ABSTRACT
This work developed a numerical model for determining the dynamic response of a concert high-rise flexible frames (tall frame) subjected to wind load in Nigeria. The model was based on the finite element idealization for flexible frames, static condensation and lumped mass parameters. A MATLAB program was developed for the model and two investigations were carried out. Firstly, an investigation was carried out to determine the level of significance of the dynamic response for design purposes in Nigeria. To this end, the developed MATLAB program was used to determine the dynamic and static responses of four structures, of various heights (29.2m to 72.4m) and floors (8 to 20) respectively. An assumed wind velocity of 15m/s was considered for safety reasons since the actual maximum wind velocity in Nigeria is less than 10m/s. The responses considered included floor translations, axial force, bending moment and shear force. The percentage difference between static and dynamic response as well as their level of significance were determined. Based on the results it was inferred that dynamic analysis of tall building below 72.4m in height and width not exceeding 13m, could be ignored in Nigeria provided that the static analysis is done properly, with appropriate code provisions. Secondly, the effect of increase in width and height on the fundamental natural frequency of tall frames was also investigated, requiring the addition of three frames, namely Frame 5, Frame 6, Frame 7 all of which have the same height of 29.2m but vary in width from 13m to 28m. Analysis results showed that the fundamental natural frequency increases with increase in width at constant height and decrease with increase in height at constant width.
TABLE OF CONTENT
Cover Text i
Title Page ii
Declaration iii
Certification v
Dedication iv
Acknowledgements vi
Table of Contents vii
List of Tables x
List of Figures xii
Abstract xiii
CHAPTER 1: INTRODUCTION
1.1 Background of Study 1
1.2 Statement of the Problem 2
1.3 Aim and Objectives 4
1.4 Scope of Work 4
1.5 Justification of Study 5
1.6 Limitation of Study 5
CHAPTER 2: LITERATURE REVIEW
2.1 Previous Reviews 6
2.2 Dynamic Loads 7
2.3 Wind Load Variation with Height 8
2.4 Difference Between Static Pressure, Total Pressure, and Dynamic Pressure 8
2.5 Dynamic Effects of Wind on Building 9
2.6 Degrees of Freedom 11
2.7 Natural Frequency and Factors Affecting It 12
2.8 Static Condensation 12
2.9 Finite Element Method 12
2.10 Lumped Parameter 13
2.11 Force of Inertia 14
2.12 MATLAB Computer (PC) Program 14
2.13 Important Deductions from Literature Review 15
CHAPTER 3: MATERIALS AND METHODS
3.1 Methodology 17
3.2 Wind Dynamics 17
3.2.1 Dynamic pressure 17
3.2.2 Vortex shedding of wind 18
3.2.3 Strouhal number 18
3.2.4 Drag and lift 19
3.2.5 Wind gust 19
3.2.6 Forcing frequency θ 19
3.2.7 Dynamic wind force on a building 20
3.2.8 Standard drag coefficient Cd for a building 20
3.3 Design Wind Load 20
3.4 Equations of Motion for a MDOF Flexible Frame using
Static Condensation 22
3.5 Free Vibration Analysis of high-rise Flexible Frames 24
3.6 Forced Vibration Analysis 25
3.7 Algorithm for Program Development 26
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Application of the Developed Numerical Model 28
4.2 Numerical Examples 29
4.3 Example 1 30
4.3.1 Building description 32
4.4 Wind Load Analysis and Direction 34
4.5 Design Wind Speed (Vs) 35
4.5.1 Dynamic wind pressure (q) 36
4.5.2 Dynamic wind load (F(t)) 36
4.5.3 Wind load distribution 37
4.6 MATLAB Program Validation 38
4.7 MATLAB Program Development 38
4.8 Results of the Numerical Example 1 39
4.8.1 Natural frequency (ω_i) for example 1 39
4.8.2 Displacements results for example 1 41
4.8.3 Internal stresses results for example 1 44
4.9 Example 2 52
4.10 Parametric Study and Comparison of Analysis Results 54
4.10.1 Effect of building height on natural frequencies 54
4.10.2 Effect of building width on natural frequencies 56
4.10.3 Horizontal translation effects 57
4.10.4 Internal stresses and percentage difference 58
4.10.4.1 Static and dynamic stresses result for frame 1 (8 Floors) 58
4.10.4.2 Static and dynamic stresses result for frame 2 (12 Floors) 59
4.10.4.3 Static and dynamic stresses result for frame 3 (16 Floors) 60
4.10.4.4 Static and dynamic stresses result for frame 4 (20 Floors) 61
4.11 Discussion of Results 62
CHAPTER 5: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 64
5.2 Recommendations 66
REFERENCES 67
APPENDIX 1: MATLAB computer program file for the study 70
APPENDIX 2: Uniformly distributed (UDL) calculation for each floor respectively 74
APPENDIX 3: Axial force checks 75
APPENDIX 4: Moment checks 76
LIST OF TABLES
4.1: Geometrical and mechanical properties of the analytical model for example1 32
4.2a: Natural frequencies ω_i (Rad/Sec x 103) 40
4.2b: Natural frequencies ω_i (√EI ) 40
4.3a: Results of static analysis of flexible frame 1 41
4.3b: Result of dynamic analysis of flexible frame 1 42
4.4a: Result of internal stresses for static analysis of frame1 44
4.4b: Result of internal stresses for dynamic analysis of frame 1 48
4.5a: Short overview of the analytical structures in height 52
4.5b: Short overview of the analytical structures in width 53
4.6a: Effect of building height on natural frequencies ω_i (rad/sec x 103) 54
4.6b: Effect of building height on natural frequencies ω_i (√EI ) 55
4.7a: Effect of building width on natural frequencies ω_i (rad/sec x 103) 56
4.7b: Effect of building width on natural frequencies ω_i (√EI ) 56
4.8: Comparison of horizontal translation 57
4.9a: Static stresses result for frame 1 (8 Floors) 58
4.9b: Dynamic stresses result for frame 1 (8 Floors) 58
4.9c: Percentage difference in frame 1 (8 floors) 58
4.10a: Static stresses result for frame 2 (12 Floors) 59
4.10b: Dynamic stresses result for frame 2 (12 Floors) 59
4.10c: Percentage difference in frame 2 (12 Floors) 59
4.11a: Static stresses result for frame 3 (16 Floors) 60
4.11b: Dynamic stresses result for frame 3 (16 Floors) 60
4.11c: Percentage difference in frame 3 (16 Floors) 60
4.12a: Static stresses result for frame 4 (20 Floors) 61
4.12b: Dynamic stresses result for frame 4 (20 Floors) 61
4.12c: Percentage difference in frame 4 (20 Floors) 61
LIST OF FIGURES
4.1: The model configuration for the study 30
4.2: The model for the study with detailed numbered elements and nodes 31
4.3: Wind action on the analytical structure 34
4.4: Wind load distribution on the analytical structure 37
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF STUDY
Tall frame constitutes the main load bearing segment of tall structure. Aside from static loads, structural frames are likewise subjected to dynamic forces which have their potential sources from earthquakes, blast, impact loads, moving loads, wind and different aggravations or other disturbances (Osadebe et al, 2005). The basic feature of every dynamic disturbances is that they create vibrations in the structure whereupon they act. Consequently, prerequisite to the design of such a structure is a good insight into its vibration movements and, in particular, the natural frequency (ω), (Anya et al, 2005). Generally, as the height of a building increases, its overall response to lateral load (such as wind and earthquake) increases. At the point when such response becomes sufficiently great such that the effect of lateral load must be explicitly taken into consideration in design, a multistory structure is said to be tall (Chinwuba, 2011).
Wind is a dynamic and random phenomenon in both time and space (Boggs, 2015). Wind in general exerts forces and moments on the building and its cladding and eventually distributes the air in and around the structure mainly named as wind pressure. Wind is characterized by its strength and direction of blowing. High speed winds or breezes of short duration are called Gusts (Swami et al, 2012). Wind impacts on buildings continue to pose danger that continues to attract the attention of scientist and researcher round the globe. This is because of its trend of impact that changes with time, a dynamic issue, and inadequate information on the response of building to wind action (Zhang et al., 1993). Wind approaching a structure is a complex phenomenon and response in various directions (Tharaka, 2017).
Tall frames are basically systems with infinite degrees of freedom. However, few simplifications are made in their dynamic investigations by considering them as multi degree of freedom (MDOF) system or framework (Anya et al, 2005). Basically, calculating wind loads is significant in design of wind force-resisting system or framework, including structural members, components, and cladding, against shear, sliding, overturning, and uplift actions (Huang, 2001).
In general, not much work has been carried out in Nigeria, on the dynamic response of high-rise buildings subjected to wind. More so, application of MATLAB to such analysis is not common in Nigeria. Hence the present study will develop a numerical model for determining the dynamic response of high-rise framed structures subjected to wind excitation using the finite element method with static condensation and lumped mass parameter. MATLAB is the software of choice.
1.2 STATEMENT OF THE PROBLEM
Wind is the most widely recognized dynamic excitation in the world, yet it is not given the desired consideration in Nigeria with regards to dynamic analysis and design of structures. It has the greatest effect on high rise buildings otherwise known as multi-degrees of freedom frames (MDOF).
Given a multi-degrees of freedom frame (MDOF), with masses lumped at the floor levels and subjected to dynamic wind load, the following parameters are considered for equilibrium of the system: effective wind load F(t), the relative displacements d(t), velocity d ̇(t) and the acceleration d ̈(t) of the masses m_j.
The forces acting on the system or frame include: md ̈ the force of inertia due motion of the masses, cd ̇ the damping force, kd the elastic resisting force from structural members, and F(t) the external force (dynamic wind load). The general differential equation of motion for forced vibration of the damped system is obtained by the algebraic sum of the four forces. For maximum response of the system, damping is neglected (cd ̇=0) and so the forced vibration response (displacements and internal stresses) is determined from the undamped equation of motion for forced vibration. The differential equation of motion for free vibration of the undamped system is obtained by setting F(t)=0, cd ̇=0, to obtain md ̈+kd=0, and solved as an eigenvalue problem to obtain the natural frequencies of the undamped structure.
The dynamic wind load F(t) is a product of the drag coefficient C_D, dynamic pressure q, and the reference area A. The dynamic pressure q is related to the wind velocity V by q=1/2 ρV^2, where ρ is the mass density of air. The drag coefficient is obtained from design tables.
The acceleration d ̈(t) is a function of the vortex-shedding frequency θ, (also known as the excitation frequency of the wind) which depends on the wind velocity V, Strouhal number S, and the width (or diameter) D of the structure (member).
The parameters outlined above shall be synthesized in a numerical model for the determination of the dynamic response of high rise framed structures or buildings subjected to wind excitation using the finite element method with static condensation and lumped mass parameter.
1.3 AIM AND OBJECTIVES
The principle aim of this research work is to develop a numerical model for determining the dynamic response of high-rise framed structures subjected to wind excitation using the finite element method with static condensation and lumped mass parameters.
The specific objectives achieve are:
i. To develop a numerical model using the FEM.
ii. To apply the developed numerical model using the FEM to this study.
iii. To validate the model by comparing analytical results obtained with the developed MATLAB program to those in published literature or results obtained with other methods.
1.4 SCOPE OF WORK
The extent of work here includes the determination of the dynamic response of tall frames with multi degree of freedom (MDOF) subjected to wind load. The static response will be considered for comparison with the dynamic responses. However, stability analysis is outside the scope of this work, resonance effect is not ignored. Natural frequencies, displacements and internal stresses responses are to be considered.
1.5 JUSTIFICATION OF STUDY
This research work has many significances around the world including Nigeria as a whole, because there is high wind storms occurrence than earthquake and just few studies of wind load analysis has been done around this area in Nigeria.
Moreover, just few studies have been done on this field in Nigeria utilizing MATLAB software. Hence, this present study will pave a way for further research works in this area using MATLAB for programming.
1.6 LIMITATION OF STUDY
The limitations of this study are:
i. The numerical model considers linear elastic responses. In effect, it can not be used for non-linear analysis.
ii. Experimental studies will not be used to validate the model, based on the cost of procuring the sophisticated equipment needed for such experiment.
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