Doctoral Thesis / Dissertation, 2014

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Excerpt

DEDICATION

ACKNOWLEDGEMENTS

LIST OF FIGURES

LIST OF TABLES

ABSTRACT

CHAPTER 1. INTRODUCTION
1. Motivation
2. Objective and Scope
3. Organization of Dissertation

THIS DISSERTATION IS SUMMARIZED FROM JOURNAL ARTICLES

CHAPTER 2. SECOND-ORDER SPREAD-OF-PLASTICITY APPROACH FOR NONLINEAR

DYNAMIC ANALYSIS OF TWO-DIMENSIONAL SEMI-RIGID STEEL FRAMES

1. Introduction

2. Nonlinear Finite Element Formulation
2.2 Nonlinear Beam-to-Column Connections
2.2.1 Modified Tangent Stiffness Matrix including Nonlinear Connections
2.2.2 Moment-Rotation Relationship of Nonlinear Connections
2.2.3 Cyclic Behavior of Nonlinear Connections

3. Nonlinear Solution Procedures

4. Numerical Examples and Discussions
4.1 Portal Steel Frame subjected to Earthquakes
4.2 Two-Story Steel Frame with Nonlinear Connections
4.3 Vogel Six-Story Steel Frame with Nonlinear Connections - A Case Study

5. Summary and Conclusions

CHAPTER 3. SECOND-ORDER PLASTIC-HINGE APPROACH FOR NONLINEAR STATIC AND DYNAMIC ANALYSIS OF THREE-DIMENSIONAL SEMI-RIGID STEEL FRAMES

1. Introduction

2. Nonlinear Finite Element Formulation
2.1 Second-Order Plastic-Hinge Beam-Column Element
2.1.1 Stability Functions accounting for Second-Order Effects
2.1.2 Refined Plastic Hinge Model accounting for inelastic effects
2.1.3 Shear Deformation Effect
2.1.4 Element Stiffness Matrix accounting for P − Δ Effect
2.2 Semi-rigid Connection Element
2.2.1 Element Modeling
2.2.2 Semi-Rigid Connection Models for Rotational Springs
2.2.3 Cyclic Behavior of Rotational Springs

3. Nonlinear Solution Procedures
3.1 Nonlinear Static Algorithm
3.1.1 Formulation
3.1.2 Application
3.2 Nonlinear Dynamic Algorithm
3.2.1 Formulation
3.2.2 Application

4. Numerical Examples and Discussions
4.1 Static Problems
4.1.1 Vogel 2-D Portal Steel Frame
4.1.2 Stelmack Experimental 2-D Two-Story Steel Frame
4.1.3 Liew Experimental 2-D Portal Steel Frame
4.2 Dynamic Problems
4.2.1 Chan 2-D Two-Story Steel Frame
4.2.2 Vogel 2-D Six-Story Steel Frame
4.2.3 Chan 3-D Two-Story Steel Frame subjected to Impulse Forces
4.2.4 3-D Two -Story Steel Frame subjected to Earthquakes
4.2.5 Orbison 3-D Six-Story Steel Frame - A Case Study

5. Summary and Conclusions

CHAPTER 4. SECOND-ORDER SPREAD-OF-PLASTICITY APPROACH FOR NONLINEAR STATIC AND DYNAMIC ANALYSIS OF THREE-DIMENSIONAL SEMI-RIGID STEEL

FRAMES

1. Introduction

2. Nonlinear Finite Element Formulation
2.1.1 The Effects of Small P-delta and Shear Deformation
2.1.3 Element Stiffness Matrix accounting for the Effect of Large P-delta
2.2 Nonlinear Beam-to-Column Connection Element
2.2.1 Element Modeling
2.2.2 Cyclic Behavior of Rotational Springs

3. Nonlinear Solution Procedures
3.1 Nonlinear Static Algorithm
3.2 Nonlinear Dynamic Algorithm

4. Numerical Examples and Discussions
4.1 Static Problems
4.1.1 Vogel Portal Steel Frame
4.1.2 Stelmack Experimental Two-Story Steel Frame
4.1.3 Vogel Six-Story Steel Frame
4.1.4 Orbison Six-Story Space Steel Frame - A Case Study
4.2 Dynamic Problems
4.2.1 Portal Steel Frame subjected to Earthquakes
4.2.3 Space Six-Story Steel Frame - A Case Study

5. Summary and Conclusions

CHAPTER 5. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

1. Summary and Conclusions

2. Recommendations

REFERENCES

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## DEDICATION

This dissertation is dedicated to my parents, my wife, and my daughters.

## ACKNOWLEDGEMENTS

First of all, I would like to express my deep gratitude and respect to my advisor, Professor Seung-Eock Kim, for his guidance and support throughout my doctoral course in South Korea. The professional knowledge and research methodology that I learned from him have been useful in order to pursue my goal in life. In addition, I wish to thank Professors Seung-Eock Kim, Hyuk-Chun Noh, Jong-Jae Lee, Dong-Joo Kim, JaeHong Lee, Nam-Shik Ahn and Ki-Hak Lee for their valuable lectures and warm-hearted treatment towards me over the years.

Besides my adviser, I wish to thank my thesis committee members for their guidance and positive comments on my thesis. I greatly appreciate Professor Nguyen-Vu Duong, Director, John Von Neumann Institute - Vietnam National University HCMC, who firstly taught me scientific research methodology.

I am grateful for the assistance from all my friends in the Vietnamese Student Association and in the Bridge and Steel Structure Laboratory at Sejong University. I can never forget about what you have shared with me in happiness. I wish you all brilliant success in your future.

The financial support given by both Sejong University and the Bridge and Steel Structure Laboratory for my doctoral course is greatly appreciated. Finally, the great motivation that helped me to overcome obstacles during the course of this research was the endless encouragement and support from my family, especially my wife and my parents. I would like to thank deeply them for this invaluable sacrifice and love. I would like to dedicate this dissertation to my parents Van-Phu and My-Le, my lovely daughters Dan-Chau and Lam-Chau and my wife Ngoc-Chi.Seoul, June 2014

## LIST OF FIGURES

Fig. 2.1 Beam-column element modeling under arbitrary loads

Fig. 2.2 Meshing of beam-column element into n sub-elements

Fig. 2.3 Illustration of meshing of element cross-section and states of fibers

Fig. 2.4 Constitutive model is assumed for steel material

Fig. 2.5 Beam-column member including nonlinear connections with eight degrees of freedom

Fig. 2.6 Modified beam-column member with conventional six degrees of freedom

Fig. 2.7. The independent hardening model

Fig. 2.8. Earthquake records

Fig. 2.9. Portal frame subjected to earthquakes

Fig. 2.10. Displacement time-history responses of portal frame under Loma Prieta earthquake

Fig. 2.11. Displacement time-history responses of portal frame under San Fernando earthquake

Fig. 2.12. Second-order inelastic responses of portal frame with and without residual stress

Fig. 2.13. Cut section A-A and fiber no. 1 is being monitored

Fig. 2.14. Plastic deformation and stress at fiber no. 1 in cut section A-A in portal frame with and without residual stress

Fig. 2.15. Plastic deformation and stress at fiber no. 1 in cut section A-A in portal frame with and without residual stress

Fig. 2.16. Two-story steel frame with nonlinear connections

Fig. 2.17. Second-order elastic responses of two-story frame for various connection 49 Fig. 2.18. Second-order inelastic responses of two-story frame for various connections

Fig. 2.19. Hysteresis loops at connection C of two-story frame

Fig. 2.20. Vogel six-story steel frame with semi-rigid connections

Fig. 2.21. Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings - without bowing effects

Fig. 2.22. Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings - with bowing effects

Fig. 2.23. Moment-rotation responses at connection C of Vogel frame under forced loadings with ω = 1.66 rad/sec in the second-order elastic analysis

Fig. 2.24. Second-order inelastic displacement responses at roof floor of Vogel frame under El Centro earthquake considering geometric imperfections

Fig. 2.25. Moment-rotation responses at connection C of Vogel frame under El Centro earthquake considering geometric imperfections in the second-order inelastic analysis

Fig. 3.1 Full plastification surfaces

Fig. 3.2 Element end force and displacement notations

Fig. 3.3 Space connection element model with zero-length

Fig. 3.4. The independent hardening model

Fig. 3.5 General characteristic of nonlinear systems

Fig. 3.6 Characteristics of GSP

Fig. 3.7 Flow chart of the GDC algorithm

Fig. 3.8 Flow chart of the proposed algorithm for dynamic analysis

Fig. 3.9. Vogel portal frame with semi-rigid connections

Fig. 3.10. Load - displacement response of Vogel portal frame (PZ: plastic-zone method, PH: plastic-hinge method)

Fig. 3.11. Stelmack experimental two-story frame

Fig. 3.12. Moment - rotation behavior of Stelmack two-story frame

Fig. 3.13. Load - displacement response of Stelmack two-story frame

Fig. 3.14. Liew SRF3 portal frame

Fig. 3.15. Moment - rotation relations of SRF3 portal frame

Fig. 3.16. Load - displacement response of Liew SRF3 portal frame

Fig. 3.17. Orbison six-story space frame with semi-rigid connections

Fig. 3.18. Load - displacement response at Y direction of Orbison six-story frame ...

Fig. 3.19. Chan 2-D two-story steel frame

Fig. 3.20. Second-order elastic responses of 2-D two-story frame

Fig. 3.21. Second-order inelastic responses of 2-D two-story frame

Fig. 3.22. Hysteresis loops at the connection C for various analyses of 2-D two-story frame

Fig. 3.23. Geometry and loads of Vogel six-story frame

Fig. 3.24. Time-displacement response by second-order elastic analysis (ω = 1.00 rad/s) .. 114 Fig. 3.25. Time-displacement response by second-order elastic analysis (ω = 1.66 rad/s) ..

Fig. 3.26. Time-displacement response by second-order elastic analysis (ω = 2.41 rad/s) .. 115

Fig. 3.27. Time-displacement response by second-order elastic analysis (ω = 3.30 rad/s) .. 115

Fig. 3.28. Time-displacement response by second-order elastic analysis under sudden load during 1s: F1(t) = 10.23kN, F2(t) = 20.44kN

Fig. 3.29. Comparing time-displacement response of the three models ( ω = 1.66 rad/s) .. 116 Fig. 3.30. Comparing hysteresis loops of the three models at connection C ( ω = 1.66 rad/s)

Fig. 3.31. Dimensions and properties of Chan space two-story frame

Fig. 3.32. Time-displacement response at node A in nonlinear elastic analysis

Fig. 3.33. Moment-rotation curve of nonlinear strong-axis spring at connection C

Fig. 3.34. Moment-rotation curve of nonlinear weak-axis spring at connection C

Fig. 3.35. 3-D two-story frame subjected to earthquakes

Fig. 3.36. Nonlinear time-history responses of 3-D two-story frame under El Centro earthquake

Fig. 3.37. Nonlinear time-history responses of 3-D two-story frame under Northridge earthquake

Fig. 3.38. Nonlinear time-history responses of 3-D six-story steel frame under Northridge earthquake

Fig. 3.39. Nonlinear time-history responses of 3-D six-story steel frame under San Fernando earthquake

Fig. 3.40. Hysteresis loops of strong-axis rotational spring at connection C for various analyses of 3-D six-story steel frame subjected to El Centro earthquake

Fig. 3.41. Hysteresis loops of strong-axis rotational spring at connection C for various analyses of 3-D six-story steel frame subjected to San Fernando earthquake

Fig. 4.1 Discretization of beam-column element

Fig. 4.2 The ECCS residual stress pattern for I-section

Fig. 4.3 Modeling of space connection element with zero-length

Fig. 4.4. The independent hardening model

Fig. 4.5. Vogel portal frame with semi-rigid connections

Fig. 4.6. Load - displacement curve of Vogel portal rigid frame with and without shear deformation

Fig. 4.7. Load - displacement curve of Vogel portal semi-rigid frame

Fig. 4.8. Stelmack two-story frame

Fig. 4.9. Moment - rotation curve of connections by Stelmack experiment and curve fitting

Fig. 4.10. Load - displacement curve of Stelmack two-story frame

Fig. 4.11. Vogel six-story frame with semi-rigid connections

Fig. 4.12. Load - displacement curve of Vogel six-story rigid frame (PZ - Plastic Zone,

RPH - Refined Plastic Hinge)

Fig. 4.13. Moment-rotation curve of semi-rigid connections for Vogel six-story frame .. 173

Fig. 4.14. Load - displacement curve of Vogel six-story frame with different beam-to- column connections

Fig. 4.15. Orbison six-story space frame with semi-rigid connections

Fig. 4.16. Load - displacement curve at Y-direction node A of Orbison 3-D six-story frames

Fig. 4.17. Load - displacement curve at Y-direction node A of Orbison 3-D six-story frames with and without initial member out-of-straightness

Fig. 4.18. Portal frame subjected to earthquakes

Fig. 4.19. Displacement time-history responses of portal frame under Loma Priete earthquake

Fig. 4.20. Displacement time-history responses of portal frame under San Fernando earthquake

Fig. 4.21. Second-order inelastic responses of portal frame with and without residual stress

Fig. 4.22. Experimental two-story steel frame

Fig. 4.23. Comparison of moment-rotation curves at connection C of Stelmace experimental frame

Fig. 4.24. Nonlinear time-history responses of six-story space steel frame subjected to El Centro earthquake

Fig. 4.25. Nonlinear time-history responses of six-story space steel frame subjected to San Fernando earthquake

Fig. 4.26. Second-order inelastic displacement responses of six-story space steel frame under two earthquakes considering geometric imperfections

Fig. 4.27. Moment-rotation responses at connection C of six-story space steel frame under two earthquakes considering geometric imperfections

## LIST OF TABLES

Table 2.1 Peak ground acceleration and its corresponding time steps of earthquake records (PEER, 2011)

Table 2.2 Comparison of first two natural periods (s) along the applied earthquake direction of portal frame

Table 2.3 Comparison of peak displacements (mm) of portal frame

Table 2.4 Periods and Rayleigh damping coefficients of Vogel frame

Table 3.1 Curve-fitting connection parameters of Liew SRF3 portal frame

Table 3.2 Peak displacements (mm) of 2-D two-story steel frame

Table 3.3 Comparison of fundamental natural frequencies ω (rad/s)

Table 3.4 Comparison of fundamental natural frequencies ω (rad/s)

Table 3.5 Parameters of semi-rigid connections follow the Kishi-Chen model

Table 3.6 Comparison of first two natural periods (s) along applied earthquake direction of 3-D six-story steel frame

Table 3.7 Comparison of peak displacements (mm) at node A of 3-D six-story steel frame

Table 4.1 Comparison of ultimate load factor of Vogel portal frame

Table 4.2 Comparison of ultimate load factor of Vogel six-story rigid frame

Table 4.3 Parameters of connections for the Chen-Lui exponential model

Table 4.4 Parameters of semi-rigid connections follow the Kishi-Chen model

Table 4.5 Comparison of periods and Rayleigh damping coefficients of portal frame

Table 4.6 Comparison of peak displacements (mm) of portal frame

Table 4.7 Comparison of peak displacements (mm) at node A of six-story space steel frame

## ABSTRACT

Nguyen Phu Cuong

Dept. of Civil & Environmental Engineering The Graduate School

Sejong University

This dissertation presents three various advanced analysis approaches which can capture accurately and efficiently the ultimate strength and behavior of steel framed structures with nonlinear beam-to-column connections subjected to static and dynamic loadings. Three major sources of nonlinearity are considered in the analyses as follows:

(1) material nonlinearity; (2) geometric nonlinearity; and (3) connection nonlinearity. Three types of nonlinear beam-column element formulation considering both geometric and material nonlinearities are coded into two nonlinear structural analysis programs:
(1) Nonlinear Structural Analysis Program (NSAP) - 2-D plastic-zone finite element;
(2) Practical Advanced Analysis Program (PAAP) - 3-D refined plastic-hinge element and 3-D plastic-fiber element. Three types of steel frames analyzed by the proposed program are: (1) rigid frames - beam-to-column connections are fully rigid; (2) linear semi-rigid frames - beam-to-column connections have constant stiffness; and (3)

nonlinear semi-rigid frames - beam-to-column connections have continuously variable stiffness. Three types of analysis can be performed are: (1) nonlinear inelastic static analysis; (2) nonlinear elastic and inelastic time-history analysis; and (3) free vibration analysis. Three main resources of damping are taken into account in the proposed programs are: (1) hysteretic damping due to inelastic material; (2) structural viscous damping employing Rayleigh damping; (3) hysteretic damping due to nonlinear beam- to-column connections.

In the first approach - using the proposed program NSAP, a displacement-based finite element procedure for second-order spread-of-plasticity analysis of planar steel frames with nonlinear beam-to-column connections under dynamic and seismic loadings is developed. A partially strain-hardening elastic-plastic beam-column element, which directly takes into account geometric nonlinearity, bowing effects, gradual yielding of material, and flexibility of nonlinear connections, is proposed. The geometric nonlinearities are captured by using linear and Hermite interpolation functions through members are divided into several sub-elements. The spread of plasticity is considered by tracing the uniaxial stress-strain relationship of each fiber on the cross section of sub-elements. Nonlinear connections are simulated by zero-length rotational springs, which are taken into account by modifying the tangent stiffness matrix of the nonlinear beam-column element. A numerical procedure based on the combination of the Newmark numerical integration method and the Newton-Raphson equilibrium iterative algorithm is proposed for solving differential equations of motion. The nonlinear dynamic behavior predicted by the proposed program compare well with those given by the commercial finite element software ABAQUS and previous studies.

In the second and the third approaches - using the proposed program PAAP, for practical advanced analysis, the geometric nonlinearity caused by the interaction between axial force and bending moment is taken into account by using the stability functions, while the material nonlinearity is captured by using the refined plastic hinge model (the Refined Plastic-Hinge element - RPH) or using the spread-of-plasticity model (the Plastic-Fiber element - PF). The benefit of employing the stability functions and the refined plastic hinge model is that it can acceptably accurately capture the nonlinear effects by modeling one or two element per member, and hence this leads to a high computational efficiency compared to the finite element method using the interpolation functions. In some cases, it needs to obtain the higher level of accuracy, and it also gives more information for the member, the spread-of-plasticity model should be used. To capture accurately the spread-of-plasticity effect, the member is monitored through integration points along the member length and fibers on the cross- section. To consider the nonlinear behavior of semi-rigid connections, a spatial connection element with three translation springs and three rotation springs is developed to simulate the beam-to-column joint.

To solve nonlinear static equilibrium equations, the modified Newton-Raphson method and the generalized displacement control method is adopted herein because of their general numerical stability and efficiency. The generalized displacement control method can accurately trace the equilibrium path of nonlinear problems with multiple limit points and snap-back points. The modified Newton-Raphson method is utilized to apply fully static loads before the dynamic analysis procedure is executed. An incremental-iterative solution algorithm based on the Hilber-Hughes-Taylor method or the Newmark direct integration combined with the Newton-Raphson equilibrium iterative method is adopted for solving equations of motion.

Two computer programs are developed: (1) Nonlinear Structural Analysis Program (NSAP) - written in the C++ programming language; (2) Practical Advanced Analysis Program (PAAP) - written in the FORTRAN 77 programming language. They are verified for accuracy and computational efficiency by comparing predicted results with those generated by the commercial finite element analysis packages of ABAQUS and SAP2000, and other results available in the literature. Through several numerical examples, the proposed program (PAAP) proves to be a reliable and efficient tool for daily practice design in lieu of using costly and time-consuming commercial software.

## 1. Motivation

It is generally recognized that steel framed structures exhibit significantly nonlinear behavior prior to achieving their ultimate load-carrying capacity or instability. Thus, a second-order inelastic analysis or an advanced analysis is the most exactly result for predicting the real performance of steel framed structures instead of using conventional analysis/design approach. Advanced analysis can efficiently capture the ultimate strength and stability of a whole structural system and its component members so that separate member capacity checks encompassed by specification equations are no longer necessary. While greater complexity is introduced in the analysis, a significant reduction in effort is achieved in the design assessment. This may be accomplished through an efficient final checking of both member and system limit states for a structure where preliminary member sizing was based on serviceability requirements. For steel framed structures, advanced analysis methods can be generally classified into two categories of plastic hinge and plastic zone approaches based on the level of refinement used to represent yielding.

The beam-column member in the plastic hinge approaches is modeled by an appropriate way to eliminate its further subdivision, and the plastic hinges representing the inelastic behavior of material are assumed to be lumped at both ends of the member. The refined plastic hinge method is one of plastic hinge approaches. In this method, the inelastic behavior in the member is modeled in terms of member forces instead of the detailed level of stress and strain as used in the plastic zone analysis, the yielding is

evaluated based on a yielding surface criteria. The principal advantages of this method are that it is simple in formulation as well as implementation and more importantly, it is relatively accurate for the assessment of strength and stability of a structural system by using the one or two elements per member in the modeling.

In the plastic zone approach, the beam-column member is discretized into several finite sub-elements along the member length, and the cross section of each sub-element is divided into several small fibers, of which the uniaxial stress-strain relationships of material are monitored during the analysis process. This method performs the spread of plasticity throughout the cross section and along the member length. Although the solution of this method is considered the “exact” solution and easily included the effects of local, flexural-torsional, and lateral-torsional buckling which are significant characteristics of steel structures, it is generally recognized to be too expensive in computational time and computer resources because a very refined discretization of the structure is necessary. Therefore, it is usually applicable only for the research purpose or analyzing of special structural details.

In conventional analysis and design of steel frames, beam-to-column connections are often assumed to be either fully rigid or ideally pinned for simplicity. Experiments demonstrated that connections behave nonlinearly in a manner between the two extremes of perfectly rigid and frictionless pinned connection (Popov, 1983; Tsai and Popov, 1990; Nader and Astaneh, 1991; Tsai et al., 1995). This means that there is a finite degree of joint flexibility at connections. Such connections are called semi-rigid connections. Semi-rigid connections show a significantly and major source of nonlinearity in the structural behavior of steel frames under static and dynamic loadings.

Thus, the real behavior of connections needs to be simulated accurately for analyzing the global responses of a whole structural system.

A Practical Advanced Analysis Program (PAAP) based on the refined plastic hinge method for second-order inelastic static and dynamic analysis of space steel frames was successfully developed by Kim et al (Kim et al., 2006). It was proved to be accurate and time-efficient to use in daily practical design. Thai and Kim (Thai and Kim, 2009; Thai and Kim, 2011) developed a nonlinear static algorithm, a type of truss element, and a type of cable element for nonlinear inelastic static and dynamic analysis of steel structures. However, this program ignores the effects of nonlinear beam-to-column connections and spread-of-plasticity. Therefore, the nonlinear behavior of beam-to- column connections and spread of plasticity should be included in the PAAP program for analyzing accurately the strength and deformations of steel framed structures. Another disadvantage of the PAAP program is that its solution algorithm for dynamic problems is the Newmark average accelerate method without numerical dissipation. Also, to assure the numerical stability of complex nonlinear dynamic problems, the nonlinear solution algorithm should be improved. In this research, the above-mentioned limitations of the PAAP program are overcome.

## 2. Objective and Scope

The overall objective of this dissertation is to develop advanced analysis methods which can capture accurately and efficiently the strength and behavior of three- dimensional semi-rigid steel framed structures subjected to static and dynamic loadings. Three nonlinear elements included both geometric and material nonlinearities are implemented into the computer program (PAAP): (1) plastic-hinge beam-column element; (2) plastic-fiber beam-column element; and (3) multi-spring connection element. Three types of analysis provided in the proposed programs are: (1) nonlinear elastic and inelastic static analysis including rigid, linear semi-rigid, or nonlinear semi- rigid connections; (2) nonlinear elastic and inelastic time-history analysis including rigid, linear semi-rigid, or nonlinear semi-rigid connections; and (3) free vibration analysis. The proposed programs can be used to assess realistically both strength and behavior of steel framed structures and their component members in a direct manner.

Specific objectives of the research are as follows:

Develope a plastic-zone method using Hermite interpolation functions for second- order inelastic analysis of plane semi-rigid steel frames subjected to dynamic loadings. A computer program named NSAP (Nonlinear Structural Analysis Program) is developed.

Develope a plastic-fiber beam-column element using stability functions. This beamcolumn element is capable of capturing accurately the second-order spread-of-plasticity behavior of space steel frames.

Develope a general multi-spring element aiming to simulate nonlinear steel beam-to- column connections. This element can exhibit the behavior of rigid, pinned, linear semirigid, or nonlinear semi-rigid connections.

Improve the nonlinear dynamic algorithm of the PAAP program becoming the Hilber-Hughes-Taylor method (Hilber et al., 1977) because it possesses unconditional numerical stability and second-order accuracy. In addition, it can induce numerical damping in the nonlinear solution that is impossible with the Newmark average acceleration method.

The present studies are limited to steel framed structures including semi-rigid connections subjected to static and dynamic loadings. The element library of the proposed program (PAAP) is limited to “line” element including three nonlinear elements: plastic-hinge beam-column element, plastic-fiber beam-column element, and multi-spring connection element.

## 3. Organization of Dissertation

This dissertation includes five chapters. Contents of these chapters are as follows:

Chapter 1 introduces the motivation, objective, scope of this research and journal articles.

Chapter 2 presents a second-order distributed plasticity approach for nonlinear time- history analysis of planar semi-rigid steel frames using a developed program based on the C++ programming language (Nonlinear Structural Analysis Program - NSAP).

Chapter 3 presents a second-order plastic-hinge approach for nonlinear static and dynamic analysis of space semi-rigid steel frames using the PAAP program based on the FORTRAN 77 programming language (Practical Advanced Analysis Program - PAAP), this program is being developed by Bridge and Steel Structures Laboratory.

Chapter 4 presents a second-order spread-of-plasticity approach for nonlinear static and dynamic analysis of space semi-rigid steel frames using the PAAP program. In Chapter 5, summary and conclusions of the present work are made and directions for future work are recommended.

## THIS DISSERTATION IS SUMMARIZED FROM JOURNAL ARTICLES

Cuong Ngo-Huu, Phu-Cuong Nguyen and Seung-Eock Kim, Second-order plastic-hinge analysis of space semi-rigid steel frames, Thin-Walled Structures 60 (2012) 98-104.

Phu-Cuong Nguyen and Seung-Eock Kim, Nonlinear elastic dynamic analysis of space steel frames with semi-rigid connections, Journal of Constructional Steel Research 84 (2013) 72-81.

Phu-Cuong Nguyen and Seung-Eock Kim, An advanced analysis method for threedimensional steel frames with semi-rigid connections, Finite Elements in Analysis and Design 80 (2014) 23-32.

Phu-Cuong Nguyen and Seung-Eock Kim, Distributed plasticity approach for time-history analysis of steel frames including nonlinear connections, Journal of Constructional Steel Research 100 (2014) 36-49.

Phu-Cuong Nguyen and Seung-Eock Kim, Nonlinear inelastic time-history analysis of three-dimensional semi-rigid steel frames, Journal of Constructional Steel Research 101 (2014) 192-206.

Phu-Cuong Nguyen, Ngoc Tinh Nghiem Doan, Cuong Ngo-Huu, and Seung-Eock Kim, Nonlinear inelastic response history analysis of steel frame structures using plastic-zone method, Thin-Walled Structures 85 (2014) 220-233.

Phu-Cuong Nguyen and Seung-Eock Kim, Second-order spread-of-plasticity approach for nonlinear time-history analysis of space semi-rigid steel frames, Finite Elements in Analysis and Design 105 (2015) 1-15.

## 1. Introduction

Conventional designs usually assume that beam-to-column connections are fully rigid or ideally pinned. This assumption causes an inaccurate prediction of the seismic response of moment-resisting steel frames because the real moment-rotation relationship of connections is a nonlinear curve, and such connections are called semirigid connections. Several dynamic tests were carried out to investigate the ductile and stable hysteretic behavior of steel frames, which is one of the important features of semi-rigid connections under cyclic and seismic loadings (Azizinamini and Radziminski, 1989; Nader and Astaneh, 1991; Nader and Astaneh-Asl, 1992; Elnashai and Elghazouli, 1994; Nader and AstanehAsl, 1996; Elnashai et al., 1998).

In order to predict actual behavior of steel frames, especially in severe loading conditions, advanced analysis methods are employed. An advanced analysis must include key factors of steel frames such as geometric nonlinearities (P - large delta and P - small delta effects), plasticity of material, nonlinear connections, geometric imperfections (out-of-straightness and out-of-plumbness), residual stress, etc., simultaneously. There are two beam-column approaches for advanced analysis of steel frame structures: (i) the plastic hinge approach (concentrated plasticity) and (ii) the distributed plasticity approach (spread-of-plasticity). In the former approach, once yielding criteria is obtained, a plastic hinge will form at one of monitored points on the

member (usually at the two ends). This method is a computationally efficient and simple way to consider the effect of inelastic material. However, the hinge methods overpredict the limit strength of structures (King et al., 1992; White, 1993; White and Chen, 1993), which can also lead to unsafe designs. What's more, it may inadequately give information as to what is happening inside the member because the member is assumed to remain fully elastic between plastic hinges. On the other hand, by the distributed plasticity approach, yielding spreads throughout the whole length and depth of members. Therefore, the distributed plasticity method is more accurate than plastic hinge methods in capturing the inelastic behavior of frame structures under severe loadings.

In the last two decades, there have not been many analytical studies about the second-order inelastic dynamic behavior of steel frames with nonlinear semi-rigid connections (Gao and Haldar, 1995; Lui and Lopes, 1997; Awkar and Lui, 1999; Chan and Chui, 2000; Sekulovic and Nefovska-Danilovic, 2008). Gao and Haldar (1995) (Gao and Haldar, 1995) presented an efficient and robust finite-element-based method for estimating nonlinear responses of space structures with partially restrained connections under dynamic and seismic loadings. Lui and Lopes (1997) (Lui and Lopes, 1997) proposed an approach for dynamic analysis of semi-rigid frames using stability functions, the tangent modulus concept, and the bilinear model for capturing the effects of geometrical nonlinearities, inelastic behavior, and connection flexibility, respectively. In 1999, Awkar and Lui (Awkar and Lui, 1999) developed the method of Lui and Lopes (Lui and Lopes, 1997) for multi-story semi-rigid frames. Chan and Chui (2000) (Chan and Chui, 2000) published a book about static and dynamic analysis of semi-rigid steel frames, in which they proposed a spring-in-series model for simulating material plasticity and nonlinear connections; both plastic hinge and refined plastic-hinge methods are presented in detail. Recently, Sekulovic and Nefovska-Danilovic (2008) (Sekulovic and Nefovska-Danilovic, 2008) applied the refined plastic hinge method and the spring-in-series concept proposed by Chan and Chui (Chan and Chui, 2000) for transient analysis of inelastic steel frames with nonlinear connections; however, their study ignored the P-small delta effects. All the above mentioned studies utilized the plastic hinge methods. Thus, analytical researches about the second-order distributed plasticity analysis of semi-rigid steel frames under dynamic loadings are uncommon.

In this paper, a sophisticated second-order spread-of-plasticity method proposed by Foley and Vinnakota (Foley and Vinnakota, 1997; Foley and Vinnakota, 1999; Foley and Vinnakota, 1999) for static analysis is developed for nonlinear inelastic time-history analysis of plane semi-rigid steel frames. An elastic-perfectly plastic model with linear strain hardening is applied to establish a new nonlinear element tangent stiffness matrix based on the principle of stationary potential energy. Accurately, to capture the second- order effects and spread of plasticity, each frame member is divided into many sub- elements along the member length and the cross-section depth. The tangent stiffness matrix of the nonlinear beam-column element directly takes into account the effects of geometric nonlinearity, gradual yielding, and flexibility of nonlinear connections. Nonlinear connections are simulated by zero-length rotational springs. The moving of the strain-hardening and elastic neutral axis, which are due to gradual yielding of the cross-section, is directly included in the element tangent stiffness matrix, and this effect is updated during the analysis process. The bowing effect, geometrical imperfections, and residual stress are also considered in this study. Three major sources of damping are

integrated in the same analysis. They are structural viscous damping, hysteretic damping due to nonlinear connections, and hysteretic damping due to material plasticity. A numerical procedure using the Newmark average acceleration method (Newmark, 1959) and the well-known Newton-Raphson iterative algorithm is proposed to solve nonlinear equations of motion. Several numerical examples are performed to illustrate the accuracy, validity, and features of the proposed second-order inelastic dynamic analysis procedure for steel frames with nonlinear flexible connections.

## 2. Nonlinear Finite Element Formulation

### 2.1 Second-Order Spread-of-Plasticity Beam-Column Element

illustration not visible in this excerpt

Fig. 2.1 Beam-column element modeling under arbitrary loads

Investigation of a typical beam-column member subjected to loads is plotted in Fig. 2.1. In order to capture the distributed plasticity, the beam-column member is divided into n elements along the member length as illustrated in Fig. 2.2; each element is divided into m small fibers within its cross section as illustrated in Fig. 2.3; and, each

fiber is represented by its material properties, geometric characteristic, area A j , and its coordinate location (y j z, j ) corresponding to its centroid. This way, residual stress is directly considered in assigning an initial stress value for each fiber. The second-order effects are included by the use of several sub-elements per member through updating of the element stiffness matrix and nodal coordinates at each iterative step.

illustration not visible in this excerpt

Fig. 2.2 Meshing of beam-column element into n sub-elements

To reduce the computational time when assembling the structural stiffness matrix and solving the system of nonlinear equations, n sub-elements are condensed into a typical beam-column member with the six degrees of freedom at the two ends by using the static condensation algorithm derived by Wilson (Wilson, 1974). A reverse condensation algorithm is used to find the displacements along the member length for evaluating the effects of distributed plasticity and the second-order effects.

In the development of the second-order spread-of-plasticity beam-column element, the following assumptions are made: (1) the element is initially straight and prismatic;

(2) plane cross-sections remain plane after deformation and normal to the deformed axis of the element; (3) out-of-plane deformations and the effect of Poisson are neglected;

(4) shear strains are negligible; (5) member deformations are small, but overall structure displacements may be large; (6) residual stress is uniformly distributed along the member length; (7) yielding of the cross-section is governed by normal stress alone; (8) the material model is linearly strain-hardening elastic-perfectly plastic; and, (9) local buckling of the fiber elements does not occur. In this study, an elastic-perfectly plastic stress-strain relationship with linearly strain hardening used by Toma and Chen (Toma and Chen, 1992) is adopted as shown in Fig. 2.4. Strain hardening starts at the strain of

illustration not visible in this excerpt

illustration not visible in this excerpt

(2.2)

The total internal strain energy of a partially strain-hardening elastic-plastic beamcolumn element can be expanded as

Fig. 2.3 Illustration of meshing of element cross-section and states of fibers

The normal stresses corresponding to the strain state of fibers are calculated as follows:

illustration not visible in this excerpt

Fig. 2.4 Constitutive model is assumed for steel material

where ε is the normal strain at any fiber within a cross section, σ is the normal stress at any fiber within a cross section, E is the elastic modulus for the material, E is the

sh

strain-hardening modulus of the material, V is the volume of fibers corresponding to their states within a cross section of an element, and subscripts e, p (y), sh stand for elastic, plastic, and strain-hardening states of fiber elements, respectively. Fig. 2.3

illustrates cross-section partitions with fiber states, in which

d and d are the

CGe CGsh

shift of the center of the initial neutral axis and the distance from the initial neutral axis to the strain-hardening neutral axis created by fibers in the strain-hardening regime, respectively.

Replacing the integrations over the volume of the element in Eq. (2.3) by integrating along the length and throughout the cross section of the element, Eq. (2.3) is expressed as

illustration not visible in this excerpt

(2.4)

where A e is the remaining elastic area, A p is the yielding area, A sh is the strain-

hardening area within a cross section, and L is the length of the element.

The normal strain of the assuming beam-column element can be predicted by the

following strain-displacement relationship (Goto and Chen, 1987)

illustration not visible in this excerpt

where u is a function describing the longitudinal displacements along the element, v is a function describing the transverse displacements, and dx is an infinitesimal length of element. In this formulation, linear shape functions and cubic Hermite shape functions are employed for longitudinal displacements and transverse displacements, respectively.

illustration not visible in this excerpt

(2.7)

Substituting Eq. (2.5) into Eq. (2.4), the total internal strain energy of the partially strain-hardening elastic-plastic beam-column element is written as

illustration not visible in this excerpt

[...]

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Title
Course
9.5
Author
Year
2014
Pages
234
Catalog Number
V304587
ISBN (eBook)
9783668031371
ISBN (Book)
9783668031388
File size
6694 KB
Language
English
Notes
Phu-Cuong Nguyen was born in Vietnam on 04-Nov-1983. He graduated B.Eng and M.Eng in Civil Engineering at Ho Chi Minh City University of Technology. He got his Ph.D in Structural Engineering and had been a Postdoctoral Fellow in the Department of Civil and Environmental Engineering at Sejong University, South Korea. His research interests have been in the areas of finite element analysis (linear/non-linear), advanced analysis of steel structures, structural connection performance, reliability-based design using advanced analysis and genetic algorithm, structural viscous damping modeling.
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Phu Cuong Nguyen (Author), 2014, Advanced Analysis for Three-Dimensional Semi-Rigid Steel Frames subjected to Static and Dynamic Loadings, Munich, GRIN Verlag, https://www.grin.com/document/304587

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