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APPLICATION OF A DIRECT VARIATIONAL PRINCIPLE IN ELASTIC STABILITY ANALYSIS OF THIN RECTANGULAR FLAT PLATES
ABSTRACT
The purpose of this work is to use the energy approach in the form of direct variational principle (Rayleigh-Ritz method) for buckling analysis of thin rectangular plates with various boundary conditions using Taylor series shape functions. To do this, thin rectangular flat plate of various boundary conditions with three dimensions Ly, Lx and t was analyzed in this research. Ly and Lx are secondary and primary in-plane dimensions respectively and t is the plate thickness. The boundary conditions covered in this research included SSSS, SSSC, SSCC, SCCC, SCSC, CCCC, SSSF, CCCF and SCFS plates. Rayleigh-Ritz method of direct variational approach for the plate analysis was adopted. The total potential energy functional of the method was derived from first principle by using equations and principles of theory of elasticity. Taylor-MacLaurin’s series was used to formulate the approximate shape functions for the plate with various boundary conditions. The shape functions from TaylorMacLaurin’s series were substituted into the total potential energy functional, which was subsequently minimized to get the stability equations. Derived Eigen-value solver was used to solve the stability equations for plates of various aspect ratios (from 0.1 to 1 at the increment of 0.1) to get the buckling loads of the plates. The buckling loads from this study were compared with those of earlier researches. The results showed that the average percentage differences recorded for SSSS, CCCC, CSCS, CSSS, and SSFS plates are 0.069%, 3.54%, 3.071%, 6.25% and 4.14% respectively. The convergence of the shape function showed that for
CSCS plate, the difference between the buckling loads when the TaylorMacLaurin’s series were truncated at m = n = 4 and m = n = 5 is 1.11%. This difference is 0.878% for CCSS plates. These differences showed that the shape functions formulated by using Taylor-McLauruin’s series has rapid convergence and very good approximation of the exact displacement functions of the deformed thin rectangular plate under in-plane loading.
TABLE OF CONTENT
CERTIFICATION II
APPROVAL III
DEDICATION IV ACKNOWLEDGEMET V
ABSTRACT VI
LIST OF FIGURES X
LIST OF TABLES XI
DEFINITION OF NOTATIONS XII
CHAPTER ONE 1
Introduction 1
1.1 Background of study 1
1.2 Statement of problem 6
1.3 Objective of study 7
1.4 Justification / contribution of the study 7
1.5 Scope of study 8
CHAPTER TWO 11
Literature Review 11
2.1 History of thin plates 11
2.2 Previous works on thin plates 14
2.3 Boundary conditions 18
2.4 Elastic buckling analysis of thin plates 19
2.4.1 Equilibrium approach 20
2.4.2 Energy approach 21
2.4.3 Direct variational principle 22
2.4.4 Rayleigh-ritz method 23
2.5 Taylor-mclaurin series 24
2.6 Buckling mode of a rectangular plate 26
2.7 Eigen-value problem 28
- 8 Solutions of rectangular thin plates of various boundary conditions. 30
CHAPTER THREE 39
Method 39
3.1 Total potential energy functional 39
3.2 Displacement function 60
3.2.1 Displacement function for ssss plate 64
3.2.2 Displacement function for cccc plate 66
3.2.3 Displacement function for cscs plate 67
3.2.4 Displacement function for csss plate 70
3.2.5 Displacement function for ccsc plate 72
3.2.6 Displacement function for ccss plate 73
3.2.7 Displacement function for csfs plate 76
3.2.8 Displacement function for ssfs plate 78
3.2.9 Displacement function for ccfc plate 79
3.3 Total potential energy functional for ssss plate 80
3.4 Total potential energy functional for cccc plate 82
3.5 Total potential energy functional for cscs plate 84
3.6 Total potential energy functional for csss plate 90
3.7 Total potential energy functional for ccsc 92
3.8 Total potential functional for ccss plate 94
3.9 Total potential functional for csfs plate 101
3.10 Total potential functional for ssfs plate 104
3.11 Total potential functional for ccfc plate 107
3.12 Minimization of total potential energy 110
3.12.1 Stability equation for ssss plate 111
3.12.2 Stability equation for cccc plate 112
3.12.3 Stability equation for cscs plate 113
3.12.4 Stability equation for csss plate 116
3.12.5 Stability equation for ccsc plate 116
3.12.6 Stability equation for ccss plate 118
3.12.7 Stability equation for csfs plate 121
3.12.8 Stability equation for ssfs plate 121
3.12.9 Stability equation for ccfc plate 123
3.13 Matrix iterative-inversion 128 3.14 Summary of stability equations 135
Program D (program for cscs plate) 137
Program E (program for ccss plate) 143
Program A (program for csfs plate) 147
Program B (program for scsf plate) 150
Program C (program for ccfc plate) 152
CHAPTER FOUR 155
Result and Discussion 155
4.1 Result of ssss thin rectangular flat plate 155
4.2 Result of cccc thin rectangular flat plate 157
4.4 Result of scsc thin rectangular flat plate 161
4.5 Result of csss thin rectangular flat plate 162
4.6 Result of scss thin rectangular flat plate 164
4.8 Result of ssfs thin rectangular flat plate 165
4.9 Result of csfs thin rectangular flat plate 166
4.10 Result of ccfc thin rectangular flat plate 167
4.11 Convergence test of the displacement function 168
4.11.1 Convergence of displacement fuction for cscs 168
4.11.2 Convergence of displacement fuction for ccss 169
4.12 Result of iteration matrix inversion method 170
CHAPTER FIVE 172
Conclusions and Recommendations 172
5.1. Conclusions 172
5.2 Recommendations 173
REFERENCES 174
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND OF STUDY
Three approaches are used in the solution of elastic stability analysis of thin plates. They are the equilibrium (Euler) approach, the energy (approximate) approach and the numerical approach. The Euler approach tends to find solution of the governing differential equation by direct integration and satisfying the boundary conditions of the four edges of the plate. The rectangular plate has four edges and the numbering of the edges is shown in figure 1:
Figure 1.1: Rectangular plate with edge numbering
Some of the boundary conditions of the edges of a rectangular plate are:
S designates simply support
C designates clamped support
F designates free support
A rectangular plate is unique from the other by the conditions of its four edges. SCFS for instance means that edges 1and 4 are simply supported, edge 2 is clamped and edge 3 is free of support. Various rectangular plates, which are distinct from one another, are shown in figure 1.2.
Direct integration leads to stability equation, from which solution is obtained. This solution is called the exact solution. Unfortunately, the method can only be used to analyze plates that are simply supported along the four edges, which is SSSS plate. When one of the edges is not simply supported, the approach becomes very tedious and difficult. In other words, it can be extremely difficult to analyze a plate that has at least one of its edges not simply supported along the four edges. Consequently, the use of other approaches has become very necessary and imperative.
SSSS plate CCCC plate
SSSC plate SSSF plate
SSCC plate CCCF plate
SCCC plate
SCFS plate
SCSC plate
Figure 1.2: Plates of various boundary conditions
Numerical approach is a good alternative to the Euler approach. Some examples of this approach include truncated double Fourier series, finite difference, finite strip, Runge-Kutta and finite element methods among others. Methods of numerical approaches have the capacity of handling plates of various boundary conditions. It has been shown from past works that in most cases, the solution from numerical approach approximate closely those of the exact approach (Ventsel and Krauthammer, 2001). The problem with these numerical solutions is that the accuracy of the solution is dependent on the amount of work to be done. For instance, if one is using finite element method, the more the number of elements used in the analysis the closer the approximate solution to the exact solution. Hence, when a plate has to be divided into several elemental plates for an accurate solution to be reached, then the extensive analysis is involved, requiring enormous time to be invested. Outside the time input, extensive analysis and the great volume of data generated will be difficult to condense into design charts and tables. Although, the data storage capacity of a microcomputer is high, the volume may exceed the memory of a particular computer, with the result that the computer goes out of memory. In view of this, band width control becomes necessary in matrix formulation. A sound knowledge in mathematics and skilful experience in computer programming is inevitable in this case. At this point one will see vividly that the problem one is trying to avoid in equilibrium approach is still found in numerical approach.
Energy approach is another method that can be used. This approach is quite different from Euler and numerical approaches. The solution from it agrees approximately with the exact solution. Typical examples of energy approaches are Ritz, Raleigh-Ritz; Garlekin, minimum potential energy etc. These methods are called variational methods. They seek to minimize the total potential energy functional in order to get the stability matrix. This functional is a function of plate deflection function. The accuracy of the solution is dependent on the accuracy of the approximate deflection function (shape function). Approximate shape function is substituted in the total potential energy functional, and the resulting equation is partially differentiated. The total potential energy will be said to be minimized when its partial derivative is equated to zero. This implies that the difference between the approximate and exact solutions is zero (Iyengar, 1988).
The more the approximate shape function gets closer to the exact shape function the more the approximate solution gets closer to the exact solution. Many scholars have used trigonometric series in this approach. For instance, trigonometric series can be used to formulate approximate shape function for a plate, whose four edges are simply supported or clamped. It can also be used for a plate whose opposite edges are clamped and the other opposite edges are simply supported. However, it is extremely difficult to formulate a shape function for plates using trigonometric series when opposite edges are clamped and simply supported like propped cantilever beams. Examples of plates, whose shape function can not be formulated using trigonometric series, include SSCC plate, SCCC plate, CSSS plate, SSCF plate, CSSF plate etc. Some other boundary conditions make it difficult to use the trigonometric series (Ugural, 1999, Iyengar, 1988, Ventsel and Krauthammer, 2001).
Because of these limitations of energy method using trigonometric series to formulate the approximate shape functions, one will be tempted to use the numerical approach. In the light of the above problems, researches in thin plate buckling are going on so as to obtain solutions that are very close approximation to exact solution, and at the same time reduce the volume of computation.
Consequently, this research sought to formulate shape functions using Taylor series. For different cases with different boundary conditions, Taylor series are used to obtain approximate shape functions. The resulting approximate shape functions are substituted into the total potential energy functional which is then minimized to get the stability equations.
The stability equations are, in most cases, Eigen-value matrix equations involving consistent mass as against lumped mass. Polynomial method of Eigen-value problem is the only method from literature review that can effectively handle
Eigen-value equations containing consistent mass. However, solution using Polynomial method becomes intractable when the size of the matrix is up to 4 x 4.
It is in attempt to address these problems that gave birth to the research topic
“Application of a Direct Variational Principle in Elastic Stability Analysis of Thin Rectangular Flat Plates”.
1.2 STATEMENT OF PROBLEM
The problems of this research are stated as:
- Dir
ect integration is somewhat easy only for the condition of SSSS plate. It is
very difficult for any other plate that is not SSSS plate. Direct integration results in exact solutions.
ii. Pre
vious research works had formulated shape functions by using trigonometric
functions in energy approach. The problem with trigonometric functions is
that they can not be used to formulate shape functions for CSSS plate, CCSC plate, and CCSS plate.
- In
the light of the above problems, numerical approaches were inevitable. The
problem of the numerical methods is the amount of work involved in the formulation, and expertise in the use of computer. The accuracy of the
numerical methods depends on the number of finite units the plate is divided
into.
- Wit
h all these problems, the present research will use energy approach in the form of a direct variational principle (Rayleigh-Ritz Method). The approximate shape function will be formulated by using Taylor series as against using trigonometric functions.
1.3 OBJECTIVE OF STUDY
The objectives of this proposed research include the following:
- To use Taylor series in Raleigh-Ritz method for buckling analysis of thin rectangular flat plate.
- To develop a new Eigen-value solver that can handle consistent matrix form of Eigen-value.
- To find how close the obtained solutions of plate buckling analysis using Taylor-MacLaurin’s series are to exact solutions.
1.4 JUSTIFICATION/CONTRIBUTION OF THE STUDY
This research has
- Contributed in addressing problem of dearth of literature in the use Taylor series approximation of shape functions.
- Contributed in addressing problem of dearth of literature for eigenvalue solvers to effectively handle consistent mass eigenvalue problem of a matrix size of up to 4 x 4 and above.
- Exposed the potentials of the use of Taylor series as against trigonometric series in analysis of plates and shells problems. Provided Solution for CSSS plate, CCSC plate, CCSS plate and CCFC plate using Raleigh-Ritz method and Taylor series.
1.5 SCOPE OF STUDY
A flat rectangular thin plate has three dimensions Ly, Lx and t. Where Ly and Lx are respectively secondary and primary in-plane dimensions and t is the plate thickness. Lx/t is used to classify a plate as thick, stiff, thin or membrane. If the ratio, Lx/t is less than ten (10) then the plate is thick. If the range, 10 ≤ Lx/t ≤ 100 holds then the plate is thin. If the ratio, Lx/t is greater than hundred (100) then the plate is a membrane. This research studied thin plates. A plate can be flat (of uniform thickness) or of varying thickness. The plate can also be rectangular, circular or any other polygonal shape. However, this research was concerned with rectangular shape. The various boundary conditions covered in this research included
SSSS, SSSC, SSCC, SCCC, SCSC, CCCC, SSSF, CCCF and SCFS plates
After defining the types of plate analyzed in this study, the stages involved in the course of executing the project were stated as thus.
Literature review and internet search were made in this proposed research. This enabled the proposed research to discover some unanswered theoretical questions, and know the extent past scholars had gone in this direction. This was to avoid repetition of a study that had been made before now. Chapter two under the title of “literature review” handled this. The literature review shall be followed by the formulation of total potential energy functional from the first principle by using the equations and principles theory of elasticity. The substitution of various boundary conditions as concerned various cases of rectangular thin plates into the Taylor series to approximate various shape functions followed. These shape functions were substituted into the formulated total potential energy functional. The resulting total potential energy functionals for various plates of different boundary conditions were minimized. This minimization gave the stability equations for various plates. All of these were done in chapter three under the title of “method”. Solution of various plates by substituting the various aspect ratios (from 0.1 to 1 at the increment of 0.1) into the stability equations was the next thing that was done. The resultant solutions were compared with the solutions from the use of trigonometric series where available. Comparison was also made with some known exact solutions. Some available results from numerical methods were used to compare with the solutions of this research. General discussions and comments were made as concerned the comparisons. All these were done in chapter four under the title “results”. The next thing that was done is drawing of conclusions based on the results and making sundry recommendations. This was treated in chapter five under the title of “conclusion”.
APPLICATION OF A DIRECT VARIATIONAL PRINCIPLE IN ELASTIC STABILITY ANALYSIS OF THIN RECTANGULAR FLAT PLATES
