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Super finite elements for nonlinear static and dynamic analysis of stiffened plate structures Koko, Tamunoiyala Stanley 1990

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SUPER FINITE ELEMENTS FOR NONLINEAR STATIC AND DYNAMIC ANALYSIS OF STIFFENED P L A T E STRUCTURES By T A M U N O I Y A L A S T A N L E Y K O K O B. Sc. (Hons), University of Ife, Nigeria, 1982 M . Eng., University of Nigeria, Nsukka, Nigeria, 1986 A THESIS SUBMITTED IN T H E REQUIREMENTS D O C T O R OF PARTIAL FULFILLMENT OF FOR T H E DEGREE OF P HILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1990 © T A M U N O I Y A L A S T A N L E Y K O K O , 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract The analysis of stiffened plate structures subject to complex loads such as air-blast pressure waves from external or internal explosions, water waves, collisions or simply large static loads is still considered a difficult task. The associated response is highly nonlinear and although it can be solved with currently available commercial finite element programs, the modelling requires many elements with a huge amount of input data and very expensive computer runs. Hence this type of analysis is impractical at the preliminary design stage. The present work is aimed at improving this situation by introducing a new philosophy. That is, a new formulation is developed which is capable of representing the overall response of the complete structure with reasonable accuracy but with a sacrifice in local detailed accuracy. The resulting modelling is relatively simple thereby requiring much reduced data input and run times. It now becomes feasible to carry out design oriented response analyses. Based on the above philosophy, new plate and stiffener beam finite elements are developed for the nonlinear static and dynamic analysis of stiffened plate structures. The elements are specially designed to contain all the basic modes of deformation response which occur in stiffened plates and are called super finite elements since only one plate element per bay or one beam element per span is needed to achieve engineering design level accuracy at minimum cost. Rectangular plate elements are used so that orthogonally stiffened plates can be modelled. The von Karman large deflection theory is used to model the nonlinear geometric behaviour. Material nonlinearities are modelled by von Mises yield criterion and associated flow rule using a bi-linear stress-strain law. The finite element equations are derived using the virtual work principle and the matrix quantities are evaluated by n Gauss quadrature. Temporal integration is carried out using the Newmark-/? method with Newton-Raphson iteration for the nonlinear equations at each time step. A computer code has been written to implement the theory and this has been applied to the static, vibration and transient analysis of unstiffened plates, beams and plates stiffened in one or two orthogonal directions. Good approximations have been obtained for both linear and nonlinear problems with only one element representations for each plate bay or beam span with significant savings in computing time and costs. The displacement and stress responses obtained from the present analysis compare well with experimental, analytical or other numerical results. 111 Table of Contents Abstract ii List of Tables ix List of Figures xiv Acknowledgements xv 1 Introduction 1 2 Literature Review 4 2.1 Analytical Methods 4 2.2 Numerical Methods 6 2.2.1 Finite Difference and Finite Element Methods 6 2.2.2 Finite Strip Method 8 3 Description of the Super Finite Elements 9 3.1 Introduction 9 3.2 Super Finite Element Discretization 11 3.3 Displacement Functions 14 3.3.1 Plate elements 15 3.3.2 Beam elements 18 3.4 Compatibility, Convergence and Order of Accuracy 21 4 Theoretical Formulation and Analysis of Problem 23 4.1 Introduction 23 iv 4.2 Equations of Mot ion 23 4.3 Finite Element Formulation 25 4.3.1 Introduction 25 4.3.2 Shape Function Matrices 26 4.3.2.1 Plate elements 26 4.3.2.2 Beam elements 28 4.3.3 Strain Displacement Relations 29 4.3.4 Constitutive Relations 30 4.3.5 JMass and Damping Matrices 34 4.3.6 Stiffness Formulation 35 4.3.7 Load Vector 37 4.3.8 Torsion Beam Element 39 4.3.8.1 Stiffness matrix for beam torsional element 40 4.3.8.2 Mass matrix for beam torsional element 44 4.4 Numerical Integration 46 4.5 Temporal Integration 48 4.6 Computer Code 51 5 Static Analysis Results 53 5.1 Introduction 53 5.2 Unstiffened Plates 54 5.2.1 Square Plate I with Simply Supported Edges 54 5.2.2 Square Plate I with Clamped Edges 59 5.3 Beams 65 5.3.1 Rectangular Beam with Simple Supports 65 5.3.2 Rectangular Beam with Clamped Ends . 72 5.4 Stiffened Plates 73 5.4.1 Clamped 2-Bay Stiffened Plate I 73 v 5.4.2 Clamped DRES Stiffened Panel 79 5.4.3 Clamped DRES1B Stiffened Panel 95 5.4.4 Simply Supported 2x2-Bay Stiffened Plate I 106 6 Vibration Analysis Results 120 6.1 Introduction 120 6.2 Unstiffened Plates 121 6.2.1 Square Plates with Various Edge Conditions 121 6.2.2 3-Bay Continuous Plate 123 6.2.3 2 x 2-Bay Continuous Plate . 124 6.3 Beams 125 6.3.1 Rectangular Beams with Various Boundary Conditions . . . . 125 6.4 Stiffened Plates 127 6.4.1 2-Bay Stiffened Plate II with Clamped Boundaries 127 6.4.2 3-Bay Stiffened Panel with Clamped Edges 130 6.4.3 DRES Stiffened Panel 133 6.4.4 2x 2-Bay Stiffened Plate 134 6.4.5 2x4-Bay Stiffened Plate with Clamped Edges 135 6.4.6 4x4-Bay Stiffened Plate 137 7 Transient Analysis Results 141 7.1 Introduction 141 7.2 Unstiffened Plates 142 7.2.1 Square Plate I with Simply Supported Edges 142 7.2.2 Square Plate I with Clamped Edges 144 7.2.3 Simply Supported Square Plate II Subject to Triangular Load 146 7.3 Beam Example 149 7.3.1 Rectangular Beam with Simple Supports 149 vi 7.4 Stiffened Plates 151 7.4.1 Clamped 2-Bay Stiffened Plate II 151 7.4.1.1 Step Load 152 7.4.1.2 Blast Load 154 7.4.2 Simply Supported 2-Bay Stiffened Plate II 155 7.4.2.1 Step Load 156 7.4.2.2 Blast Load 158 7.4.3 HOB 315 Loading on Clamped DRES Stiffened Panel 160 7.4.4 Clamped DRES1B Stiffened Panel 162 7.4.5 Clamped 2x2-Stiffened Plate II 166 7.4.6 Clamped 4x4-Bay Stiffened Plate 169 8 Summary and Conclusions 183 Bibliography 187 A Shape Functions 194 B Strain-Displacement Matrices 198 C [fi] Matrices for Plate and Beam Elements 202 D Formulas for J, I z z , J0 and T 205 vn List o f Tables 4.1 Strain rate parameters for steel and aluminum 33 4.2 Sampling points and weights for Gaussian integration 47 5.1 Linear elastic response of simply supported Square Plate I 55 5.2 Linear elastic response of clamped square plate I 59 5.3 Linear elastic response of simply supported rectangular beam . . . . . 68 5.4 Linear elastic response of clamped rectangular beam 72 5.5 Linear elastic response of 2-Bay Stiffened Plate I 77 5.6 Deflections and strain energy in linear elastic DRES stiffened panel . 87 5.7 Deflections and strain energy in linear elastic DRES1B panel 98 5.8 Nonlinear elastic response of DRES1B panel 103 5.9 Nonlinear elastic-plastic response of DRES1B panel 103 5.10 Nonlinear elastic response of 2x2-Bay Stiffened Plate I 115 5.11 Nonlinear elastic-plastic response of 2x2-Bay Stiffened Plate I . . . . 115 6.1 Eigenvalues of square plates with various edge conditions 122 6.2 Natural frequencies of 3-bay continuous plate 124 6.3 Natural Frequencies of 2x2-Bay Continuous Plate 126 6.4 Natural frequencies of rectangular beams 127 6.5 Comparison of net number of variable in 2-bay stiffened plate . . . . 129 6.6 Natural frequencies of 2-bay stiffened panel 130 6.7 Comparison of net number of variable in 3-bay stiffened plate . . . . 132 6.8 Natural frequencies of 3-bay stiffened panel 133 6.9 Natural frequencies of DRES stiffened panel 134 viii 6.10 Natural frequencies of 2 x 2-bay stiffened plates 135 6.11 Natural frequencies of 2 x 4-bay stiffened plate 137 6.12 Net number of variables in 4x4-bay stiffened plate 139 6.13 Natural frequencies in 4x4-bay stiffened plate 140 ix List of Figures 3.1 Schematic representation of an orthogonally stiffened plate 10 3.2 Assemblage of plate and beam elements 12 3.3 The super finite elements 13 3.4 Shape of the Lagrange polynomials 17 3.5 Shape of the Hermitian polynomials 18 3.6 Shape of the <j> function 19 3.7 Shapes of the sin 27r£ and sin47r£ functions 20 4.1 Bi-linear stress-strain relationship 31 4.2 Stress-strain relation of strain-rate sensitive material 34 4.3 Shapes of typical blast loads 38 4.4 Torsion beam element 41 4.5 Beam cross-section 48 5.1 <rx stress in linear elastic simply supported Square Plate I 56 5.2 Central deflection in linear elastic-plastic simply supported Square Plate I 57 5.3 Central deflection in nonlinear analysis of simply supported Square Plate I 58 5.4 ax at 0.04 N / m m 2 in simply supported Square Plate I 60 5.5 o~x at 0.2 N / m m 2 in simply supported Square Plate I 61 5.6 crx stress in linear elastic clamped Square Plate I 63 5.7 Linear elastic-plastic response of clamped Square Plate I 64 5.8 Large deflection response of clamped Square Plate I 65 x 5.9 ax at 0.2 N / m m 2 in nonlinear elastic clamped Square Plate I 66 5.10 ax at 0.8 N / m m 2 in nonlinear elastic clamped Square Plate I 67 5.11 Linear elastic-plastic response of simply supported rectangular beam . 69 5.12 Large deflection response of simply supported rectangular beam . . . 70 5.13 Stresses in large deflection analysis of simply supported rectangular beam 71 5.14 Large deflection response of clamped rectangular beam 74 5.15 Stresses in large deflection analysis of clamped rectangular beam . . . 75 5.16 Configuration of 2-Bay Stiffened Plate I 76 5.17 In-plane displacement in 2-Bay Stiffened Plate I 78 5.18 Panel centre displacement in 2-Bay Stiffened Plate I 79 5.19 Stiffener centre displacement in 2-Bay Stiffened Plate I 80 5.20 Displacement shapes along CB of 2-Bay Stiffened Plate I 81 5.21 Configuration of DRES stiffened panel 82 5.22 Super element models for DRES stiffened panel . 83 5.23 ADINA discretization of DRES stiffened panel 84 5.24 Displacement of point D in DRES stiffened panel 85 5.25 Displacement of point C in DRES stiffened panel 86 5.26 Normal stress perpendicular to stiffener at 0.5 psi . 89 5.27 Normal stress parallel to stiffener at 0.5 psi 92 5.28 Configuration of DRES1B panel 96 5.29 Models for DRES1B panel 97 5.30 Linear elastic in-plane displacements in DRES1B panel . . '. 99 5.31 Linear elastic bending displacements in DRES1B panel 100 5.32 Linear elastic normal stresses in DRES1B panel 101 5.33 Nonlinear elastic in-plane displacements in DRES IB panel 102 5.34 Nonlinear elastic bending displacements in DRES1B panel 104 xi 5.35 Nonlinear displacements of points D and E in DRES1B panel 105 5.36 Large deflection elastic-plastic displacement profiles in DRES IB panel 107 5.37 Nonlinear elastic normal stresses in DRES1B panel 108 5.38 Normal stress ax in nonlinear elastic DRES1B panel 109 5.39 Normal stress ay in nonlinear elastic DRES1B panel 110 5.40 Details of 2x2-Bay Stiffened Plate I I l l 5.41 Bending displacements in linear elastic 2x2-Bay Stiffened Plate I . . 112 5.42 In-plane displacements in linear elastic 2x2-Bay Stiffened Plate I . . 113 5.43 Normal stress o~x in linear elastic 2x2-Bay Stiffened Plate I 114 5.44 Normal stress o~y in linear elastic 2 x 2-Bay Stiffened Plate I 116 5.45 Nonlinear elastic-plastic u displacement profile along y = 15 in in sim-ply supported 2x2-Bay Stiffened Plate I 117 5.46 Nonlinear elastic-plastic v displacement profile along x — 7.5 in in simply supported 2x2-Bay Stiffened Plate I 118 5.47 Nonlinear elastic-plastic w displacement profile along x = 7.5 in in simply supported 2x2-Bay Stiffened Plate I 119 6.1 Configuration of 3-bay continuous plate 123 ^ 6.2 Configuration of 2 x 2-bay continuous plate 125 6.3 Configuration and discretizations for 2-Bay Stiffened Plate II 128 6.4 Configuration and discretizations for 3-bay stiffened plate 131 6.5 Configuration and discretization for 2x4-bay stiffened plate 136 6.6 Configuration of 4x4-bay stiffened plate 138 7.1 Dynamic relaxation response of linear elastic simply supported Square Plate I 144 7.2 Transient response of simply supported Square Plate I 145 7.3 Transient elastic response of clamped Square Plate I 146 xii 7.4 Transient nonlinear elastic-plastic response of clamped Square Plate I 147 7.5 Details of Square Plate II 148 7.6 Central displacement history of simply supported Square Plate II . . 149 7.7 Transient linear elastic response of simply supported rectangular beam 151 7.8 Transient linear elastic-plastic response of simply supported rectangu-lar beam 152 7.9 Loads applied to 2-Bay Stiffened Plate II 153 7.10 Linear elastic response of clamped 2-Bay Stiffened Plate II due to step load 155 7.11 Nonlinear elastic response of clamped 2-Bay Stiffened Plate II due to step load 156 7.12 Nonlinear elastic-plastic response of clamped 2-Bay Stiffened Plate II due to step load 157 7.13 Panel centre displacement of clamped 2-Bay Stiffened Plate II due to blast load 158 7.14 Stiffener mid-point displacement of clamped 2-Bay Stiffened Plate II due to blast load 159 7.15 Linear elastic response of simply supported 2-Bay Stiffened Plate II due to step load 160 7.16 Panel centre displacement in nonlinear elastic analysis of simply sup-ported 2-Bay Stiffened Plate II due to step load 161 7.17 Stiffener mid-point displacement in nonlinear elastic analysis of simply supported 2-Bay Stiffened Plate II due to step load 162 7.18 Panel centre displacement in nonlinear elastic-plastic analysis of simply supported 2-Bay Stiffened Plate II due to step load 163 7.19 Stiffener mid-point displacement in nonlinear elastic-plastic analysis of simply supported 2-Bay Stiffened Plate II due to step load . . . . . . . 164 xm 7.20 Panel centre displacement in simply supported 2-Bay Stiffened Plate II due to blast load 165 7.21 Stiffener mid-point displacement in simply supported 2-Bay Stiffened Plate II due to blast load 166 7.22 Blast load on D R E S Stiffened Panel 167 7.23 Panel centre displacement of blast loaded D R E S Stiffened Panel . . . 168 7.24 Stiffener mid-point displacement of blast loaded D R E S Stiffened Panel 169 7.25 Discretizations of D R E S I B panel 170 7.26 Displacements of points D and E in D R E S 1 B panel due to blast load 171 7.27 Displacements of points C and F in D R E S 1 B panel due to blast load 172 7.28 Displacement along y = 24in in D R E S 1 B panel due to blast load . . . 173 7.29 Displacement along G D in D R E S 1 B panel due to blast load 174 7.30 Displacement along B C in D R E S I B panel due to blast load 175 7.31 Configuration of 2 x 2 Stiffened Plate II 176 7.32 Linear elastic response of 2 x 2 Stiffened Plate II - Step Load 177 7.33 Nonlinear response of 2 x 2 Stiffened Plate II - Step Load 178 7.34 Response of points A , B and G in 4 x 4 Bay Stiffened Plate - Rectan-gular Pulse 179 7.35 Displacement profiles along H G F in 4 x 4 Bay Stiffened Plate - Rect-angular Pulse 180 7.36 Displacement profiles along D B A in 4 x 4 Bay Stiffened Plate - Rect-angular Pulse 181 7.37 F ina l displacement profiles along beams D B A and E C B in 4 x 4 Bay Stiffened Plate - Rectangular Pulse 182 xiv Acknowledgements The author wishes to express his gratitude to his supervisor Dr. M . D. Olson for his guidiance, advice and valuable time during the preparation of this thesis. The valuable suggestions offered by Dr. D. L. Anderson at various stages of the work are appreciated. The author would also like to thank Dr. R. Houlston of the Defence Research Establishment, Suffield, Alberta for providing the ADINA results for the DRES IB panel. The author is also appreciative of the many useful discussions he had with Dr. J . Jiang who also provided some of the comparison dynamic finite strip results. The support and encouragement offered by P. Kumar, R. B. Schubak, Dr. J . D. Dolan, Dr. A. Filiatrault and all the author's friends are also much appreciated. Financial support of the Natural Sciences and Engineering Research Council of Canada in the form of a Research Assistantship from the department of Civil Engi-neering, University of British Columbia is gratefully acknowledged. xv Chapter 1 Introduction Stiffened plates are structural components consisting of plates1 reinforced by a sys-tem of orthogonal beams2 (or ribs) to enhance their load carrying capacities. These structural components find wide application in various kinds of structures. For ex-ample, in naval architecture stiffened plates are used in the construction of the hulls of ships while in the aircraft industry they are used in constructing the fuselage of aircraft. Stiffened plate construction is also found in bridges, buildings, railway cars, large transportation carrier panels and storage tanks. There are situations in which a stiffened plate structure might be subjected to complex loads such as air blast pressure waves from external or internal explosions, water waves, collisions or simply large static loads. These loads can induce large deformations which stress the material well over the elastic limit to cause significant plastic deformations in the structure. Hence in the design/analysis of the structures geometric and material nonlinearities must be taken into account. The static or dynamic large deflection elastic-plastic analysis of stiffened plates is a difficult task. Solution of the problem by analytical techiques is practically impos-sible due to the intractable task of integrating the governing differential equations of motion. On the other hand the problem can be solved by numerical methods based 1 Plates are flat surface structures whose thicknesses are small compared to their other dimensions. Of particular interest in this study are rectangular thin plates whose thicknesses are less than a tenth of the least other dimension. 2 A beam is a structure whose length is large compared to its other dimensions and carries load primarily by bending. 1 Chapter 1. Introduction 2 mainly on the finite element technique. Currently, there are available some com-mercial finite element programs capable of performing nonlinear analysis of stiffened plates. However, with these regular finite element programs, the modelling requires many elements with a huge amount of input/output data and very expensive com-puter runs. This type of analysis is impractical at the preliminary design stage. The work presented in this thesis is aimed at improving the situation by introduc-ing a new analysis/design philosophy. It is the objective of this thesis to develop a simplified numerical formulation which is capable of representing the overall response of the complete structure with reasonable accuracy but with a sacrifice in local de-tailed accuracy. This will result in a relatively simple model which requires much reduced input data and run times and makes it feasible to carry out design oriented analyses. On the basis of the above philosophy, new plate and stiffener beam finite ele-ments are developed for the nonlinear static and dynamic analysis of stiffened plate structures. Using polynomial as well as continuous analytical displacement functions, the elements are specially designed to contain all the basic modes of deformation response which occur in stiffened plate structures and are called super finite elements since only one plate element per bay or one beam element per span is needed to achieve engineering design level accuracy. Rectangular plate elements are used so that orthogonally stiffened plates can be modelled. Following a brief literature review, presented in Chapter 2, the new super elements are described in Chapter 3. The super element discretization and the displacement functions are discussed and the chapter ends with a discussion on the justification for the choice of the displacement functions using a study of the compatibility, conver-gence and order of accuracy analysis of the elements. In Chapter 4 the finite element matrix quantities are derived for the super el-ements. The mathematical derivations conducted in the chapter include details of Chapter 1. Introduction 3 the Gaussian integration, the Newton-Raphson iterative scheme and the implicit Newmark-/3 temporal integration procedure. The chapter also highlights some de-tails of the computer code developed to implement the theory. The numerical investigations carried out to verify the new formulation are pre-sented in Chapters 5, 6 and 7. Chapter 5 deals with the static response, Chapter 6 the vibration analysis and Chapter 7 is devoted to transient applications. Unstiffened plates, beams and plates stiffened in one or two mutually perpendicular directions are considered. For the static and transient applications, various combinations of geometric and material nonlinearities have been incorporated but only linear elastic vibrations have been carried out in Chapter 6. Finally, Chapter 8 gives the summary and conclusions derived from the present study and ends with some suggestions for future research. Chapter 2 Literature Review 2.1 Analytical Methods The classical thin plate theory is well known for the linear elastic analysis of isotropic unstiffened plates. For large deflection analysis of such plates the large deflection theory by von Karman is usually employed. The applications of these methods to analysis of plate structures is well documented in [1]. A comprehensive study of the historical development of analytical procedures for the analysis of stiffened plates has been presented by Troitsky [2]. It is clear from the review that the first known attempt at the analysis of stiffened plate structures is due to Boobnov in 1902. In his work Boobnov applied stress analysis procedures to two-way stiffened plates and treated the problem as a beam on an elastic foundation and obtained design charts for stiffened plate structures. The development of the orthotropic plate theory by Huber in 1914 made significant contributions to the analysis of stiffened plates. Although the theory was developed for naturally orthotropic plates1 it can also be applied to stiffened plates by treating them as being equivalent to an orthotropic plate of constant thickness. In order for the orthotropic theory to apply it is required that 1 Natural orthotropy is used to describe the situation in which a body possesses different elastic properties in orthogonal directions. On the other hand the elements of the body may be arranged in proper geometric configurations such that it exihibits different properties in orthogonal directions. This is called structural or technical orthotropy. Orthogonally stiffened plates are examples of such structures. 4 Chapter 2. Literature Review 5 1. The material of the stiffened plate follow Hooke's law. 2. Plane sections remain plane before and after bending and the out-of-plane dis-placements be small compared to the plate thickness. 3. The stiffeners be evenly and closely spaced. 4. The stiffeners be disposed symmetrically with respect to the mid-plane of the plane. The last assumption is difficult to meet in practice since for most stiffened plate structures the stiffeners (or ribs) are placed asymmetrically with respect to the mid-plane of the plate. As a consequence, the inherent assumption of a strain-free mid-plane is grossly violated by such structures and hence the orthotropic plate theory cannot be applied rigorously to these structures unless certain modifications are made. A more rigorous extension of the orthotropic plate theory to eccentrically stiffened plates has been conducted by Pfluger in 1947. Using the force-displacement relations for a plate element with ribs on one side he obtained an eighth order differential equation involving only a displacement component. This method has been employed by several investigators and Clifton et al [3] have developed an exact theory, based on the method, for plates with ribs of open or closed box sections. An important feature associated with stiffened plates subjected to bending loads is the shear lag phenomenon. This pheneomenon describes the situation where the normal stress is maximum at the stiffener location and shows a lag with increasing distance from the stiffener. The analysis of this behaviour has been given considerable attention by some investigators and some of these have been discussed by Troitsky [2] and Timoshenko and Goodier [4]. There are very few analytical solutions of stiffened plates exhibiting geometric non-linear behaviour. Troitsky [2] has reported on the large deflection theories developed by Vogel in 1961 and Abdel-Sayed in 1963. These theories are, essentially, extensions Chapter 2. Literature Review 6 of the large deflection theory for orthotropic plates, following the von Karman large deflection theory for isotropic plates. Solutions exist to only a few problems. Modern structures have to be designed into the plastic range to take advantage of the extra load-carrying capacity afforded by the ductility of the material. However, incorporating the theory of plasticity into a stiffened plate theory presents enormous difficulties if an analytical solution is sought. The problem can be simplified by assuming a rigid-plastic material behaviour. The yield line analysis is based on this assumption and proceeds with assumed collapse mechanisms. With this method solution to some plate structures have been obtained [5]. A major difficulty in the use of analytical methods is due to the onerous task of integrating the governing differential equations of motion. The situation is worse when geometric and/or material nonlinearities are involved. However, with numerical techniques it has been possible to obtain fairly easily approximate solutions suitable for engineering purposes. These numerical methods are discussed in the next section. 2.2 Numerical Methods 2.2.1 Finite Difference and Finite Element Methods The finite difference and finite element methods are well known in the analysis of engi-neering problems. In the finite difference method, the governing differential equations of motion are replaced by a set of difference equations written for a finite number of grid points into which the domain of the problem is divided. The resulting set of algebraic equations are solved simultaneously for the finite number of unknown parameters at the grid points and this represents an approximation to the exact so-lution [6] . Webb and Dowling [7] have applied the method to the large deflection elasto-plastic analysis of discretely stiffened plates. In the finite element method, the governing equations are also replaced by a set Chapter 2. Literature Review 7 of algebraic equations which are obtained by discretizing the continuum into a finite number of elements. The accuracy of the solution usually depends on the number of elements and the order of the trial polynomial functions used to approximate the displacement/stress variations within each element. The finite element method is by far the most versatile of all numerical methods as it can handle problems with complicated geometry, boundary conditions or loadings very easily. Most importantly, with finite elements the ribs of a stiffened plate need not be symmetrically placed with respect to the midplane of the plate or be densely and equally spaced since the method is quite capable of simulating the response of plates with discrete stiffeners easily. Several researchers have applied the finite element method to linear elastic static and dynamic analysis of stiffened plates. Extension of the method into the large deflection elastic-plastic range have also been investigated [8,9,10]. Several all purpose finite element programs have been developed in recent times. Some of these programs have capabilities for static and dynamic large deflection elastic-plastic analysis of stiffened plates. For example the VAST (Vibration and Strength Analysis Program) and ADINA (Automatic Dynamic Incremental Nonlinear Analysis) programs have recently been used to analyze stiffened plates subjected to air blast loading [11]. Inspite of the existence of these finite element programs there are very few publi-cations dealing with orthogonally stiffened plates subjected to large static or dynamic loads capable of inducing geometric/material nonlinearities. The reason for this is that a complete nonlinear finite element analysis requires the use of huge input data and very expensive computer runs. It is for this reason special finite elements such as finite strips have been developed to analyze certain classes of problems. These are discussed in the following subsection. Chapter 2. Literature Review 8 2.2.2 Finite Strip Method The finite strip method, developed by Cheung [12], is suitable for the analysis of structures with regular boundaries. The structure is divided into a finite number of strips and, unlike the finite element method in which polynomial functions are used in all directions, the finite strip method uses continuously differentiable analytical functions in one direction and polynomials in other directions. Furthermore, the con-tinuous functions are stipulated to satisfy a priori the kinematic boundary conditions at the ends of the strips. Details of the application of the method to static and dynamic analysis of plate structures are well documented in [13]. The extension of the method to nonlinear anal-ysis of unstiffened plates has also been investigated by some researchers [14,15,16,17]. Only recently, the method has been extended to static and dynamic large deflection elastic-plastic analysis of stiffened plates [18,19,20]. However, these applications are restricted to one-way stiffened plates. Attempts have been made to model orthog-onally stiffened plate using compound strips [21,22] but these have been limited to linear elastic problems. The major advantage of the finite strip method over the more versatile finite element method is its simplicity. It requires smaller amount of input data and core memory and uses much reduced run times compared to the finite element method. In the present study, the advantages of the finite element and finite strip methods are combined to develop a relatively simple formulation capable of representing the response of stiffened plates subject to intense static or dynamic loads with reasonable accuracy. The method uses special finite elements called super finite elements which are macro elements having both polynomial and continuous analytical displacement functions in all in-plane directions. The description of the new elements is presented in Chapter 3. Chapter 3 Description of the Super Finite Elements 3.1 Introduction The class of structures which are of interest in this work include rectangular plates reinforced by stiffener beams placed in one or two mutually perpendicular directions, as typified by the structure in Figure 3.1. Each panel bay (eg. A B C D ) is modelled by a rectangular plate element and the stiffeners (such as A B , BC) are treated as beam elements running along the edges of the panels. The displacement fields for the plate and beam elements have been carefully chosen to simulate all possible linear and nonlinear deformation modes for the elements acting together as in stiffened plates or separately as simple or continuous beams or plates with all possible boundary conditions. The elements are termed super finite elements since only a single element is required to model the basic response. The details of the elements and the associated displacement fields are presented in this chapter. In Section 3.2 the super element discretization is presented and the degrees of freedom associated with each element are described. Section 3.3 gives details of all the displacement fields for each element and Section 3.4 highlights the justification for the choice of the displacement functions. 9 Chapter 3. Description of the Super Finite Elements 10 y. v i A i : B : J i D i : c : 1 z ,w Plan ^ p | a t e c. beams J Jl X, u elevation Figure 3.1: Schematic representation of an orthogonally stiffened plate Chapter 3. Description of the Super Finite Elements 11 3.2 Super Finite Element Discretization Figure 3.1 shows details of an orthogonally stiffened plate. By the present formulation each panel bay is represented by a super plate element and each stiffener span by a super beam element. A panel bay and two adjacent beams are isolated in Figure 3.2 to illustrate the assemblage of the elements in the structure. Details of the nodal variables associated with each element are presented in Figure 3.3. The plate element has 9 actual nodes, numbered 1 to 9. The 'extra nodes' labelled uio> u n , •••  v i 5 a r e actually the amplitudes of the trigonometric functions used to model the in-plane displacements (described in Section 3.3) and these are lumped at the mid-side and central nodes labelled 5 to 9. Each of the four corner nodes has six variables — the two in-plane displacements, u, v; the out of plane displacement, w; the two slopes, wx, wy and the twist, wxy, where wx = dw/dx, etc. The positive directions of u, v and w are shown in Figure 3.2. Each of the mid-side nodes numbered 5 to 8 also has six variables — u, v, w and the normal slope wx or wy, together with two additional in-plane variables. For example, the mid-side node, numbered 5, has the variables u, v, w, wy, uw and Ui3; node 6 has the variables u, v, w, wx, vn and v14; and similarly for nodes 7 and 8. The central node, numbered 9, has seven degrees of freedom —u, v, w, u u , v12, ui5, and v15. Over all, the plate element has 55 degrees of freedom as shown in Figure 3.3(a). The beam element (Figure 3.3(b)) has 3 actual nodes numbered 1 to 3. In analogy to the plate element, the 'extra nodes' labelled u4 and u5 are amplitudes of the in-plane displacement and these are taken as variables at the middle node. For bending and axial action alone, a beam in the ^-direction has three variables u, w, and wx at the end nodes and four variables u, w, u 4 and u5 at the middle node. However, for problems in which there is significant presence of torsion in the stiffener beams, additional variables are included to approximate the torsional rotation, 6 and lateral Chapter 3. Description of the Super Finite Elements 12 Figure 3.2: Assemblage of plate and beam elements Chapter 3. Description of the Super Finite Elements 13 4 •wio 1*14 i * n —• • 7 • " 1 5 U12 n • • • 9 • v12 • VX5 1 1*13 Degrees of Freedom: Jy) wxy X, u At nodes 1,2,3,4 - u,v,w,wx,wy At nodes 5,7 - u,v,w,wy; plus (1*10,1413) and ( i 4 n , i 4 i 4 ) , respectively At nodes 6,8 - u,v,w,wx; plus (I ; II , I>14) and ( ^ 1 0 , ^ 1 3 ) , respectively At node 9 - u,v,w; plus i * 1 2 , v12, W i s , U 1 5 (a) plate element 1 1*5 i * 4 3 2 • • — • • • Degrees of Freedom: At nodes 1,2 - u,v,w,wX,9,9X At node 3 - u,v,w,9; plus 1*4,1*5 (b) beam element in x direction Figure 3.3: The super finite elements Chapter 3. Description of the Super Finite Elements 14 bending displacement, v. The additional variables are v, 9, 9X at the end nodes and v and 6 at the middle node. For beam problems, the nodal variables are located along the beam centroidal axis and in this case torsional displacements have been ignored so that the beam element has a total of 10 variables. However, in stiffened plate structures, ,all nodal variables are assumed to be located at the centroidal plane of the plate. This way, no additional degrees of freedom are introduced at a beam-plate connection [23]. In structures with negligible torsional displacements, the stiffener beam element still has 10 variables but the number is increased to 18 if torsion is significant. Inter-element continuity1 between two adjacent plate elements or a plate and a beam element is ensured at the three nodes along the plate edge, while continuity between two adjacent beam elements is provided at the beam end nodes. 3.3 Displacement Functions The displacement fields for the super elements have been carefully chosen so that all possible displacement modes in a stiffened plate structure can be modelled fairly ac-curately with only one super element per bay or span. To achieve this goal continuous analytical functions (usually, trigonometric and hyperbolic functions) are 'smeared' with the usual finite element polynomial functions in a fashion similar to the finite strip formulation, except that, in this case, no boundary conditions are satisfied a priori and the analytic and polynomial functions all run in the two in-plane direc-tions. The super elements thus combine the simplicity of the finite strip method and the versatility of the regular finite element method. The displacement fields chosen for each element are described in the following subsections. 1 The continuity is defined in Section 3.4. Chapter 3. Description of the Super Finite Elements 15 3.3.1 Plate elements Figure 3.3(a) shows a typical super plate element of length a and width b. The nodal variables are as indicated in the figure and the element has 55 degrees of freedom. The displacement fields associated with the element are given by u 10 u = Nfm + sin 27r£[L 1 (7 ? ) ,L 2 (77) , L3(T))\ < u12 + sm4ir£[Ll(v),L2(-n),L3(y)} < Ul3 Ul5 (3.1) v = N-Vi + [LT_(0, L2((), L3(£)] sin 2wn I VlO v v12 / } + + [L1(aL2(aL3(()]sm4Trrl{ Vl3 '14 Vl5 (3.2) w = N ^ + <l>U)[Hi(v),H2(V),Ha(r,),H4(V)]< Wy 5 Wy7 } + +[H1((),Hi{C)tH3{aHA((Mv) w$ w. xS w6 Wx6 + # 0 M K (3.3) Chapter 3. Description of the Super Finite Elements 16 where i — 1, 2, . . . , 9; j = 1, 2, . . . , 16; u, v are the two in-plane displacements; w is the flexural bending (out-of-plane) displacement and £ = x/a and 77 = y/b. The quadratic Lagrange interpolation polynomials are given by: HO = n 2 - H + 1 L2{() = 2 e - i (3.4) These quadratic polynomials are shown in Figure 3.4. The cubic Hermitian polyno-mials are given by: H2(() = a{i - 2e + e) HsU) = 3(2 - 2£ 3 (3.5) and their shapes are as shown in Figure 3.5. (j> 1S the first symmetric vibration mode of a clamped beam and is given by: 4>(0 — [o;(sinh fi( — sin/x£) + (cosh//£ — cos/z£)]/<£> (3-6) where p = 4.7300407448, cos p — cosh fl sinh p — sin p anc (p = a(sinh0.5/x — sin0.5/i) + (cosh0.5/i — cos0.5/x) The shape of the (f> function is shown in Figure 3.6. Chapter 3. Description of the Super Finite Elements 17 1.2 °"20.0 0.5 1.0 x/a Figure 3.4: Shape of the Lagrange polynomials iV", TV" are products of the Lagrange interpolation polynomials and JVj" are prod-ucts of the Hermitian polynomials. These are given explicitly in Appendix A. u,, are the nodal variables in the x, y directions, ipj are the corner node lateral displace-ments, slopes and twists; all attached to the midplane of the plate. The corner node lateral displacement vector is given by {%/J}t = [wl,wxl,wyl,wxy-L,w2,wX2,wy2,wxy2, (3.7) w3, wx3,wy3,wxy3, w4, wx4,wy4, wxy4] where wx = dw/dx, etc. The shapes of the trigonometric functions — sin 2 ^ and sin47r£ — used to model the in-plane displacements are shown in Figure 3.7. Chapter 3. Description of the Super Finite Elements 18 x/a Figure 3.5: Shape of the Hermitian polynomials 3.3.2 Beam elements Figure 3.3(b) shows a super beam element of length a, with its degrees of freedom. The membrane and flexural displacement fields referred to the centroidal axis of a beam in the ^ -direction are given by u3 > + u 4 sin 2TV( + u5 sin 4ir£ + +e[H[{(),H'2(t),H'3(t),H'4(t))l Wx2 + e4>\£)wz (3.8) Chapter 3. Description of the Super Finite Elements 19 ™ = [ t f i ( 0 , # 2 ( 0 , # 3 ( 0 . - ^ ( 0 M w2 wx2 (3.9) where the primes denote differentiation with respect to x and e is the distance between the centroidal axis of the beam and the mid-plane of the plate. The effect of torsion and lateral bending in the stiffener beam element has been included in some cases and the rotation, 6 and lateral displacement, v fields are approximated, respectively, by Chapter 3. Description of the Super Finite Elements 20 1.5 1.0 h 0.5 0.0 -0.5 h -1.0 -1.5 sin(27rx/a) sin(47rx/a) 0.0 0.5 x/a Figure 3.7: Shapes of the sin27r£ and sin4.7r£ functions 1.0 * = [ffl(0,tf2(0,#s(0,#4(0] 01 9X1 9x2 ( \ Vl + ^ 3 » = [ii(0.ij(0.i»(0] V2 { V3 ) (3.10) (3.11) where 6i,9xi, ...v3 are the beam nodal rotation, twist or lateral displacement vari-ables as illustrated in Figure 3.3(b) Chapter 3. Description of the Super Finite Elements 21 3.4 Compatibility, Convergence and Order of Ac-curacy Consider the w displacement field for the panel A B C D in Figure 3.1. The basic response of the panel with clamped boundaries all round is represented by the last term 0(^)^>(T/) in Equation (3.3). Then to allow for support movements and to ensure compatibility of displacements and slopes between adjacent panels the 16 degree of freedom plate bending element, developed by Bogner et al [ 24 ] , having the bi-cubic polynomial functions (first term of Equation 3.3) is employed. The other two terms in Equation 3.3 are included firstly to match the plate and beam displacements along the plate edges and secondly to allow various boundary conditions to be modelled along the edges. The w displacement field is C 1 continuous and hence will ensure convergence of the solution of linear plate bending problems with the error in strain energy being of order 0(l4) where Z is a characteristic element dimension. Bi-quadratic shape functions are used for the in-plane displacements u and v in order to obtain an order of accuracy in energy consistent with the out of plane dis-placements, C° continuity between adjacent elements also being ensured. The sine terms in the in-plane displacement fields are included to capture nonlinear geomet-ric effects — sin 2 7 r £ being suitable for simply supported boundaries and sin 4 7 r £ for clamped boundaries [18,19]. These functions are essential in providing a good ap-proximation to the distribution of membrane stresses in large deflection elastic-plastic analyses as demonstrated by some of the results of the analyses in Sections 5.2.2 and 7.2.3. The sine functions are multiplied by quadratic shape functions in the other direction to maintain the order of accuracy, ensure compatibility between plate and beam displacements and also to capture shear lag effects. In analogy to the plate, the basic response of the beam element A B with clamped Chapter 3. Description of the Super Finite Elements 22 boundaries is represented by (f>(£). Then to allow for arbitrary end motion and to en-sure compatibility between elements, the cubic polynomials are included. A consistent order of accuracy (0(Z4)) in strain energy is also ensured. The choice of the in-plane displacement field also ensures compatibility between beam and plate displacements, continuity of displacements between adjacent beam elements and a consistent order of accuracy. The basic rotational response of the stiffener beam element A B with clamped ends is also approximated by <^ >(£) to correspond to the wy displacement field along the edge A B of the plate element A B C D . Then for arbitrary end rotation and for compatibity between elements the cubic Hermitian polynomials are included in Equation 3.10. This rotation field is also C 1 continuous and the error in the linear torsional strain energy is of order 0(/ 4 ) . The lateral bending displacement field, v is chosen to be quadratic for compatibility with the in-plane displacement field in the plate. Con-sequently, it does not provide C 1 continuity which is normally required for beam bending. However, this is probably of little consequence since the effect of lateral bending is expected to be very small. Chapter 4 Theoretical Formulation and Analysis of Problem 4.1 Introduction The theoretical formulation and method of analysis of the problem is presented in this chapter. First, the governing equations of motion are derived in Section 4.2. Then in Section 4.3 the finite element formulation is introduced. Here, the super elements de-scribeed in Chapter 3 are used in conjuction with the strain-displacement and consti-tutive relations to derive the finite element matrix quantities. The Newton-Raphson iterative scheme used to obtain the tangent stiffness matrix is also presented. The nu-merical integration scheme used to evaluate the matrix quantities is briefly discussed in Section 4.4, while Section 4.5 highlights the temporal integration scheme. Finally, Section 4.6 focuses on some important aspects of the computer implementation. Because the analysis procedures presented in this chapter are widely available in the literature, only brief summaries are presented to highlight the important aspects that apply to this work. 4.2 Equations of Motion The governing equations of motion are obtained via the principle of virtual work. If a deformable body subject to a set of arbitrary loading and boundary conditions is 23 Chapter 4. Theoretical Formulation and Analysis of Problem 24 in equilibrium, the principle of virtual work stipulates that Wint + Wext = 0 (4.1) where Wint and Wext are kinematically admissible small variations in the internal virtual work and external virtual work, respectively. When the applied loading is temporal in nature d'Alembert's principle is employed to include the inertial forces. Thus ignoring body forces the equations of motion can be expressed as Jv[{d}Tp{d} + {d}TKd{d} + {e}T{*})dV - Js{d}T{q}dS = 0 (4.2) where {c}, {e} are the stress and strain vectors, p the mass density, is a viscous damping parameter, q is the applied surface traction, V and S are, respectively, the volume and surface area and {d} is the displacement vector. The tilde ( * ) is used to denote a virtual change in the given quantity with respect to a generalized displacement and the superimposed dot [( ' ) = (d/dt)} denotes differentiation with respect to time. For static loads, the velocity and acceleration terms {d} and {d} in Equation (4.2) vanish. In Equation (4.2) the internal virtual work Jv{e}T{cr}}dV has been stated in terms of the true (Eulerian) stresses o~ij which are defined in the deformed configuration. This is made possible by the assumption that the strains considered in this work are small and hence the true stresses cr^ are approximately equal to the Kirchhoff stresses Sij which are defined in the undeformed configuration [25]. Equation (4.2) can be specialized for plate or beam structures by appropriately defining the stress, strain and displacement terms. For plate structures, UV = f ( { d } ) = k x , e y , 7 * J (4.3) Chapter 4. Theoretical Formulation and Analysis of Problem 25 where <rx, cry are the stresses in the x, y directions, respectively, rxy is the shear stress, ex, ey are the strains in the x, y directions and *yxy is the engineering shear strain. For a beam structure spanning the x-direction, {cr} and {e} contain only the terms ax and ex, respectively for planar bending only. If the beam spans the y-direction then the stress and strain terms are ay and ey, respectively. The governing equations of motion, Equations (4.2), are highly nonlinear due to the presence of geometric and material nonlinearities. Since analytical solution of these nonlinear equations is in most cases impossible, a numerical solution technique has to be employed. Here, the super elements developed in Chapter 3 are utilized and the finite element formulation of the problem using these elements is discussed in detail in Section 4.3. 4.3 Finite Element Formulation 4.3.1 Introduction In this section the finite element relationships are derived for the super elements devel-oped in Chapter 3. Using Equations (4.2) the matrix quantities (load, mass, damping and stiffness matrices) are calculated for each element type — plate or beam — and these are added together in the usual finite element fashion to obtain the global ma-trices. First, the shape function matrices are developed in Section 4.3.2 and these are used to obtain the nonlinear strain-displacement relations in Section 4.3.3. The nonlinear constitutive relations employed in the work are then discussed briefly in Section 4.3.4. Section 4.3.5 is devoted to the evaluation of the mass and damping matrices. Using the strain-displacement relations described in Section 4.3.3 in con-junction with the constitutive relations in Section 4.3.4 the stiffness quantities are derived in Section 4.3.6, where the Newton-Raphson iteration scheme is also high-lighted. The load vector for each element type is derived in Section 4.3.7. The Chapter 4. Theoretical Formulation and Analysis of Problem 26 torsional stiffness and mass matrices for the stiffener beam elements are treated sepa-rately in Section 4.3.8 since the effect of stiffener beam torsion has not been included in all cases. 4.3.2 Shape Function Matrices 4.3.2.1 Plate elements The displacement fields for the plate elements are given by Equations (3.1), (3.2) and (3.3). These equations can be written collectively as u v w = [N]{Se} (4.4) where {8e} is the element nodal displacement vector which for the plate element is given by (4.5) {6e} T = l u i , V l , W l , W x l , W y l , W x y l , U 2 , V 2 , W 2 , W x 2 , W y 2 , W x y 2 U 3 , V 3 , W3, W x 3 , W y 3 , W x y 3 , UA, V 4 , W 4 , W x 4 , Wy4, Wxy4 u5,v5, w5,wy5, uw,u13,u6, v6, w6, wx6, vn,v14 u7, v7, wr, wy7, uu, u14,u8, v8, w8, wx8, vw,vl3 u9, v9, w9, u12,v12, u15, v15\ and [TY] is the shape function matrix which for the super plate element is given by [N} = 7Y" 0 0 0 0 O i V j O 0 0 0 0 0 Nf 0 0 0 0 0 7V2V 0 0 0 0 0 0 Ng Ng Ng Ng 0 0 Ng Ng Ng Ng Chapter 4. Theoretical Formulation and Analysis of Problem 27 0 0 0 0 N? 0 0 The shape functions, i V " , . . . in Equation (4.6) are defined exphcitly in A p -pendix A . In the regular finite element method, a shape function for any nodal variable has a unit value at that node and zero value at all other nodes. In the present formulation only the functions TV", . . . , Ng\ N",... ,N% and N™, ... , N™ satisfy this requirement and.hence, in a finite element analysis, the magnitudes of the nodal variables corre-sponding to these shape functions will represent the magnitudes of the corresponding displacements at the respective nodes. For example, in any analysis, the magni-tudes of Wi, wxi, wyi,wxyi represent the actual out-of-plane displacement, slopes and twist at node 1; and similarly for other corner nodes. Also, the magnitudes of Ui, u2, • •. ,UQ\ VI, v2, • • • ,v9 give the actual in-plane displacements at the respective nodal positions. However, the remaining functions N?0,... , JV^; Nf0, ..., 7V"5 and N™7, ... , N™5 do not satisfy the requirement of having a unit value at one node and zero at all other 0 0 0 0 0 N% 0 0 0 0 0 0 0 0 0 0 Nv4 0 0 0 0 0 Nw i V 1 0 Nw J v 12 0 0 Nw i v 1 3 •i v14 Mw i V 1 5 Nw 0 0 0 i V i o J V 1 3 iV 6" 0 0 0 0 0 Nv5 0 • 0 0 0 0 NX 0 0 • ( V i i J V 1 4 0 Nw i \ 1 7 Nw 0 0 0 0 i V 1 9 i V 2 0 0 0 0 0 0 -<vii J V 1 4 N% 0 0 0 0 0 TV? 0 0 0 0 0 0 0 Nv i V 1 0 i v 1 3 0 N?i • i v22 0 0 0 0 ^ 3 Nw 0 0 N% 0 0 J V 12 0 Nu i V 1 5 0 0 0 0 Nv - i v12 0 Nv J V 1 5 0 0 ^25 0 0 0 0 Chapter 4. Theoretical Formulation and Analysis of Problem 28 nodes. This is due to the presence of the analytical functions in these displacement functions. The variables corresponding to these shape functions merely represent displacement amplitudes at the respective nodes. To obtain the actual displacements at these nodes, in any analysis, the contributions from other shape functions have to be included. For example, w5, wy5 do not automatically represent the out-of-plane displacement and normal slope, respectively, of node 5. The displacement and normal slope at this node will include contributions from these amplitudes as well as those from other shape functions at that point. 4.3.2.2 B e a m elements The in-plane and out-of-plane displacement fields for the beam element (Equations (3.8) and (3.9)) can be combined as u w = [N){Se} where the element displacement vector in this case is given by (4.7) { £ e } T = l u 1 , W 1 , W x l , U 2 , W 2 , W x 2 , U 3 , W 3 , U 4 , U 5 \ and the shape function matrix for the super beam element is (4.8) [N] = N™ Ng1 N™ N™ iV 5 m N™ N? N™ N? N 10 0 Nl 0 Nh3 NbA 0 Nb5 0 0 (4.9) The shape functions appearing in Equation (4.9) are presented in Appendix A. The superscripts m and b denote membrane and bending, respectively. Again, the function Nb does not rigourously satisfy the requirements of a shape function and hence the variable corresponding to this function, w3, does not auto-matically represent the displacement at node 3, as discussed for the plate element Chapter 4. Theoretical Formulation and Analysis of Problem 29 case. 4.3.3 S t r a i n Disp lacement Re la t ions In this study, it is assumed that the plates are thin and the beams slender so that the effect of shear deformation in these structures is negligible. Large deflection effects are taken into account by including first order nonlinearities in the strain displacement relations (following von Karman theory). By this theory it is assumed that the deflections are equal to or larger than the plate or beam thickness, but still small relative to the other dimensions (a or 6) of the plate or beam. For the plate the strain-displacement relations are du d2w 1 dw 2  £ x = dx~~Z~dx^ + ^~dx~> dv d2w 1 dw ^ , £» = * - * V + 2 ( * ) ( 4 1 0 ) ^ du ' dv d2w dw dw ^xy x y dy dx dxdy dx dy where ex, ey and exy are the strain components in the x — y plane and ^ x y is the engineering shear strain. For a beam spanning the x or y direction the relevant relation is the first or second of Equations (4.10). Using these equations and taking a small variation with respect to the generalized displacements, the virtual strain vector {e} in Equation (4.2) can be related to the virtual displacements, {8e}, symbolically as {i_} = [[B] + [Co(6.)]]{6e} where [B] is the linear strain-displacement matrix given by (4.11) Chapter 4. Theoretical Formulation and Analysis of Problem 30 \B\ -2- 0 -z&-dx dx2 jsi-9 2 0 £ a_ ay JL JL . dy dx -2z dx&y [N] (4.12) [Co] is the nonlinear strain-displacement matrix given by [Co] dNT" dNV dx dx dN? dNJ dy dy , dN™ dNJ dNV dm I 1 1— J 1 1— \ w . \ dx dv dx dv I 1 (4.13) dx dy dx dy ' 3 i,j = 1, 2, .. .25. The relevant expressions for beam elements are obtained by drop-ping the appropriate terms in Equations (4.12) and (4.13). See details in Appendix B. 4.3.4 C o n s t i t u t i v e Re la t ions The assumption of elastic material behaviour is no longer adequate in the design of modern structures since emphasis is placed on savings in weight. The theory of plasticity provides more realistic estimates of the load carrying capacities of struc-tures. To keep pace with modern trends the material of the stiffened plate structure is assumed to be elastic-plastic with a bilinear stress-strain relationship. Figure 4.1 shows the elastic-plastic (hardening) model (for uniaxial conditions) adopted in this study. E and ET are, respectively, the elastic and plastic modulii; c>o, e0 the uniaxial yield stress and strain, respectively; and ee, ep, respectively, rep-resent the elastic and plastic strains. For the situations considered the elastic and plastic strains are assumed to be of the same order of magnitude. In plasticity theory three important aspects of material behaviour, namely, the yield criterion, the flow rule or normality condition and the hardening rule, must be specified to define the constitutive relations. These are described briefly in the following paragraphs. The yield criterion specifies the state of stress which causes yielding of the ma-terial. For a multiaxial state of stress this is equivalent to the state in which an Chapter 4. Theoretical Formulation and Analysis of Problem 31 Figure 4.1: Bi-linear stress-strain relationship equivalent effective stress first exceeds the uniaxial yield stress of the material. The yield function, F has the general form F({*},K) = 0 (4.14) where K is a positive parameter and in general K and F({a}) depend on the existing level of plastic strain and the plastic strain history. The von Mises yield criterion is adopted here since it predicts well the behaviour of metals [26,27,28]. This criterion assumes that yielding occurs when the distortion or shear strain energy equals the distortion energy at yield in simple tension and the yield condition is given by \ " * » ) a + (*2 - + ( '3 - * i ) 2 ] = al (4.15) where, C j , <r2, cr3 are the principal stress components in three dimensions. The flow rule or normality condition characterizes the behaviour of the material Chapter 4. Theoretical Formulation and Analysis of Problem 32 after the onset of yielding. An associated flow rule is employed. This assumes that the incremental strain vector at any point on the yield surface is normal to the yield surface at that point, as opposed to a non-associated flow rule where the strain vector takes any other direction. The associated flow rule results in an incremental plasticity theory. Most structural materials exhibit strain hardening behaviour, in that, the yield surface changes as yielding progresses. Hardening rules describe how the yield sur-face changes with yielding. Two commonly used hardening models are the kinematic and isotropic hardening models. In the kinematic hardening model, the yield sur-face maintains its original size but translates in stress space as yielding progresses. This model accounts for the Bauschinger effect, which is a situation in which the yield surface appears to translate in stress space under cyclic loading situations. In the isotropic hardening model the yield surface retains its initial shape but expands uniformly about the origin of the stress space. Although this model ignores the Bauschinger effect it is widely used in engineering practice due to its simplicity and is employed in this formulation. Based on the plasticity theory described above the incremental stress-strain rela-tions can be written as [29] {da} = [DT]{de} (4.16) where {da} and {de} are the incremental stress and strain vectors, respectively, and [DT] is the elastic-plastic constitutive matrix given by [DT] = [D] - [D]{A}{A}T[D}[E' + {A} r[I>]{A}]- 1 (4.17) [D] is the elasticity matrix, {A} = dF/d{a}, F is the yield function, E" = ET/(1-^-) is obtained from the bi-linear stress-strain curve of Figure 4.1 and E, Ej are the elastic and plastic modulii, respectively. Chapter 4. Theoretical Formulation and Analysis of Problem 33 The stress increment due to a given strain increment can now be obtained from Equation (4.16) and the state of stress at the end of any iteration step is the sum of the stress increment and the stress from the previous iteration. Yielding takes place when the equivalent effective stress, cre exceeds the uniaxial yield stress, <r0. For plate elements, °l = <Tx<7y + <r\ + 3rx 2 y and for beams cre = crx (or ay), depending on the beam direction. Once the yield stress is exceeded, the stresses are scaled back to the yield surface and the elastic-plastic constitutive relation (Equation (4.17)) is used from then on-wards unless unloading occurs (ae < cr0) when the elastic constitutive relations are employed. Many metals exhibit strain rate sensitivity under dynamic loading. An increase in strain rate leaves the elastic and plastic modulii practically unchanged but produces a rise in the yield stress as illustrated in Figure 4.2. The Cowper-Symonds relationship [30] provides an empirical dependence of the yield stress on strain rate: 1 + o-o e l / » 2 " — where cr^, a0 are the instanteneous and static yield stresses, respectively, sx and s2 are strain rate parameters which for steel and aluminum have the typical values shown in Table 4.1. Table 4.1: Strain rate parameters for steel and aluminum Aluminum Steel 6500 s"1 40 s- 1 S2 4 5 Chapter 4. Theoretical Formulation and Analysis of Problem 34 a e Figure 4.2: Stress-strain relation of strain-rate sensitive material Strain rate effects can be easily incorporated in elastic-plastic subroutines [20,27], although these have not been included in the elastic-plastic algorithm in this work. However, strain rate effects have been taken into account in some cases by upgrading the static yield stress in accordance with Equation (4.18). 4.3.5 Mass and Damping Matrices Having established the shape functions, strain-displacement and stress-strain rela-tions, attention will now be focused on the derivation of the finite element matrix quantities. In this section the mass and damping matrices will be derived. Using Equation (4.11) and the substitution {d} = [N]{Se}, Equation (4.2) can now be written for each element as Chapter 4. Theoretical Formulation and Analysis of Problem 35 {Se}TJv [[N]Tp{N]{Se} + [Nf Kd[N]{Se} + [[B] + [C0}f{*}} dV = {Se}T ^[NfqdS (4.19) Since {8e}T is arbitrary, Equation (4.19) can be written as K K U + [ce]{Se} + / [[B] + [C0]}T{<r}dV = {p} (4.20) where [me] = / [Nfp[N]dV (4.21) is the element consistent mass matrix, with p expressed in units of mass per volume and [ce] = Jv[N]T nd[N}dV - (4.22) is the element consistent damping matrix. For simplicity the damping matrix is assumed to be proportional to the mass matrix. The stiffness matrix and consistent load vector, {p} are derived in Sections 4.3.6 and 4.3.7, respectively. 4.3.6 Stiffness Formulation Suppose that the element is subject to an applied static load. Then Equation (4.20) takes the form / [[B] + [Co\f {o-}dV = {p} (4.23) This is a nonlinear equation and has to be solved iteratively. Here, the Newton-Raphson iterative scheme is used. First, the stress vector, {cr} has to be expressed in terms of the displacements and then the left hand side of Equation (4.23) can be expressed as a function, {$} = {$((5)}, which is a function of the generalized Chapter 4. Theoretical Formulation and Analysis of Problem 36 displacements. The equation can now be expanded in a Taylor series about a known solution {S0} so that Equation (4.23) is expressed as {*(*,)} + 8 W g { e } ({S}-{S0}) + . ..={?} (4.24) Ignoring higher order terms and defining the incremental nodal displacement vector as {AS} = {S} — {S0}, Equations (4.24) can be recast as [kT}{AS} = {p} - {$(S0)} (4.25) where [kT] = <9{$}/d{S} is the tangent stiffness matrix, obtained by differentiating {$} with respect to the displacement vector and utilizing the constitutive relations (Equations (4.16) and (4.17)). = I (HB] + lCn]]TlDT][\B] + [Coll + Vt])dV (4.26) where For plate elements the terms of [fi] are given by fi _ dN? ON?^ dN^dN?^ (8N?dN? dN? 8N?\ 13 dx dx dy dy v \ dx dy dy dx ) x y where i,j = 1,2, . . . ,55 and r, s = 1,2, . . . ,25 are selected to correspond to the appropiate w variables corresponding to the i, j indices, with the terms corresponding to the u and v variables being equal to zero. No summation is intended for any index or subscript in Equation (4.28). Chapter 4. Theoretical Formulation and Analysis of Problem 37 For a beam element in the x-direction the entries in [fl] are expressed as dNb dNb where i,j = 1, 2 , . . . , 10; r, s = 1, 2, . . . , 5 are selected to correspond to the appropriate w variables, with those corresponding to the u variables being zero. Details of the [fl] matrices are given in Appendix C. 4.3.7 Load Vector In this thesis it is assumed that the structure is subject to uniformly distributed loads only. That is, the spatial variation of the load q is not considered. In the dynamic realm, q is a function of time. Blast type dynamic loads are considered, and in general, the temporal variation of the load can be expressed as q(t) = qm(l - i / r ) e x p ( - A 1 i / r ) 0<t<r (4.30) or q(t) = qm(l-t/T)^'1 0<t<T (4.31) where qm is the peak pressure, t is the time variable, r the duration of the load and Ai, A 2 are decay parameters. The shapes of the blast loads for some values of A 2 are as shown in Figure 4.3. It is also possible to account for blast loads of arbitrary time histories. In this case, q(t) is specified by discrete data points with linear interpolation between points as in shown the figure. With the loading characteristics determined the element consistent load vector is given by {p} = Js[N]TqdS (4.32) where q is the uniformly distributed load. For plate elements q is in units of force per unit area and the integration in Equation (4.32) is performed over the plate surface Chapter 4. Theoretical Formulation and Analysis of Problem 38 Figure 4.3: Shapes of typical blast loads Chapter 4. Theoretical Formulation and Analysis of Problem 39 area, S. For beam elements q is in units of force per unit length and the integration is carried over the beam length. In a stiffened plate the load is usually applied on the plating so there is no loading on the stiffener beams. Only vertical loads (loads in the out-of-plane direction) are considered in this work, so the consistent load vector will not contain entries in the locations corresponding to the in-plane variables. The intergral in Equation (4.32) is evaluated exactly and the load vector for each element type is given below: Plate Elements M T = f|f LO, 0,36, 6, 6,1,0, 0 , 3 6 , - 6 , - 6 , - 1 , 0, 0, 36, - 6 , - 6 , 1 , 0, 0, 36, 6, - 6 , - 1 , 0, 0, 72*, 12*, 0, 0, 0, 0, 72*, - 1 2 * , 0, 0, 0, 0, 72*, - 1 2 * , 0, 0, 0, 0, 72*, 12*, 0, 0, 0,0,144*2, 0,0,0, 0J (4.33) where a, b are the length and width, respectively, of the plate element, q is the value of the load level and the constant factor * is given by y 1 l * = / )d( — [a(cosh p + cos p — 2) + (sinh fi — sin u)) Jo <pp Beam Elements W T = § L0, 6,1, 0, 6, - 1 , 0,12*, 0, 0J (4.34) 4 . 3 . 8 Torsion Beam Element So far, the formulation has been based on the assumption that torsional effects in the stiffener beams are negligible. However, there are several situations in which this assumption introduces significant error in the response analysis. In particular, ignoring stiffener torsion has negligible effect on the static response results since only Chapter 4. Theoretical Formulation and Analysis of Problem 40 symmetric loads are considered, but has significant effect on the linear vibration response of stiffened plates (see Chapter 6 for details of the response results). It has, therefore, been considered necessary to include the effect of stiffener beam torsion for dynamic problems. In this section, the beam torsional stiffness and mass matrices are derived explicitly from the strain energy and kinetic energy integrals. These matrices are calculated in a separate subroutine in the computer program and the beam torsional stiffness and mass matrices are conceptually added to the global stiffness and mass matrices in the usual finite element fashion. It should be noted that only linear torsional displacements have been considered on the asumption that these displacements are small. Also, coupling of bending and torsional deformations has not been considered in the formulation for simplicity. A more rigorous formulation might consider the effect of finite bending on the twisting of the beam and also an investigation of lateral buckling of the beam. 4.3.8.1 Stiffness matrix for beam torsional element Consider a beam element in the a; —direction. The degrees of freedom for torsion and lateral bending are as shown in Figure 4.4. The variable v represents displacement in the y — direction (lateral displacement). Consider the lateral movement of the beam (Figure 4.4). The beam centroidal displacement vc is given by vc = vT + e6 (4.35) where vr is the lateral displacement at the top of the beam and is equal to the v-displacement along the edge of the plate. Along an edge of the plate v is quadratic (see Equation (3.2)), hence is chosen to be quadratic to ensure compatibility of displacements. Using Equations (3.10) and (3.11) the torsional rotation and lateral Chapter 4. Theoretical Formulation and Analysis of Problem 41 displacement fields can be expressed as 0 < > = VT Hi H2 H3 HA 4>(£) 0 0 0 0 0 0 0 0 Lx L2 L3 Oi 0x1 02 0x2 03 Vl V2 V3 (4.36) V, , 0 ! , 0 x 1 V 3 , 0 J V*2, 0 2 . 0 x 2 Figure 4.4: Torsion beam element The shape functions in Equation (4.36) have been described in Chapter 3. Note also that for a beam in the x-direction, 9 = wy. The strain energy in the beam due to torsion, lateral bending and warping (in-duced by torsion) is given by Chapter 4. Theoretical Formulation and Analysis of Problem 42 U = \CJ r 92xdx + \EIZZ [\vc,xx)2dx + \ET r 92xxdx Z Jo Z Jo Z Jo = \GJ [ae2xdx + \EIZZ f (4xi + 2evTtXXetXX + e2e2xx)dx + ^ ET f e2xxdx Z Jo Z Jo Z Jo (4.37) where a is the beam length, G is the shear modulus, J the torsional constant, T the warping constant and I Z Z is the moment of inertia about the z — z axis. The symbolism ( > x ) implies differentiation with respect to x. Formulas for the calculation of J, T, I Z Z for various beam cross-sections are given in [31] and the relevent ones are quoted in Appendix D. Equation (4.35) has been used to obtain the expanded form of Equation (4.37). By taking the first variation of the strain energy with respect to the generalized displacements the torsional stiffness matrix is obtained as [k] = GJ [ae2xdx + EIZZ r v2xxdx+EIzze I" vTiXX6iXxdx + E(e2IZZ+T) f 92xxdx Jo Jo Jo Jo (4.38) Utilizing the shape functions in Equation (4.36) and carrying out the integrations in Equation (4.38) the torsional stiffness matrix can be written as [k] = [k,] + [k2] + [*3] + [fc4] (4.39) Chapter 4. Theoretical Formulation and Analysis of Problem 43 where [*i] = GJ 6 a 6 a 0 5 10 5 x10 a 10 2a2 15 a 10 a2 30 pxa 6 a 6 a 0 5 10 5 10 a 10 a2 30 a 10 2a2 15 -pia 0 0 -uta Pi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16EIZ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 - 2 0 0 0 0 0 1 1 - 2 0 0 0 0 0 - 2 - 2 4 [*s] = EIzze 0 0 0 0 0 0 0 0 0 0 0 0 0 - 4 - 4 8 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 - 8 0 0 0 0 0 0 0 0 0 - 4 0 4 0 0 0 0 0 - 4 0 4 0 0 0 0 0 8 0 - 8 0 0 0 0 (4.40) (4.41) (4.42) Chapter 4. Theoretical Formulation and Analysis of Problem 44 [*4 E(e2Izz + V) 12 6a -12 6a 0 0 0 0 6a 4a 2 — 6a 2a 2 0 0 0 0 -12 — 6a 12 — 6a 0 0 0 0 6a 2a 2 — 6a 4a 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where px = 0.5231643586, p2 = 4.877697585, and u3 - 198.4629301. 4.3.8.2 M a s s m a t r i x for b e a m tors iona l e lement The kinetic energy in the beam due to torsion and lateral bending is given by T = \w*p (\jc62 + Av2c)dx (4.44) I Jo where J c is the polar moment of inertia about the centriod, A is the area of the beam cross section, CJ is the natural frequency of the beam and p the mass density. Substi-tuting Equation (4.35) into Equation (4.44), and defining Jo as the polar moment of inertia about the top of beam the kinetic energy can now be expressed as T = \ L O 2 P [a[Jo02 + A(vT + 2evT8)]dx (4.45) 2 Jo Hence, by variational techniques the torsional beam mass matrix is obtained as pa /»a y»a [m] = pj0 / 02dx + pA vTdx + pAe / vT6dx (4.46) Jo Jo Jo Utilizing the shape functions in Equation (4.36) and carrying out the integrations in Equation (4.46) the torsional mass matrix can be written as [m] = [ m i ] + [m2] + [m3] (4.47) Chapter 4. Theoretical Formulation and Analysis of Problem 45 where, [ m i ] = pJo 13a 35 11a2 210 9a 70 13a2 420 p4a 0 0 0 11a2 210 a 3 105 13a2 420 a 3 140 p5a2 0 0 0 9a 70 13a2 420 13a 35 11a2 210 p4a 0 0 0 13a2 420 a 3 140 11a2 210 a 3 105 -p5a2 0 0 0 p4a p5a2 p4a -p5a2 p6a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \m2 pAa ~3u~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 - 1 2 0 0 0 0 0 - 1 4 2 0 0 0 0 0 2 2 16 [ m 3 ] = pAae 0 0 0 0 0 l l a2 60 a 60 a 3 0 0 0 0 0 a2 60 0 a 2 15 0 0 0 0 0 a 60 l la 60 1 3 0 0 0 0 0 0 a2 60 a 2 15 0 0 0 0 0 p7a p7a psa 11a2 60 a 2 60 a 60 0 p7a 0 0 0 a 60 0 11a 60 a 2 60 pTa 0 0 0 a 3 a 2 15 1 3 a 2 15 p8a 0 0 0 (4.48) (4.49) (4.50) where p4 = 0.2615821802, p5 = 0.0562872124, p6 = 0.396478085, p7 = 0.0364333308 and p8 = 0.4502976989. Chapter 4. Theoretical Formulation and Analysis of Problem 46 4.4 Numerical Integration The well known Gaussian integration scheme is adopted in the evaluation of the volume integrals in Equations (4.21), (4.22) and (4.26). Let / = f(£,7]X) be the integrand in each of the volume integrals, then by the integration scheme the volume integrals can be presented as j f fQ f(t,V,C)dtdVdC = llllllji^o^dfjdc ~ E E E ^ W ( 6 , % , a ) (4.5i) k j i where £, fj, £ are the transformed normal coordinates which vary from —1 to +1, £i, Vj> Ck are the sampling points in the x, y, z directions, respectively, W{, Wj, are the wieghting factors in the respective directions. The weights and sampling points are selected to minimize the error in the integral. The scheme employed depends on the form of the integrand, / . In any one direction, n sampling points can integrate a polynomial of order (2n — 1) exactly [29]. However, for the problem at hand this formula cannot be applied as the integrand, in general, consists of products of polynomial and circular or hyperbolic functions. Based on previous experience [18,19,32] and a numerical experiment conducted in this work, 5 integration points are employed in each in-plane direction. In the out-of-plane direction a 2-point integration scheme is used for elastic analysis. However, for plastic analysis it is necessary to upgrade the number of sampling points to 4, in order to capture the plastic stress distribution across the plate or beam thickness [33]. The location and weights of the sampling points are presented in Table 4.2. A more comprehensive list can be found in [34]. For beams of I or T cross-sections an additional point is introduced for each flange. In the evaluation of the volume integrals for a beam element, the integrand is evaluated at the sampling point in the beam direction and this is multiplied by the Chapter 4. Theoretical Formulation and Analysis of Problem 47 Table 4.2: Sampling points and weights for Gaussian integration / + 1 / (0 t f = £ w / ( 6 ) 1=1 n ±£ W 2 0.57735 02691 89626 1.00000 00000 00000 4 0.86113 63115 94053 0.33998 10435 84856 0.34785 48451 37454 0.65214 51548 62546 5 0.90167 98459 38664 0.53846 93101 05683 0.00000 00000 00000 0.23692 68850 56189 0.47862 86704 99366 0.56888 88888 88888 corresponding weighting coefficient and a weighted area. The weighted area is equal to the product ^hwtwW{] where hw, tw are, respectively, the height and thickness of the beam web and Wj is the weighting factor corresponding to the sampling point in the thickness direction. For an extra point in a flange, the weighted area is simply equal to the area of that flange. If the beam cross-section is rectangular or has equal flanges, the centroidal axis coincides with the normal coordinate, £ axis. But for T or I sections with unequal flanges these axes do not coincide and it is necessary to transform the normal coor-dinate variable about the beam centroidal axis. Consider the beam shown in Figure 4.5. Let A A be the centroidal axis of the beam section and B B the centroidal axis of the web alone. Also, let ( c , £ be the running variables measured with respect to the axes A A and B B , respectively. Then, Cc = C - d But d = ( C 2 - / 2 ) - y = h - C l - h - \ { h - h - h ) Chapter 4. Theoretical Formulation and Analysis of Problem 48 Hence, Cc = C + ^ ( 2 C l - & - / ! + / , ) with £ = ^Chw. For the extra point in a flange, the variable is measured from the beam centroid to the middle of the flange. B h w 2 L B Figure 4.5: Beam cross-section 4.5 Temporal Integration The element matrix quantities are assembled conceptually in the usual finite element fashion to obtain the global equations of motion: [M}{6} + [C}{6} + Jv[[B] + [C0]}T{a}dV = {P} (4.52) where, [M], [C] are the global mass and damping matrices, respectively, and {P} is the global load vector. Equations (4.52) are nonlinear ordinary differential equations in time. Chapter 4. Theoretical Formulation and Analysis of Problem 49 In this section attention is focused on the time integration of the equations of motion (Equations (4.52)). For nonlinear problems, direct integration, or step-by-step, methods are preferred to modal or other methods. In the direct integration methods, the time derivatives appearing in the governing equations are replaced by a finite difference approximation. Two main groups of direct integration methods are available. These are the explicit and implicit schemes. Explicit schemes only require satisfaction of the equations of motion at the previous time step in order to advance the solution to the next time step. In contrast, implicit schemes require information from both the previous and next time steps. There are several direct integration methods available in the literature [27,35,36] and, in general, the choice of the scheme to use depends on the nature of the problem at hand. In this thesis, the implicit Newmark-/3 method [37] is employed. The method is second order accurate and is unconditionally stable for linear problems, in that the solution does not grow without bound even when a large time step is used. Unconditional stability of the method also applies for nonlinear problems, although there is no rigorous proof in this case [27]. As a consequence of this condition the method allows for the use of large time steps. Hence, time steps could simply be based on the lowest fundamental frequency (highest period) rather than on the highest frequency (which requires more effort to evaluate accurately) as would be the case if a conditionally stable explicit scheme is used. This is particularly useful in the present work since one of the aims of the study is to develop a simple formulation which requires reduced run times. The Newmark-/? method is also self starting, in that it requires only the initial data to commence the solution and is thus a single step method - a desirable feature. The method has also been applied successfully to many nonlinear structural problems [27]. Chapter 4. Theoretical Formulation and Analysis of Problem 50 One major disadvantage of an implicit integration scheme such as the Newmark-(3 method is that it requires more storage space and computational effort in each time step. However, the use of large time steps compensates for this deficiency. The implementation of the integration scheme now follows. The governing difference equations are given by [M}{6}n+1 + [C}{8}n+1 + f({S}n+1) = {P}n+1 (4.53) Wn+l = + A«[(l - 7 ) { * } B + 7tf}n+l] (4-54) Wn + i = Wn + (At){6}n + (At)2[(± - (3){6}n + 0{8}n+1] (4.55) where [M], [C] are the global mass and damping matrices and the subscripts refer to the time step number, At is the time step interval, 7, f3 are parameters depending on the integration scheme. For the Newmark scheme adopted here, 7 = 0.5 and f3 = 0.25. The nonlinear term /({£} n +i) represents the internal force vector at the (n + l)th time step. Using a Taylor's series expansion this term can be expressed as /(Wn+i) = / ({*}„) + [KT]{AS}N+1 (4.56) where \KT\ is the global tangent stiffness matrix and { A S } n + 1 = {*}n+i - {S}n (4.57) is the incremental displacement vector. The acceleration vector is obtained from Equation (4.55) as {8}n+l - f3(Aty which on substitution into Equation (4.54) gives {A8}-(At){6}n-^(l-2(3){6}T (4.58) W» + i = ^ ) { M > » + i + ^ g 2 ^ } " + ^2/3 7 ) W " ( 4 - 5 9 ) Chapter 4. Theoretical Formulation and Analysis of Problem 51 Substituting Equations (4.55), (4.58) and (4.59) into Equation (4.53), the difference equation can be written as [K]{A6}n+1 = {P}n+l (4.60) where, [K] is an effective stiffness matrix given by [K} = 1 W] + ^ [C] + [KT] (4.61) J3(At)21 1 (3At and {P} is the effective load vector given by {p} = ,{Pu1-m) + [ M ] ( ^ { ^ + ^ { % ) + + [ C ] ( ^ p W n + At2^i{S}n) (4.62) For linear problems, [KT] = [K] and f({S}n) = [K]{6}n and the incremental displacement term {A6}n+i in Equation (4.60) has to be replaced by {c^} n + i and the effective load vector in this case is given by 4.6 Computer Code A computer code named NAPSSE (Nonlinear Analysis of Plate Structures by Su-per Elements) has been written to implement the theory. The program is written in F O R T R A N language. It is presently functioning on an I B M 3081K main frame computer and a GA-386L microcomputer. The program has the capabilities to perform static, vibration and transient analy-sis of unstiffened plates, beams and plates stiffened in one or two orthogonal directions. Chapter 4. Theoretical Formulation and Analysis of Problem 52 For static and transient analyses all possible combinations of material and geometric nonlinearities can be specified. For these nonlinear analyses, the solution is obtained by an iterative procedure and convergence is achieved if the maximum norm, defined by \SAi/Ai\max, or the Euclidean norm, defined by £ i=? (£A; ) 2 / D|=i (A t - ) 2 , is less than an acceptable tolerance specified by the user, where, A ; is the solution of the nodal displacement variable i, SAi the correction factor for that variable and n the total number of nodal variables. Since not all nodal variables represent the actual displacements at the correspond-ing nodal points, the program furnishes additional output for the displacements at specified locations within the plate or beam elements. These displacements have been computed using the nodal solution vector in conjunction with the values of the re-spective shape functions at the predetermined locations. The program can also give information on stresses and strains at the Gauss points, if required. Chapter 5 Static Analysis Results 5.1 Introduction In this chapter the super element formulation discussed in Chapter 4 is used to inves-tigate the response of various plate structures subjected to applied static loads. The governing equations of motion for this case reduce to where {P} is the global load vector, and the equations are solved by Newton-Raphson iteration as discussed in Chapter 4. Although problems with nonlinear geometric and/or material behaviour are of primary interest, the linear elastic responses have also been investigated to provide a fuller understanding of the response characteristics of the new elements. In this case, the governing equations solved are where [K] is the global stiffness matrix and no iteration is required here. The applied loading is assumed to be uniformly distributed in all cases. For linear elastic analysis the load is applied in one load step while for nonlinear analysis the load is applied in several steps and accumulated up to the full load. In keeping with the objectives of this work only one element is used to represent a panel bay or beam span as the case may be. However, some exceptions to this (5.1) [K]{8} = {P} (5.2) 53 Chapter 5. Static Analysis Results 54 general rule have been made, in a few instances, to study the convergence properties of the super elements or to employ discretizations similar to those used by other investigators. The problems analyzed are categorized according to structure type. First, unstiff-ened plates are analyzed in Section 5.2 and then beams acting alone, with no plating, are considered in Section 5.3. Finally, Section 5.4 focuses on the response of plates stiffened in one or two mutually perpendicular directions. 5.2 Unstiffened Plates 5.2.1 Square Plate I with Simply Supported Edges The dimensions and material properties of the square plate are as given below: dimensions = 100 mm x 100 mm x 1 mm elastic modulus, E = 205,000 N / m m 2 Poisson's ratio, v = 0.3 The plate is simply supported all round. A uniform pressure load is applied and the structure is modelled by one super plate element. This model has 55 gross and 15 net degrees of freedom. The boundary conditions are applied as follows: • At the four corner nodes all variables except the twist variable are constrained. • At the four mid-side nodes, all variables except the normal slope (e.g. wy5 at node 5) are set equal to zero. • At the middle node (node 9) all variables are left free. (See Figure 3.3 for location of nodes and variables). Thus the structure is constrained against in-plane motion. Chapter 5. Static Analysis Results 55 First, a linear elastic analysis is carried out for an applied load of 0.1 N / m m 2 . The panel centre deflection and strain energy are presented in Table 5.1 together with those obtained by the finite strip [19] (one mode solution) and exact [1] methods. It is observed that the super element analysis is in excellent agreement with the finite strip solution. However, it over estimates the exact central deflection by 1.2% and under estimates the exact strain energy by only 0.4%. This excellent agreement of the one-super element solution with both the finite strip and exact analyses demonstrates its viability. Table 5.1: Linear elastic response of simply supported Square Plate I Central Strain Deflection (mm) Energy (Nm) Present 2.1887 0.4517 (% Error) (1.15) (-0.4). Finite Strip 2.1890 0.4512 Exact 2.1639 0.4535 Contours of the crx stress at the bottom surface of the plate are plotted in Fig-ure 5.1. Note that the stress values have been computed only at the Gauss points and these have been extrapolated linearly (depth wise) to the bottom or top surface. Only the stresses computed at the 25 in-plane sampling points are used to plot the contours. The force boundary conditions are not exactly satisfied at the boundaries, in that the stresses do not come out as zero along the edges as expected. However, the stress distribution away from the boundaries comes out as expected and the peak stress at the panel centre predicted by the super element is about 300 N / m m 2 compared to the exact value of 287.4 N / m m 2 . As expected in an approximate method based on a displacement approach, the error in the stress (4.4%) is bigger than that in the displacement. Since both the structure and loading are symmetric, the o~y stress Chapter 5. Static Analysis Results Figure 5.1: o~x stress in linear elastic simply supported Square Plate I Chapter 5. Static Analysis Results 57 contour is also represented by Figure 5.1 with the x and y axes interchanged. A small deflection, elastic-perfectly plastic (Ej = 0) analysis is carried out with the yield stress being 300 N / m m 2 . The load-displacement plot is shown in Figure 5.2 along with the finite strip and yield line results. The super element collapse load is quite close to the finite strip prediction and is about 11% higher than the yield fine solution. 0.25 0.20 0.00 S.S. Square Plate I Small Deflection, Elastic-Plastic Analysis Present Finite Strip Yield Line 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Central displacement (mm) Figure 5.2: Central deflection in linear elastic-plastic simply supported Square Plate I Large deflection elastic and elastic-plastic central displacement responses are pre-sented in Figure 5.3. For the elastic-plastic case, Ex = 1025 N / m m 2 and <r0 = 210 N / m m 2 . Good agreement is observed between the present and the one mode finite strip solutions. The stiffening action due to the inclusion of the nonlinear strain displacement terms is clearly demonstrated, in that at higher loads the central dis-placement from the large deflection elastic analysis is smaller than that from the Chapter 5. Static Analysis Results 58 linear elastic analysis. However, as expected, the solution shows some softening when material nonlinearities are introduced. 0.5 (0 O 0.1 0.0. 0.0 S.S. Square Plate I Large Deflection Analysis Elastic/ / ' ' / '/'/ P l a s t i c Present Analysis / / / i J / »J / Finite Strip * / / *// •// */s * f *J *f / X *^ 1 ' 1 1 1 Linear Elastic i 0.5 1.0 1.5 Central deflection (mm) 2.0 Figure 5.3: Central deflection in nonlinear analysis of simply supported Square Plate I The o~x stress contours at load levels of 0.04 N / m m 2 and 0.2 N / m m 2 for the large deflection elastic case are plotted in Figures 5.4 and 5.5, respectively. At 0.04 N / m m 2 all of the bottom surface is in tension (positive stress), with the maximum tensile stress being about 60 N / m m 2 at the panel centre. However, the top surface is mostly in compression (negative stress) and is only in tension near the middle of the two edges parallel to the y-axis. The maximum compressive stress at the centre is 25 N / m m 2 and the maximum tensile stress at the edge is 7.5 N / m m 2 . At 0.2 N / m m 2 , membrane action predominates and both the top and bottom surfaces are now in tension. The largest amount of membrane stretching occurs along the middle axis parallel to the x-axis and a maximum tensile stress of 170 N / m m 2 is noticed near the Chapter 5. Static Analysis Results 59 third points along this axis. 5.2.2 Square P l a t e I w i t h C l a m p e d E d g e s The same square plate discussed in Section 5.2.1 is analyzed again, but this time with all edges clamped. The structure is again modelled by one super plate element and all variables are restrained along the plate edges, so that the net number of variables in the analysis is 7. A l l these variables are located at the middle node (node 9 in Figure 3.3). The linear elastic central displacement and strain energy responses at a load of 0.1 N / m m 2 are shown in Table 5.2 along with the finite strip and exact response results. It is observed that the one super element solution is in excellent agreement with the one-mode finite strip solution [18] but over estimates the exact central deflection by 5.8% and under estimates the exact strain energy by 6.2%. These errors are significantly larger than the corresponding ones in the simply supported case due to the fact that only one shape function, namely, Ng5 contributes to the response. Table 5.2: Linear elastic response of clamped square plate I Central Strain Deflection (mm) Energy (Nm) Present, 1 element 0.7102 0.0972 (% Error) (5.8) (-6.2) Present, 2 elements 0.6890 0.1000 (% Error) (2.6) (-3.5) Present, 4 elements 0.6699 0.1027 (% Error) (-0.2) (-0.9) Finite Strip 0.6915 0.1004 Exact 0.6715 0.1036 Although the errors in the present analysis are quite acceptable for engineering design purposes, improved solutions are sought by the use of finer meshes in order to Chapter 5. Static Analysis Results 60 Chapter 5. Static Analysis Results 6 1 Chapter 5. Static Analysis Results 62 study the convergence properties of the super elements. In Table 5.2 the solutions obtained using 2 or 4 super elements are also presented. Note that in these cases only one element is actually used to model one half or a quarter of the structure, with symmetry conditions applied along the centre lines. The improvement in the solution with element refinement is clearly evident. The bottom surface normal stress, ax, contours from the one element solution are shown in Figure 5.6. The normal stress is compressive near the edges and tensile in the middle part. A t the edges (x = 0 and x = 100 mm), the super element predicts a maximum compressive stress of about 170 N / m m 2 which is significantly less than the exact maximum compressive stress of 307.8 N / m m 2 . This large error in the boundary stress is due to the coarseness of the one element solution. However, the exact maximum tensile stress at the plate centre is 138.6 N / m m 2 and from the figure it is seen that the corresponding super element result is only slightly in error. The small deflection, elastic-perfectly plastic (E? = 0 and o~0 = 300 M P a ) central displacement response is shown in Figure 5.7 along with the finite strip and yield line analyses results. The super element predicts a collapse load of 0.415 M P a compared to 0.400 M P a from the one mode finite strip analysis and 0.360 M P a from the yield line analysis. Large deflection, elastic-plastic central displacement responses are presented in Figure 5.8. Here, ET = 1025 N / m m 2 and <r0 = 210 N / m m 2 . Good agreement is observed between the super element and finite strip solutions. To study the effect of the sine terms in the in-plane displacement fields (see Equa-tions (3.1) and (3.2)) the large deflection elastic-plastic analysis is performed again without the sin2^, sin4^, sm2v and sin4r/ terms in the in-plane displacement fields. A t a load of 0.8 M P a the predicted panel centre deflection is 2.08 m m compared to 2.41 mm when all the in-plane displacement functions have been included. Thus in this particular case the central displacement is stiffer by 14% if the sine terms are Chapter 5. Static Analysis Results 63 Figure 5.6: ax stress in linear elastic clamped Square Plate I Chapter 5. Static Analysis Results 64 0.50 0.40 -'c? 0.30 0_ •o CO o 0.20 0.10 0.00 0.00 Clamped Square Plate I Small Deflection, Elastic-Plastic Analysis Present Finite Strip Yield Line 10.00 20.00 30.00 40.00 Central displacement (mm) 50.00 Figure 5.7: Linear elastic-plastic response of clamped Square Plate I ommitted in the formulation. This discrepancy is dependent on the parameters of the problem and in general it is expected to be more for problems with more geometric and material nonlinearities. It is important to note that without the sine functions in the in-plane displacement fields the predicticted in-plane displacements are identically zero everywhere. This, of course, is grossly incorrect. Hence to model the correct be-haviour of the structure in the present formulation, the in-plane displacement fields must contain the continuous sine functions. The large deflection elastic normal stress, ax contours at load levels of 0.2 N / m m 2 and 0.8 N / m m 2 are plotted in Figures 5.9 and 5.10. At 0.2 N / m m 2 , the central portion of the bottom surface is in tension while the parts close to the plate boundaries are in compression. The situation is reversed at the top surface. At 0.8 N / m m 2 , when membrane action is fully developed, the same trend is observed although the area over Chapter 5. Static Analysis Results 65 Figure 5.8: Large deflection response of clamped Square Plate I which tensile stresses act increases due to the membrane action and the maximum tensile stress at the centre is about 500 N/mm 2 . It is expected that with further increase in the load level all of the plate will be in tension as observed in the simply supported case. 5.3 B e a m s 5.3.1 Rectangular Beam with Simple Supports In the analysis of beams acting alone, the eccentricity e in Equation (3.8) is set equal to zero and the nodal degrees of freedom are assumed to be associated with the beam centroidal axis. The rectangular beam previously solved by Abayakoon et al [19] with the finite strip method has been chosen to evaluate the performance of the Chapter 5. Static Analysis Results 66 Chapter 5. Static Analysis Results 67 Bottom ° - ° 100.0 Stresses are in N/mm 2 Figure 5.10: o~x at 0.8 N /mm 2 in nonlinear elastic clamped Square Plate I Chapter 5. Static Analysis Results 68 super beam elements acting alone. The beam is simply supported and the dimensions and material properties are: length, a width, b depth, h elastic modulus, E plastic modulus, ET yield stress, cr0 = 500 mm = 10 mm = 10 mm = 220,000 N / m m 2 = 0 300 N / mm The beam is subjected to a uniform line load. One super beam element is used to model the beam. This model has 10 gross and 6 net degrees of freedom. The linear solution central deflection, maximum normal stress and strain energy at a load of 10 k N / m are displayed in Table 5.3. The super element results are very close to both the finite strip and exact solutions. The super element strain energy is on the low side of the exact answer by only 0.14%, while the maximum stress is over estimated by 2.3%. Table 5.3: Linear elastic response of simply supported rectangular beam Central Strain Maximum Deflection (mm) Energy (Nm) Stress (N/mm 2 ) Present 44.486 70.924 1917.4 (% Error) (0.22) (-0.14) (2.3) Finite Strip 44.560 70.924 1935.1 Exact 44.389 71.023 1875 A small deflection, elastic-perfectly plastic analysis is performed and the displace-ment response is presented in Figure 5.11 along with the yield line and finite strip results. The present analysis is in agreement with the finite strip analysis and both of these methods predict a higher collapse load than the yield line method in which elastic deformations are ignored. Chapter 5. Static Analysis Results 69 The large deflection, elastic and elastic-plastic central displacement responses are plotted in Figure 5.12. It is seen that the present analysis result compares very well with the Timoshenko [1] and finite strip [19] results for the elastic case. Also, for the elastic-plastic material case, the results from the present analysis agree well with the other comparison results [19,38]. Z TJ CO Q 3.50 3.00 -2.50 2.00 * 1.50 0.00_ S.S. Rectangular Beam Small Deflection, Elastic-Plastic Analysis "0.00 10.00 20.00 30.00 40.00 Central displacement (mm) Figure 5.11: Linear elastic-plastic response of simply supported rectangular beam Figure 5.13 shows the variation of the normal stress along the beam at various load levels. The beam bottom surface is in tension at all load levels and the maximum stress at mid-span increases from 270 N / m m 2 at 2 k N / m to 895 N / m m 2 at 18 k N / m . However, at the top surface, the beam is in tension near the ends and in compression near the middle. Figure 5.12: Large deflection response of simply supported rectangular beam Chapter 5. Static Analysis Results 200.0 100.0 CO Q. ~ 0.0 to 6 CO o z -100.0 --200.0 --300.0. 0.0 Top Stresses q = 2 kN/m q = 10 kN/m q = 18 kN/m 100.0 200.0 300.0 Distance (mm) 71 400.0 500.0 1000.0 800.0 -CO ~ 600.0 h | 400.0 -o z 200.0 -0.0. 0.0 q = 18 kN/m q = 10 kN/m q = 2 kN/m Bottom Stresses 100.0 200.0 300.0 Distance (mm) 400.0 500.0 Figure 5.13: Stresses in large deflection analysis of simply supported rectangular beam Chapter 5. Static Analysis Results 72 5.3.2 Rectangular Beam with Clamped Ends The same beam example considered in Section 5.3.1 is analyzed again but with the beam ends fixed. A l l dimensions and material properties are the same. One super beam element is also used to model the structure and in this case there are only 4 net degrees of freedom. The structure is first subjected to a line load of 10 kN/m and the results obtained in a linear elastic analysis are compared with the exact and finite strip results in Table 5.4. The results obtained from the super element are in good agreement with the other two solutions. The super element strain energy is only 0.8% lower than the exact solution. As expected, higher errors are observed in the stress results. The maximum bottom stress at mid-span is 7.7% higher than the exact value while the maximum bottom stress at the clamped end is about 18% lower. Note that the maximum stress at the support has been obtained by linear extrapolation (length-wise) of the stresses obtained at the Gauss points. This is partly responsible for the relatively higher error in the maximum support stress. Only the bottom surface stresses are presented in the table. Table 5.4: Linear elastic response of clamped rectangular beam Central Strain Max. mid-span Max. end Displ. (mm) Energy (Nm) Stress (N/mm 2 ) Stress (N/mm 2 ) Present 8.975 11.738 673.2 -1026 (% Error) (1.1) (-0.8) (7.7) (-17.9) Finite Strip 8.987 11.754 697.4 -1147 Exact 8.878 11.837 625.0 -1250 The large deflection elastic and elastic-plastic central displacement responses are plotted along with the finite strip and yield line solutions in Figure 5.14. Excellent agreement is observed between the super element and finite strip, Timoshenko [1] and Chapter 5. Static Analysis Results 73 F E N T A B [39] solutions. The variation of the normal stress along the beam is plotted at three load levels — 10 kN/m when membrane action is just beginning, 40 kN/m when significant mem-brane action has taken place and 80 kN/m when membrane action is fully developed — in Figure 5.15. At all three load levels the bottom surface is in tension and the maximum tensile stresses at the middle of the beam at these loads are 664 N / m m 2 , 1340 N / m m 2 and 1560 N/mm 2 , respectively. At the top surface, at 10 kN/m, the middle portion is in compression while the portions close to the ends are in tension. But as the membrane force increases due to increase in load a larger portion of the top surface is now in tension and at 40 kN/m the whole beam section goes into tension. 5.4 S t i f f e n e d P l a t e s 5.4.1 Clamped 2-Bay Stiffened Plate I The details of this problem are presented in Figure 5.16. Al l edges are clamped and using symmetry, one half of the structure is modelled by one plate element and one beam element. Note that the symmetry line divides the beam into two, and hence the relevant beam section properties have to be halved in the computations. The super element model has 55 gross and 11 net degrees of freedom. Some of the linear elastic response results at a load of 0.001 N / m m 2 are given in Table 5.5 along with those from a finite strip analysis using 8 strips for one bay of the structure. It is seen that the super elements overestimate the panel central deflection by 2.2 percent and underestimate the strain energy by 3.7 percent, with respect to the one mode finite strip solution. The distribution of the in-plane displacement v (parallel to the stiffener) along y = 0.125 m is plotted in Figure 5.17. The discretization employed here consists of 2 super elements (1 plate and 1 beam) for one quarter of the structure. The present result Chapter 5. Static Analysis Results 120.0 100.0 -80.0 — 60.0 T3 CO Q 40.0 -20.0 -Figure 5.14: Large deflection response of clamped rectangular beam Chapter 5. Static Analysis Results 600.0 400.0 CO Q . f 200.0 h I CO o z 0.0 -200.0 h -400.0 Top Stresses q = 80 kN/m q = 40 kN/m q = 10 kN/m 0.0 100.0 200.0 300.0 Distance (mm) 400.0 500.0 CO C L CO o Z 1800.0 1600.0 1400.0 1200.0 1000.0 800.0 600.0 400.0 200.0 - q = 80 kN/m Bottom Stresses ^ = A W W ™ i i 1 , 1 . 0.0 100.0 200.0 300.0 Distance (mm) 400.0 500.0 Figure 5.15: Stresses in large deflection analysis of clamped rectangular beam Chapter 5. Static Analysis Results 76 E E o o in y.v 2 Panels at 500 mm centres = 1000 mm z, w 3 mm 10 mm- LT Young's Modulus = 71,700 N/mm Hardening Modulus = 358.5 N/mm2 Yield Stress = 284 N/mm2 Poisson's ratio = 0.33 x, u 21 mm Figure 5.16: Configuration of 2-Bay Stiffened Plate I Chapter 5. Static Analysis Results 77 Table 5.5: Linear elastic response of 2-Bay Stiffened Plate I Panel Center Deflection (mm) Stiffener Center Deflection (mm) Strain Energy (Nm) Present 0.471 0.042 0.335 Finite Strip 0.461 0.052 0.348 is the solid line, while the finite strip comparison result is shown dashed. The super elements capture the shear lag effect but the predicted peak in-plane displacement at the stiffener location is not as high as that predicted by the finite strips. This is due to the fact that for the super elements the shear lag is modelled by a quadratic shape function across the whole panel whereas for the finite strip case the shear lag is represented more accurately by 8 strips parallel to the stiffener. By employing a discretization which has 2 super plate elements parallel to the stiffener and 1 super beam element (shown dotted), it is seen that the in-plane displacements from the superelements approach the finite strip solution. Figures 5.18 and 5.19 show the load-displacement relationships at the panel centre and stiffener mid-point (points A and B in Figure 5.16), respectively, in a large deflection analysis. Note that the super element model used in this case consists of 1 plate element and 1 beam element for one half of the structure. For elastic large deflection analysis, the predictions from the super elements are quite close to those by the finite strip method at low loads. However, at higher loads when membrane action is fully developed, the deflections predicted by the super elements are larger than those from the one mode finite strip analysis. At a load of 0.8 N / m m 2 , for example, the error in the super element panel centre deflection compared to the finite strip solution is about 15%. However, at the same load level, the strain energy from the present analysis is on the stiff side by only 0.8 percent. This can be explained by making reference to Figure 5.20 where the displacement Chapter 5. Static Analysis Results 78 2.0 1.5 -2 super elements for quarter of structure ?1 .0 - 3 super elements for quarter of structure E, Finite Strip Distance from clamped edge (m) Figure 5.17: In-plane displacement in 2-Bay Stiffened Plate I profiles at three load levels — 0.001 N /mm 2 when the response is purely linear elastic; 0.04 N / m m 2 when some membrane action has taken place; and 0.40 N /mm 2 when membrane action is fully developed — are presented. It is seen that at all three load levels, the super element solutions are stiffer along most of the profile, and are only more flexible near the panel centre. The super element results for large deflection elastic-plastic analysis also compare well with the finite strip solution at low loads. At higher loads the results from the present analysis tend to be more flexible than those from finite strips, although the predictions from both analyses are still close. Chapter 5. Static Analysis Results 79 1.00 0.80 «T 0.60 0_ T3 CO O 0.40 0.20 0.00 0.0 Large Deflection Analysis Elastic Present Analysis Finite Strip Linear Elastic Plastic 5.0 10.0 15.0 20.0 Panel centre displacement (mm) 25.0 Figure 5.18: Panel centre displacement in 2-Bay Stiffened Plate I 5.4.2 Clamped DRES Stiffened Panel Figure 5.21 shows details of the problem configuration. The panel was constructed at the Defence Research Establishment in Suffield, Alberta (DRES) [40] and a static analysis of the panel has been carried out using an earlier version of the general purpose computer program ADINA [41]. The material properties are E = 30 x 106 psi, ET = 0.178 x 106 psi, <r0 = 45 x 103 psi and v = 0.3. The panel is clamped all round and the section A B C D is modelled by one plate element and one beam element for half of the stiffener BC as shown in Figure 5.22. Symmetry conditions are invoked along B C , CD and DA respectively. The model has 55 gross and 21 net degrees of freedom. This model, called Model 1, is used to enable direct comparison with the ADINA analysis for which only the portion A B C D was modelled using 964 triangular elements, as shown in Figure 5.23. This representation Stiffener centre displacement (mm) Figure 5.19: Stiffener centre displacement in 2-Bay Stiffened Plate I assumes that the panel is effectively infinitely long with identically repeated bays and stiffeners. Some of the linear elastic response results at a load of 50 psi are presented in Table 5.6 along with the ADINA solutions. The super elements under estimate the panel centre and stiffener midspan deflections by 7.3 and 7.5 percent, respectively, as compared \o the ADINA analysis. The strain energy from the ADINA analysis was not available. Load deflection plots for the panel centre (point D) and stiffener mid-span (point C) from a large deflection analysis are shown in Figures 5.24 and 5.25, respectively. For elastic analysis the displacement responses obtained from the super finite elements are stiffer than the ADINA solutions. But for plastic analysis, the super elements tend to be more flexible than the ADINA solution at higher loads. However, Chapter 5. Static Analysis Results Finite Strip Present Analysis Figure 5.20: Displacement shapes along CB of 2-Bay Stiffened Plate I er 5. Static Analysis Results 0.25 in in 2.93 i r H h" I T 0.53 in T 5.645 in Figure 5.21: Configuration of DRES stiffened panel Chapter 5. Static Analysis Results 83 Figure 5.22: Super element models for DRES stiffened panel Figure 5.23: ADINA discretization of DRES stiffened panel Chapter 5. Static Analysis Results 60.0 Elastic Plastic Panel centre deflection (in) Figure 5.24: Displacement of point D in DRES stiffened panel Chapter 5. Static Analysis Results Figure 5.25: Displacement of point C in DRES stiffened panel Chapter 5. Static Analysis Results 87 Table 5.6: Deflections and strain energy in linear elastic DRES stiffened panel Panel Centre Deflection (in) Stiffener Center Deflection (in) Strain Energy (kip-in) Present 5.19 0.37 900 ADINA 5.60 0.40 in both cases comparison with the ADINA result is good and the displacement error in the super element analysis is never more than 17 percent. The super element elastic-plastic curves tend to branch out from the elastic curves earlier than the ADINA ones. This is probably due to the difference in the yield criteria: the present analysis uses the von Mises yield criterion at each Gauss point, whereas the ADINA analysis for this example employed the Ilyushin criterion. In Figures 5.24 and 5.25 the deflection plots obtained from a model which uses one super plate element for the middle panel and two super beam elements for the two middle stiffeners are also presented. This model, named Model 2 (see Figure 5.22) is used to conform with the idea of using only a single element per bay. For both elastic and elastic-plastic large deflection analyses the panel centre and stiffener mid-point deflections predicted by Model 2 are larger than those predicted by Model 1. As explained in the previous example this is only a localized phenomenon. The strain energy for elastic problems from Model 1 is on the flexible side of that from Model 2, as expected theoretically. In Figures 5.26 and 5.27, top surface stress contours from a large deflection elastic analysis using Model 1 are plotted at three load levels — 0.5 psi when the response is linearly elastic; 20 psi when considerable membrane action has taken place and 50 psi when membrane action is fully developed. Stresses in directions perpendicular and parallel to the stiffeners are considered and contours are presented for a quarter of the middle bay (portion A B C D ) . The top line represents the axis of the stiffener, Chapter 5. Static Analysis Results 88 the right line the clamped edge and the others represent lines of symmetry. The contours for the normal stress perpendicular to the stiffener direction are presented in Figure 5.26. At 0.5 psi the stress contours obtained from the super elements are very similar to those from ADINA. At 20 psi the stress contours from the super elements are quite similar to those from ADINA except that the maximum stress predicted by the super elements is about 60 ksi compared to 80 ksi for ADINA. However, the contours obtained from the present analysis seem more complete than those from ADINA and besides, the effect of membrane stretching is more evident in the present study. At 50 psi the super elements predict a maximum stress of about 90 ksi compared to ADINA's 120 ksi. The contours from the present study seem reasonable but are not as similar to the ADINA stress contours as for the previous load levels, although the ADINA plot does not look complete. Figure 5.27 shows the contours for the normal stress parallel to the stiffener di-rection. There is very good agreement between the stress contours from the super element and ADINA analyses at 0.5 psi. The stress free contour lines are almost identical and the predicted maximum stress near the stiffener is about 1 ksi from both methods. However, at the clamped edge, the super elements are only able to predict a maximum stress of about 1.25 ksi compared to 3.5 ksi from the ADINA solution. This is expected since the super element discretization is too coarse to properly model the sharp stress gradient near the clamped boundary. At 20 psi and 50 psi the comparison between the present and ADINA results is poor, although the ADINA plot is incomplete. The super elements are, again, unable to model the sharp stress gradients near the clamped edge and the predicted maximum stresses at the clamped boundary are considerably lower than the ADINA prediction. For example, at the 50 psi load level ADINA predicts a maximum stress of about 200 ksi while the super element prediction is only about 70 ksi. Chapter 5. Static Analysis Results 89 A D I N A CODE STRESS(ps i ) I 2 3 4 5 6 7 8 - 2 0 0 0 - I 0 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 P R E S E N T Figure 5.26(a)•• Normal stress perpendicular to the stiffener at 0.5 psi . Chapter 5. Static Analysis Results 90 A D I N A CODE STRESS(ps i ) 0 I 2 3 4 5 6 7 8 9 -IOOOO 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 0 0 0 0 P R E S E N T Figure 5 . 2 6 ( b ) : Normal stress perpendicular to the stiffener at 20 ps i . Chapter 5. Static Analysis Results 91 P R E S E N T CODE STRESS(ps i ) 1 0 2 2 0 0 0 0 3 4 0 0 0 0 4 6 0 0 0 0 5 8 0 0 0 0 6 1 0 0 0 0 0 7 1 2 0 0 0 0 Figure 5 .26 (c ) : Normal stress perpendicular to the stiffener at 5 0 p s i . Chapter 5. Static Analysis Results 92 Figure 5.27(a) = Normal stress parallel to stiffener at 0.5psi. Chapter 5. Static Analysis Results 93 5 2 0 0 0 0 6 2 5 0 0 0 7 3 0 0 0 0 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 PRESENT Figure 5.27(b) : Normal stress parallel to stiffener at 2 0 p s i . Chapter 5. Static Analysis Results 94 ADINA CODE STRESS(psi) l 2 3 4 5 6 7 8 9 0 IOOOO 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 1 0 0 0 0 0 1 5 0 0 0 0 2 0 0 0 0 0 P R E S E N T Figure 5.27(c) : Normal stress parallel to stiffener at 50psi . Chapter 5. Static Analysis Results 95 5.4.3 Clamped DRES1B Stiffened Panel So far, only one-way stiffened plate examples have been considered. This example attempts to investigate the application of the super elements to two-way orthogonally stiffened plates. The DRES1B panel is made up of the 5-bay DRES stiffened panel discussed in Section 5.4.2 with an additional T-beam located along G G ' of the panel as shown in Figure 5.28. The beam dimensions are the same as in the DRES panel and the material properties are E = 30 x 106 psi, ET — 0.178 x 106 psi, cr0 = 45 x 103 psi and v = 0.3. In analogy to Model 1 for the DRES panel, a quarter of the two middle bays is modelled by 3 super elements — 1 plate and 2 beam elements — to enable direct comparison to a new version ADINA analysis in which the portion A B C D and the beams CD and BC are modelled by 220 four node rectangular shell elements. The two discretizations are sketched in Figure 5.29. The super element model has 55 gross and 21 net degrees of freedom. Linear elastic displacement and strain energy responses at a load of 50 psi are presented in Table 5.7 along with the ADINA solution. The super element results compare very well with those from the ADINA analysis, with the latter being slightly on the flexible side as the ADINA model has a finer mesh. In Table 5.7 the DRES1B panel solutions are also compared with those of the DRES panel to show the effect of the additional stiffener in the transverse direction. Clearly, the maximum deflection in the system is reduced from about 5.2 in to about 4.2 in and the overall strain energy is reduced by 36 percent. The in-plane displacements parallel to the x- and y-direction stiffeners are shown in Figure 5.30. In Figure 5.30(a) the u displacement from the super element analysis is plotted along a line parallel to and 9 in from A D . The plot exhibits the shear lag phenomenon, in that the displacement has a peak at the cross beam but dies out away from the stiffener. The corresponding ADINA result is not available but in er 5. Static Analysis Results y.v 5 Panels at 36 In centre = 180 in B F E F G C D C G" A' * 0.25 in -Hh- 0.28 in t t t f 2.93 in -H h* - 0.53 in Section G-G' T 5.645 in I 0.25 in • H £ 2 8 i n 0.53 In 1 ^Ufe in 5.645 in Section A-A* Figure 5.28: Configuration of DRES1B panel Chapter 5. Static Analysis Results Model 1 Figure 5.29: Models for DRES IB panel Chapter 5. Static Analysis Results 98 Table 5.7: Deflections and strain energy in linear elastic DRES1B panel DRES IB Panel DRES Panel Present ADINA Present Deflection at C (in) 0.37 0.45 0.37 Deflection at D (in) 0.38 0.49 5.19 Deflection at E (in) 4.23 4.20 4.73 Deflection at F (in) 0.21 0.30 0.21 Strain Energy (kip-in) 580 900 Figure 5.30(b) the v displacement profiles along EF from the two analyses are pre-sented and an excellent agreement between the super element and ADINA solutions is observed. An excellent agreement is also observed for the bending displacement, w profiles along CD and E F as shown in Figure 5.31. The linear elastic normal stress profiles near the beam axis, BC are plotted in Figure 5.32 together with the ADINA prediction. The stresses are computed at the Gauss points and have been extrapolated to the top surface. Note that for the present analysis the stresses at only the 5 Gauss points close to BC have been used for the plotting. The profiles from the super elements and ADINA follow a similar trend for both crx and cry, but in general, the ADINA predicts higher peaks than the super elements. Next, a large deflection analysis is carried out with elastic or elastic-plastic ma-terial behaviour. The in-plane displacements along EF for the elastic material case are compared with the A D I N A results in Figure 5.33. Both analyses predict a simi-lar trend for the u and v displacements, with the ADINA being on the flexible side. A reasonably good agreement also exists for the w displacement profiles along CD and E F plotted in Figure 5.34. The maximum panel displacement at E was reduced from 4.2 in for the small deflection case (see Figure 5.31) to about 0.9 in due to the nonlinear geometric effects. er 5. Static Analysis Results 99 0.0 8.0 16.0 24.0 32.0 40.0 48.0 Distance from edge CD (in) (a), u displ. along x=9 in from edge AD Figure 5.30: Linear elastic in-plane displacements in DRES IB panel Chapter 5. Static Analysis Results 100 Figure 5.31: Linear elastic bending displacements in DRES1B panel The load-deflection responses at points C, D, E and F are given in Tables 5.8 and 5.9 for elastic or elastic-plastic material behaviour. The super finite element (S.F.E) results compare very well with the ADINA solution. The nonlinear load-deflection response results for points D and E from the present and ADINA analyses are also presented in Figure 5.35. The results from both analyses are very close and follow the same trend. The ADINA solution is again more flexible than the super element solution at most load levels. For large deflection elastic analysis, the deflections at D are always less than those at E since point D lies along a stiffener. However, for large deflection elastic-plastic analysis, the deflections at D become larger than those at E for very high loads. This happens at a load level of about 43 psi in the present analysis, but a little later, at about 49 psi in the ADINA solution. Chapter 5. Static Analysis Results 101 500.0 400.0 « 300.0 x-stress along BC In L.E. DRES1B panel " Present " o — ADINA 16.0 24.0 32.0 Distance from edge CD (in) 40.0 48.0 150.0 100.0 tn | 50.0 0.0 y-stress along BC in L.E. DRES1B panel -50.0 • E i -Present ADINA 0.0 8.0 16.0 24.0 32.0 Distance from edge CD (in) 40.0 48.0 Figure 5.32: Linear elastic normal stresses in DRES1B panel Chapter 5 40.0 30.0 20.0 o 10.0 o o o Static Analysis Results 0.0 -10.0 -20.0. . u displ. along EF at 50 psi DRES1B Panel * * / / / / /' / ' / * ^X / * / * / y / / // / / // // - // Large Deflection Elastic \ \ \ \ V \ \ % \ \ \ \ v \ \ X % * • \ ^ * \ \ ^ \ \ \ \ Present • ADINA ...J i_ i i i i 0.0 3.0 6.0 9.0 12.0 Distance from BC (in) 15.0 18.0 5.0 •) o I ' i i i i i i i i i i l ' 0.0 3.0 6.0 9.0 12.0 15.0 18.0 Distance from BC (in) Figure 5.33: Nonlinear elastic in-plane displacements in DRES1B panel Chapter 5. Static Analysis Results 103 Table 5.8: Nonlinear elastic response of DRES1B panel Load Displ. at C (in) Displ. at D (in) Displ. at E (in) Displ. at F (in) (psi) S.F.E. ADINA S.F.E. ADINA S.F.E. ADINA S.F.E. ADINA 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 0.038 0.039 0.275 0.022 10.0 0.075 0.094 0.078 0.099 0.404 0.417 0.044 0.058 15.0 0.113 0.117 0.497 0.066 20.0 0.150 0.188 0.155 0.197 0.573 0.599 0.088 0.115 25.0 0.187 0.193 0.639 0.109 30.0 0.224 0.281 0.231 0.295 0.698 0.737 0.130 0.171 35.0 0.261 0.269 0.753 0.151 40.0 0.297 0.374 0.307 0.391 0.803 0.855 0.172 0.226 45.0 0.333 0.343 0.851 0.193 50.0 0.369 0.465 0.381 0.486 0.896 0.960 0.214 0.281 Table 5.9: Nonlinear elastic-plastic response of DRES1B panel Load Displ. at C (in) Displ. at D (in) Displ. at E (in) Displ. at F (in) (psi) S.F.E. ADINA S.F.E. ADINA S.F.E. ADINA S.F.E. ADINA 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 0.038 0.047 0.039 0.049 0.275 0.279 0.022 0.029 10.0 0.075 0.094 0.078 0.099 0.404 0.417 0.044 0.058 15.0 0.113 0.141 0.117 0.148 0.497 0.517 0.066 0.086 20.0 0.155 0.188 0.160 0.198 0.577 0.604 0.091 0.115 25.0 0.201 0.239 0.216 0.250 0.654 0.685 0.126 0.146 30.0 0.276 0.316 0.283 0.329 0.733 0.778 0.169 0.198 32.5 0.340 0.347 0.785 0.207 33.0 0.395 0.410 0.854 0.256 35.0 0.470 0.488 0.478 0.509 0.862 0.927 0.276 0.321 37.5 0.653 0.662 0.956 0.369 38.0 0.721 0.740 1.081 0.473 40.0 0.887 0.895 0.897 0.915 1.063 1.188 0.488 0.583 45.0 1.373 1.292 1.382 1.315 1.265 1.426 0.724 0.830 49.0 1.575 1.598 1.597 1.005 50.0 1.774 1.639 1.783 1.663 1.479 1.636 0.930 1.045 Figure 5.34: Nonlinear elastic bending displacements in DRES IB panel In Figure 5.36 the large deflection elastic-plastic displacement profiles along AD and B C are plotted at three load levels — 20 psi when yielding has just begun, 40 psi when considerable yielding has taken place and the displacement at D is still less than that at E and finally, at 50 psi when the displacement at D has exceeded that at E. At 20 psi the displacement profiles predicted by the super element and A D I N A analyses are quite close, with the ADINA solution on the flexible side. At this load level, the displacements along BC are much smaller than those along A D . At higher load levels the predictions from both analyses are still close but it is clear that the displacements along BC approach those along AD. This phenomenon might be due to the fact that as yielding progresses, with increased loading, the stiffness of the beams reduces significantly so that their stiffening action becomes insignificant and the entire panel deforms effectively as an orthotropic plate. Consequently, the Chapter 5. Static Analysis Results 60.0 i Displacements (in) Figure 5.35: Nonlinear displacements of points D and E in DRES1B panel 105 Chapter 5. Static Analysis Results 106 displacement at point D becomes larger than that at point E at higher loads. Stresses at the top surface of the panel near the beam axis B C are displayed in Figure 5.37, for the large deflection elastic case at a load level of 50 psi. Clearly, the profiles from both the present and ADINA analyses follow the same trend, but the super elements grossly underestimate the maximum stresses since they do not have enough modes to model the steep stress gradients near the beam B C . This situation is similar to that observed for the clamped Square Plate I case where the present analysis grossly underestimates the stresses near the boundaries but gives a good prediction of stresses away from the boundaries. The x- and y- normal stress contours at the panel top surface are plotted in Figures 5.38 and 5.39. Unfortunately, the corresponding ADINA results are not available for comparison. 5.4.4 Simply Supported 2x2-Bay Stiffened Plate I Details of this problem are presented in Figure 5.40. The material properties are: E = 30,000 ksi, Ej — 180 ksi, cr0 = 45 ksi and v — 0.3 and the structure is subjected to a uniform pressure load. Al l the stiffened plate boundaries are simply supported. This problem has previously been solved by Rossow and Ibrahimkhail [42] using a constraint method approach in finite element analysis. One quarter of the symmetric structure is modelled, using one super plate element and two super beam elements. The structure is unconstrained against in-plane motion and hence the boundary conditions are applied as follows: • along x = 0; u j+ 0, v = w = wy = 0 • along y = 0; u = 0, v ^ 0, w = wx = 0. Thus this model has 25 net degrees of freedom. A linear elastic analysis is performed for an applied load of 10 psi. In Figure 5.41 the deflected shapes along x = 7.5 in and x — 15.0 in are plotted. It can be seen from £s & A Def lect ions along B C , Present Def lect ions along AD, Present G G O Deflect ions along B C . ADINA Deflect ions along AD, ADINA Figure 5.36: Large deflection elastic-plastic displacement profiles in DRES1B panel Chapter 5. Static Analysis Results 200.0 150.0 -0.0. DRES1B Panel Large Deflection Elastic Analysis 50 psi 0 - "Q. Present ADINA '0.0 8.0 16.0 24.0 32.0 40.0 Distance from edge CD (in) —[j 48.0 100.0 DRES1B Panel 30 o U Large Deflection Elastic Analysis 60.0 -& 40.0 --20.0 -40.0. • re-present ADINA 50 psi 0.0 8.0 16.0 24.0 32.0 40.0 Distance from edge CD (in) 48.0 Figure 5.37: Nonlinear elastic normal stresses in DRES IB panel Chapter 5. Static Analysis Results 109 50 psi Stresses are in ksi Figure 5.38: Normal stress ax in nonlinear elastic DRES IB panel 5. Static Analysis Results Stresses are in ksi Figure 5.39: Normal stress o~y in nonlinear elastic DRES IB panel Chapter 5. Static Analysis Results y.v o CO o CO B Y 1 Y t r — X A G A' — X F B' 0.25 in 5 in —11— 0.5 in Section XX 0.25 in i _ 3 in J —I— 0.5 in Section YY x,u 15 in 15 in Figure 5.40: Details of 2x2-Bay Stiffened Plate I Chapter 5. Static Analysis Results 112 the figure that the comparison between the super element and constraint method solutions is quite good. The super elements predicts a panel centre displacement of 0.067 in while the constraint method gives 0.062 in. The strain energy from the present analysis is 239.1 in-lb. Constraint Method -80.0 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Distance from edge y=0 (in) Figure 5.41: Bending displacements in linear elastic 2X 2-Bay Stiffened Plate I The in-plane displacement distributions obtained from the present analysis only are presented in Figure 5.42. The displacements are plotted along the two orthogonal axes passing through the centre of the bay (that is, along x = 7.5 in and y = 15 in). The bottom surface normal stresses are presented in Figures 5.43 and 5.44 along with the constraint method results. In general, the super element plots are similar to the comparison results, but the discrepancies between the two methods are more pronounced here than in the displacements. I I I I I I I I I I 0.0 0.3 0.5 0.8 1.0 Non-dimensional distance from edge to stiffener Figure 5.42: In-plane displacements in linear elastic 2x2-Bay Stiffened Plate I A large deflection analysis is now carried out, first with the in-plane displace-ments unconstrained at the boundaries as discussed above and secondly, with them constrained along the boundaries. For the latter case all in-plane displacement vari-ables at the boundaries are set equal to zero and the net degrees of freedom is 21. Unfortunately no comparison results are available for the nonlinear problem. The displacement responses at points F and G for the elastic and elastic-plastic cases are presented in Tables 5.10 and 5.11, respectively. Consider the elastic case. At a load of 10 psi the imposition of in-plane constraint results in a reduction of 6.1% in the displacement of point F. This difference in the deflection of point F for the unconstrained and constrained conditions increases with load, as membrane action developes, and at 100 psi when membrane action is fully developed, the difference in the solutions for the two boundary conditions increases Chapter 5. Static Analysis Results 114 20.0 10.0 0.0 •10.0 -20.0 --30.0 0.0 2x2 - Bay Stiffened Plate Linear Elastic Analysis SN Along y = 15 in Along y = 30 in Present Constraint Method 5.0 10.0 Distance from edge x = 0 (in) 15.0 Figure 5.43: Normal stress o~x in linear elastic 2x2-Bay Stiffened Plate I to 24.6%. The same behaviour is observed in the deflection of point G and here, the imposition of in-plane constraint reduces the deflection by 22.2% and 34.7% at 10 psi and 100 psi, respectively. The overall strain energy also shows the same trend and at 100 psi the strain energy is reduced by about 25% due to the in-plane constraint. For the elastic-plastic case, the stiffening action induced by in-plane constraint is more pronounced at higher loads when considerable plastification has taken place. For example, at 100 psi the deflections of points F and G are reduced by about 45% and 62%, respectively. These results demonstrate that in performing a large deflection analysis of a structure with simply supported boundaries the correct in-plane boundary conditions must be apphed. Chapter 5. Static Analysis Results 115 Table 5.10: Nonlinear elastic response of 2x2-Bay Stiffened Plate I Load No In-plane Constraint With In-plane Constraint (psi) Displacements (in) Strain Energy Displacements (in) Strain Energy At F At G (in-kips) At F At G (in-kips) 10 0.066 0.009 0.24 0.062 0.007 0.22 20 0.126 0.018 0.91 0.112 0.013 0.79 30 0.177 0.028 1.93 0.150 0.020 1.62 40 0.222 0.037 3.24 0.182 0.026 2.65 50 0.261 0.047 4.81 0.210 0.032 3.84 60 0.297 0.057 6.61 0.234 0.038 5.18 70 0.329 0.066 8.62 0.256 0.044 6.67 80 0.360 0.077 10.8 0.276 0.050 8.28 90 0.388 0.085 13.2 0.295 0.056 10.0 100 0.415 0.095 15.8 0.313 0.062 11.9 Table 5.11: Nonlinear elastic-plastic response of 2x2-Bay Stiffened Plate I Load No In-plane Constraint With In-plane Constraint (psi) Displ. at F (in) Displ. at G (in) Displ. at F (in) Displ. at G (in) 10 0.066 0.009 0.062 0.007 20 0.126 0.018 0.112 0.013 30 0.177 0.028 0.150 0.020 40 0.222 0.037 0.182 0.026 50 0.265 0.050 0.211 0.033 60 0.312 0.070 0.238 0.042 65 0.340 0.090 0.252 0.048 70 0.369 0.112 0.268 0.066 75 0.407 0.151 0.284 0.074 80 0.464 0.232 0.301 0.098 85 0.523 0.315 0.323 0.120 90 0.587 0.413 0.355 0.183 95 0.658 0.527 0.374 0.214 100 0.727 0.637 0.393 0.244 Chapter 5. Static Analysis Results 116 10.0 5.0 0.0 CO & -5.0 0* -10.0 --15.0 2x2 - Bay Stiffened Plate I Linear Elastic Analysis Along x = 7.5 in Along x = 15 in \ Present Constraint Method -20.0 0.0 5.0 10.0 15.0 20.0 25.0 Distance from edge y = 0 (in) 30.0 Figure 5.44: Normal stress ay in linear elastic 2x2-Bay Stiffened Plate I The u, v and w displacement profiles at 100 psi in the large deflection, elastic-plastic case are plotted along the stated axes in Figures 5.45, 5.46 and 5.47, respec-tively. The effect of in-plane constraint is clearly displayed by these profiles. Chapter 5. Static Analysis Results 117 40.0 30.0 2x2 - Bay Stiffened Plate I Large Deflection Elastic-Plastic Analysis Profile along y = 15 in at 100 psi ^ 20.0 o o o 10.0 Unconstrained Constrained 0.0 -10.0 0.0 5.0 10.0 Distance from edge x = 0 (in) 15.0 Figure 5.45: Nonlinear elastic-plastic u displacement profile along y supported 2 x 2-Bay Stiffened Plate I = 15 in in simply Chapter 5. Static Analysis Results 118 15.0 2x2 - Bay Stiffened Plate I Large Deflection Elastic-Plastic Analysis Profile along x = 7.5 in at 100 psi Unconstrained Constrained -5.0 -10.0 0.0 5.0 10.0 15.0 20.0 Distance from edge y = 0 (in) 25.0 30.0 Figure 5.46: Nonlinear elastic-plastic v displacement profile along x = 7.5 in in simply supported 2x2-Bay Stiffened Plate I Chapter 5. Static Analysis Results 119 Figure 5.47: Nonlinear elastic-plastic w displacement profile along x = 7.5 in in simply supported 2x2-Bay Stiffened Plate I Chapter 6 Vibration Analysis Results 6.1 Introduction Prior to the presentation of the transient applications of the super elements, the adequacy of the consistent mass matrices derived in Chapter 4 is investigated by per-forming vibration studies of various stiffened plate structures. Only linear undamped vibrations are considered and, in this case, the governing equations of motion may be expressed as (-u, 2 [M] + [K]){6} = {0} (6.1) where u is the frequency, [M], [K] are the global mass and stiffness matrices, re-spectively, {<!>} represents the global displacement amplitude vector and {0} is a null vector. First, results are presented for unstiffened plates with various boundary condi-tions in Section 6.2. The vibration response of beams acting alone is investigated in Section 6.3, while the results for stiffened plates are discussed in Section 6.4. In all the examples illustrated here only one super element per bay or span is used, unless otherwise noted. 120 Chapter 6. Vibration Analysis Results 121 6.2 U n s t i f f e n e d P l a t e s 6.2.1 Square Plates with Various Edge Conditions This problem has been analyzed by several investigators [43] using exact or other approximate methods. It is presented here to assess the performance of the super element formulation against other theories. In each case, only one super plate element is used to model the structure. The gross number of variables in each case is 55 and the net number of variables varies from 55 for the case with all edges free to 7 for the case with all edges clamped. How-ever, these net numbers include in-plane variables which uncouple from the bending displacements that are of interest here. So strictly, the clamped case, for example, has only one variable while the all sides free case has 25 flexural vibration modes. The natural frequencies are presented in terms of the non-dimensional parameter A, given by where D = Eh3/12(l — u2) is the plate flexural rigidity, h is the plate thickness, p the mass density, u> the frequency in rad/sec, and a the side length of the plate. The non-dimensional frequencies for the various combinations of edge conditions are presented in Table 6.1. The edge conditions are described by a combination of letters — C for clamped, F for free and S for simply supported, starting from the left edge and moving clockwise. For the all sides simply supported case the exact eigenvalues are obtained from the formula [44] where a, b are the plate dimensions and m, n = 1,2,... are the numbers of half waves in the in-plane directions of the plate. The other comparison results are as compiled by Durvasula et al [43] based on the works in [45,46,47,48]. A = £ > 7 T 4 Chapter 6. Vibration Analysis Results 122 Table 6.1: Eigenvalues of square plates with various edge conditions Case Reference A Mode 1 Mode 2 C C C C Present Ref. [43] 13.37 13.40 c c c s Present 10.45 74.83 Ref. [43] 10.49 41.62 s c s c Present 8.63 41.59 Ref. [43] 8.64 31.13 cess Present 7.55 70.96 Ref. [43] 7.60 37.13 sssc Present 5.75 38.07 Ref. [43] 5.77 27.75 C F C F Present 5.07 7.25 Ref. [43] 5.14 7.27 ssss Present 4.00 35.38 Ref. [43] 4.00 25.00 F F F F Present 2.02 4.11 Ref. [43] 2.04 4.34 SCSF Present 1.66 11.60 Ref. [43] 1.65 11.22 The fundamental eigenvalues obtained from the super element formulation are very close to the other results in all nine cases considered. In all cases, the error in the first eigenvalue from the super element is less than 1.4 percent. However, the error in the second or higher modes is significant in some of the cases. This is expected because of the limited number of variables available in the super element. In particular, for the C C C C case, since there is only one bending degree of freedom, no higher modes are obtained at all. However, as the net number of variables increases, as in the F F F F case for example, the error in the higher mode eigenvalues decreases. Chapter 6. Vibration Analysis Results 123 6.2.2 3-Bay Continuous Plate Figure 6.1 shows the configuration of the problem. The long sides A B and DC are simply supported and the two other edges are both either clamped or simply supported. The structure is modelled by 3 super plate elements and the gross number of variables for this model is 129. When all the sides are simply supported the net number of bending displacement variables is 21 and the number is 15 for the case with edges AD and BC clamped. Figure 6.1: Configuration of 3-bay continuous plate The natural frequencies are presented in terms of the non-dimensional parameter A , given by where w is the natural frequency in radians per second, D is the plate flexural rigidity, h the plate thickness, p the mass density and I is as defined in the figure. Chapter 6. Vibration Analysis Results 124 In Table 6.2 the first three natural frequencies obtained from the super elements are compared with a finite strip analysis by Wu and Cheung [49], for the two different boundary conditions along edges AD and BC. Al l frequencies predicted by the super elements are in excellent agreement with those from the finite strip analysis. Table 6.2: Natural frequencies of 3-bay continuous plate Mode Number A ' AD and BC Simply Supported AD and BC Clamped F.S. [49] Present F.S. [49] Present 1 12.94 12.94 12.99 12.98 2 19.74 20.20 20.84 22.84 3 21.61 21.54 25.89 25.67 6.2.3 2 x 2-Bay Continuous Plate The configuration of the plate, as well as the material properties are as shown in Figure 6.2. The structure is modelled by four plate elements — one for each bay. This model has 154 gross and 48 or 32 net degrees of freedom for simply supported or clamped boundaries, respectively. An equivalent model, one for which a plate element is used to represent a quarter of the structure, could be used to reduce the total number of variables to 55 but this requires four runs for the various symmetry conditions along the centre lines. In Table 6.3, the first six natural frequencies obtained from the present analysis are compared to those obtained by the use of continuous beam functions in a finite strip analysis [49], for clamped and simply supported boundaries. The first four frequencies from the super elements are quite close to the finite strip solutions, with the error in those modes being less than 2%. However, the error increases from the fifth mode up. This is expected as the super element representation does not have Chapter 6. Vibration Analysis Results 125 Line supports — • £ £ I £ Figure 6.2: Configuration of 2 x 2-bay continuous plate enough modes to properly represent the higher modes. In this work, only the first few modes are of interest and hence the representation is quite adequate. 6.3 Beams 6.3.1 Rectangular Beams with Various Boundary Condi-tions This example is used to illustrate the vibration characteristics of the super beam elements acting alone. The beam dimensions are: length, a = 500 mm; width, b = 10 mm; depth, h = 10 m m and the material properties are: E = 220,000 M P a ; p = 7900 k g / m 3 . Various boundary conditions are considered and in each case the structure is modelled by one super beam element. Define the non-dimensional Chapter 6. Vibration Analysis Results 126 Table 6.3: Natural Frequencies of 2x2-Bay Continuous Plate Mode Number A = UJI2^ /(ph/D) S.S. Case Clamped Case F.S. [49] Present F.S. [49] Present 1 19.74 20.05 27.11 27.65 2 23.67 23.67 31.92 31.90 3 23.68 23.67 31.96 31.90 4 27.11 27.11 36.17 36.09 5 49.35 58.71 60.78 83.14 6 49.74 60.67 61.62 85.38 parameter A = u> y/pA^/EI, where, p is the mass density, A the beam cross-sectional area, u> the frequency in rad/sec, a the beam length and EI is the beam flexural rigidity. In Table 6.4 the frequencies obtained for five end conditions are compared with the exact results [50]. The boundary conditions are specified by the letters: C for clamped, F for free and S for simply supported. The super element and exact first mode frequencies for all cases are in excellent agreement. Since the super element uses the exact first vibration mode for the clamped beam, the super element fundamental frequency for the CC case is expected to be exact but the result obtained is slightly in error by 0.13%. This is probably due to errors in numerical integration. The second mode frequencies are not as good as the first mode frequencies because of the limited number of variables in the analysis. Improved higher mode frequencies can be obtained by using two or more super elements. Chapter 6. Vibration Analysis Results 127 Table 6.4: Natural frequencies of rectangular beams Case Reference A Mode 1 Mode 2 F F Present 22.45 91.65 Exact 22.40 61.70 CC Present 22.37 Exact 22.40 61.70 CS Present 15.44 73.80 Exact 15.40 50.00 SS Present 9.87 50.20 Exact 9.87 39.50 C F Present 3.52 22.13 Exact 3.52 22.00 6.4 Stiffened Plates 6.4.1 2-Bay Stiffened Plate II with Clamped Boundaries Figure 6.3 shows details of the problem. One quarter of the structure is represented by two elements — one plate element and one stiffener beam element. This model has 55 gross degrees of freedom and is used to enable direct comparison with the finite element solution given in [51], where only a quarter of the structure was modelled by 21 elements — eighteen 36 degree of freedom high precision triangular elements and three 18 degree of freedom refined beam elements. The two models are also shown in Figure 6.3. Four symmetry conditions are considered and the net degrees of freedom for the four cases are presented in Table 6.5. It can be seen from the table that there is a significant reduction in the problem size by using the super elements as compared to the regular finite elements. The first six natural frequencies from the super elements are compared to the finite element and experimental results in Table 6.6 for two cases of rib sizes - one, the full rib case, for which the stiffener dimensions are ha = 12.7 mm, ts = 6.35 mm; and Chapter 6. Vibration Analysis Results r 203 mm E E E = 68,900 MPa Density = 2670 kg/ms Poisson's ratio = 0.3 X,U I • 1.37 mm t8= 6.35 mm 1 h 8 = 12.7 mm C •ymmafry A C aymm«try A Super Element Grid Finite Element Grid Figure 6.3: Configuration and discretizations for 2-Bay Stiffened Plate II Chapter 6. Vibration Analysis Results 129 Table 6.5: Comparison of net number of variable in 2-bay stiffened plate Symmetry Case Net number of variables Present F.E. [51] symm-symm 16 94 symm-anti 15 88 anti-symm 15 85 anti-anti 12 80 the reduced rib case for which ha = 9.65 mm and ta = 4.83 mm. The experimental results are obtained from real-time holograhic interferometry [51]. The symmetry conditions apphed along the centre lines C A and AD in the super element analysis are denoted by the symbols.— A for anti-symmetric and S for symmetric — in the second column. Thus AS, for example, stands for anti-symmetry conditions along line C A and symmetry conditions along line A D . The frequencies in parentheses represent the results obtained from the super element analysis with the torsional stiffness of the beams ignored. The frequencies for modes 2, 4 and 6 are insensitive to the presence of the torsional beam element. These modes correspond to situations in which symmetry conditions are apphed along line AD (beam axis) such that the rotation (wx) along this line is zero. However, it is seen that the natural frequencies for modes 1, 3 and 5 are significantly lower in the analysis with no stiffener beam torsion than the case in which torsion is included. These modes correspond to situations in which anti-symmetric conditions are apphed along line A D . For these situations significant beam rotations occur, which ought not to be ignored. Hence, the effect of beam torsion has been included in all the dynamic analysis results that follow. A l l frequencies from the super elements are very close to the experimental and regular finite element results. For example, for the full rib case, the first frequency from the present analysis is only 6.9 and 2.6 percent higher than the experimental Chapter 6. Vibration Analysis Results 130 Table 6.6: Natural frequencies of 2-bay stiffened panel Mode Number Symmetry Condition Natural Free uencies (Hz) Full Rib Reduced Rib Present F.E. [51] Exp. Present F .E. [51] Exp. 1 SA 736.8 (597.8) 718.1 689 679.1 (597.8) 670.7 627 2 SS 769.4 (769.4) 751.4 725 716.9 (716.9) 724.0 662 3 A A 1019.6 (899.5) 997.4 961 990.1 (899.5) 977.2 924 4 AS 1032.3 (1032.3) 1007.1 986 1022.9 (1022.9) 1002.1 953 5 SA 1483.7 (1376.3) 1419.8 1376 1469.3 (1376.3) 1408.7 1370 6 SS 1488.3 (1488.3) 1424.3 1413 1442.3 (1442.3) 1414.1 1338 and finite element results, respectively. As expected, the present analysis is on the stiff side of the experimental results. Also, the present representation, being coarser, is generally on the stiff side of the regular finite element solution. However, it is significant that the error in the super element predictions is very small, in spite of the fact that the present analysis uses significantly fewer variables. 6.4.2 3-Bay Stiffened Panel with Clamped Edges The structure is made up of a 203 mm square plate reinforced by two equally spaced ribs as shown in Figure 6.4. The plate thickness is 1.27 mm. For the full ribs hs — 17.8 mm, ts = 2.29 mm and for the reduced ribs case hs = 12.7 mm, t„ = 1.85 mm. The material properties are: E = 68,900 MPa, p = 2670 kg/m 3 and u = 0.3. This problem has previously been solved by Olson and Hazel [51] using the regular finite element method. The super element and regular finite element discretizations used to model the ipter 6. Vibration Analysis Results ' y,v a/3 j - a/3 J a/3 _ | E E 8 II CO x,u 1.27 mm ! L h 8 T symmetry clamped symmetry clamped Super Element Grid Finite Element Grid Figure 6.4: Configuration and discretizations for 3-bay stiffened plate Chapter 6. Vibration Analysis Results 132 Table 6.7: Comparison of net number of variable in 3-bay stiffened plate Symmetry Case Net number of variables Present F.E. [51] symm-symm 36 104 symm-anti 35 101 anti-symm 33 102 anti-anti 30 99 structure are also displayed in the figure and the net numbers of variables associated with the two models for the four symmetry cases are compared in Table 6.7. Again, there are significantly fewer variables for the super element model than the regular finite element model. The natural frequencies obtained from the super elements are presented in Ta-ble 6.8 along with the finite element and experimental results for both cases of rib sizes. Stiffener beam torsional effects have been included in all analyses. The super element predictions are in agreement with the experimental and finite element results except for mode 1 in the full ribs case which is 18% stiffer than the experimental result. It is not clear why the error in this mode is much bigger than in other modes. However, the viability of the super element formulation is clearly exhibited in that most of the super element solutions are close to the experimental and regular finite element results, even though the method uses significantly fewer variables. In general, the super element analysis is stiffer than the experiment, except for mode 3 in the reduced ribs case and mode 4 in the full ribs case, contrary to what is expected of a displacement based theory. Since this phenomenon was also noticed in the finite element solution it is likely that the experimental procedure did not measure the frequencies of those modes accurately. Chapter 6. Vibration Analysis Results 133 Table 6.8: Natural frequencies of 3-bay stiffened panel Mode Number Symmetry Condition Natural Frequencies (Hz) Full Rib Reduced Rib Present F.E. [51] Exp. Present F.E. [51] Exp. 1 SS 1072.8 965.3 909 938.5 928.6 859 2 SA 1334.2 1272.3 1204 1178.6 1205.1 1044 3 SS 1410.3 1364.3 1319 1182.4 1229.8 1292 4 AS 1483.2 1418.1 1506 1330.4 1274.6 1223 5 A A 1649.2 1602.9 1560 1569.8 1557.4 1503 6 AS 1730.5 1757.1 1693 1674.5 1714.5 1650 6 .4.3 DRES Stiffened Panel The panel configuration is the same as that in Figure 5.21, except that in this case the beams have a depth of 5.85 in and a flange width of 2.95 in. The material properties are the same, and in addition, the mass density is 0.733 x 10~3 lb-s 2 / in 4 . Using symmetry, half of the structure is modelled by 3 plate elements and 2 beam elements. The natural frequencies obtained are presented in Table 6.9 along with those computed by A D I N A and finite strips [11,20]. In the finite strip analysis 11 strips are used to model one half of the structure. Two symmetry conditions are considered in the present case so both symmetric and anti-symmetric modes are captured. Only symmetric mode results are available from the other two methods. The mode 1 and 3 frequencies obtained from the superelements are significantly lower than those obtained from the finite strip or ADINA programs. It is suspected that this anomaly might be due to the significant presence of torsion in the stiffener beams for these modes. To investigate this, a one mode Ritz approximation of the fundamental frequency, using the approximating function sin 7r£(l — C O S 27r7;) is carried out to check the results obtained from the program. The result obtained from this analysis — 29.8 Hz — seems to corroborate the super element solution. Since these Chapter 6. Vibration Analysis Results 134 Table 6.9: Natural frequencies of DRES stiffened panel Mode Number Natural Frequencies (Hz) Present F.S. [20] ADINA [11] 1 30.25 40.33 41.01 2 33.32 3 37.59 41.70 41.99 4 41.73 5 43.44 42.50 42.80 two solutions are based on the same theory, it does seem that the discrepancy between the super element solution and the finite strip and ADINA solutions might be due to the difference in the modelling of the stiffeners — the super element uses beam theory while the others use plane stress elements to model the beams. However, the frequencies of the fifth mode from all three analyses are quite close and the super element solution is a little on the stiff side as expected. 6.4.4 2 x 2-Bay Stiffened Plate The structure is made up of a clamped rectangular plate stiffened by two identical, eccentric, orthogonally placed and centrally located rectangular beams. The plate dimensions are: a = 304.5 mm, b — 203.0 mm, (a/b = 1.5) h = 1.37 mm and the beams are 6.35 mm wide and 11.33 mm deep. The material properties are: E = 71 x 109 N / m 2 , p = 2700 kg /m 3 and v = 0.3. This example was introduced by Balendra and Shanmugam [52] as an extension of the 2-bay example of Section 6.4.1 with the same width, plate thickness, etc. They have used the grillage method in their analysis. For the present analysis symmetry is invoked and a quarter of the structure is modelled using 1 plate and 2 beam elements. The first five fundamental frequncies are given in Table 6.10. The letters in parentheses represent the symmetry conditions Chapter 6. Vibration Analysis Results 135 applied along the longer and shorter centre lines, respectively. The letter, A stands for anti-symmetry and S for symmetry. The first frequency from the super elements compares well with the grillage method solution and is only 6.3 percent on the stiff side. No comparisons exist for the higher modes. Note that the first four frequencies are very closely spaced. Table 6.10: Natural frequencies of 2 x 2-bay stiffened plates Mode Number Frequencies (Hz); a/b — 1.5 Frequencies (Hz); a/b = 1.0 Present Grillage Method [52] Present 1 846.1 (AA) 796.1 1149.6 (SS) 2 846.8 (SS) 1152.5 (AA) 3 849.4 (AS) 1161.9 (AS) 4 862.0 (SA) 1161.9 (SA) 5 1448.0 (SS) 2042.1 (SS) In Table 6.10 the frequencies obtained for the case in which a = b — 203 mm (i.e. a/b — 1.0) are also presented without comparison results. However, the stiffening action of the additional cross beam is clearly displayed, in that, these frequencies are significantly higher than those for the full ribbed 2-bay stiffened plate (see Table 6.6). Again, the first four frequencies are all closely spaced. 6.4.5 2x4-Bay Stiffened Plate with Clamped Edges A schematic representation of the structure is shown in Figure 6.5. The plate is clamped all round and the material properties are: E = 71 x 109 N / m 2 , p = 2700 kg /m 3 and v = 0.3. Due to symmetry only one quarter of the structure is modelled with six super elements — 2 plate and 4 beam elements — as shown in the figure. This model has 92 gross number of variables and the net degrees of freedom is never more than 16 for the four symmetry conditions. The first six natural frequencies are presented in Table 6.11. The first frequency Chapter 6. Vibration Analysis Results E E 8 CM 9> 8 E E io CO CM y.v 4 at 76.125 mm centres = 304.5 mm 1.37 mm J T i 12.7 mm 1.37 mm J 2.117mm -*IU-Section AA (Typical) A • f r B l» B x,u T 6.35 mm—*-l r e -section BB I 12.7 mm symmetry clamped Super Element Grid Figure 6.5: Configuration and discretization for 2x4-bay stiffened plate Chapter 6. Vibration Analysis Results 137 compares well with that obtained by the grillage method [52], with the super elements being on the stiff side by 9.4 percent. Unfortunately, there are no comparison results for higher modes but the results obtained look reasonable. Table 6.11: Natural frequencies of 2 x 4-bay stiffened plate Mode Natural Frequencies (Hz) Number Present Grillage Method [52] 1 1113.5 1017.8 2 1314.1 3 1327.3 4 1434.8 5 1489.9 6 1526.8 6.4.6 4x4 -Bay Stiffened Plate The 4 x 4-bay stiffened plate is made up of a 4 m square plate reinforced by three T-beams in each orthogonal direction as shown in Figure 6.6. The plate is 7 mm thick with all edges clamped. The material properties are: E = 210, 000 x 106 N / m 2 , p = 7900 kg/m 3 and v = 0.3. Using symmetry one quarter of the structure is modelled by 4 super plate elements — one for each bay, and 8 super beam elements — one for each span. This super element model has 154 gross degrees of freedom and the net number of variables for the four symmetry cases are presented in Table 6.12. Four vibration analysis runs are performed to capture all the various combina-tions of symmetry conditions about the two centre lines, D A B . The first 32 natural frequencies are presented in Table 6.13. In the table, the symbols in parenthesis rep-resent the symmetry conditions applied about the centre lines. The letter A stands for anti-symmetry and S for symmetry. Notice that the first 16 frequencies are very Chapter 6. Vibration Analysis Results 138 • y.v 4 Panels at 1000 mm centres = 4000 mm a 7 mm I 1 _! 6.5 mrfl^lr* •I 1 183 mm 10.7 mm 128 mrrrH K Typical side view Figure 6.6: Configuration of 4x 4-bay stiffened plate Chapter 6. Vibration Analysis Results 139 closely spaced as expected of a structure with 16 similar bays. Note that the first mode frequency for one clamped bay is 62.73 Hz. This is slightly higher than the fre-quency for mode 16 as expected since mode 16 is essentially a half wave per bay with zero slope along all sides but with support movements along the stiffener locations. It is anticipated that the frequencies of higher modes, which have 2 half waves per bay, will be badly predicted by the present grid since the super element does not include this type of mode when the boundaries are clamped. However, for many practical blast loaded structures, these higher modes do not contribute significantly to the final response, therefore, the present grid is quite adequate for the objective of this work. No comparison results are available for this problem. Table 6.12: Net number of variables in 4x4-bay stiffened plate Symmetry Case Net number of variables symm-symm 80 symm-anti 78 anti-symm 78 anti-anti 73 Chapter 6. Vibration Analysis Results 140 Table 6.13: Natural frequencies in 4x4-bay stiffened plate Mode No. Frequency (Hz) Mode No. Frequency (Hz) 1 57.86 (SS) 17 115.69 (SS) 2 61.04 (AA) 18 208.02 (SA) 3 61.45 (SA) 19 208.02 (AS) 4 61.45 (AS) 20 277.36 (AA) 5 61.98 (SA) 21 292.41 (AA) 6 61.98 (AS) 22 322.68 (SA) 7 62.05 (AA) 23 322.68 (AS) 8 62.14 (SS) 24 323.45 (SS) 9 62.24 (AA) 25 323.51 (SS) 10 62.38 (SA) 26 344.92 (AA) 11 62.38 (AS) 27 345.05 (AA) 12 62.41 (SS) 28 358.24 (SS) 13 62.51 (AA) 29 384.72 (SA) 14 62.53 (SA) 30 384.72 (AS) 15 62.53 (AS) 31 387.60 (SA) 16 62.55 (SS) 32 387.60 (AS) C h a p t e r 7 T r a n s i e n t A n a l y s i s R e s u l t s 7.1 Introduction The super elements are now applied to the analysis of stiffened plate structures sub-jected to transient loads. Since the ultimate aim of this work is to develop a simplified numerical method for blast loaded stiffened plates, most of the applied loads are of the blast-type described in Chapter 4. These loads usually rise instantaneously to a peak intensity and then decrease monotonically to zero in a very short time. They are assumed to be uniformly distributed over the surface of the structure. The equations of motion to be solved in this case are [M]{S} + [C]{6} + j [[B] + [C0])T{<r}dV = {P} (7.1) where [M], [C] are the global mass and damping matrices, respectively and {£}, {P} are the global displacement and load vectors, respectively. The equations are solved by the implicit Newmark-/3 method with Newton-Raphson iteration within each time step as discussed in Chapter 4. In all the transient example problems presented in this chapter the effect of structural damping has been ignored, except in the one example in which dynamic relaxation is carried out to obtain a static solution using critical damping. Also, the effect of stiffener beam torsion has been included in all the stiffened plate examples. To study the complete transient behaviour of the super elements the linear elastic 141 Chapter 7. Transient Analysis Results 142 as well as large deflection elastic and elastic-plastic responses are investigated. How-ever, in some of the examples in which the structures are subjected to intense blast loads only the large deflection elastic-plastic response has been investigated, since such structures usually exhibit both geometric and material nonlinearities. As in the previous two chapters, only one element is used to represent a panel bay or beam span, except in a few cases in which finer meshes are used to study the convergence properties of the super elements or to employ discretizations similar to those used by other investigators. Some of the results are compared to finite strip results, which in all cases are one mode (in the strip direction) results, except stated otherwise. Again, the problems are categorized according to structure type. First, unstiffened plates with various boundary conditions are analyzed in Section 7.2 and then a beam example is considered in Section 7.3. Finally, Section 7.4 is devoted to the response of stiffened plates subject to various dynamic loads. 7.2 Unstiffened Plates 7.2.1 Square Plate I with Simply Supported Edges A transient analysis is now performed on Square Plate I with simply supported edges. The dimensions and material properties of the plate are: dimensions = 100 mm x 100 mm x 1 mm elastic modulus, E = 205,000 N / m m 2 plastic modulus, ET = 0 yield stress, <x0 = 210 N / m m 2 Poisson's ratio, v = 0.3 Note that this is the same square plate for which a static analysis was performed in Chapter 5. The plate is subjected to a step load of 67.2 x l O - 3 MPa and the structure Chapter 7. Transient Analysis Results 143 is modelled by one super plate element. This model has 55 gross and 15 net gross degrees of freedom. The boundary conditions are applied as in the static case: • at the four corner nodes all variables except the twist variable are constrained to be zero • at the four mid-side nodes, all variables except the normal slope (e.g. wy5 at node 5) are set equal to zero • at the middle node (node 9) all variables are left free. j That is, in-plane constraint is provided at the boundaries. From the vibration analysis, the fundamental frequency cu0 of the plate is obtained as 3042.9 rad/sec, so that the fundamental period T 0 is 2.07 msec. First, the linear elastic static deflection due to a 67.2 X l 0 - 3 MPa uniform load is computed using the dynamic relaxation technique. In the analysis the damping factor is taken as 2u>o- To study the effect of the time step size on the solution three different time steps — At = 0.4 msec % To/5, A i = 0.2 msec % T o /10 and At = 0.1 msec T 0 / 20 — are employed and the panel centre displacement histories for the three cases are shown in Figure 7.1. The results from the three cases are almost indistinguishable, and the static displacement from all three cases is about 1.47 mm which compares well with the exact solution of 1.45 mm. This demonstrates that dynamic analysis will converge to the correct static solution in the steady state. The results also show that a time step size as large as At % T 0 /5 is quite adequate for linear elastic analysis. In Figure 7.2 the central displacement histories for the elastic, linear and nonlinear geometry responses, due to the 67.2 x 1 0 - 3 M P a step load, are compared to the finite strip results [53]. A time step size of 0.04 msec (~ To/50) is used for the super element analysis. The super element and finite strip results are in excellent agreement in both cases. The stiffening action due to nonlinear geometric effects is Chapter 7. Transient Analysis Results 144 1.6 Time (msec) Figure 7.1: Dynamic relaxation response of linear elastic simply supported Square Plate I clearly demonstrated, in that the peak central displacement and fundamental period are reduced by about 56% and 42%, respectively. The apphed load is not high enough to cause significant yielding of the plate material and hence the large deflection, elastic-plastic response does not differ much from the large deflection elastic case and is not presented separately, here. 7.2.2 S q u a r e P l a t e I w i t h C l a m p e d Edges The Square Plate I is analyzed again, with all the plate edges clamped. The super element model has 7 net degrees of freedom. A step load of 268.8 x 1 0 - 3 MPa is apphed. For linear elastic analysis a time step of 0.05 msec (< T o/20) is used, while a time step of 0.025 msec ( < T o/40) is employed for nonlinear analysis. Chapter 7. Transient Analysis Results 145 3.5 g 2.0 E « 1.5 Q. W b 15 " 1.0 c O 0.5 0.0 -0.5. S.S. Square Plate I - Step Load Elastic Response >w Present — / \ Finite Strip — / \ Linear — / \ Nonlinear / . ' ~ ~ s / J x If \ % /f \ * ~ I' \ \ * \ \ I f / N. \ / X \ * / * \ \ / ' \ \ / ' \ \ / ' \ / X \ * / \ \ - / \ \ / * \ / / ' \ / / ' \ / y* ^x S i . i . i . i . 1. , — '0.0 0.5 1.0 1.5 2.0 2.5 Time (msec) Figure 7.2: Transient response of simply supported Square Plate I 3.0 The linear and nonlinear elastic reponses from the super element and finite strip analyses are shown in Figure 7.3. The agreement between the two methods is ex-cellent. The presence of geometric nonlinearities causes a 52.5% reduction in the maximum displacement amplitude and a 42% reduction in the fundamental period. The nonlinear elastic-plastic response is presented in Figure 7.4 together with the finite strip response. The results from both analyses are very similar. Due to the ma-terial nonlinear behaviour a 9% increase is observed in the peak central displacement amplitude. Both methods predict a steady central displacement of about 1.4 mm. On the I B M 308IK mainframe computer the large deflection elastic analysis takes about 6 sec of cpu time per load step while the large deflection elastic-plastic analysis takes about 12 cpu sec per load step. Chapter 7. Transient Analysis Results 146 5.0 4.0 E r 3.0 c E CD i 2.0 a to O S 1.0 O 0.0 -1.0 Clamped Square Plate I - Step Load - Elastic Analysis - Present — Finite Strip /* \ Linear £r m vy ft ^» A /' \ r : r rNv Nonlinear -— ^r. 1 , 1 , 1 , 1 , 1 , 1 . 0.0 0.2 0.4 0.6 0.8 Time (msec) 1.0 1.2 1.4 Figure 7.3: Transient elastic response of clamped Square Plate I 7.2.3 Simply Supported Square Plate II Subject to Trian-gular Load Details of the problem are given in Figure 7.5. The plate has a length of 20 in and a thickness of 0.1 in. The material properties are: E = 107 psi, ET — 0, <r0 = 40,000 psi, p = 0.25 x 1 0 - 3 lb-s 2 / in 4 and v = 0.3. The plate boundaries are simply supported with rigid restraint against in-plane motion. The load history is triangular in shape and varies from 200 psi to 0 in a short duration of 2 msec. The structure is first modelled by one super plate element with 15 net variables. A linear eigenvalue analysis is carried out and the fundamental frequency is obtained as 47.54 Hz, which is equal to that obtained by the finite strip method. A large deflection elastic-plastic analysis is now carried out with a time step of 0.1 msec which is approximately equal to T o/200, where T0 is the fundamental period. The Chapter 7. Transient Analysis Results 147 Time (msec) Figure 7.4: Transient nonlinear elastic-plastic response of clamped Square Plate I panel central displacement history is plotted in Figure 7.6 along with the ADINA [54] and finite strip [20] results. The finite strip analysis uses 2 in-plane displacement modes, 8 strips and a time step size of 0.02 msec. The one super element representation gives a reasonable estimate of the response, although it is stiffer than the ADINA and finite strip representations. To see the effect of element refinement on the solution, the analysis is repeated by using one superele-ment to represent one quarter of the plate (equivalent to using 4 super elements). As can be seen from Figure 7.6, this results in an improvement in the solution and the super element solution is now closer to the finite strip analysis. This indicates that the present solution will converge to the correct solution as the number of elements increases. Note that the displacements are extremely high with the panel centre dis-placement being greater than 20 times the plate thickness. Hence for much of the Chapter 7. Transient Analysis Results 148 Poisson's ratio = 0.3 Time (msec) Figure 7.5: Details of Square Plate II response the plate behaves as a plastic membrane. This is a severe test of the super element deformation modes. In Figure 7.6 the response obtained from the one super element solution without the sine functions in the in-plane displacements is also plotted (shown chain-dotted). This model is generally much stiffer than the ADINA and complete super element solutions and it predicts a permanent displacement of about 1 in compared to 2.4 in and 1.8 in, respectively, for the ADINA and complete one super element analyses. Thus if for purposes of argument the ADINA solution is taken as the correct one, then the one superelement solution with and without the sine functions are in error by 25% and 58%, respectively. Clearly, the latter is not within design level accuracy. This example illustrates the importance of modelling the in-plane displacements cor-rectly in the presence of severe nonlinearities. The superelement model with the sine Chapter 7. Transient Analysis Results 149 functions is better able to model the in-plane displacements, and hence the membrane stresses, than the one without the sine functions, which predicts very poor in-plane displacements and membrane stresses. It takes about 20 and 25 sec per load step, respectively, for the 1 and 4 super element solutions on the I B M 3081K main frame computer. Time (msec) Figure 7.6: Central displacement history of simply supported Square Plate II 7.3 Beam Example 7.3.1 Rectangular Beam with Simple Supports The beam is simply supported (and axially constrained) and the dimensions and material properties are: Chapter 7. Transient Analysis Results 150 length, a = 30.0 in width, b = 1.0 mm depth, h = 2.0 mm elastic modulus, E = 30 x 107 psi plastic modulus, ET = 0 yield stress, a0 = 50 x 104 psi density, p = 0.733 x 10~3 lb -s 2 / in 4 collapse load, P0 = 2a0b(h/af = 444.4 lb/ in step load = 0.75Po = 333.3 lb/in static deflection, A = 0.1758 in The beam is subjected to a step load of 333.3 lb/in. The linear eigenvalue analysis gives the fundamental frequency as 203.9 Hz, so the fundamental period is 4905 ^sec. Using one beam element and a time step size of 250 psec (ss T o/20) a small deflection elastic analysis is performed and the mid-span deformation history is compared with the F E N T A B [39] solution in Figure 7.7. Note that the F E N T A B analysis uses 12 cubic beam elements and the explicit central difference scheme with a time step size of 6.948 psec . The correlation between the two solutions is quite good. The small deflection, elastic-perfectly plastic central displacement response using the same time step is presented in Figure 7.8 along with the finite strip and F E N T A B results. The one super element solution is in good agreement with the finite strip analysis, but is stiffer than the F E N T A B solution. However, the present analysis result improves significantly when two super elements are employed in the analysis. In this case, the super element and F E N T A B results are almost identical. The loss of stiffness due to plastification is clearly exhibited, in that the maximum displacement amplitude and fundamental period are, repectively, 44% and 20% higher than the linear elastic quantities. Chapter 7. Transient Analysis Results 151 Figure 7.7: Transient linear elastic response of simply supported rectangular beam 7.4 Stiffened Plates 7.4.1 Clamped 2-Bay Stiffened Plate II The transient response of the clamped 2-Bay Stiffened Plate II, for which a vibration analysis was carried out in Chapter 6, is now investigated. The problem configuration and material properties are presented in Figure 6.3. Only the full-ribbed structure is considered here. The plate boundaries are fully clamped. For plastic analysis ET = 0.005.&7 = 344.5 MPa and er0 = 284 MPa. In the present analysis only one half of the structure is modelled using one super plate element and one super beam-element. Note that the symmetry line divides the beam into two, and hence the beam section properties such as J , Jo, T, etc. have to be halved in the computations. The super element model has 55 gross and 11 net degrees of freedom. Chapter 7. Transient Analysis Results 152 Figure 7.8: Transient linear elastic-plastic response of simply supported rectangular beam Two loading cases are considered. In the first case the structure is subjected to a step load of 0.3 MPa. In the second case the stiffened plate is subjected to a blast load given by the exponential variation in Equation (4.30), (that is q(t) = qm(l - t/r)exp(-\1t/T)), with qm = 0.6 MPa, r = 1 msec and A x = 0.28. These loads are sketched in Figure 7.9. The response of the stiffened plate to the two loads are presented in the following subsections. 7.4.1.1 Step Load Recall that the fundamental frequency of the structure is 736.8 Hz, so To = 1.36 msec. A time step size of 0.025 msec is employed for both linear and nonlinear analyses, although larger time step sizes could well be used. Chapter 7. Transient Analysis Results 153 Pressure (MPa) 0.3 Time (msec) Step Load Pressure (MPa) 1.0 Time (msec) Blast Load Figure 7.9: Loads applied to 2-Bay Stiffened Plate II The linear elastic panel centre (point B) and stiffener mid-point (point A) dis-placement responses are given in Figure 7.10 along with the finite strip response [53]. The finite strip model uses four plate strips and one beam element for one half of the structure and employs the implicit Newmark integration scheme with a time step size of 0.025 msec. This finite strip model uses strips for the plate and beam elements for the stiffeners and employs quadratic functions for the in-plane displacements. The super element prediction is in excellent agreement with the one mode finite strip solution. Good agreement is also observed for the large deflection elastic response presented in Figure 7.11. However, as can be seen from the figure, for the response at A the super elements display some higher modes which have not been captured by the finite strips. Due to the nonlinear geometric effects, the fundamental period of the Chapter 7. Transient Analysis Results 154 structure is reduced by more than 40% and the panel centre maximum displacement amplitude is reduced from 12.1 mm to 4.1 mm. On the other hand, the peak stiffener displacement increases slightly from 1.64 mm to 2.23 mm. This behaviour is different from what is observed for other stiffened plate analysis results for which both the nonlinear elastic panel centre and stiffener mid-point displacents are smaller than the corresponding linear elastic displacements. The phenomenon is probably due to the change in the proportions of the load carried by the plate and beam components of the structure as their stiffnesses change due to geometric nonlinearities, with beam stiffness showing less change than the plate stiffness. The large deflection elastic-plastic behaviour is displayed in Figure 7.12. Again the super elements agree well with the finite strips. The inclusion of nonlinear material behaviour causes only slight increases in the displacements of points A and B: the peak displacements of A and B are increased by only 12% and 0.5%, respectively, with reference to the large deflection elastic case. Note that the increase in the peak displacements due to material nonlinearities is more pronounced in the beam than in the plate due to the fact that the beam shows less stiffening due to geometric nonlinear effects. The higher modes seem to have been filtered out by the material yielding. The large deflection elastic-plastic super element solution takes about 12 sec per load step on the IBM 3081K computer. 7.4.1.2 Blast Load Using a time step size of 0.025 msec the large deflection elastic-plastic response of the 2-Bay Stiffened Plate II is simulated with one super plate element and one super beam element for one half of the structure. The panel centre and stiffener mid-point displacement responses are presented in Figures 7.13 and 7.14. The super element and finite strip predictions are very close. By the super element analysis the peak displacement amplitudes of points A and B are 5.04 mm and 4.24 mm, respectively. Chapter 7. Transient Analysis Results 14.0 155 12.0 10.0 E E, 8.0 ¥ o E 6.0 8 Q. 4fj b 2.0 0.0 -2.0 Clamped 2-Bay Stiffened Plate I Linear Elastic Analysis 1 • Step Load Present B / \ • Finite Strip — / \ B: at half panel centre \ A: at stiffener midpoint - / A 1 , 1 , 1 , 1 , 1 , 1 , 1 , 0.0 0.2 0.4 0.6 0.8 1.0 1.2 .1.4 1.6 Time (msec) Figure 7.10: Linear elastic response of clamped 2-Bay Stiffened Plate II due to step load The final permanent displacements of both points are approximately equal at close to 2 mm. It takes about 13 cpu sec per load step on the IBM 3081K main frame computer. 7.4.2 Simply Supported 2-Bay Stiffened Plate II Transient analysis is now performed on the 2-Bay Stiffened Plate II with simply supported edges. The plate configuration and material properties are as given for the clamped case. The structure is again subjected to the two transient loads presented in Figure 7.9 and the response of the structure to these loads is discussed in the following subsections. Chapter 7. Transient Analysis Results 156 5.0 •1.0 Clamped 2-Bay Stiffened Plate II - Step Load Large Deflection Elastic Analysis Present Finite Strip J i L 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (msec) Figure 7.11: Nonlinear elastic response of clamped 2-Bay Stiffened Plate II due to step load 7.4.2.1 Step Load Using symmetry, one half of the structure is modelled by 1 plate and 1 beam element. Restraint is provided against in-plane displacements along the boundaries and hence the super element model has 15 net degrees of freedom. A step load of 0.3 MPa is apphed. The first 5 frequencies are obtained as 508.6, 510.3, 805.1, 815.8 and 1060.2 Hz and hence the fundamental period of the structure is 1.966 msec. A time step size of 0.05 msec (~ T o/40) is used for all analyses. Linear elastic displacement responses are displayed in Figure 7.15. The super element predictions are in excellent agreement with those from the finite strip analysis [53]. The super element peak panel centre displacement is 24.61 mm and this is only Chapter 7. Transient Analysis Results 157 5.0 Clamped 2-Bay Stiffened Plate II - Step Load Large Deflection Elastic-Plastic Analysis Present B/\\ Finite Strip 4.0 0.0 " 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time (msec) Figure 7.12: Nonlinear elastic-plastic response of clamped 2-Bay Stiffened Plate II due to step load 2% lower than the corresponding finite strip value, while the peak stiffener mid-point displacement amplitude, 7.04 mm, is 7.5% higher than the finite strip result. Compared to the clamped case it is seen that the simply supported boundaries result in a 51% increase in the panel centre displacement amplitude. In Figures 7.16 and 7.17 are shown the large deflection elastic displacement re-sponses of the panel centre (point B) and stiffener mid-point (point A), respectively. The super element and finite strip predictions follow a similar trend. The peak dis-placement of point B has now reduced from 24.61 mm, for the linear case, to 4.71 mm for the nonlinear one. Also, due to geometric nonlinearities the peak stiffener mid-point displacement is reduced to 4.58 mm. The super elements again predict the presence of some higher modes which are not shown by the finite strips. Chapter 7. Transient Analysis Results 6.0 158 Clamped 2-Bay Stiffened Plate II - Blast Load Present Finite Strip Figure 7.13: Panel centre displacement of clamped 2-Bay Stiffened Plate II due to blast load Figures 7.18 and 7.19 show the large deflection elastic-plastic displacement re-sponses of points B and A, respectively. Again the super element and finite strip solutions compare very well. The nonlinear material behaviour only slightly increases the peak panel centre displacement to 4.74 mm and the stiffener mid-point peak dis-placement to 5.39 mm. A permanent displacement of about 3.5 mm is predicted by the super elements. Some of the higher modes present in the elastic analysis seem to have been filtered out by the material yielding. 7.4.2.2 Blast Load A nonlinear elastic-plastic analysis is performed on the simply supported 2-Bay Stiff-ened Plate II subject to the blast load shown in Figure 7.9. The time step size used Chapter 7. Transient Analysis Results 159 Time (msec) Figure 7.14: Stiffener mid-point displacement of clamped 2-Bay Stiffened Plate II due to blast load in the analysis is 0.05 msec. The analysis is first carried out with the in-plane dis-placements constrained along the boundaries and secondly with them unconstrained. The panel centre and stiffener mid-point displacement responses are shown in Fig-ures 7.20 and 7.21. In each of these figures the solid line represents the response for constrained conditions and the dashed line the response for unconstrained conditions. The imposition of in-plane constraint results in 49% and 45% reductions in the peak panel centre and stiffener mid-point displacements, respectively. Also, these peak displacements occur earlier for the in-plane constrained case than for the in-plane unconstrained case. For example, the peak panel centre displacement occurs at 0.3 msec and 0.75 msec for the constrained and unconstrained problems, respectively. Some permanent displacements are noticed at the two points of interest for both the constrained and unconstrained conditions. Chapter 7. Transient Analysis Results 30.0 160 25.0 20.0 E E, ¥ 15.0 o E | 10.0 Q. CO O 5.0 0.0 -5.0. 0.0 0.5 S.S. 2-Bay Stiff. Plate II Linear Elastic Analysis - Step Load . B >v V X \ Present Finite Strip §] if il ti A: Stiffener midpoint B: Panel centre ti it / A 11  m * " ™  m ™ V if *'/ - / / ^^^^ \ \ \ \ 1.0 1.5 Time (msec) 2.0 2.5 Figure 7.15: Linear elastic response of simply supported 2-Bay Stiffened Plate II due to step load For each of the large deflection elastic-plastic analyses discussed here it takes about 18 sec per load step on the I B M 3081K main frame. 7.4.3 HOB 315 Loading on Clamped DRES Stiffened Panel The panel configuration is as shown in Figure 5.21. A l l dimensions are the same except, in this case, the depth and flange width of the T-beams are 5.85 in and 2.95 in, respectively. Also, here cr0 = 54.4 x 103 psi and p = 0.733 x 1 0 - 3 lb-s 2 / in 4 . The panel is clamped all round and one quarter of the middle bay, shown as A B C D in Figure 5.21 is modelled by one plate element and one beam element. This is Model 1 in Figure 5.22. The structure is subjected to an intense blast load, named HOB 315, whose history Chapter 7. Transient Analysis Results 6.0 161 S.S. 2-Bay Stiffened Plate II - Step Load Large Deflection Elastic Analysis 1.0 1.5 Time (msec) Present Finite Strip 2.0 2.5 Figure 7.16: Panel centre displacement in nonlinear elastic analysis of simply sup-ported 2-Bay Stiffened Plate II due to step load is shown in Figure 7.22. The symbols on this figure represent data points for the piece-wise linear representation used in the program. A transient large deflection elastic-plastic analysis is performed with a time step size of 250 ^sec and the panel centre and stiffener mid-point displacement responses are presented in Figures 7.23 and 7.24, respectively, along with the finite strip [20] predictions. Note that the finite strip analysis uses 16 strips (this version of the finite strip analysis uses strips for both the plate and stiffeners) for half of the structure and a time step of 5 psec with an explicit time integration scheme. The permanent displacements predicted by the two methods are very close and from both figures it is clear that the panel centre and stiffener mid-point end up with almost the same permanent deflection. The analysis takes about 26 sec per load step on the IBM 3081K computer. Chapter 7. Transient Analysis Results 162 5.0 -2.0 S.S. 2-Bay Stiffened Plate II - Step Load Larege Deflection Elastic Analysis 0.0 0.5 1.0 1.5 Time (msec) Present Finite Strip 0 2.0 2.5 Figure 7.17: Stiffener mid-point displacement in nonlinear elastic analysis of simply supported 2-Bay Stiffened Plate II due to step load 7.4.4 Clamped DRES IB Stiffened Panel The configuration of the DRES IB Stiffened Panel has been presented in Figure 5.28. Al l material properties are the same, except tr 0 = 54.4 x 103 psi, and p = 0.733 x 1 0 - 3 lb-s 2 / in 4 . The structure is subjected to the HOB 315 blast load shown in Figure 7.22. Two super element discretizations are employed. In the first model, called Model 1, the portion A B C D is modelled by one super plate element and two super beam elements (for half of beams BC and CD, respectively), with symmetry conditions along the center lines B C , CD and A D . Model 1 is used to enable direct comparison to the ADINA solution for which only the portion A B C D is modelled with 220 four node rectangular shell elements (see Figure 5.29). To include the effects of end restraint ignored by Model 1 another discretization, called Model 2 is used. In this model Chapter 7. Transient Analysis Results 163 S.S. 2-Bay Stiffened Plate II - Step Load Large Deflection Elastic-Plastic Analysis Time (msec) Figure 7.18: Panel centre displacement in nonlinear elastic-plastic analysis of simply supported 2-Bay Stiffened Plate II due to step load 3 plate and 5 beam elements are used to model a quarter of the structure (portion A D G H ) , with symmetry conditions along GD and A D . For the sake of completeness Model 1 and the ADINA discretization are presented again along with Model 2 in Figure 7.25. A linear free vibration analysis is first carried out using Model 2 and the first ten natural frequencies obtained, in Hertz, are: 48.7, 50.2, 50.9, 51.6, 51.9, 52.3, 53.2, 53.4, 53.4 and 53.7. It is seen that first ten frequencies are very closely spaced. The A D I N A solution is not available for comparison. The transient large deflection elastic-plastic response of the structure due to the blast load is now evaluated. A time step of 250 psec is used for both the present and A D I N A analysis. The displacement histories at points D and E are presented in Figure 7.26. It is observed that the solutions obtained from Models 1 and 2 compare Chapter 7. Transient Analysis Results 164 Time (msec) Figure 7.19: Stiffener mid-point displacement in nonlinear elastic-plastic analysis of simply supported 2-Bay Stiffened Plate II due to step load very well with the ADINA solution. Initially, the displacement at E is greater than that at D as expected but as plastification of the structure progresses with time the stiffened panel behaves more like a smeared orthotropic plate so that point D being at the plate center ends up with a larger permanent displacement than point E. The same phenomenon is also observed for the displacements of points C and F presented in Figure 7.27. Figure 7.28 gives some insight into the behaviour of the structure as the response progresses. The figure shows the bending displacement profile parallel to the long side along an axis at y = 24 in at different times using Model 2. At 2 msec the stiffening action of the stiffeners is still very significant and the deflections at the stiffener locations are much smaller than in the panel (as evidenced by the bumps in the figure). However, at 4 msec when significant yielding of the material has occurred Chapter 7. Transient Analysis Results 165 12.0 0.0 S.S. 2-Bay Stiffened Plate II - Blast Load Without In-plane Constraint 0.5 With In-plane Constraint 1.0 1.5 Time (msec) 2.0 2.5 Figure 7.20: Panel centre displacement in simply supported 2-Bay Stiffened Plate II due to blast load the deflections of the stiffeners approach those of the panel. At 6 msec the stiffeners have completely lost their stiffnesses and the displacement profile is similar to that expected from an unstiffened plate. The displacement profiles along the cross beam GD and the longitudinal beam BC are plotted in Figures 7.29 and 7.30, respectively, at various times. Consider the displacement profiles along GD. Each profile resembles a plastic hinge mechanism used in rigid-plastic collapse analysis. At 2 msec the plastic hinge is formed away from the centre of the beam (that is x = 90 in in the figure). However, as time progresses the hinges tend to move towards the centre. This behaviour is similar to the classical travelling hinge mechanism associated with dynamic plastic response of beams subject to intense pressures [55,56]. The same behaviour is exhibited by beam B C in Figure 7.30, although it is not as pronounced there. Chapter 7. Transient Analysis Results 166 14.0 S.S. 2-Bay Stiffened Plate II • Blast Load Without In-plane Constraint 1.0 1.5 Time (msec) 2.5 Figure 7.21: Stiffener mid-point displacement in simply supported 2-Bay Stiffened Plate II due to blast load On the I B M 3081K main frame computer the super element analysis using Model 1 takes about 22 sec per load step while it takes about 7 min per load step on the GA-386L microcomputer. Although no direct comparison can be made, due to the differences in the computers used for the analysis, it is instructive to note that the ADINA analysis takes about 1 min and 20 min per load step, respectively, on the FPS 364 and Honeywell main frame computers. These illustrations clearly exhibit the savings in time derivable by the use of the super element formulation. 7.4.5 Clamped 2x2-Stiffened Plate II The 2x2 Stiffened Plate II is made up of the 2-Bay Stiffened Plate II discussed in Section 7.4.1 with an additional orthogonal cross beam of the same dimentions. Chapter 7. Transient Analysis Results 167 1,000 r Time (msec) Figure 7.22: Blast load on DRES Stiffened Panel Figure 7.31 shows the configuration of the structure. The material properties are: E = 68, 900 MPa, ET = 3,445 MPa, <r0 = 284 MPa, p = 2670 kg /m 3 and v = 0.3. A l l the stiffened plate boundaries are clamped. Note that this is the same stiffened plate discussed in Section 6.4.4, for the case with a/b = 1. The structure is subjected to the 0.3 MPa step load also shown in Figure 7.31. No comparison results are available for this example. Using symmetry, one quarter of the structure is modelled by one super plate element and two super beam elements. This model has 16 net degrees of freedom. From the vibration analysis carried out in Chapter 6 the fundamental period of the structure is 0.87 msec and the time step sizes used for the analyses are: 0.04 msec (% TQ/20) for linear elastic analysis and 0.02 msec (% T o/40) for nonlinear analysis. The linear elastic displacement responses of points A and B are presented in Chapter 7. Transient Analysis Results 168 16.0 14.0 HOB 315 Load on DRES Stiffened Panel Present Finite Strip 4.0 6.0 Time (msec) 8.0 10.0 Figure 7.23: Panel centre displacement of blast loaded DRES Stiffened Panel Figure 7.32. Due to the additional beam the peak panel displacement amplitude is reduced from 11.92 mm, for the 2-Bay Stiffened Plate II, to 5.58 mm, here. The stiffening effect of the beams is very significant, in that the displacement at B is much smaller than that at A . The nonlinear elastic and elastic-plastic displacement responses are displayed in Figure 7.33. Nonlinear geometric effects account for a 52% reduction in the panel centre peak displacement amplitude. The stiffening effect due to geometric nonlin-earities is also manifested in a 43% reduction of the fundamantal period. On the other hand, material nonlinearities do not have any significant effect on the response because the applied load is not high enough to cause significant yielding. On the I B M 3081K computer it takes about 1.5, 5.8 and 12.0 sec/load step, Time (msec) Figure 7.24: Stiffener mid-point displacement of blast loaded DRES Stiffened Panel respectively, for the linear elastic, large deflection elastic and large deflection elastic-plastic analyses. 7.4.6 Clamped 4x4-Bay Stiffened Plate This structure is made up of a 4 m square steel plate of 7 mm thickness reinforced by 6 identical T-beams as shown in Figure 6.6. The plate is clamped all round and the material properties are: E = 210,000 x 106 N / m 2 , ET = 1,250 x 106 N / m 2 , (T0 = 375 x 106 N / m 2 , p = 7900 kg /m 3 and v = 0.3. The stiffened plate is subjected to a 2 MPa rectangular pulse load which acts over a duration of 2 msec. The vibration analysis of this structure has been carried out in Chapter 6, with a discretization which employs 4 super plate and 8 super beam elements for one quarter of the structure. From that analysis the fundamental period is 17.28 msec. Hence a Chapter 7. Transient Analysis Results B S u p e r E l e m e n t M e s h M o d e l 1 A D I N A M e s h Super Element Mesh Model 2 A 15 E E ST symmetry Figure 7.25: Discretizations of DRES IB panel Chapter 7. Transient Analysis Results 171 Figure 7.26: Displacements of points D and E in DRES1B panel due to blast load time step size of 0.15 msec (< T o/100) is employed in the transient analysis. For this analysis both geometric and material nonlinearities are taken into account. Displacement responses at points A, B and G are presented in Figure 7.34. The solid lines represent the responses from the present analysis, while the dashed lines are the responses obtained from a rigid-plastic beam grillage solution [57]. The points A and B displacement response predictions from the two analyses are quite close with the present analysis being slightly more flexible. The beam grillage solution for point G is not available. Note that point A is at the midspan of the middle beams and point G is at the centre of the bay closest to the centre of the entire structure. Initially, the super element displacement of G is bigger than that at A as expected but as time goes on the displacement at point A overtakes that at G and comes to rest at a permanent Chapter 7. Transient Analysis Results 172 Figure 7.27: Displacements of points C and F in DRES IB panel due to blast load displacement of 288 mm compared to 231 mm for point G. This phenomenon can again be explained by making reference to the displacement profiles along HGF at various times as shown in Figure 7.35. It is evident that as yielding of the stiffened plate progresses a time is reached when the beams lose their stiffnesses and the structure behaves more like an unstiffened plate. Hence, point A being located at the centre of the entire structure ends up with a larger deformation than point G. The displacement profiles along beams DBA are plotted in Figure 7.36. The super element profiles are very similar to the rigid-plastic beam grillage profiles. Indeed, the travelling hinge behaviour is exhibited by the two solutions, in that the profiles resemble plastic collapse modes with the hinges formed initially away from the centre line but moving towards the centre with time. Figure 7.37 shows the final permanent displacement profiles along beams D B A Chapter 7. Transient Analysis Results 173 10.0 £ 8.0 HOB 315 Load on DRES1B Panel t = 6 msec 30.0 45.0 60.0 Distance from edge (in) 75.0 90.0 Figure 7.28: Displacement along y = 24in in DRES1B panel due to blast load and E C B . The excellent comparison between the super element and beam grillage solutions is clearly evident from this figure. The super element solution comes to rest after 8.4 msec, while the beam grillage analysis comes to rest after 7.9 msec. The super element solution takes about 2 min per load step on the IBM 308IK main frame computer. Chapter 7. Transient Analysis Results Figure 7.29: Displacement along GD in DRES IB panel due to blast load Chapter 7. Transient Analysis Results Figure 7.30: Displacement along BC in DRES IB panel due to blast load Chapter 7. Transient Analysis Results 176 y.v E E 8 E E to w <D C (0 CL CM 2 Panels at 101.5 mm centres = 203 mm 1 B y. x,u 1.37 mm 6.35 mnT"*" ' 12.7 / mm Typical view CO — 0.3 a. Time (msec) Step Load Figure 7.31: Configuration of 2x2 Stiffened Plate II Chapter 7. Transient Analysis Results 177 6.0 Time (msec) Figure 7.32: Linear elastic response of 2x2 Stiffened Plate II - Step Load Chapter 7. Transient Analysis Results 178 Figure 7.33: Nonlinear response of 2x2 Stiffened Plate II - Step Load Chapter 7. Transient Analysis Results 179 350.0 300.0 _ 250.0 E E, ? 200.0 o E | 150.0 a <n ° 100.0 50.0 Rect. Pulse on 4x4 Stiffened Plate Point A Present Beam Grillage 4.0 6.0 Time (msec) 8.0 10.0 Figure 7.34: Response of points A , B and G in 4x4 Bay Stiffened Plate - Rectangular Pulse Chapter 7. Transient Analysis Results 180 300.0 Red. Pulse on 4x4 Bay Stiffened Plate .-250.0 E E, O 200.0 I O) c •1150.0 t = 8.4 msec 0.5 1.0 1.5 Distance from fixed edge (m) 2.0 Figure 7.35: Displacement profiles along HGF in 4x4 Bay Stiffened Plate - Rectan-gular Pulse Chapter 7. Transient Analysis Results 181 E E £ Q O) c o CO CD E § Q. CO 5 350.0 300.0 250.0 200.0 150.0 100.0 50.0 Rect. Pulse on 4x4 Bay Stiffened Plate Present Beam Grillage t = 8.4 msec 0.5 1.0 1.5 Distance from fixed edge (m) 2.0 Figure 7.36: Displacement profiles along DBA in 4x4 Bay Stiffened Plate - Rectan-gular Pulse Chapter 7. Transient Analysis Results 182 E E, ? CD E 8 co Q. (0 350.0 300.0 250.0 200.0 Red. Pulse on 4x4 Bay Stiffened Plate Present Beam Grillage Beams DBA Beams E C B 0.5 1.0 1.5 Distance from fixed edge (m) 2.0 Figure 7.37: Final displacement profiles along beams D B A and E C B in 4x4 Bay Stiffened Plate - Rectangular Pulse C h a p t e r 8 S u m m a r y a n d C o n c l u s i o n s New plate and beam elements called super finite elements have been developed for the large deflection elastic-plastic analysis of stiffened plate structures subject to static and dynamic loads. The displacement fields of the elements are represented by polynomial as well as continuous analytical functions, and the elements have been specially designed so that only one plate element per bay or one beam element per span is needed to model the response. Large displacements have been taken into account by including the first order nonlinear terms in the strain-displacement relations (following von Karman theory). However, for torsion and lateral bending in the stiffener beam elements, only linear effects have been considered. Material nonlinearities are modelled by von Mises yield criterion and associated flow rule using a bi-linear strain-hardening law. The finite element equations are derived using the virtual work principle and the matrix quan-tities are evaluated by Gauss quadrature. Temporal integration has been performed by the Newmark beta method with Newton-Raphson iteration within each time step. The new formulation has been applied to the static, vibration and transient analy-sis of unstiffened plates, beams and plates stiffened in one or two orthogonal directions. Good approximations are obtained in most cases using only a single super element per bay or span. In the static case, the results for beams and unstiffened plates are very good for all types of analysis (linear or nonlinear) in that the super element solutions compare very well with analytical or other numerical solutions. 183 Chapter 8. Summary and Conclusions 184 The results for stiffened plates also compare well with other numerical solutions. For linear elastic analysis the displacements predicted by the super elements are quite close to the finite strip or ADINA solutions. Also, using one super element per bay a reasonable prediction of the shear lag effect is obtained, although this is not as good as the more detailed finite strip or ADINA solutions. However, it has been shown that the super element solution will converge to the right answer if two or more super elements are employed per bay. The linear elastic stresses obtained from the super element analysis are in agreement with the ADINA predictions for the 5-Bay DRES and the 2 x 5-Bay DRES1B panels. The in-plane and out-of-plane displacement results for large deflection analyses are very good for the cases in which comparison results are available. However, some of the maximum stresses are underestimated and, in general, the predicted stress distributions are not as accurate as the displacement results. The super element linear elastic response of the 2 X 2-Bay Stiffened Plate I com-pares favorably with another solution based on regular finite elements. No comparison results are available for the nonlinear case but the results obtained look reasonable and will be useful to future researchers. Natural frequency results for beams and unstiffened plates obtained by the new formulation are in excellent agreement with exact or other numerical methods. For the two and three bay stiffened panels the super element predictions, with significantly reduced number of variables, agree well with the experimental and finite element results. The super element solution is, in general, on the stiff side of the experimental and finite element solutions and the importance of including the effect of beam torsion has been demonstrated by the results. The super element fundamental frequencies obtained for the 2 x 2 - and 2 X 4-bay stiffened plates are only slightly stiffener than a solution based on the grillage method. The natural frequencies of a 4 X 4-bay stiffened plate have also been presented with no comparison results. Chapter 8. Summary and Conclusions 185 The super element transient displacement responses of various structures to com-plex loads such as air-blast pressure waves have also been investigated. For beams and unstiffened plates the super element results are in excellent agreement with either the finite strip or other finite element analyses. For the simply supported Square Plate II subjected to a very intense triangular blast load, the one super element response, provides a reasonable estimate but is on the stiff side of the finite strip and A D I N A analyses. However, it has been shown that by employing more super elements the present analysis wil l converge to the other results. The transient response of several stiffened plates have been investigated. First, the response of the 2-Bay Stiffened Plate II with clamped and simply supported boundaries subjected to a step load and a moderate blast load has been investigated. For the step load case, linear as well as nonlinear geometric/material behaviour has been included and the super element displacement responses agree well with the finite strip solution. For the blast load case, only large deflection elastic-plastic analysis has been performed. The results obtained in this case also compare well with the finite strip analysis. The super element responses of the 5-Bay D R E S and 2x5-Bay D R E S I B panels to an intense blast load are also in good agreement with the finite strip and A D I N A predictions, respectively. Reasonable results are also obtained for the transient response of the 2x2-Stiffened Plate II, although no comparison results are available for this problem. Finally, the large deflection elastic-plastic response of the 4 x 4 - B a y Stiffened Plate to a rectangular pulse has been obtained by the present formulation. The response obtained from the super elements agree reasonably well with a rigid-plastic beam grillage solution. From the numerical investigations conducted in this study it is seen that the super elements provide a relatively simple modelling of stiffened plate structures as evidenced by the fact that, in most cases, the input file contains less than 50 lines. Yet, the results obtained are very similar to those obtained from other methods for Chapter 8. Summary and Conclusions 186 which more complex models, requiring huge data input, are used. Also, most of the transient analyses have been carried out with very large time step sizes and besides the super element analysis takes much reduced run times for both static and dynamic applications. This conforms with the objectives of the present study and indeed the quality of the results has demonstrated the viability of the new super elements in the nonlinear analysis of plate structures. It would be worthwhile to extend the new formulation to other structural me-chanics problems. The super elements formulation can be extended to the analysis of stiffened box and unstiffened or stiffened shell structures. The super elements can also be extended for the nonlinear analysis stiffened structures made of composite materials. 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[32] Kumar, P., "Large Deflection Elastic-Plastic Analysis of Cylindrical Shells using the Finite Strip Method", M.A.Sc Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada, 1989. [33] Wu, R. W. H. and Witmer, E. A.,"Nonlinear Transient Responses of Structures by the Spatial Finite Element Method", AIAA Journal, Vol. 11, No. 8, 1973, pp. 1110-1117. [34] Abramowitz, M . and Stegun, I. A . , Handbook of Mathematical Functions, Dover Publications Inc., N . Y . , 1972. [35] Belytschko, T. and Hughes, T. J . R., Editors, Computational Methods for Tran-sient Analysis, Vol. 1, Elsevier Science Publishers B. V. , 1983. Bibliography 191 [36] Cook, R. D., Malkus, D. S. and Plesha, M . E. , Concepts and Applications of Finite Element Analysis, 3rd. Edition, John Wiley and Sons, New York, 1989. [37] Newmark, N . M . , "A Method of Computation for Structural Mechanics", Journal of the Engineering Mechanics Division, ASCE, Vol. 85 No. EM3, Proc. Paper 2094, 1959, pp. 67-94. [38] Soreide, T. H., Moan, T. and Nordsve, N . T., 'On the Behaviour and Design of Stiffened Plates in Ultimate Limit State', Journal of Ship Research, Vol. 22, No. 4, December 1978, pp. 238-244. [39] Folz, B. R., FENTAB - Finite Element Non-linear Transient Analysis of Beams - Version 1.0, Department of Civil Engineering, University of British Columbia, Vancouver, Canada, 1986. [40] Houlston, R. and Slater, J . E. , 'A Summary of Experimental Results on Square Plates and Stiffened Panels Subjected to Air-Blast Loading', 57th Shock and Vibration Bulletin, Part 214, pp. 55-67, 1987. [41] Bathe, K. J., 'ADINA - A Finite Element Program for Automatic Dynamic Incremental Non-Linear Analysis ', Report 82^8-1, Acoustics and Vibration Laboratory, Dept. of Mechanical Eng., Massachusetts Institute of Technology, Cambridge, Massachusetts, 1975. [42] Rossow, M . P. and Ibrahimkhail, A. K. , "Constraint Method Analysis of Stiffened Plates",Computers and Structures, Vol. 8, pp. 51-60, 1978 [43] Durvasula, S., Burbure, R. N . and Srinivasan, S., "Natural Frequencies of V i -bration of Isotropic Flat Plates - I", Report No. AE 190 S, Department of Aero-nautical Engineering, Indian Institute of Science, Bangalore-12, 1968. Bibliography 192 [44] Timoshenko, S., Vibration Problems in Engineering, 2nd. Edition, Van Nostrand Co. Inc., New York, 1937. [45] Hearmon, R. F. S., "The Frequency of Vibration of Rectangular Isotropic Plates", Journal of Applied Mechanics, Vol. 19, Transactions, A S M E , Vol. 74, 1952, p. 404. [46] Warburton, G. B. , " The Vibration of Rectangular Plates", Proceedings of the Institution of Mechanical Engineers, Vol. 168, 1954, p. 371. [47] Kaul, R. K. and Cadambe, V. , "The Natural Frequencies of Thin Skew Plates", The Aeronautical Quarterly, Vol. 7, 1956, p. 337. [48] Fletcher, H. J. , "The Frequency of Vibration of Rectangular Plates", Journal of Applied Mechanics, Vol. 26, Transactions, ASME, Vol. 81, 1959, p. 290. [49] Wu, C. I. and Cheung, Y . K.; "Frequency Analysis of Rectangular Plates Con-tinuous in one or two Directions", Earthquake Eng. and Structural Dynamics, Vol. 3, 1974, pp. 3-14. [50] Thomson, W. T., Theory of Vibration with Applications, 2nd. Edition, Prentice-Hall, Inc., New Jersey, 1981. [51] Olson, M . D. and Hazell, C. R.; "Vibration Studies on Some Integral Rib-Stiffened Plates", of Sound and Vibration, 50(1), 1977, pp. 43-61. [52] Balendra, T. and Shanmugam, N . E.; "Free Vibration of Plated Structures by Grillage Method", J. of Sound and Vibration, 99(3), 1985, pp. 333-350. [53] Jiang, J. , Olson, M . D. and Anderson, D. L., "Nonlinear Transient Analysis of Cylindrical Shell Structures by the Finite Strip Method", Structural Research Series, Report No. 35, Department of Civil Engineering, University of British Columbia, Vancouver, May 1990. Bibliography 193 [54] Stagliano, T. R. and Mente, L. J., 'Large Deflection Elastic-Plastic Dynamic Structural Response of Beams and Stiffened or unstiffened Panels - A Compari-son of Finite Element, Finite Difference and Model Solutions', Nonlinear Finite Element Analysis and ADINA, Proc. ADINA Conference, 1979. [55] Vaziri, R., Olson, M . D. and Anderson, D. L., "Dynamic Response of Axially Constrained Plastic Beams to Blast Loads", Int. J. Solid Structures, Vol. 23, No. 1, 1987, pp. 153-174. [56] Schubak, R. B., Anderson, D. L. and Olson, M . D., "Simplified Dynamic Analysis of Rigid-Plastic Beams", Int. J. Impact Engineering, Vol. 8, No. 1, 1989, pp. 27-42. [57] Schubak, R. B., "Nonlinear Analysis of Beams, Grillages and Stiffened Plates under Blast Loads", Ph. D. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada, In preparation, 1990. [58] Olson, M . D. and Lindberg, G. M . , "Free Vibrations and Random Response of an Integrally-Stiffened Panel", Aeronautical Report LR-544, National Aeronautical Establishment, Ottawa, Oct. 1970. Appendix A Shape Functions Using the notations u = Y.\UNiui> v = T . l L N i v i a n d w = £ " i WjVj t h e s h a P e functions for the plate elements are given by = HOMv) m = HOHv) = HOHv) Nl Nt = Ht)Hv) = LA{()L7{r,) = H()L3(V) Nt = HOHv) N?0 — sin 2IT£LI(TI) = sin 2TT(L2{TJ) Nu = sin27r^L3(n) Nu = sin 47r£Z, 1(77) Nu J V 1 4 = sin47r^L2(^/) 194 Appendix A. Shape Functions 195 Ngs = 811147^3(77) = for * < 9 ^ 0 = 2^(0 s i n 2 ^ Nvn = L2ti)sm2TrV Nv12 = L3(Osm2irr] ^ 3 = I 1 (Os in47r r / Nv14 = I 2 ( O s i n 4 7 T 7 / ^ 5 = ^ 3 ( 0 ^ 4 ^ Ng = Hi(()Hi(rj) Ng Ng = Hx{i)H2(V) Ng = H2(OH2(v) Ng = Hs(t)H1(7,) Ng = ff4(0#ifo) Ng = H3(t)H2(V) Ng = H4(t)H2(V) Ng = H3(()H3(V) = HA(0Hs(v) Nw = H3(Z)H4(V) Nw J V 12 = HA{()HA{n) Appendix A. Shape Functions 196 = H^OHM N™ = H2(t)Hs{V) JVTe = H3(t)H<(V) N?7 = 4>(t)Hiiv) AT- = 0(O#afo) i V - = i/ 3(OM) i V - = Hi(t)d>(V) iVTs = 0 ( 0 ^ ) Shape Functions for Beam Element in z-direction The shape functions for a beam element in the x-direction are given by = M O N? = * m ) NT = eH'2(C) N™ = HO NT NT Appendix A. Shape Functions 197 NT = : HO NT = N? = sin 27r£ Ko = sin 4"7r£ Nbt = m) Nb = m) Nb = m) Nb = Nb5 = where the superscripts m and b stand for membrane and bending, respectively. For a beam in the y-direction replace ^ by 77. Appendix B Strain-Displacement Matrices The linear and nonlinear strain-displacement matrices, [B] and [C0], discussed in Chapter 4 are presented here. For the super plate elements, these are 3 x 55 matrices and 1 x 10 for beam elements. Details of the [B] Matrix For plate elements, the [B] matrix is given by dx 0 a2 AT™ z d x 2 d 2 N ? — 7 Z d x 2 d2N™ Z d x 2 d 2 N ? Z d x 2 [B] = 0 ~~dy~ d2N™ z dy2 a 2 A ™ — 7 — Z dy2 d2N™ Z dy2 a 2 AT™ Z dy2 dN* dx dxdy ft2 N w 2z 2 dxdy dxdy 2 z d 2 N X dxdy dx 0 d2N™ z dx2 a 2 A ™ z 8x2 a 2 A™ Z dx2 d2N™ Z dx2 0 dN% d2N™ a2jv™ ~ n d2N™ Q2NW dy Z dy2 Z dy2 Z dy2 Z dy2 dN% dy dNl dx 2zd2N* dxdy dxdy a2 ATW 2z 7 dxdy 2zd2N* dxdy dN% dx 0 a2 A ™ Z dx2 d2N™ Z d x 2 d2N™ — y Li. Z d x 2 a 2 TV.™ — y - L i Z d x 2 0 dNl dy a 2 A™ Z dy2 d 2 N ? n  Z dy2 d2Nf. Z dy2 a 2 AT™ Z dy2 dNf dy dNl dx d2N™ 2 2 dxdy a 2 AT™ dxdy a 2 NW 0 - d J V11 dxdy r)2 / V w A Z dxdy dN% dx 0 a2 A7 3 Z dx2 Z d x 2 d2N™. 7 15 Z d x 2 Z d x 2 0 8N% dy d2N™ Z dy2 Z dy2 a 2 A ™ — 7 L5-Z dy2 Z dy2 aA™ dy d N \ dx 0 ~ ° JV13 dxdy a 2 NW dxdy a 2 iv" aaiay a 2 NW dxdy 198 Appendix B. Strain-Displacement Matrices 199 dNl dx 0 92 A ™ . 7 1 ' Z dx2 Z dx2 d AT™ 10 dx 0 dNl By 92JV™ Z By2 a 2 AT™. y . La. Z By2 0 0 dN£ By dNl dx pS i\j~w 2z 17 dxdy 2z 19 dxdy By aw-9y dN£ dx 0 Z dx2 Z dx2 0 0 0 dNl By d2N™ Z dy2 2 By2 By dN£ By dNl dx dxdy 2zd2N™ dxdy 9 N U dx a dN? dx 0 d2N™ — y -Al. Z dx2 d2N?-y AA. Z dx2 8N» dx as 0 dNl By d2N™ — 7 Z By2 82N™ y AA. Z dy2 0 0 dNf By dNl dx dxdy i v22 dxdy By S A T -By dN% dx 0 d2N™ y AA. Z dx2 82N™ Z dx2 0 0 0 dNl By a 2 AT™ y Z dy2 a 2 A ™ Z dy2 9y dN% By dNl dx 2zd2N" dxdy 2zd2N" dxdy dx dx 8Ng dx 0 d2N™ ~ 2fj Z dx2 ^ 0 ox S A T " 0 0 dNl By Z By2 n d N h By 0 aA75 dN% dy dNl dx dxdy 9A 7"" dN* dy dx By dx For a b e a m element i n the cc-direction the [B] m a t r i x is given by [B} = dLx dx d2H2 dx2 dLi dx , {e-z) d2H3 dx2 (e-z) d2H4 dU d2(f) dL^ dL% dx2 ' dx ' dx2' dx ' dx where, £4 (1; ) = sin2-7Tc; and L5(£) = s in 4 ^ . Fo r a beam i n the y-d i rec t ion , replace x by y a n d £ by 77. Appendix B. Strain-Displacement Matrices 200 Details of the.[C0] M a t r i x For plate elements the [Co] is given by [Co} = n n dN? dN? dN? dN? dN? dN? dN? dN? 0 0 - d t ~ d t w i ~ d t ^ t w J - e t ^ - w i -i£-^rwi n n dN? dN? dN? dN? dN? dN? dN? dN? 0 0 -eZ—BZ-Vj -af-dy1-™! ~ d t ^ W i ~dt^yl-Wi n n ,dN?dm' dN? dN? dN? dN? dN? dN? 0 0 gmn(-^-^-)Wj g r n r i { ^ - ^ - ) W j g m n ^ ^ ) w . g m n ( _ ± . _ l _ ) w n n dN? dN? dN? dN? dN? dN? d N ? dN? 0 0 -at-atwi -at~dtwJ ~ d ^ ^ w J dx* al w i n n dN? dN? dN? dN? dN? dN? dN? dN? 0 0 -et-btwi ~dt^-wi -bf-btw3 -at-stwi n n „ i d N f d N V . ( d N ? d N ? . f d N ? d N ? . d N ? dN? . 0 0 5 m n ( ^ ^ K - flfmn(^-^-K- S m n ^ ^ K 0 0 ~dt^twi -air-nt^i - d ^ ^ t w J -bir-dtwJ 0 Q d N ^ d N l dN^dNJ^ dNJ^dNl d N ? 2 d N ? U U dy dy W3 By dy W3 dy dy W 0 dy dy W 3 0 0 a (^^L)w- n f ^ ^ W a (^^)w a ffi^W U U 9mn{QXm d X r i )Wj 9mn{ d X m d X n )Wj 9mn\. 8 x m 8 x n )W3 9mn{ Q X m 8 x n )Wj n n dN?3 dNT dN?. dN? dN?. dN? d N ? R dN? II I) J—W- " t—1tl- "i L . M . IS. J—tn • U U 9 x 9 x 9 x dx 9 x 9 x W J dx dx W3 n n S A T 3 a A 7 9ATJ". SAT™ dN? 8 N ? . dN? 0 0 -af^-Vi ~ & t ^ w i -dt~atwi - b t ~ b t w i U U 9 m n \ d X m d x n )Wj g m n { d X m g X n )Wj 9mn{ g X m g X n )Wj 9rnn{ 8 x m g X n )U>3 0 0 dN?- dN? — — w • 9 x 9 x 3 8N?. dN? — — w • dx dx 3 0 0 0 0 9AT™ dN? dy dy 3 dN?. dN? —— —*—w • dy By 3 0 0 0 0 3N?r dN? 9mn{ 8 x m Q X n )Wj dN?s dN? 9™A d x m d x n )W3 0 0 0 0 dN?- dN? — ——w • dx dx 3 dN?n dN? — -—— w • dx dx 3 0 0 0 0 dN?. 9N? dy 8y 3 d N w dNV dy dy 3 0 0 0 0 9 ™ \ d x m 9 x n )W3 9mn{ d X m 8 x n )Wj 0 0 Appendix B. Strain-Displacement Matrices 201 0 0 0 0 6A"?. SAT —— ——W ' dx dx 3 dN™ BN* 121 dy dy dN£ BN™ dx dx d N w BNV dy dy •ID; 0 0 0 0 n n (aN?idNF\ ,dN™BN™ 0 0 0 0 SAT, 9 N r 2 i '—in • dx dx W3 dN™ dN? dy dy •? , 8N™ BN 8N™ BNV 121 dx dx dN?. BNf dy dy •? 8N™ SAT 0 0 a (^22.—i-W o f ^ 2 4 — - M 0 0 0 0 0 0 0 0 0 0 SA™ SAT 2 i Z _ yj . dx dx J dN™ BN™ 2 i i _ yj . By dy 3 0 0 0 0 0 0 0 0 o o 3 - ( S f - ^ K - o o o o where m, n = 1, 2; j = 1, 2, . . . 25 and Wj consists of the corner flexural displace-ments given in Equation (3.7). gmn is defined as 9mn 1 8-mn where 8mn is the Dirac delta, so that 1 if m / n 0 if m = n The terms xm define the two coordinate directions, such that, xx represents the x-axis and x2 the y-direction. For a beam in the x-direction the [Co] is given by [Co} = dH\dH) dHb2dH] n dE\dE\ dE\dE\ dE\dE\ dx dx v dx dx ' dx dx ^ dx dx 3' ' dx dx where, j = 1, 2, . . . 5 and Wj contains the terms Wi, wxi, w2, wx2 and ws. A p p e n d i x C [fi] M a t r i c e s f o r P l a t e a n d B e a m E l e m e n t s The matrix [ft] appears in the tangent stiffness matrix expressed in Equation 4.26. It is symmetric and has 55 x 55 terms for plate elements and 10 x 10 terms for beam elements. The terms in the [ft] matrix for plate elements hace been expressed by Equa-tion 4.28 as ON? dN? ON? dN? (dN? dN? dN? dN?\ 13 dx dx dy dy y \ dx dy dy dx J x y where i , j = 1, 2,. . . , 55 and r, s = 1,2,..., 25. The subscripts r, s corresponding to the indices i, j, respectively, can be obtained from the following chart: 202 Appendix C. [Q] Matrices for Plate and Beam Elements 203 i or j r or s i or j r or s i or j r or s 1 0 19 0 37 0 2 0 20 0 38 0 3 1 21 13 39 21 4 2 22 14 40 22 5 3 23 15 41 0 6 4 24 16 42 0 7 0 25 0 43 0 8 0 26 0 44 0 9 5 27 17 45 23 10 6 28 18 46 24 11 7 29 0 47 0 12 8 30 0 48 0 13 0 31 0 49 0 14 0 32 0 50 0 15 9 33 19 51 25 16 10 34 20 52 0 17 11 35 0 53 0 18 12 36 0 54 0 55 0 For example, using the chart, the subscript corresponding to i = 4 is r — 2 and that corresponding to j = 33 is s = 19, so that the term ft4|33 is given by _ dNg 8N™ dNg_dN^ (dNg_dN^ dN? dN?9  4'33 dx dx dy dy y \ dx dy dy dx Similarly, Appendix C. [ft] Matrices for Plate and Beam Elements 204 Q5117 = ? ^ . ? ^ n < r + ^ 1 ^ 1 ^ | (MndNZ | dN™dN™\ dx dx dy dy y \ dx dy dy dx J x y and so on. If any of the subscripts r, s corresponding to i, j is zero, then the ftjj term is zero. Thus, ftl,12 = ^37,27 = ^55,20 = ^44,44 = 0 and similarly for other terms. For a beam element in the x-direction the entries in the [ft] matrix are given by Equation 4.29 as dNbdNb iiij — — —c x dx dx where i,j = 1, 2, . .. , 10; r,s = 1, 2, . . . , 5 and the shape functions Nb have been described in Appendix A. The values of r, s corresponding to i, j in this case can be obtained from the chart below: i or j 1 2 3 4 5 6 7 8 9 10 r or s 0 1 2 0 3 4 0 5 0 0 The terms in [ft] are obtained by following the procedure outlined above for the plate element case. Some examples are given below: ft 2,5 ft8,3 — ^6,6 — dN' dNb3 dx dx dNb5 dNb dx dx dN\ dN\ dx dx ftl,9 — ftl0,8 — ft4,7 — ft9,9 — 0 Other terms can be similarly obtained. For a beam in the y-direction, replace x by y in the relevant expressions above. A p p e n d i x D F o r m u l a s f o r J , Izz, Jo a n d F In the formulas presented here, the symbols in the following figure are used: Rectangular Cross-section l-Beam Cross-section 3 - , - f , Torsional Constant J For rectangular cross sections, the torsional constant is given, approximately, by [58] T bh2 64a 4 , (vb\ J — r~ tanh — 3 Tr 5 \2hJ For thin I or T sections, the torsional constant is given, approximately, by J = \(bj! + b2rf + hwtl) 205 Appendix D. Formulas for J, Izz, Jo and T 206 Moment of Inertia, Izz For beams of I cross sections, the moment of inertia about the z — z axis is given by Izz=^(hb31 + f2b32+twhl) The moment of inertia about the z — z axis for beams of T or rectangular cross sections can be obtained from the above expression by eliminating the appropiate flanges. Polar Moment of Inertia, J 0 For I cross sections, the polar moment of inertia about the mid-plane of the plating is given expressed as Jo = ^ (6 l / 1 3 +fe 2 / 2 3 + ^ ^ ) + ^ ( < p + / l ) 2 + fe2/2{^ + |(< P-/2)} 2 + +twhw{fi + l(hw + tp)}2 + ±(hb\ + f2b\ + twhl) where, tp is the thickness of the plating which the beam reinforces. The same formula can be used for rectangular or T cross sections by eliminating the appropiate flanges. Warping Constant, T For thin-walled I beam sections the warping constant is given, approximately, by ~ n I2{hb\ + f2b\) Note that in the above expression, the tangential component of T has been ne-glected as it is very small compared to the normal component, Tn. For T and rect-angular shapes the normal component, T n is zero and hence T = 0 for these cross sections. 

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