- : Format
- : Pages
- :
- : Chapters
- Click to DOWNLOAD Materials
APPLICATION OF SHEAR DEFORMATION THEORY IN THE ANALYSIS OF THICK RECTANGULAR PLATES USING POLYNOMIAL DISPLACEMENT FUNCTIONS
ABSTRACT
This research work presents the application of shear deformation theory in the analysis of thick rectangular plates with various boundary conditions using polynomial displacement functions. Total of twelve boundary conditions were considered and they are SSSS, CCCC, CCSS, CSCS, CCCS, CSSS, SSFS, CCFC, CSFS, SCFS, SCFC and CCFS. Total potential energy equation of rectangular thick plate was formulated from first principle based on the theory of elasticity. This equation was subjected to direct variation to obtain three simultaneous governing equations for the determination of displacement coefficients. The obtained governing simultaneous equations herein are typical of Hooke’s law. Shape (profile) equation for vertical shear stress through the thickness of the plate was formulated from first principle. From this profile equation, the deformation line equation (called function of z or s) was obtained in line of the works of Timoshenko and Woinowsky-Krieger 1970. The stiffness coefficients of thick plate with various boundary contions were determined using polynomial displacement functions. A functional excel worksheet was developed to ease the numerical analysis of thick plates with various boundary conditions. Numerical problems for plate with various boundary conditions were carried out using the developed functional excel worksheet. The obtained results for stiffness coefficient functions from this study agree with the values from previous works on classical plate theory. The result for displacements and in plane stresses decrease with increase in span to depth ratio (α) while the out of plane stresses increase with increase in span to depth ratio (α). However, the obtained non-dimensional values of vertical shear stress (̅𝝉̅𝒙𝒛̅̅ ) were used to delineate the boundary between thick and thin plate based on span to depth ratio. The values of non-dimensional vertical shear stress (̅𝝉̅𝒙𝒛̅̅ ) between span to depth ratios (α) of 40 and 1000 were approximately constant and equal to values obtained from classical plate theory (CPT), therefore this can be idealized to be thin plate. There were minimal variations of the values of the vertical shear
stress (̅𝝉̅𝒙𝒛̅̅ ) of plate, whose span to depth ratio (α) falls between 20 and 30 and the values differ from that of classical plate theory, so the plate can be taken to be moderately thick. Furthermore, values of the vertical shear stress (̅𝝉̅𝒙𝒛̅̅ ) of plate whose span to depth ratio (α) falls between 4 to 15 vary significantly with span to depth ratio. Therefore, the plate can be taken to be thick. The obtained results from present study for vertical shear stress (̅𝝉̅𝒙𝒛̅̅ ) for a square plate of span to depth ratio of 4, 10, 100 and 1000 are 0.3906, 0.3920, 0.3920 and 0.3923 respectively while the results from previous works are 0.3849, 0.3909, 0.3909 and 0.3921. A close look at the results show that the results from the present study are in excellent agreement with that of previous works, hence the study herein is reliable and adequate for thick plate analysis.
Keywords: shear deformation, vertical shear stress, deflection, displacement, potential energy, shape function, stiffness coefficient.
TABLE OF CONTENTS
CERTIFICATION II
DEDICATION III
ACKNOWLEDGEMET IV
LIST OF TABLES VIII
ABSTRACT XXXIII
1.4 Significance of the Study 7
CHAPTER TWO: LITERATURE REVIEW 9
2.1 Plate Types and Theories: 9
2.2 Classical Plate Theory: 11
2.3 Thick Plate Theories / Shear Deformation PlateTheories: 12
2.3.1 First Order Shear Deformation Theories 12
2.3.2 Higher Order Shear Deformation Theories: 14
2.4 Polynomial Shape Functions 14
2.5 Previous Works on Higher Order Shear Deformation Theories (HSDT): 16
3.1 Formulation of Direct Governing Equations 39
3.1.2 Strain – displacement relations (Kinematic relations) 42
3.1.3 Constitutive Relations 44
3.1.4 Stress – displacement equations 44
3.1.5 Total Potential Energy 45
3.1.6 Direct Governing Equation 48
3.2 Formulation of Polynomial Shear Deformation Function F(z) 50
3.3 Determination of Stiffness Coefficient Using Polynomial Displacement Functions 52
3.3.1 All Edges Simply Supported (SSSS) Rectangular Plate 52
3.3.2 All Edges Clamped (CCCC) Rectangular Plate 58
3.3.3 Clamped – Clamped – Simply – Simply Supported (CCSS) Rectangular Plate 63
3.3.4 Clamped – Simply Supported – Clamped – Simply Supported (CSCS) Rectangular Plate 68
3.3.5 Clamped – Clamped – Clamped – Simply Supported (CCCS) Rectangular Plate 73
3.3.6 Clamped – Simply – Simply – Simply Supported (CSSS) Rectangular Plate 78
3.3.7 Simply – Simply Supported – Free – Simply Supported (SSFS) Rectangular Plate 83
3.3.8 Clamped – Clamped –Free Edge – Clamped (CCFC) Rectangular Plate 92
3.3.9 Clamped – Simply – Free Edged – Simply Supported (CSFS) Rectangular Plate 98
3.3.10 Simply Supported – Clamped – Free Edged – Simply Supported (SCFS) Rectangular Plate 104
3.3.11 Simply Supported – Clamped – Free Edged – Clamped (SCFC) Rectangular Plate 110
3.4 Determination of Displacements and Stresses in Rectangular Thick Plate 123
3.5 Excel Worksheet for the Analysis of Thick Plate 127
CHAPTER FOUR: RESULTS AND DISCUSSION 129
4.1 Presentantion of Results 129
4.1.1 Direct Governing Equation 129
4.1.2 Polynomial Shear Deformation Functions 130
4.1.3 Stiffness Coefficient Functions of Rectangular Thick Plates 130
4.1.4 Displacements and Stresses of Rectangular Thick Plates: 134
4.1.4.1 Displacements and Stresses of Rectangular Thick SSSS Plates 135
4.1.4.2. Displacements and Stresses of Rectangular Thick CCCC Plates 136
4.1.4.3 Displacements and Stresses of Rectangular Thick CCSS Plates 137
4.1.4.4 Displacements and Stresses of Rectangular Thick CSCS Plates 138
4.1.4.6 Displacements and Stresses of Rectangular Thick CSSS Plates 140
4.1.4.7 Displacements and Stresses of Rectangular Thick SSFS Plates 141
4.1.4.8 Displacements and Stresses of Rectangular Thick CCFC Plates 142
4.1.4.9 Displacements and Stresses of Rectangular Thick CSFS Plates 143
4.1.4.10 Displacements and Stresses of Rectangular Thick SCFS Plates 144
4.1.4.11 Displacements and Stresses of Rectangular Thick SCFC Plates 145
4.1.4.12 Displacements and Stresses of Rectangular Thick CCFS Plates 146
4.1.5 Functional Excel Worksheet for Rectangular Thick Plate Analysis 147
4.2.1 Direct Governing Equation: 147
4.2.2 Polynomial Shear Deformation Functions: 148
4.2.3 Stiffness Coefficient Functions of Rectangular Thick Plates: 148
4.2.4 Displacements and Stresses of Rectangular Thick Plates: 150
4.2.5 Functional Excel Worksheet Program for Rectangular Thick Plate Analysis: 155
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS 157
5.3 Contribution to Knowledge: 158
CHAPTER ONE INTRODUCTION
1.1 Background of Study
Plates have wide application in engineering constructions, specifically in aeronautical, mechanical, marine, and civil engineering for the construction of aircraft, bridges, ships, vehicles, satellites, platforms, tug boats, badges, building floors and roofs, silos, telecom mask, shear walls, computer hard-disk drives and other complex structures (Birman, 2011; Volmir, 1963; and Amabili, 2008). A plate is a very important structural element that has been in existence for ages. A plate is a flat structural element having two surfaces bound together by a plane which is known as plate thickness (t). The thickness of a plate (t) is small compared with the in-plane surface dimensions ‘a’ and ‘b’ (Shufrin et al., 2008). The thickness is usually constant but may be variable and is measured normal to the middle surface of the plate. When the plate thickness is divided equally by a plane parallel to its surface, this plane is referred to as the middle surface (Ugural, 1999, Ezeh et al., 2013), (see Figure 1.1).
Figure 1.1: An Element of a Rectangular Thick Plate showing Middle Surface dimensions
The rectangular plate has four edges and these edges are numbered from the top in anti clockwise direction as shown in Figure 1.2.
Figure 1.2: Rectangular plate with edge numbering
In engineering applications, plates are often constrained at the edges with different boundary conditions. The boundary conditions of a rectangular plate are simply supported, clamp support and free of support. The designation and symbol for the edge conditions are as shown in Figure
1.3.
Edge Condition
Free Edge (F)
Simply Support (S)
Clamped Edge (C)
Figure 1.3: Edge conditions of rectangular plate showing sections and plan view
The combination of these boundary conditions in a plate results to plates of different boundary conditions as shown on Figure 1.4.
Figure 1.4: Plan view of rectangular plates with various boundary conditions
Plates as structural elements/members are subjected to transverse loads. These loads are normal to the plate’s midplane. Plates resist transverse loads by means of bending. Pure bending forces are usually applied perpendicular to the plane of the plate only. Therefore a plate resists the applied load by means of bending in two directions and twisting.
Plates may also be subjected to in-plane loading or direct forces which act in the middle plane of the plate’s surface as shown in Fig. 1.5. These applied forces have significant contribution to the bending of plates. These forces and their corresponding stresses are known as the in-plane forces and the in-plane stresses respectively.
Figure 1.5: An Element of a Rectangular Thick Plate Subjected to Applied Load
The aim of plate theory is to evaluate the deformations and stresses in a plate subjected to loads. A flat plate, like a straight beam carries lateral load by bending. The analyses of the effect of load on plates based on thickness to breadth ratio, can be categorized into two namely: thick plate and thin plate analyses. If the thickness to width ratio of the plate is less than 0.05 (that is t/a < 0.05) and the maximum deflection is less than one tenth of the thickness (that is w/t < 0.1), then the plate is classified as a thin plate. When the thickness to width ratio is more than 0.05 (t/a 0.05), the plate is taken as a thick plate and the effect of shear deformation is included in its analysis.
In the classical plate theory, it is assumed that the plane cross sections initially normal to the plate’s midsurface before deformation remain plane and normal to the midsurface after deformation. This is the result of neglecting the transverse shear strains. However, in thick and moderately thick plates, significant transverse shear strains occur (Reisner, 1945), and the theory gives inaccurate results for thick plates. So, it is obvious that the shear strains have to be taken into account. There are numerous theories of plates that include the transverse shear strains. One of them is the Reissner and Midlin theory, known as the first-order shear deformation theory, which defines the displacement field as linear variations of midplane displacements.This theory, in which the relationship between the resultant shear forces and the shear strains is obtained by using shear correction factors, has some advantages due to its simplicity and low computational cost. Some other plate theories, namely the higher-order shear deformation theories, include the effect of transverse shear strains.
For example, the theory developed by Reddy (1984) allows not only for the transverse shear strains, but also for parabolic variations in the strains across the plate thickness, and thus there is no need to use shear correction coefficients in computing the shear stresses. Refined plate theory assumed that the plane cross sections that are initially normal to the plate midsurface before bending are no longer normal and straight to the midsurface after bending due to transverse shear deformation. Hence, it takes into consideration the effect of transverse shear strain. Works on refined plate theory have been characterized by the use of trigonometric displacement function. Many scholars have obtained the closed form solutions and others have obtained approximate solution using assumed displacement functions in energy method. However, one thing that is common in them all is the use of trigonometric displacement functions to approximate the deformed shapes of the plates. (Chikalthankar et al., 2013; Sayyad, 2011; Akavci, 2007; Sayyad & Ghugal, 2012; Sadrnejad et al., 2009; Daouadji et al., 2013; Hashemi & Arsanjani, 2005; Reddy, 2014; Shimpi & Patel, 2006; Murthy, 1984; Daouadji, et al., 2012; Zhen-qiang, et al.1994). Others have applied the assumed polynomial displacement functions in numerical methods like finite element method and differential quadrature element methods (Matikainen et al., 2009; Goswami & Becker, 2013, Wu & Liu, 2001).
1.2 Statement of Problem
The main reason engineers resort to thin plate analysis in the face of its numerous short coming is because of the complexity involved in handling double Fourier series of thick plate analysis. The idealization of a thick plate as a thin plate by most engineers because the difficulty of handling double Fourier series of thick plate analysis always underestimates the stresses in the plate. The consequencies of using these erroneous stresses in design and construction is structural failure and sometimes total collapse. Previous researchers have delved into different aspects of thick plate analysis such as: pure bending (Ghugal & George, 2010; Sayyad & Ghugal, 2012; Sayyad et al., 2016; etc), buckling (Avalos & Larondo, 1995; Wang et al., 2001; Kim al., 2009; Ibearugbulem et al., 2014; etc), free vibrations (Guruwamy & Yang, 1979; Gupta & Ansari, 1998; Wu & Liu; 2001; Sayyad & Ghugal, 2012; etc), isotropic plates (Raju & Rao, 1996; Sayyad, 2011; Sayyad & Ghugal 2012, etc), orthotropic plates (Gupta & Lai, 1985; Shimpi & Patel, 2006; Chikathanker et al., 2013; etc), anisotropic plates (Murthy, 1984; Setoodeh & Karami, 2004; Azhari & Kassaei, 2004; etc), graded laminated plates (Karama & Mistov, 2003; Goswami & Becker, 2013; Daouadji et al., 2013, Reddy, 2014; etc). One common observation is that most of these works are based mainly on trigonometric displacement functions. In the course of the development of refined plate theory, the assumption that the shear deformation line does not vary linearly with the depth of the plate was introduced. This according to many scholars helps to ensure that the vertical shear stress across the plate section does not remain constant, but varies parabolically with zero values at both the top and bottom surfaces (Ambartsumian, 1958; Murty, 1984; Touratier, 1991; Karama & Mistou, 2003). They came up with different shear deformation line functions, here-in-after called F(z). However, their F(z) functions were not strictly based on the vertical shear stress mathematical formulation. If we follow the work of Timoshenko and Woinowsky-krieger, (1970), we shall note that maximum shear stress occurs at the mid surface (where z = 0) and the value of the maximum shear stress is one and half of nominal vertical shear stress (1.5 [0.5qa/t] , where q is the normal uniform distributed load on the plate). With most of the F(z) functions from the literature, we may obtain good profile (curve) for the deformation line and shear stress distribution across the section, but the midsurface value of shear stress may not coincide with that by Timoshenko and Woinowsky-krieger, (1970). One can scarely see work on thick plate based on polynomial dispacement functions.This gap in literature is worth filling, hence the need for this present research work, which presents a new polynomial shear deformation function for thick plates analysis.
1.3 Objective of Study
The main objective of this study is the application of shear deformation in the analysis of thick plates using polynomial displacement functions with various boundary conditions. To acheive this main objective, the following specific objectives are set aside:
- To develop a direct governing simultaneous equations for thick plate analysis.
- To mathematically formulate a polynomial shear deformation function, F(z) in line with the works of Timoshenko and Woinowsky-Krieger (1970).
- To determine the stiffness coefficients of thick plate using polynomial displacement functions.
- To obtain the in-plane and out of plane stresses of rectangular thick plate.
- To develop a functional excel worksheet programm for the analysis of thick plates.
1.4 Significance of the Study
The importance of thick plate and the need to provide more economic and simplified approach of thick plate analysis is the driving factor of this research work. The integration of double Fourier series is quite involving unlike orthogonal polynomials that can be easily integrated. The use of this approach (orthogonal polynomial) will eliminate the difficulties associated with the use of double Fourier series in analyzing thick plates. Polynomial displacement functions can be used successfully to solve any boundary condition of thick rectangular plate; a feat that could not be easily achieved using trigonometric shape functions.
This work will contribute in addressing the problem of dearth of literature in the use of polynomial displacement functions. Scholars and practicing engineers can now assess and apply this simplified approach of analyisis of thick plates to carry out analysis and designs. It will enable scholars/engineers to generate financial benefit because they now have better and simplified approach to anlyze thick plates and carry out design of steel structures, which previously they contracted out to few knowledgeable individuals. This will inturn increase the demand of thick plates and hence boost the economy
1.5 Scope of Study
This study is limited to the bending analysis of rectangular isotropic thick plates using thick plate theory and its assumptions and Ritz energy method. Twelve boundary conditions of rectangular thick plates were considered. The boundary conditions are: (i) all four edges simply supported (ssss), (ii) all four edges clamped (cccc), (iii) adjacent edge clamped and the other adjacent edge simply supported (ccss) and (iv) opposite edge clamped and the other opposite edge simply supported (cscs). Others are (v) three edges clamped and one edge simply supported (cccs), (vi) one edge clamped and other three edges simply supported (csss), (vii) free of support at the third edge and the other three edges simply supported (ssfs) and (viii) free of support at third edge and the other edges clamped (ccfc). The remaining cases are: (ix) clamped at first edge, simply supported at second and fourth edges and free of support at the third edge (csfs), (x) simply supported at first and fourth edge, clamped at second edge and free of support at third edge (scfs), (xi) simply supported at first edge, clamped at second and fourth edge and free of support at third edge (scfc) and (xii) clamped at first and second edge, free of support at third edge and simply supported at fourth edge (ccfs). The in-plane and out of plane displacements (u, v and w) and stresses (x, y, τxy and τxz) were determined for the twelve boundary conditions.
Furthermore, this work deloveloped a functional excel worksheet that eases the analysis of thick rectangular plates.
APPLICATION OF SHEAR DEFORMATION THEORY IN THE ANALYSIS OF THICK RECTANGULAR PLATES USING POLYNOMIAL DISPLACEMENT FUNCTIONS
