Frequency Domain Analysis for Geometric Nonlinear Seismic Response of Tall Reinforced Concrete Buildings

This paper aims to study the second-order geometric nonlinearity effects of P-Delta on the dynamic response of tall reinforced concrete buildings due to a wide range of earthquake ground motion forces, including minor earthquake up to moderate and strong earthquakes. The frequency domain dynamic analysis procedure was used for response assessment. Reinforced concrete building models with different heights up to 50 stories were analyzed. The finite element software ETABS (version 16.0.3) was used to analyze reinforced concrete building models. The study reveals that the percentage increase in buildings' sway and drift due to P-Delta effects are nearly constant for specific building height irrespective of the seismic design category assigned to the building. Generally, increase in building lateral displacement and story drift due to P-Delta effects for all seismic design categories is less than 2% for 10 story buildings, whereas this increase for 20 stories or taller buildings is significant with a maximum value around 16% for 50 story building. As for column forces, the study shows that, generally, columns bending moment increases and shear force decreases when P-Delta effects accounted for. In conclusion, the study recommended that the effects of P-Delta need to be addressed for all SDCs allowed by ASCE7-10 and the most important factor to abandonment P-Delta effects is the building height limit.


INTRODUCTION
To determine design forces resulting from loads acting on a building there are, generally, three types of analysis that can be carried out, as follows, Powell, 2010:  The first type is small displacements analysis, in this type, equilibrium is considered in the undeformed position, and the compatibility relationships are assumed to be linear.In this case, geometric nonlinearity is neglected. The second type is large displacements analysis.In this type, equilibrium is considered in the deformed position, and the compatibility relationships are nonlinear.In this case, geometric nonlinearity is considered with no approximations. The third type is the P-Delta analysis.In this type, equilibrium is considered in the deformed position with some minor approximations, and the compatibility relationships are assumed to be linear.In this case, geometric nonlinearity is considered approximately.P-Delta analysis is more efficient computationally than large displacements analysis.For most structures, it is a loss of computer time to consider for true large displacements.P-Delta effect is the additional overturning moments due to lateral movement of a story mass to a deformed position.The second order effect of vertical loads acting upon a laterally displaced structure is termed the P-Delta effect, where P is the total vertical load, and Delta is the lateral displacement relative to the ground.In reality, when horizontal loading acts on a building and causes it to drift, the resulting eccentricity of the gravity loading from the axes of the walls and columns produces additional external moments to which the structure responds by drifting further.The additional drift induces additional internal moments sufficient to equilibrate the gravity load moment, Smith and Coull, 1991.
To better understand the seismic-induced response of high-rise buildings, a plenty of studies have been carried out.Most recently, Dhawale and Narule, 2016, studied the P-Delta effect on high rise R.C. framed buildings with a different number of stories.All analyses (Linear static analysis without P-Delta effect and nonlinear static analysis with P-Delta effect) carried out in software SAP 2000-V12.The results showed that it is essential to consider the P-delta effect for 25 story building.Pillai and Chandran, 2016, focused on the effectiveness of P-Delta analysis in the design of tall slender reinforced concrete structures.The researchers analyzed building models with different story heights.The stability of tall structures to lateral forces with and without considering P-Delta effects is carried out using ETABS 2015 Structural analysis software.The results showed that the P-Delta effects significantly influence the displacement and have a higher value than linear static analysis and that P-delta is essential for stories higher than 15 stories.Bondre and Gaikwad, 2016, compared different methods in terms of their efficiency and accuracy to recognize in what way the P-Delta effects determine the variation of responses of the structure such as bending moments, displacements and shear forces against linear static analysis.They studied 12 cases for buildings with different heights.They performed linear static and P-Delta analysis separately using STAAD pro software.The results showed that P-Delta effects significantly influence the structural components and get a higher value than the linear static analysis.

OBJECTIVES OF THE STUDY
According to ASCE7-10, ASCE7-10, 2010, all structures shall be assigned to a Seismic Design Category (SDC) which is a classification assigned to a structure based on its Risk Category and the severity of the design earthquake ground motion at the site.This study aims to study the significance of P-Delta effects on the dynamic response of tall reinforced concrete buildings when assigned to different SDCs allowed by ASCE 7-10.To achieve this goal, the dynamic response of these buildings is examined due to a wide spectrum of earthquake ground motion forces, including minor earthquake up to moderate and strong earthquakes.Moreover, dynamic response is examined for linear and nonlinear analyses with P-Delta effects using frequency domain analysis.Maximum story displacement, story drift, columns bending moments and shear forces were investigated for five building models with different heights and ground motion forces.The parameters adopted here include a number of building stories and the seismic design category assigned for response analysis.

PROBLEM ASSESSMENT 3.1 Description of Building Models
The finite element software ETABS "Extended 3D Analysis of Building Systems", CSI, 2015, is used in this research to investigate the structural behavior of the modeled reinforced concrete building prototypes.Building models adopted throughout the present study are essentially multistory reinforced concrete buildings with a different number of stories.Fig. 1 shows a typical view of the 3D model of the building and plans view of typical story details.The structural system has been assumed as a dual system consists of a central core of shear wall structure and interior and exterior columns arranged in a rectangular 6x6 meter grid and the exterior columns are connected by edge beam to form moment resisting frames in the two orthogonal directions.The plan of the multi-storey RC building is square 36 meter by 36 meters with columns and shear walls.The floor system for the building models has been assumed to be a reinforced concrete flat plate of 220mm thick.Five buildings models with a different number of stories and heights have been adopted including; ten (G+9), twenty (G+19), thirty (G+29), forty (G+39), and fifty (G+49) stories.Table 1 shows loads data and parameters for gravity loads, and dynamic seismic load cases respectively.On the other hand, Table 2 present section properties for the columns and shear walls for the five-building prototypes and for all stories where C1 represent the square columns, C2 represent the corner columns, and C3 represents the rectangular columns.All beams have been assumed to have 30 cm by 110 cm cross-section and coupling beams between shear walls have been assumed to have 110 cm depth and the same thickness of shear walls that make up the central core.Section properties shown in Table 2 were based on strength and serviceability requirements stipulated in the relevant specification, ASCE 7-10, 2010.

Analysis Procedure
Based on the structure's seismic design category (SDC), structural system, dynamic properties, and regularity the structural analysis for the seismic response evaluation permitted by the ASCE 7-10 shall consist of one of the types listed below: 1. Equivalent Lateral Force Analysis, 2. Modal Response Spectrum Analysis, and 3. Seismic Response History Procedure, Equivalent lateral force analysis is a simple procedure uses an estimated fundamental period and the anticipated maximum ground acceleration, together with other relevant factors to determine maximum base shear.On the other hand, Modal Response Spectrum Analysis (RSA) is a more refined procedure in which the modal frequencies of the structure are analyzed in the frequency domain and then used with conjunction with earthquake design spectra to estimate the maximum modal response, Paz, 2004.
The response spectrum predetermined as one of the most acceptable and feasible techniques that deal with the applications of structural dynamics efficiently.Therefore, in order to investigate the role of different earthquake ground force intensities on the seismic response of tall RC buildings when P-delta effect included in the analysis, the seismic performance of high rise RC buildings is analyzed in this study using Modal Response Spectrum Analysis procedure (RSA).Table 3 listed parameters adopted for seismic analysis applicable to response spectrum analysis.

Seismic Analysis Data
Table 4 shows the seismic coefficients for the Seismic Design Categories (SDCs) and site class D implemented in the numerical analyses, while Fig. 2 shows the design response spectrum for the adopted SDCs.The seismic spectral response acceleration parameters (S S and S 1 ) are selected so that the seismic coefficients in Table 4 represent average values for the corresponding SDC according to ASCE7-10.

ANALYSES RESULTS
In the following summary of the analyses results for the different building models due to different earthquake ground excitations to highlight the influence of the P-Delta effects on the dynamic response of high rise reinforced concrete buildings.Results are presented in terms of story displacements, story drifts, column moment and column shear.

Stability and P-Delta Effect
In building code for minimum design loads for buildings and other structures, ASCE7-10, 2010, P-Delta effects on story shears and moments, the resulting member forces and moments, and the story drifts induced by these effects need not to be considered where the stability coefficient (θ) as determined by Eq. ( 1) is equal to or less than (0.10): where: Px = the total vertical design load at and above Level x, where computing Px, no individual load factor need exceed 1.0 Δ = the design story drift occurring simultaneously with Vx Ie = the importance factor.Vx = the seismic shear force acting between Levels x and x -1.h sx = the story height below Level x.Cd = the deflection amplification factor in Table 12.2-1 of the ASCE 7-10 The stability coefficient (θ) must not exceed θmax determined as follows: Where (β) is the ratio of shear demand to shear capacity for the story between levels (x) and (x -1).This ratio is permitted to be conservatively taken as 1.0.When the stability coefficient (θ) is greater than (0.10) but less than or equal to (θ max ).The incremental factor related to P-Delta effects on displacements and member forces shall be determined by rational analysis.Alternatively, it is permitted to multiple displacements and member forces by [1.0/ (1 -θ)].Where (θ), is greater than (θ max ), the structure is potentially unstable and shall be redesigned, ASCE7-10, 2010.
In this study, section properties for building models compiled in Table 1 were selected to satisfy strength and serviceability requirements.Accordingly, stability coefficient (θ) have been calculated for all building stories and the resulting maximum value for (θ) for each building model is shown in Table 5.It is observed that all building models satisfy the stability criterion for ASCE 7-10.Results for incremental factor [1.0/ (1 -θ)] related to P-Delta effects on displacements and member forces allowed by ASCE 7-10 to be compared with the calculated values for P-Delta effect shown in the following sections.

Buildings Displacement and Story Drift
This subsection summarizes models' responses in terms of building's top displacement and story drift.Table 6 shows results of top story displacement and maximum story drift, respectively, for linear and nonlinear dynamic analyses for all building models and for different seismic design categories and the percentage increase in buildings sway and drift when P-Delta effects included in the analyses.Fig. 3 and Fig. 4 show schematically comparison between maximum top story displacement and maximum story drift, respectively, for cases of analyses of with and without P-Delta effects for SDC A, SDC B, SDC C, and SDC D. These figures and tabulated values for all models response reveal that taller buildings display fewer oscillations than their shorter counterparts for a given time period and that peak values of response are, generally, greater for taller buildings.Moreover, the nonlinear response for building's sway and drift are larger as opposed to linear analysis and that percentage increase due to P-Delta effects are almost the same for each building height irrespective of the seismic design category assigned to the building.Generally, buildings response in terms of lateral sway and story drift increases as P-Delta accounted for and as seismic excitation force, i.e. the seismic design category assigned, increased.
Finally, results presented indicate that for 10 story building the increase in building response due to P-Delta effects is around 1%, whereas an increase of about 5% to 16% is encountered for buildings with 20 stories and up to 50 stories.

Columns Moment and Shear Force
As in subsection 4.2, the same building models and analysis procedure are applied here with only one exception, an investigation for linear and nonlinear with P-Delta effects frequency domain analyses to focus on the effect of P-delta analysis on the response values for columns bending moments and shear forces.To achieve this goal, the column indicated in the plan of the building as shown in Fig. 5 has been examined to determine the P-Delta effect when seismic forces due to different earthquake intensities applied in the X-direction.
Below are the graphs in Fig. 6 for the five-building models and for different SDC depicting results of column bending moment variation due to P-Delta effect when analyzed under linear and nonlinear frequency domain analyses.The same results shown in these figures are compiled in Table 7 in which the variation percentage in column bending moment and shear force when P-Delta effects included in the analyses are presented.
Column moment results presented in Fig. 6 and Table 7 illustrate that nonlinear P-Delta analysis yields larger response values and, generally, column moment increases when P-Delta effects accounted for in the analysis.Generally, 10 story building exhibit the least increase in column moment due to P-Delta effects and that for taller building up to 50 stories a maximum increase of about 8% in column moment is encountered.Results presented reveal that there is no general trend for the percentage increase variation to be expected regarding different seismic design categories (SDC) implemented in the analyses.
As for column base shear results, Table 8 demonstrates that column base shear due to nonlinear analysis is, generally, smaller than that of linear analysis.This result might be attributed to the fact the more flexible buildings' structure it becomes due to nonlinear behavior and the more time it requires to complete a cycle of lateral sway which leads to decrease of base shear values.Generally, a maximum decrease in column base shear values of about 8% is observed.As for moment values, shear results indicate that no general trend for the percentage variation to be expected due to different seismic design categories (SDC) implemented in the analyses.

Figure 1 .
Figure 1.Typical view of the building 3D model and a plan view of the typical story.

Figure 3 .
Figure 3. Maximum story displacement for linear and nonlinear analyses.

Figure 4 .
Figure 4. Buildings G+29 and G+49 story drift due to linear and nonlinear analyses.

Figure 5 .
Figure 5. Location of the studied column.

Figure 6 .
Figure 6.Column bending moment for linear and nonlinear analyses.

Table 2 .
Section properties for building models.
* C denotes the specified concrete compressive strength for 150mm cube at 28 days, expressed in N/mm 2

Table 3 .
Parameters used for the dynamic response spectrum analysis.
Figure 2. Response spectrum curves for SDC A, SDC B, SDC C, and SDC D.

Table 5 .
Maximum stability coefficient (θ) for the adopted building models.

Table 6 .
Top story displacement and maximum story drift for different SDC.

Table 7 .
Column bending moment and shear force for different SDC.