indian Seismic codes.

2 State of Development

The performance-based design approach had been in development since the end of the ’60s, and due to the unexpected economic loss in Loma Prieta Earthquake (1989) and followed by Northridge Earthquake (1994), the recommendation for the performance based was developed by the SEAOC committee under Vision 2000 [7]. The capacity spectrum method was one of the earliest methods to be applied in PBD for assessing the seismic vulnerability of structures. It is a graphical method that plots the relation between base shear and displacement. This method was used in the early ’70s as a pilot project at the Puget Sound Naval Shipyard [8] supporting its correlation between ground motion and building performance by stating the hinge states from IO to CP and beyond due to which it serve as a verification design document in Tri-services [9]. The pushover curve which is a plot of base shear and displacement gives the capacity of the structure. With some modification over time, the pushover curve was converted to an acceleration displacement response spectrum which merges the base shear and displacement with (a) spectral acceleration versus spectral displacement representation, (b) spectral acceleration versus time and (c) spectral displacement versus time. N2 method and coefficient displacement method are the two advancements after the capacity spectrum method under performance-based design methodology [10]. The method for converting the base shear and displacement into spectral acceleration was covered using the relation given in ATC40 [11]. Spectral acceleration and displacement can be calculated through Eqs. 1 and 2.

$$S_{a} = \frac{{\frac{{V_{B} }}{W}}}{{\left( {\frac{{M_{K} }}{M}} \right)}}$$
(1)
$$S_{a} = \frac{{\Delta {\text{rooftop}}}}{{P_{K} \Phi_{{K{\text{rooftop}}}} }}$$
(2)

VB represents the base shear, W is the seismic weight of the structure, Δ is the displacement of the rooftop, PK is the modal participation factor and Φk rooftop is the modal amplitude at the rooftop.

Sa versus Sd will give the capacity curve, and for the demand curve, the response spectrum graph having Sa and T is also converted by relation given in Eq. 3 and [12].

$$S_{d} = \frac{{T^{2} }}{{4\pi^{2} }}S_{a}$$
(3)

T was represented by radial lines in the transformed plot. The response spectrum curve gives the demand curve. The intersection of the capacity curve and demand curve gives the performance point. The damping of spectra can vary from 5 to 20% or more. The pushover analysis was not introduced in Indian Seismic Code, but ATC 40 procedure can be adapted by using the value of Ca and Cv parameters concerning response spectra used in the Indian Seismic code for a different type of soil [13]. A case study model S6B3 was used from an original paper [14]. The CSM graphs will be discussed later in the paper.

3 Case Study

The model S6B3 consist of six storey and three bay with a spacing of 4 m in each direction. The building is an office building with an importance factor of 1.5 on medium soil having a floor height of 3 m with a mean live load of 4kN/m2. Lateral design load will be calculated from the equivalent static method [1, 11]. ATC40 guidelines will be followed and adapted in the Indian code for finding the performance point. The 4% of height was selected as target displacement [1, 11]. M25 grade and Fe 415 grade steel were used. Beams of size 300 × 400 mm were used. The column was designed separately for 1–3 floors and 4–6 floors with external and internal columns as given in Table 1. In this case study, we will use the modified value of Ca and Cv as per medium soil and Indian response spectra for 5% damping and check the structure for 0.004H permissible limit for Indian code and verify the structure hinges for collapse prevention parameter 0.04H. Property modifiers for beam and column as per Indian seismic codes were applied. It is very important after gravity load to add a new load case as gravity with nonlinear behaviours which will be force controlled, and after that push load cases can be used which will start from unstressed gravity load case. The plan of six storeys three-bay (S6B3) is shown in Fig. 1.

Table 1 Column size detail for S6B3
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Fig. 1
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Plan of S6

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In Sap2000, pre-target displacement of 0.04H can be changed with the user-calculated target displacement. The push cases will be deformation controlled. Hinges can be user defined or predefined based on the designer. In this study, hinges were defined at 5 and 95% at both i and j node of the beam and column while for beams M3 hinge were assigned and for column P-M2-M3, hinges were used as a relationship to describe the post-yield backbone behavior of beam and column [15].

4 Results and Discussion

Pushover curve—Graph of base shear and displacement for the X- and Y-axis was discussed in Fig. 3. The target displacement was revised from 0.004H to 0.04H to consider more displacement effect as a result we can get the results for larger displacement and larger rotations in hinges. Figure 2 states the hinge result of the bottom node of frame object 214 hinges 2 (214H2).

Fig. 2
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214H2 P-M2-M3 hinge result

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Fig. 3
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Pushover curve X-axis and Y-axis

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P-M2-M3 hinges were assigned for a column with M3 as hinge degree of freedom resulting in plastic rotation of 0.0304 radians at Step 27 of Push X stating the degradation of fibres after CP point. Beam hinges were found to be between IO-LS state, change in a property of column in fourth storey result in loss of stiffness in columns.

Pushover results also consider the hinge state and results for various push steps which will be discussed after covering the capacity spectrum graph. The top node of the structure (Node81) was pushed up to target displacement which results in loss of stiffness in members while exposing the ductile capacity of the structure. The S6B3 model performs satisfactorily till the permissible limit given by IS codes and follows the strong column weak beam methodology. At Step 11, base shear was 2949 kN and displacement was 76.46 mm with only four hinges which change their state from IO to LS as given in Table 2. According to IS code, the structure performance was elastic, but for global results, performance point must be checked thoroughly.

Table 2 Pushover capacity curve
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Capacity spectrum method shows the performance point by superpositioning the pushover curve given by green colour to the ADRS representation of damped responded spectrum by 5, 10, 15, 20% indicated by four red lines known as the demand curve as shown in Fig. 4. The Sa and Sd values correspond to the 5% damped response spectra for X and Y push [16]. The performance point for X (2694.003 kN, 63.779 mm) and Y (2677.617 kN, 64.827 mm) was given in the form of base shear and displacement (V, D point). Ca value was taken as 2.5(Z/2) and Cv value 1.39/T(Z/2) for medium soil for Indian Code [13]. T is the modal fundamental period of building corresponding to mode 1. Z is the zone factor 0.36. The 1.36/T value was referred from the value of the average response spectrum for medium soil when the period was between 0.55 and 4 s.

Fig. 4
figure 4

ATC40 capacity spectrum for Push X and Push Y

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FEMA 356 Coefficient method [17] in Fig. 5 shows the bilinearisation of the pushover curve which represents the maximum displacement and force as a performance point for the structure. For Push X and Push Y, the value of performance point was (2297.31 kN, 210.587 mm) and (2533 kN, 158 mm). These values are discussed below in hinge results as shown in Fig. 6.

Fig. 5
figure 5

FEMA356 coefficient method for Push X and Push Y

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Fig. 6
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Push X hinge formation at Step 27 and 35

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In Step 27, the hinges in the column of four storeys change their state to IO-LS at one end, and the far end column reached beyond C point in which the column degrades its strength abruptly with very minute rotation and displacement. The maximum performance point values approximately correspond to Step 27 to 29 value. The hinge state for the column is shown in Fig. 1. Step 35 consists of the response due to ultimate displacement which results in the collapse of the column at four storey; hence, the moment redistribution due to storey 4 columns was already at its peak from Step 27. The base shear can be checked for displacement and its hinge state from Table 2.

5 Conclusion

For designing an earthquake-resistant important structure, performance-based design is a perfect fit, as in this case the structure was design only for elastic base shear with the use of Indian seismic code and later on verified for a global response with nonlinear static analysis which shows the structure ductility performance, with the response of hinges generated at the nodes of beam and column in the relative distance of 0.5 and 0.95. These methods with the help of hinges show the potential beams, the column for retrofitting in the existing structure and redesigning in case of a new building to achieve the proper performance. Only selected frame members to be properly modelled as in our case label 195, 214, 217, and 199 need to be retrofitted to impart additional strength and ductility. The hinge results give a good indication of moment and rotation with each increment of push. As the hinge behaviour is deformation controlled and up to Step 26 shows satisfactory ductile behaviour while with the adaptation of Ca and Cv values, the performance point shifts its self-according to the Indian response spectra. The structure was not analysed for the nonlinear time-history method, but in the case, it should be assigned with more than seven ground motions to get proper supporting results.