The P-delta effect, which is a second order effect, is experienced by a structure mostly when the structure is subjected to lateral forces like earthquake and wind. In addition to earthquake and wind loads, when the structure is subjected to various live load patterns other than symmetrical loading, the structure experiences side sway or lateral translation. In this thesis, P-delta effects due to symmetrical and asymmetrical live loads is studied on a three-bay, two-story steel moment frame. The main objective of this research is to investigate P-delta effects on the column which is subjected to critical load case. The effect is analyzed based on the value of the moment magnification factor, B2. Besides, the story drift due to asymmetrical loading is also investigated. For the purpose of this study, “Approximate Second-Order Analysis” specified in “Appendix 8” of AISC Steel Construction Manual (AISC 2011) is used to amplify the first order effects to account for the second-order effects in the steel moment frame. Additionally, computer software is used to calculate the first order moments and axial forces.

Symmetrical steel moment frames that are subjected to sidesway forces due to unsymmetrical live loads will undergo sidesway. The P-delta effects on a moment frame under the influence of sidesway forces is studied. The effective length method is used for the second-order analysis specified in the American Institute Steel Construction - Load and Resistance Factor Design (AISC-LRFD). This study investigates the P-delta effects on a multi-story, multi-bay steel moment frame subjected to sidesway forces caused by various unsymmetrical live load patterns. The study focuses on the interaction of axial and bending moment in the columns. The actual response of a moment frame is estimated by amplifying the results of a first-order elastic analysis using moment magnification factors. The moment magnification factors for each story of the steel moment frame are summarized.

During the months I have been at Virginia Tech, I have experienced the most exciting time of my life. There are many persons who helped me to pursue my Master's degree. I would like to take this opportunity to express my appreciations to them. I would like to thank my advisor and committee chairman, Dr Finley A. Charney. He has supported me for the entire duration of this project with all of his efforts. Without his assistance, it would have been very difficult for me to learn so many concepts in the field of nonlinear dynamic analysis and practical earthquake engineering. I would also like to acknowledge my other committee members, Dr. Raymond Plaut and Dr. W. Samuel Easterling, for taking the time to review the thesis and providing valuable insights and feedback on this thesis. I would like to thank my father, Chan Kwok Fung, who encouraged me to pursue my Master Degree, and my mother, Yu Yuk Ping, who brought me to life. I would like to thank my sister, Doris Chan, and my girlfriend, Ka Man Chan, for supporting and encouraging me during the past two years at Virginia Tech. Finally, I would like to give thanks to the rest of my family, friends, professors, and fellow graduate students for their help and encouragement during my stay at Virginia Tech. / Master of Science

The main objective of the thesis is evaluating and comparing Second-Order Elastic Analysis Methods defined in two different specifications, AISC 2005 and TS648 (1980). There are many theoretical approaches that can provide exact solution for the problem. However, approximate methods are still needed for design purposes. Simple formulations for code applications were developed, and they are valid as acceptable results can be obtained within admissible error limits. Within the content of the thesis, firstly background information related to second-order effects will be presented. The emphasis will be on the definition of geometric non-linearity, also called as P-&amp / #948 / and P-&amp / #916 / effects. In addition, the approximate methods defined in AISC 2005 (B1 &ndash / B2 Method), and TS648 (1980) will be discussed in detail. Then, example problems will be solved for the demonstration of theoretical formulations for members with and without end translation cases. Also, the results obtained from the structural analysis software, SAP2000, will be compared with the results acquired from the exact and the approximate methods. Finally, conclusions related to the study will be stated.

This thesis presents the results of basically two separate studies. The first study involved identifying structural and earthquake parameters that influenced seismic structural collapse. The parameter study involved nonlinear dynamic analyses using single-degree-of-freedom (SDOF) bilinear models. Four parameters were associated with the SDOF models — the lateral stiffness, the post-yield stiffness ratio, the yield strength, and the stability ratio (P-Delta effects). Then, three parameters were associated with the ground motions — the records themselves, the lateral ground motion scales, and the vertical ground motion scales. From the parameter study, it was found that the post-yield stiffness ratio augmented by P-Delta effects (rp) in conjunction with the ductility demand was the best predictor of collapse. These two quantities include all four structural parameters and the seismic displacement demands. It was also discovered in the parameter study that vertical accelerations did not significantly influence lateral displacements unless a given combination of model and earthquake parameters was altered such that the model was on the verge of collapsing. The second study involved Incremental Dynamic Analysis (IDA) using bilinear SDOF models representative of low rise buildings in both the Western United States (WUS) and the Central and Eastern United States (CEUS). Models were created that represented three, five, seven, and nine story buildings. Five sites from both the WUS and CEUS were used. Four different damage measures were used to assess the performance of the buildings. The IDA study was primarily interested in the response of the structures between the earthquake intensities that have a 10 percent probability of occurring in 50 years (10/50) and 2 percent probability of occurring in 50 years (2/50). The results showed that all structures could be in danger of severe damage and possible collapse, depending on which damage measure and which earthquake was used. It is important to note, though, that the aforementioned is based on a damage-based collapse rule. The damage-based rule results were highly variable. Using an intensity-based collapse rule, proved to be more consistent. Due to the nature of the bilinear models, only those structures with negative rp values ever collapsed using an intensity-based collapse rule. Most of the WUS models had positive rp values and many of the CEUS models had negative rp values. While many of the CEUS structures had negative rp values, which made them prone to collapse, most of the CEUS structures analyzed did not collapse at the 2/50 intensity. The reason was that the periods of the CEUS models were much longer than the approximate periods that were required to determine the strengths. Consequently, the strength capacity of most of the CEUS models was much greater than the seismic strength demands. While many of the CEUS models did have sudden collapses due to the large negative rp values, the collapses happened at intensities that were generally much higher than the 2/50 event. In the IDA, it was also shown that vertical accelerations can significantly affect the ductility demands of a model with a negative rp post-yield stiffness ratio as the earthquake intensity approaches the collapse intensity. Since IDA is concerned with establishing collapse limit states, it seems that the most accurate collapse assessments would include vertical accelerations. / Master of Science