# A Simple Approach for the Design of Ductile Earthquake-Resisting Frame Structures Counting for P-Delta Effect

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## Abstract

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## 1. Introduction

## 2. The P-Delta Inelastic Analysis

#### 2.1. The SDOF System

- $\theta $ is the interstory drift sensitivity coefficient or the stability coefficient;
- ${P}_{tot}$ is the total gravity load above the considered story in the seismic design combination;
- ${d}_{r}$ is the design interstory drift, evaluated as the difference of the average lateral displacements at the top and bottom of the story under consideration;
- ${V}_{tot}$ is the total seismic story shear;
- and $h$ is the story height.

- ${d}_{s}$ is the displacement of a point of the structural system induced by the seismic design actions;
- ${q}_{d}$ is the displacement behavior factor, assumed equal to q unless otherwise specified;
- ${d}_{e}$ is the displacement of the same point of the structural system, as determined by a linear analysis based on the design response spectrum.

#### 2.2. The P-Delta Effect and the Behavior Factor (q)

- (a)
- Apply simultaneously the lateral and vertical loads and implement the stiffness matrix according to Equation (2).
- (b)
- Apply the vertical loads in the system and use the resulting stiffness matrix in a second step where the lateral loads are then applied. This approach is the key to the here-proposed method.

#### 2.3. The P-Delta Effect on MDOF Systems

- A simple response spectrum analysis is performed including the geometrical nonlinearities derived from the implementation of a $P$-$\Delta $ analysis.
- The deformations derived from simple response spectrum analysis are multiplied with the normal forces and then manually added to the moments obtained from the RS analysis. No moment redistribution is applied here.
- A simple response spectrum analysis is performed including the nonlinearities derived from $P$-$\Delta $ analysis assuming that the structure behaves inelastically, according to the approach presented in Section 2.2 for the SDOF systems. The vertical load in the $P$-$\Delta $ forming stiffness matrix process is multiplied with the q factor.
- The results obtained from the response spectrum analysis are amplified by the $1/\left(1-\theta \right)$ factor, assuming the maximum value is obtained for the instability coefficients at each story, as is recommended in [29].
- The results obtained from the response spectrum analysis are amplified by the $1/\left(1-{\theta}_{i}\right)$ factor, estimated to each story. At present, there is no reference to implement this approach, thenceforth the meaning is only to provide a full picture to the expected amplification effect from the $P$-$\Delta $ analysis, comparable here with the physics approach of the second method.

## 3. The Three-Step Procedure

- Step 1. The inputs for the structural analysis are selected: the peak ground acceleration (PGA) and the behavior factor (q). The selected behavior factor should reflect also the expected ductility of the structure, thenceforth it should neither underestimate nor overestimate such value in order to obtain a realistic design.
- Step 2. Perform a nonlinear analysis considering the geometrical nonlinearity. The vertical forces in this analysis are magnified with the behavior factor in order to consider the amplification of the P-Δ effects as a result of the inelastic displacements. This analysis will be used as a starting point for the application of successive analysis.
- Step 3. Perform linear analysis, Response Spectrum analysis, or equivalent seismic analysis using the stiffness matrix of the previous analysis for the determination of the seismic actions. Many software programs, like SAP2000 [30] or other commercial packages [31,32,33], allow running an analysis using the stiffness matrix of a previous analysis. Hence such a requirement is mandatory for the implementation of the present approach.

## 4. Case Studies

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Lateral force—displacement relationship with and without $P$-$\Delta $ effect; (

**b**) design approaches under $P$-$\Delta $ effects.

**Figure 4.**Illustration of the q factor in $P$-$\Delta $ effect in a cantilever: (

**a**) applied loads; (

**b**) moment diagram with $P$-$\Delta $ effect; (

**c**) deformed configuration due to lateral loads; (

**d**) axial load in the element; (

**e**) moment diagram with implicit $P$-$\Delta $ effect in the stiffness matrix; (

**f**) moment diagram with implicit $P$-$\Delta $ effect in the stiffness matrix considering the amplification from the behavior factor.

**Figure 5.**Ten-story steel frame resisting structure [28]; (

**a**) sections; (

**b**) dead load $35kN/m$;(

**c**) live load $14.6\mathrm{kN}/\mathrm{m}$; (

**d**) deformed shape of the first mode $T=2.27\mathrm{s}$; (

**e**) deformed shape of the second mode $T=0.79\mathrm{s}$.

**Figure 6.**Illustration of the q factor and the ground acceleration influence in the stability factor, $\theta $ for a 10-story steel resisting frame.

**Figure 7.**Influence of the q factor and ground acceleration in the $P$-$\Delta $ effect of nine-story steel resisting frame structure. Illustration of different approaches for the estimation of the maximum acting moment in the story columns.

**Figure 8.**Floor plans and elevations of the considered 3-, 9-, and 20-story buildings [29].

**Figure 9.**Influence of the q factor and ground acceleration in the $P$-$\Delta $ effect of three-story steel resisting frame structure. Illustration of different approaches for the estimation of the maximum acting moment in the story columns.

**Figure 10.**Influence of the q factor and ground acceleration in the $P$-$\Delta $ effect of nine-story steel resisting frame structure. Illustration of different approaches for the estimation of the maximum acting moment in the story columns.

**Figure 11.**Influence of the q factor and ground acceleration in the $P$-$\Delta $ effect of 20-story steel resisting frame structure. Illustration of different approaches for the estimation of the maximum acting moment in the story columns.

**Figure 12.**The difference between the proposed method and the stability factor method in estimating the design resisting moment.

Structure | Period [s] | |
---|---|---|

Mode 1 (Present Study) | Range of Mode 1 From [29] | |

LA 3-Story | 0.9 s | 0.85–1.03 s |

LA 9-Story | 2.03 s | 1.97–2.34 s |

LA 20-Story | 3.65 s | 3.45–3.98 s |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Shehu, R.; Angjeliu, G.; Bilgin, H.
A Simple Approach for the Design of Ductile Earthquake-Resisting Frame Structures Counting for P-Delta Effect. *Buildings* **2019**, *9*, 216.
https://doi.org/10.3390/buildings9100216

**AMA Style**

Shehu R, Angjeliu G, Bilgin H.
A Simple Approach for the Design of Ductile Earthquake-Resisting Frame Structures Counting for P-Delta Effect. *Buildings*. 2019; 9(10):216.
https://doi.org/10.3390/buildings9100216

**Chicago/Turabian Style**

Shehu, Rafael, Grigor Angjeliu, and Huseyin Bilgin.
2019. "A Simple Approach for the Design of Ductile Earthquake-Resisting Frame Structures Counting for P-Delta Effect" *Buildings* 9, no. 10: 216.
https://doi.org/10.3390/buildings9100216