Failure Analysis of Stiffened and Unstiffened Mild Steel Plates
Subjected to Blast Loading.
by
Julien R. Fagnan
B.Sc, The University of Alberta, 1982
M. Eng., The University of Alberta, 1984
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
DEPARTMENT OF CIVIL ENGINEERING
We accept this thesis as conforming
to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
© Julien R. Fagnan, 1996 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 of CNtL /5/JSWggg;/J^
The University of British Columbia Vancouver, Canada
DE-6 (2/88) Abstract
The failure of unstiffened and stiffened mild steel plates subjected to blast loads is investigated. The three failure modes identified in the literature, as the load intensity increases, are; mode I (large ductile deformation), mode II (tensile-tearing) and mode III
(transverse shear). The prediction of failure for a large structure subjected to air blasts is a complex problem involving non-linear response in both geometry and material properties. In addition, the effect of high rates of strain on material properties is not well understood. An accurate analysis using finite elements currently requires a very detailed element grid with a large number of elements and the associated volume of input and output data. In the preliminary stages of design this is not practical. The present work develops simplified analysis tools to allow engineering level accuracy for preliminary design of blast loaded plates.
The analysis tools consist of an analytical formulation used to predict when material separation has occurred, by either tensile tearing (mode II) or shear failure (mode III), in conjunction with a numerical formulation which models the large inelastic (mode I) behaviour of the plates when subjected to air blasts. This numerical formulation is capable of modelling a large structure with relatively few elements and yet obtain a reasonable overall accuracy but with a resulting loss of detail at the local level.
The numerical formulation employed for the square plates employs existing super finite elements. These elements contain all the basic modes of deformation response (mode I) which occur in orthogonally stiffened plates. This allows stiffened plate structures to, be modelled with only one plate element per bay and one beam element per stiffener span, greatly simplifying the input and output data, A regular finite element formulation is also developed to model axisymmetric plates. ' Both these formulations use von Karman large deflection theory and the von Mises yield criterion and associated flow rule to model geometric and material non-linearities respectively. Numerical and temporal integrations are carried out using Gauss quadrature and the Newmark-P method respectively with Newton-Raphson iteration at each time step.
The analytical formulation used in conjunction with these numerical models takes
advantage of the overall accuracy of the deformed profile of the plates in calculating the in-
plane strain at each time step. For mode II failure, the total strain is calculated by estimating the local bending and axial strain in the critical regions and avoiding the less accurate grid
dependent local strains. The total strain is then compared to a maximum allowable fracture
strain.
For the mode III failure of axisymmetric plates, equilibrium is used to calculate the
support reaction at the plate boundary at each time step. The calculated shear stress is then
compared to a maximum allowable shear stress to determine if shear failure has occurred. The interaction between mode II and III failure is taken into account via an interaction relationship. Once material separation has occurred, post failure analysis continues as the plate translates freely through the air.
The results from the computer models for both the axisymmetric and square stiffened
and unstiffened plates are compared to experimental data available in the literature. The
numerical results show good comparison to the experimental results.
iii Table of Contents
Abstract
List of Tables
List of Figures
Acknowledgement
1. Introduction
2. Literature Review
2.1 Introduction
2.2. Available Fracture Models
2.2.1 Introduction
2.2.2 Fracture Mechanics
2.2.3 Void Coalescence Mechanisms
2.2.4 Damage Mechanics
2.2.5 Forming Limit Curves
2.2.6 Fracture Limit Strain
2.2.7 Strain Energy Density
2.2.8 Stress Based Failure Curves
2.2.9 Transverse Plastic Shear Hinge
Selected Fracture Models
2.3.1 Mode II Failure
2.3.2 Mode III Failure
3. Numerical Formulations
3.1 Introduction
iv 3.2 Approximations 14
3.3 Equations of Motion 15
3.4 NAAPFE 16
3.4.1 Introduction 16
3.4.2. Finite Element Discretization 17
3.4.3 Displacement Functions 19
3.4.4 Strain Displacement Relations 21
3.4.5 Continuity and Order of Convergence 22
3.4.6 Constitutive Relations 23
3.4.7 Mass and Damping Matrices 28
3.4.8 Load Vector 29
3.4.9 Stiffness Matrix 32
3.4.10 Numerical Integration 34
3.4.11 Temporal Integration 36
3.4.12 Computer Implementation .37
3!5' NAPSSE 38
3.5.1 Introduction 38
3.5.2 Finite Element Discretization 39
3.5.3 Displacement Functions 42
3.5.3.1 Plate Elements 42
3.5.3.2 Beam Elements 44
3.5.4 Strain Displacement Relations 45
3.5.5 ' Computer Implementation 46
Analytical Failure Formulations 47
4.1 Introduction 47
4.2 Mode II Failure Model 48
V 4.2.1 Introduction
4.2.2 Membrane Strain
4.2.3 Bending Strain
4.2.4 Extension to 2D Problems
4.3 Mode III Failure Model
4.4 Mode II and Mode III Interaction
4.5 Implementation in Axisymmetric Plates
4.5.1 Introduction
4.5.2 Mode II
4.5.2.1 Membrane Strain
, 4.5.2.2 Bending Strain
4.5.3 Modem
4.5.4 Interaction •
4.5.5 Post-Failure Analysis
4.6 Implementation in Rectangular Plates
4.6.1 Introduction
4.6.2 Mode II
4.6.2.1 Membrane Strain
4.6.2.2 Bending Strain
4.6.2.3 Biaxial Strain Limit
4.6.3 Modelll
Square Plates
5.1 Introduction
5.2 Experiments
5.3 Modelling the Experiment
5.4 Preliminary Studies
vi 5.5 Results and Discussion 80
5.5.1 Predictions 80
5.5.2 Comparisons 86
6. Circular Plates 90
6.1 Introduction 90
6.2 Testing of Finite Element Formulation 90
6.2.1' Introduction 90
6.2.2 Static, Linear Elastic Geometry and Material 91
6.2.3 Eigenvalues 95
6.2.4 Dynamic Elastic 97
6.2.5 Static, Non-linear Geometry, Linear.Material 98
6.3 Experiments 99
6.4 Modelling the Experiment 101
6.5 Results and Discussion 102
6.5.1 Predictions 102
6.5.2 Comparisons 122
7: Stiffened Square Plates , 134
7:1 Introduction 134
. 7.2 Experiments 134
7.3 Modelling the Experiment 136
7.4 Results and Discussion 137
^ 7.4.1 Predictions 137
7.4.2 Comparisons 142
8. Conclusions 148
Bibliography 152
vii List of Tables
5.1 Uniform pressure load for given impulse (square plate). 69
5.2 Preliminary investigation results. 73
5.3 Comparison of membrane strains. 79
5.4 Strains at mode II failure. .85
6.1 Comparisons of exact static results with NAAPFE. 92
6.2 Variation of element forces with grid type: simply supported. 94
6.3 Variation of element forces with grid type: clamped. 96
6.4 Eigenvalues. 97
6.5 Dynamic step load. 98
6.6 Non-linear geometry - linear material. .99
6.7 Residual elastic vibration. 105
7.1 Material properties for stiffened plate specimens. . 136
7.2. Predicted mode II failure results (o0 = 265 MPa). 141
viii List of Figures
2.1 Generalised forming limit diagram. 8
2.2 Transverse velocity field. 11
3.1 Discretization of circular plate. 18
3.2 Degrees of freedom for circular plate element. 19
3.3 Bilinear stress-strain relationship. 24
3.4 Strain rate effect. 26
3.5 Loading functions. 31
3.6 Discretization of rectangular stiffened plate. 40
3.7 Degrees of freedom for rectangular plate and beam elements. 41
4.1 Typical centreline deformation profiles for unstiffened square plates. 49
4.2 Generalised stiffener displacement. 64
5.1 Layout of explosives for square plate tests. 68
5.2 Finite element grids for preliminary studies. 70
5.3 Effect of grid size on centreline failure profiles. 72
5.4 Effect of grid size on centreline strain distribution. (20 Ns) 75
5.5 Effect of grid size on centreline strain distribution. (25 Ns) 76
5.6 Effect of grid size on centreline strain distribution. (40 Ns) 77
5.7 Effect of impulse on centreline strain distribution. 78
5.8 •- Time history of midpoint displacement. 81
5.9 Effect of strain rate on centreline deformation profiles. 81
5.10 Effect of impulse on yield stress, strain rate and time to first yield. 82
5.11 Predicted transient centreline deformation profiles. 83
ix 5.12 Predicted final permanent centreline deformation profiles. 84
5.13 Strain perpendicular to boundary at failure. 86
5.1.4 Comparison of experimental and predicted centreline profiles. 87
5.15 Midpoint deflection versus impulse. 88
6.1 Layout of explosives for circular plate tests. 100
6.2 Models of pressure loading. 101
6.3 Time history of midpoint displacement. 104
6.4 Predicted transient centreline deformation profiles. 106
6.5 Time history of strains. 108
6.6 Distribution of finite element radial strains. 109
6.7 Total radial strains as a function of element size. 110
6.8 Time history of strain and stress ratios. 112
6.9 Time history of interaction models. 113
6.10 Failure time versus impulse. 115
6.11 Predicted transient centreline deformation profiles. (30 Ns) 117
6.12 Predicted transient centreline deformation profiles. (40 Ns) .118
6.13 Midpoint deflection versus impulse. 119
6.14(a) Stress and strain ratio versus impulse. (Linear model) 121
6.14(b) Stress and strain ratio versus impulse. (Quadratic model) ~ •. 122
6.1-5 Kinetic energy ofthe severed plate versus impulse. 123
6.16(a) Comparison of experimental and predicted centreline profiles. (13.3 Ns) 126
6.16(b) Comparison of experimental and predicted centreline profiles. (26.2 Ns) ... 127
6.16(c) Comparison of experimental and predicted centreline profiles. (30.3 Ns) 128
6.16(d) Comparison of experimental and predicted centreline profiles. (39.6 Ns) 129
6.16(e) Comparison of experimental and predicted centreline profiles. (49.6 Ns) 130
6.17(a) Midpoint deflection versus impulse, (linear interaction model) 131
X 6.17(b) Midpoint deflection versus impulse, (quadratic interaction model) 132
7:1 Stiffened plate specimen. 135
7.2 Effect of impulse on yield stress, strain rate and time to first yield. 138
7.3 Predicted transient centreline deformation profiles 139
7.4 Predicted final permanent centreline deformation profiles. 140
7.5 Predicted failure profile versus impulse. (3x4 mm stiffener) 141
7.6 Comparison of experimental and predicted mode I profiles at 10 Ns 143
7.7(a) Permanent midpoint deflection versus impulse. (3x0 mm stiffener) 144
7.7(b) Permanent midpoint deflection versus impulse. (3x2 mm stiffener) 145
7.7(c) Permanent midpoint deflection versus impulse. (3x4 mm stiffener) 145
7.7(d) Permanent midpoint deflection versus impulse. (3x5 mm stiffener) 146
7.7(e) Permanent midpoint deflection versus impulse. (3x9 mm stiffener) 146
xi Acknowledgement
The financial support of the Canadian Department of National Defence, through a contract with the Defence Research Establishment Suffield, is gratefully acknowledged. The faculty and fellow students at the University of British Columbia provided valuable advice and support. In particular, the patience and support of Dr. M.D. Olson was significant. Finally the experimental support provided by Dr. G.N. Nurick was important to the completion of my program.
xii Chapter 1
INTRODUCTION
The design of metal structures is a diverse and important component of structural engineering.
Common examples include offshore structures, ships, vehicles, aircraft and storage structures.
These structures may be subjected to large transient dynamic loads such as wave impact, vehicular collision and air blasts caused by military or industrial explosions. When sufficiently high intensities of load are applied, failure of the structure by fracture of the material may occur. This problem is one of great complexity and it is desirable to develop tools which will allow for the efficient design of safe and reliable structures. In particular, in the early stages of design, a simplified method with reasonable overall accuracy is very useful..
A number of structural computer programs are available for the transient dynamic elastic-plastic analysis of structures. These programs are able to predict quite accurately the large deflection non-linear response of metal structures. However, the prediction ofthe failure of the structure by fracture of the material is not yet well understood. Several methods have been developed to attempt to predict the fracture of ductile metals. These methods generally involve investigating the behaviour of imperfections in the material under applied loading. This may require a detailed analysis of the structure at the local level. Such an analysis may involve considerable time and expense as well as a large amount of input and output data when designing large and complex structures.
The purpose of the research outlined in this proposal is to present an approximate procedure to predict the failure due to fracture of metal structures subjected to air blast loading. This approximate procedure should have sufficient accuracy to develop a preliminary
1 Chapter 1. Introduction 2
design for metal structures and to highlight the critical areas for design. A more detailed analysis of those critical areas may then be undertaken as the design is refined.
Two separate computer models are developed in this work. One may be applied to orthogonally stiffened or unstiffened rectangular plates and the other to axisymmetric plates.
The same philosophy is employed for each model but the implementation in each case is distinct.
Chapter 2, which follows, presents the results of a literature survey which investigates the possible alternatives for the prediction of fracture in metal structures. Chapters 3 and 4' then present the details of the numerical and analytical formulations respectively which were
employed in this work. Chapters 5, 6 and 7 then present the numerical investigations and
comparisons with the experimental data. Chapter 5 investigates isotropic square plates,
Chapter 6 axisymmetric plates and Chapter 7 square stiffened plates. Chapter 8 then
' summarises the results of this work and discusses future research goals. Chapter 2
LITERATURE REVIEW
2.1 Introduction
Numerous experimental and theoretical investigators have explored the problem of the large inelastic deformation of beams and plates subjected to transverse impulsive loads [1-5].
However, only a few investigations have included loads large enough to cause actual failure of the beams or plates by rupture or tearing. The first such study mentioned in the literature was by Menkes and Opat [6] who conducted a series of experiments on clamped aluminium
6061 T6 beams subjected to surface explosive charges. As the intensity of the impulse was increased by using greater amounts of explosive, three distinct modes of failure where identified:
mode I: large inelastic deformation;
mode II: tearing (tensile failure) in outer fibers, at or over the support;
mode III: transverse shear failure at the support.
The degree of damage for mode I failure of the uniformly loaded beams is characterised by the magnitude of the central permanent deflection; the deflection increases with increasing impulse. The threshold for mode II failure occurs at the impulse which first causes tearing. As the impulse is increased further, the net permanent deformation decreases and there is an overlap of modes II and III. Finally, a pure well defined shear failure occurs when the severed beam shows no significant permanent deformation in the region away from the supports.
A study carried out by Teeling-Smith and Nurick [7] on explosively loaded clamped circular plates found that the same three failure modes also applied to that case. An
3 Chapter 2. Literature Review 4
investigation of explosively loaded square plates carried out in conjunction with this work again found the three failure modes to be generally applicable [8]. That investigation showed that the mode II failure was more complex for the square plates with failure occurring first at the mid-sides of the supports and progressing to the corners with increasing impulse. In a more detailed study of fully clamped square plates, Nurick and Shave [9] suggest that mode II be subdivided into three phases:
mode II*: tensile tearing in the outer fibers over a portion of the support;
mode Ha: mode II failure with increasing midpoint deflection;
mode lib: mode II failure with decreasing midpoint deflection.
Mode lib as proposed by Nurick and Shave [9] corresponds to mode II proposed by Menkes and Opat [6]. Mode Ha only occurs in square plates and not in beams or circular plates.
As stated earlier, prediction of model failure (large inelastic deformation) is already well established and has been investigated by many methods including mode approximation, finite difference, finite elements and rigid plastic analysis. The emphasis of this research has been on the prediction of mode II and mode III behaviour. Mode II failure is associated with large plastic deformation of the structure. Material separation occurs over the supports and is characterised by ductile fracture. Mode III failure is described by a shear failure at the supports which occurs with no significant deformation of the structure. Despite the different failure mechanisms for each mode, the overlap of mode II and III as noted by Menkes and
Opat [6] suggests that there is no sharp distinction between the two modes and that there is an interaction between them.
Several approaches are currently available for the theoretical analysis Of the failure of metal structures In the following sections these approaches will be reviewed Section 2.2 reviews the models available in the literature for the prediction of the fracture of metals and
Section 2.2 presents the models that were chosen for the investigations into the mode II and
III failure of blast loaded metal structures Chapter 2. Literature Review 5
2.2 Available Fracture Models 2.2.1 Introduction
In this section, several models developed to predict the "failure of metal structures will be reviewed and evaluated in light of the objectives of this research which are to develop an approximate procedure that provides preliminary design level accuracy in the prediction of failure of metal structures when subjected to air blast loading.
2.2.2 Fracture Mechanics
Metal structures all have, to some degree, pre-existing flaws or stress concentrations. These may cause the initiation and propagation of cracks which decrease the residual strength of the structure. Fracture mechanics concepts permit the study of the fracture resistance of metal structures. •
The basic equations of fracture mechanics were developed in 1921 by Griffith [10]. A transverse crack is assumed to exist in a stressed plate. If the length of the crack increases by a small amount da then a portion of the strain energy stored in the plate will be released. Griffith stated that the crack will grow if the energy released upon crack growth is equal to all the energy required for crack growth. The energy consumed by the growth of the crack is called the crack resistance, denoted by R.
Connolly et al. [11] modified a non-linear plane strain/stress dynamic transient finite element program to allow for through thickness crack propagation along a line of symmetry.
They used this program to model the dynamic fracture of some wide plate tests. A simple crack propagation routine was incorporated into the existing program. The crack is advanced one element at a time by releasing the nodal reactions incrementally during time stepping. This is done by applying equal and opposite forces to the nodes, thus providing an external work Chapter 2. Literature Review 6
component which equals the energy released by the crack during extension. Their study showed good comparisons with experimental results.
Broek [12] has discussed the crack propagation of stiffened sheet structures. The use of a curve describing the crack resistance R as a function of crack length is proposed. The deformation of fasteners and stringer eccentricity must be taken into account. The author states that the R curve needs to be better understood.
The use of a failure model based on fracture mechanics would require a finite element grid with a fine mesh so that an accurate representation of the stress field is obtained. In the preliminary design phase when the plate thickness and stiffener locations have not been finalised, the effort in generating detailed grids and analysing the results may not be warranted.
2.2.3 Void Coalescence Mechanisms
Another procedure which examines the behaviour of small imperfections in the metal matrix is the study of the growth and coalescence of voids. In contrast to fracture mechanics where stress intensity factors at crack tips are used to study crack growth, void coalescence mechanisms assume a distribution of voids in the matrix which grow and join together to initiate fracture.
McClintock [13], Gurson [14] and Theocaris [15] among others, have presented a method of predicting ductile fracture due to the growth of holes. Cylindrical holes are assumed to exist in the material. The condition of fracture is that the growth of the holes is
such that they touch. This condition may be described in terms of a relative growth factor giving the increase of the semi-axis of the hole relative to the corresponding hole spacing. The continuum mechanics problem is reduced to the deformation of a hole in an infinite medium.
Progressive softening of the material due to increasing porosity causes a localisation of Chapter 2. Literature Review 7
deformation in narrow shear bands. The subsequent coalescence of voids initiates an extension of existing cracks.
This analysis again requires a complete knowledge of the history of stress and strain as well as a finite element grid with a fine mesh to accurately represent the local stresses at the microscale level. As was the case for fracture mechanics, the analysis of large structures using void coalescence poses difficulties because of the need for accuracy at the microscale level.
2.2.4 Damage Mechanics
Damage models of fracture relate identified fracture inducing damage quantities to material separation or fracture. Material microscopic separation will occur when the accumulated damage reaches a critical value. In contrast to the microscale models of fracture mechanics and Void coalescence, damage mechanics may be evaluated at the macroscale; that is the level of the constitutive equations for strain behaviour. A damage variable is defined as an effective surface density of cracks.
A model for ductile fracture based on a continuum damage variable has been developed by Lemaitre [16]. The critical value of the damage variable can be calculated from the uniaxial case. The damage variable is a function of the accumulated plastic strain and the uniaxial yield strain and fracture strain as well as Poisson's ratio.
2.2.5 Forming Limit Curves
The sheet-metal forming industry has provided the impetus for several investigations into the biaxial strain limits of stretched metal sheets. Marciniak and Kuczynski [17] analysed the loss of stability for sheet metal subjected to a range of biaxial tension ratios. They concluded that fracture depends only on a local discontinuity or concentration of strains.
Hecker and Ghosh [18] produced an experimentally determined curve in principal strain space which indicated the onset of localised necking. This curve was obtained by using a Chapter 2. Literature Review 8
hemispherical punch to stretch a flat specimen. Strain ratios were varied by using different lubricants between punch and plate and by using less than a full width plate. The resulting curves had a characteristic shape as shown in Figure"2.1. The curves were shifted up or down vertically, depending of the type of metal tested.
The preceding two papers are limited to quasi-static or low strain rates. Marciniak,
Kuczynski and Pokora [19] studied the influence of strain rate sensitivity on the forming limit diagram. They concluded that the strain-rate sensitivity of the material increases the value of the limit strain and influences the shape of the forming limit curve.
Atkins and Mai [20] developed a relationship which allowed the fracture strains in
Major Strain
; /
/ Symmetric
Minor Strain
Figure 2.1 Generalised forming limit curve Chapter 2. Literature Review 9
sheet metal forming to be determined from independent measurements of the specific essential work of fracture(i?) instead of from in-situ strain measurements, which are usually employed in the construction of forming limit diagrams. The authors' method consisted of testing pre- notched specimens under uniaxial tension conditions and using the results to experimentally calculate the work of fracture.
2.2.6 Fracture Strain Limit
The biaxial strain limits of the forming limit curves may be simplified to the one dimensional case. This would result in a failure criteria based on the uniaxial fracture strain. Jones [21, 22] used a rigid plastic analysis to predict the onset of mode II behaviour for the experimental results of Menkes and Opat [6]. Plastic hinges, located at the boundaries and at the midspan of the beam, were subjected to both axial and bending strain. A kinematically admissible displacement field allowed the total strain in the plastic hinges to be calculated. When the total strain exceeded the fracturestrai n limit, failure occurred. Inscribing and circumscribing square yield curves were used to approximate the maximum normal stress yield curve. The resulting values for the onset of mode II failure bounded the experimental results.
Ratzlaff and Kennedy [23] studied the ultimate strength of steel plates subjected to uniformly distributed static transverse pressure loads. The steel plates, which had an aspect ratio of 1/3, were modelled with a finite element program which employed a two dimensional through thickness mesh with 116 elements to represent the half-width of a unit length of the plate.
The strain at the edge ofthe plate was calculated by considering the geometry of the plate at the boundary. A similar procedure was employed by McDermott et al. [24] to describe the geometry at the plate edge when analysing the damage to ship hulls due to collisions. The calculated strain was again compared to a maximum strain limit. Chapter 2. Literature Review 10
2.2.7 Strain Energy Density
In this method, the energy absorbed at a point in the structure is used as a measure of fracture toughness. This energy may be calculated as
The value of W is compared to the critical value of the strain energy density, Wcr, which is determined from a uniaxial tension test when the strain reaches the rupture strain e^:
Link [25] used the strain energy density (SED) method in conjunction with a finite element program to analyse the problem of the transversely loaded plate as presented by
Ratzlaff and Kennedy [23]. The finite element mesh employed for this analysis was the same
116 element discritization which was employed by the original authors. The analysis ofthe failure load was within 15% of the test failure load.
. Shen and Jones [26] also used a SED method to analyse the experimental results of
Menkes and Opat [6]. In their rigid plastic analysis, they included an interaction yield surface which combined the effects of bending moment, axial tensile force and the transverse shear force.
2.2.8 Stress Based Failure Curves
Duffey [27] and Lindholm et al. [28]considered stress based failure criteria. They examined the von Mises and Tresca failure surfaces in stress space and then mapped them into strain space. The resulting von Mises failure surface bared little resemblance to the experimental results of the sheet metal forming industry. On the other hand, the Tresca failure surface in strain space did have excellent qualitative agreement with the experimentally measured biaxial strain-failure data.
In their study of rectangular plates, Ratzlaff and Kennedy [23] included an analysis of the shear capacity ofthe plate. They discussed the similarity of the shearing of metal plates in Chapter 2. Literature Review 11
metal forming operations with the shearing of high-strength pretensioned bolts in a lap joint.
They concluded that an extension of the von Mises yield criterion to ultimate strength
conditions, tmax = Guit / ^3, would provide a reasonable failure criterion.
2.2.9 Transverse Plastic Shear Hinge
Jones [21,22] also developed a rigid-plastic model to predict the mode III threshold.
This model used the concept of a transverse plastic shear hinge to allow shear sliding at the
supports of a rigid plastic beam. Using the velocity field shown in Figure 2.2 and conservation
of momentum, an expression was developed for the time required for the shear hinges to come to rest following the impulse. By integrating the velocity over the time to rest, the distance
travelled by the hinge may be calculated. If this distance is greater than the beam depth then
mode III failure occurs. Jones was able to predict quite reasonably the threshold impulse for
mode IIFbehaviour.
Initial beam position
Figure 2.2 Transverse velocity field. Chapter 2. Literature Review 12
Duffey [27] then discussed extending Jones' rigid-plastic model to the problem of dynamic rupture of shells. For the mode III threshold the beam analysis developed by Jones was shown to be an upper bound to the shell problem and good results could be obtained by using the Jones method on a shell strip. The method was generalised to include the effects of non-impulsive dynamic loading and flexible stiffeners.
2.3 Selected Fracture Models
2.3.1 Mode II Failure
The preceding fracture models offer many potential approaches to the problem of predicting the failure of blast loaded stiffened or unstiffened plates. Within the constraint of efficient preliminary investigations, however, the options are reduced. The procedure adopted in this study for the prediction of mode II failure is based on the maximum allowable strain. The strain at the boundary will be calculated by taking account of the deformed geometry of the plates in a manner similar to that employed by Jones [21,22]. The deformed geometry will be evaluated through the use of a finite element model. Complete details of the procedure will be presented in Chapters 3 and 4.
2.3.2 Mode III Failure
The prediction of mode III failure will be based on an ultimate stress criteria. This procedure similar to the approach taken by Ratzlaff and Kennedy [23], will calculate the shear stress at the boundary for each time step. This will then be compared to an ultimate shear strength to determine if failure has occurred. The details of the mode III failure model will be presented in
Chapter 4. Chapter 3
NUMERICAL FORMULATIONS
3.1 Introduction
The philosophy of this work is to develop a tool to allow the preliminary design of metal
structures composed of thin stiffened plates. The structures under consideration are subjected to intense air blasts sufficient to cause failure of the structure. The tool consists of numerical and analytical computer models which provide a prediction of the response of the structure to given loads with engineering design level accuracy.
Two computer models have been developed and modified to analyse metal structures.
These models account for large elastic-plastic deformation at high rates of loading. A computer code called NAPSSE (Non-linear Analysis of Plate Structures by Super Elements) was developed by Koko [31,32] to model mode I behaviour of rectangular unstiffened plates or othogonally stiffened plates and stiffeners subjected to blast loading. In this work, the NAPSSE code has been extended to include strain rate sensitivity and the prediction of mode II behaviour. A second computer code, NAAPFE (Non-linear Analysis of Axisymmetric Plates by Finite Elements) has been developed to model the mode I, U and HI response of axisymmetric plates subjected to blast loading. The NAAPFE code includes post-failure deformation of the torn plate in free flight.
This chapter presents the details of the derivation ofthe finite element formulation for
NAAPFE and NAPSSE. The analytical failure models will be presented in Chapter 4. Section 3.2 discusses some of the approximations and limitations made in developing the numerical models.
Section 3.3 uses the principle of virtual work to develop the equations of motion for the numerical
13 Chapter 3: Numerical Formulations 14
formulations. Finally, the details of the finite element formulation for NAAPFE and NAPSSE are presented in Sections 3.4 and 3.5 respectively.
3.2 Approximations
The simplified approach taken in this work carries with it some limitations. The structures under consideration are typically composed of stiffened plates, It is assumed that the plates are thin and the beams are slender. The limited experimental evidence suggests that failure generally occurs in the plates rather.than in the stiffeners if they are present. Therefore Kirchoff thin plate theory and
Bernoulli beam theory are used and shear deformation is considered negligible.. This simplification still allows good engineering design level accuracy for the range of problems of interest.
Another major limitation in this work involves the pressure-time history ofthe loading. In order to test the models developed herein, they were used to model experimental results published in the literature. Unfortunately, no detailed data was available on the pressure-time history of the load imparted to the structure for experiments where mode U and mode ITJ behaviour was detected. Such data was available for rectangular plates where mode I response occurred and the numerical model has already been compared favourably to those results by Koko [31]. For the experimental results used in this work, the total impulse imparted to the structure was however, carefully measured and is used to derive an approximate pressure loading. The implications of this will be discussed in detail in Chapter 6.
The failure of these structures involves very short intense loading. The material therefore undergoes a large amount of straining at a very high rate particularly in the experiments used as a
comparison for this study. The material properties: yield stress, oy; and fracture strain, Sf, of steel are known to be significantly affected by high rates of strain as discussed by Jones [36]. Chapter 3. Numerical Formulations 15
These uncertainties in material properties and pressure loading combine to make accurate modelling of the problem difficult and support the use of a simplified approach for the preliminary engineering design phase.
3.3 Equations of Motion
Two separate numerical formulations are used in this study, one for axisymmetric plate problems and the other for rectangular stiffened and unstiffened plates and beams. The governing equations of motion for both formulations are developed using the principle of virtual work.
Given an elastic-plastic structure in equilibrium being acted upon by a known loading, the principal of virtual work states that
W = 0 (31) where W is a kinematically admissible small variation in virtual work denned as
W= [qldldA + f FddV- f cj^e.dV ' J A iv iV. 1 3 '
where di are virtual displacements, sy- are virtual strain, qt are surface tractions, FT are body forces, Cy are the internal stresses, A is the loaded surface area and Vis the volume, di and sy
are compatible deformations arid qi% FT and Cy are in equihbrium.
When d'Alembert's principle is used to express the inertial forces due to transient loading, viscous loading is assumed, and body forces are ignored , then the virtual work may be expressed in a matrix form as
T T T 33 {dfp [d] + [d} Kd {d} + {e } {a}]dV - j {q}dA = 0 ( ) Iv A {3} where {s } is the virtual strain vector, {oj is the stress vector, {d } is the virtual displacement vector, {d } and {d} are the velocity and acceleration vectors respectively, p is the mass density, Chapter 3. Numerical Formulations 16
Kd is the viscous damping parameter and {q} is the surface traction vector. In cases where the
loading may be considered static the first two terms drop out.
The internal virtual work term, j" {£• }r{o~} the true stresses, cry defined in the deformed configuration. The Kirchoff stresses are, however, defined in the undeformed configuration. These stresses are approximately equal for small strains. Due to the non-linear geometric and material behaviour of the blast loaded plates, analytical solutions of Equations (3.3) are not feasible and numerical solutions are sought by deriving finite element formulations for both the axisymmetric and rectangular plate cases. This equation will be . developed further in Sections 3.4 and 3.5 specifically for axisymmetric and rectangular plates respectively. • •. . • 3.4 NAAPFE 3.4.1 Introduction A finite element program has been developed to model an axisymmetric plate subjected to an air blast pressure pulse. NAAPFE (Non-Linear Analysis of Axisymmetric Plates using Finite Elements) models the circular plate as a one dimensional problem taking advantage of the axisymmetries of the loading and the plate. In this section, the details ofthe finite element formulation are derived. The discretization of the axisymmetric plate is described in Subsection 3.4.2 followed by the displacement functions in Subsection 3.4.3. The strain displacement relations are presented in Subsection 3.4.4 with a discussion on continuity and order of convergence in Subsection 3.4.5. The non-linear rate dependent constitutive relations are then discussed in Subsection 3.4.6. Using Equation (3.3) the mass and damping matrices are evaluated in Subsection 3.4.7 as well as the load vector in Subsection 3.4.8. Equation (3.3) also serves as a basis for the derivation of the stiffness matrix in 17 Chapter 3. Numerical Formulations Subsection 3.4.9 where the Newton-Raphson iteration scheme is presented. The numerical integration scheme is presented in Subsection 3.4.10. Next, Subsection 3.4.11 discusses the temporal integration scheme using an implicit Newmark-P method: Finally, Subsection 3.4.12 dicusses the implementation of this formulation in a computer program. 3.4.2 Finite Element Discretization The circular plate is discretized by a series of annular ring elements as shown in Figure 3.1. Because both the plate and the loading are assumed to be axisymmetric, the two dimensional plate of uniform thickness may be reduced to a one dimensional problem. Each annular plate element may then be represented as a one dimensional beam element as shown. The beam element of length a, has three nodes as shown in Figure 3.2. The end nodes, labelled 1 and 2, each have 3 degrees of freedom; the radial in-plane displacement w, the lateral displacement w and the slope 6= wr= dw/dr. The mid node, labelled 3, only has one degree of freedom, the in plane displacement u. Each element therefore has a total of 7 degrees of freedom. The positive directions of u and w are shown in Fig. 3.2, as well as the global co-ordinates r and z where the radial distance r is measured from the centre of the plate and the lateral djstance z is measured from the mid depth of the plate. The dimension r0 for each element is the distance from the centre of the plate to the first node ofthe element. Chapter 3. Numerical Formulations Chapter 3. Numerical Formulations 19 Z,W 1 L Degrees of Freedom Nodes 1,2 - u,w,wr Node 3 - u Figure 3.2: Degrees of freedom for circular plate element. 3.4.3 Displacement Functions The displacement fields for the one dimensional element are based on the Lagrangian and Hermitian interpolation polynomials The radial and lateral displacement fields associated with each element are given by Equations (3.4) and (3.5) below. (3.4) ^ 5 J (3.5) Ox 0, Chapter 3: Numerical Formulations 20 These equations are expressed in terms of the local variable £ where (3.6) 4 a and r is the global co-ordinate measured from the centre ofthe plate, r0 is the global co-ordinate distance from the centre of the plate to the first node of the element and a is the element length. The quadratic Lagrange interpolation polynomials in Equation (3.4) are given by Equation (3.7) below L, =2f- 3£ + l (3.7) • •• - i^-ic-e) ,••• • • and the cubic Hermitian polynomials in Equation (3.5) are given by • #,.= l-3£2 + 2£3 (3:8) H2=a{$-2e + V) //, = 3£2-2£3 . Equations (3.4) and (3.5) can be expressed in terms of a shape function matrix [N] as 1111 (3.9) where {4} is the element nodal displacement vector given by S = (3.10) { eY' =[«„w],0„«2,w2,^2,ii3] and the shape function matrix is given by (3.11) IM 0 . 0 L2({) 0 0 L&) 0 Hfe) • H2(Z\ 0 H^)-'.H4{4) 0 Chapter 3. Numerical Formulations 21 3.4.4 Strain Displacement Relations . As discussed earlier, the shear deformations are assumed to be negligible. Large deformations effects are taken into account by using von Karman theory which includes first order non-linearity's in the strain displacement relations. This theory assumes that the deflections are larger than the plate thickness but small relative to the plate length. Equations (3.12) express these strain displacement relations. • 2 du d ' ^'w+UdwY (312) dr dr1 2\dr) u z dw r dr rre=o where er, £e and yrg are the strain components in the plane of the plate. If a linear matrix operator [L] is defined as d d2 (3.13) dr ~Zdr2 1 -z d r r dr 0 0 then the linear strain displacement matrix [i?] is given by (3.14) and using Equation (3.11) this may be expanded as Chapter 3. Numerical Formulations 22 (4£-3)/a -z(l2£-6)/a2 -z(6£-4)/a (4£-l)/V (3:i5) [B]= (2£2-3£ + l)/r -z(6f--6£)/ar- -*(l-4> + 3£2)/r (2{2-{)/r 0 0 0 0 -z(6-12£)/a2 -z(6£-2)/a (4-%g)/a' -z(6{-6£2)/ar -z(3{2 -2£)/r (4£-4£2)/r 0 0 0 If the element strain vector {fj is defined as (3.16) X re then Equation (3.12) may be expressed in matrix format using Equations (3.9),(3.14) and (3.16) as {£j=[[B].[cfe)]]{^: • where the non-linear terms in Equation (3.12) are expressed in matrix format by defining [C(4)] as . • (3-18) w. 2 dr dr ' [m}=0 , . 3.4.5 Continuity and Order of Convergence The shape functions described in Section 3.4.3 were chosen to provide sufficient, continuity and a consistent order of accuracy. Using the strain displacement relations given in the preceding section, the element strain energy for a linear axisymmetric plate bending problem is given by 3 2 (3.19) TT nEh fo+a 2 1 2v 12 r 2 u ~ U =. — (w +— w~ +—yvrM!r+—\rur H v2vuur\)rdr t r\ /t 2 \ ! \ rr 2 ' rr r i 2 L ' r J/ 12(1 - V )Jr" r r h r Chapter 3. Numerical Formulations 23 where h is the thickness of the plate, a is the element length, r0 is the radial co-ordinate of the first node, v is the Poisson's ratio, E is the elastic modulus and the subscripts denote derivatives with respect to the global variable r. . The lateral displacement field w as given in Equation (3.5) is C1 continuous and cubic, therefore the solution will converge, as the finite element grid is refined, with an error in strain energy of the order of a4 in the out of plane displacements, where a is the element length. The in-plane radial displacement u is C° continuous and quadratic. This results in an order of convergence in strain energy consistent with the out of plane displacement. 3.4.6 Constitutive Relations This study is concerned with the prediction of the response of metal plates subjected to intense air blasts or explosions sufficient to cause failure. Because of the ductile material involved it is expected that the plate will undergo a significant amount of plastic straining before failure occurs. The constitutive relations must adequately model this behaviour. After the onset of first yield, structural materials typically strain harden, that is the yield stress increases as additional loading occurs thereby expanding the yield surface. Therefore the material is assumed to follow an elastic- plastic (hardening) constitutive relation represented by a bilinear stress strain relationship under uniaxial loading as shown in Figure 3.3. a0 and e0 are the uniaxial yield stress and strain respectively, & and % are the elastic and plastic strains and E and ET are the elastic and tangent modulii. Modelling of the strain hardening behaviour may typically be accomplished using an isotropic or kinematic hardening model. The latter model attempts to account for the Bauchinger effect which is important in cyclic loading cases. The loading.in this study is confined to large explosive blasts and is considered monotonic with very small residual vibrations. Therefore the isotropic hardening model is adopted. Chapter 3. Numerical Formulations 24 Figure 3.3: Bilinear stress-strain relationship. In order to define the constitutive relations, the yield criterion must first be quantified. That is, given a multiaxial state of stress, when does the elastic material yield and behave plastically. The von Mises criterion has been shown to be successful at modelling the behaviour of ductile metals. This criterion is based on the assumption that yielding will occur when the distortion or shear strain energy equals the distortion energy at yield in simple tension. This may be expressed as Chapter 3. Numerical Formulations 25 1 •(3:20) of where oi, a% and 03 are the three principle stress components and cr0 is the uniaxial yield stress. Once the material has yielded, its response is governed by a plastic stress strain relation or flow rule. A flow rule associated with the von Mises yield criterion is chosen. The yield criterion may be represented as a surface in three dimensional stress space and the incremental strain vector at any point on this surface is normal to the surface at that point. This leads to an incremental plasticity theory. Two factors cause the yield surface to change; strain hardening and strain rate. Each one will cause the yield surface to expand. In order to simplify the constitutive relations, reduce computing time and determine the location of the yield surface, the strain rate and strain hardening effect are assumed to be independent. The blast loading causes high strain rates to occur very early in the time history ofthe deformation and significantly expands the yield surface prior to yield while strain hardening only alters the yield surface once yield has occured. Therefore, the strain rate effect at a point is only considered prior to first yield and the hardening effect is only considered after yield has occurred at that point: The effect of the strain rate on the stress strain relation of a rate sensitive material is shown in Figure 3.4. As the strain rate increases, the yield stress increases whereas the elastic and plastic modulii are unaffected. An empirical relationship has been developed by Cowper-Symonds and is used here to model the strain rate sensitivity and is given by - (3.21) * a = a 1 +— s O 0 where o*0 and a0 are the instantaneous and static yield stresses respectively, and S\ and s2 are strain 1 rate parameters. The typical values of si and s2 for mild steel are 40 sec" and 5 respectively. Chapter 3. Numerical Formulations 26 The plasticity theory described in this section can now be formulated using incremental stress strain relations. The incremental stress {do} and strain {de} vectors are related by the elastic- plastic constitutive matrix [DT ] as follows: {d e Figure 3.4: Strain rate effect. Chapter 3. Numerical. Formulations 27 where [DR ] is defined as r T (323) [AJ = \D\ - [D}{A}{\} \D}[A +{A} [D]{M] ' and (3.24) dsr {de} dsg dYre In Equation (3.23) [D] is the elasticity matrix, {A}={ cFld{ o)} where F is the yield function given r by Equation (3.20) and A is the slope of the stress-plastic strain relationship in a uniaxial test given by • . . •' . E (3.25) A = T ^ ET E - . where E and £r are the elastic and plastic modulii respectively as defined in Figure 3.4. The use of the constitutive relations in the axisymmetric finite element formulation may be summarised as follows. The stress increment resulting from a given strain increment is calculated using Equation (3.22). The stress vector is then calculated at the end of the current iteration by adding the incremental stress vector to the stress vector fromth e previous iteration. The updated stress vector is then used to calculated the equivalent effective stress, a^, given by of=of-cvo,+^+3^ . (3.26) When this effective stress exceeds o~0, the uniaxial dynamic yield stress, for the first time, initial yield occurs. From this point the strain rate effect on the yield stress is no longer evaluated and the failure surface increases due to strain hardening only. The current stresses are then scaled back to the yield surface and the elastic-plastic constitutive relations are employed fromthi s point unless unloading occurs. When unloading does occur, the elastic constitutive relation is employed until yielding occurs again. Chapter 3. Numerical Formulations 28 3.4.7. Mass and Damping Matrices The mass and damping matrices required for the finite element formulation of this problem may be derived by expressing the equation of motion (Equation (3.3)) in terms of the nodal variables. Using Equation (3.17) the virtual element strain vector {^j can be expressed in terms ofthe virtual displacements by &HmcA'.W>), (327) where [C0(5eYJ = 2[C(5e)] as given in Equation (3.18).and from Equation (3.9), the general element displacement {de} may be expressed in terms of the nodal displacements {Se} by R} = MR} '. <328 ) where [N] is the shape function matrix defined in Equation (3.11). Equation (3.3) may then be expressed for each element as T + r r T T (329) [M «W [^]^[^]{^}+[[^HC(^)]]0 {-} ]dV={l} i[N] {o}dA where the virtual displacement j<5e | is arbitrary. From Equation (3.29) the consistent mass matrix for an element is denned as T M=l[N] p[N]dV (3-30) The integration over the volume, carried out using cylindrical co-ordinates is expressed as Ki=pcr£"MrM— (33l) Rearranging Equation (3.6) in terms of the variable r gives r = r0+%a (3.32) Substituting for the variable r and the shape function matrix [N] in Equation (3.31) and performing the integration results in , Chapter 3. Numerical Formulations 29 56 0 0 -14 0 0 28 (3.33) 0 156 22a 0 54 -13a 0 0. 22a 4a2 0 13a -3a2 0 Inahp -14 .. 0 0 56 0 0 28 K] = 420 0 54 13a 0 156 -22a 0 0 -13a -3a2 0 -22a 4a2 0 28 0 o 28 0 0 224 " 7 0 0 -7 0 . 0 0 " 0 36 7a 0 27 -6 a 0 0 7a la- 0 7a =la2 0 2 a -7 0 0 49 0 0 28 0 27 7a 0 120 -15a 0 . 0 -6a -a2 0 -15a ia^ 0 2 0 0 0 28 0 0 112 From Equation (3.29) the consistent damping matrix for an element is defined as v\=[M*mdv (3 34) In this formulation the damping matrix is assumed to be proportional to the mass matrix. 3.4.8 Load Vector From Equation (3.29) the consistent load vector for an element is defined as (3.35) Because this finite element formulation is derived on the basis of an axisymmetry, the loading must also be axisymmetric, that is, q is not a function of 6. The blast loading in this work is assumed to cause only transverse pressure to be imparted to the plate structure. Therefore the pressure vector is expressed as Chapter 3. Numerical Formulations 30 •{«}> Two methods for evaluating the load vector have been formulated. The first assumes that the load q is independent of r as well as 9. That is, the load is uriiformly distributed over the entire plate. Because the loading is transient however, q is a function of time, t. The variation with time is expressed in one of three ways. 1. Exponential variation given by q{t) = qm(\-t /r)exp(-A}t /z) 0 2. . Logarithmic variation given by (3.37) ?(') = Figure 3.5 shows the plots of the loading function q(f) with respect to time for each of these three cases and shows the influence of the parameters A,i and A.2. Expressing Equation (3.35) in terms of cylindrical co-ordinates results in |,j j j "|.\| fa)rJ,,W <»8> Equation (3.38) may be evaluated exactly for a given function q(t) by substituting for the shape function matrix to give . 0 ' 0. "" (3.39) 1/2 3/23/20 a/2 a/30a/30 > > {pe} = 2naq 0 • +a- 0 1/2 7/20 -a/2 -a/20 0 0 where q = q(t). Chapter 3. Numerical Formulations Figure 3.5: Loading functions Chapter 3. Numerical Formulations 32 The second method for evaluating the load vector allows the load q to be a function of the radius, (i.e. q = q(r,t)). This is accomplished through the use of an external module by specifying the pressure at a gauss point directly as a function of time and its radial co-ordinate. 3.4.9 Stiffness Matrix The governing equation of motion for the structure (Equation (3.3)) has now been simplified to M!- For a static loading case, the load q is not a function of time and the vectors {8} and {8} are equal to zero, therefore reducing Equation (3.40) to f.li''i-M]X'/r !/•: V ""' For an elastic-plastic material under large displacements this equation is a non-linear function of the displacements {8} and must be solved iteratively. If the stress vector {a} is expressed in terms of the displacements {8} then Equation (3.41) may be expressed as a function of the displacements to give {F(8)} = {p} (3.42) A truncated Taylor series expansion of the function F about a known solution {S0} modifies the equation to (3.43) !••!!•• !/'! 3{S) If the tangent stiffness matrix, [Ky] is defined as .' (3:44) R . MF(80)\ {s)={s0 Chapter 3. Numerical Formulations 33 and the known terms of Equation (3.43) are brought to the right side, then that equation may be rewritten as where {AS} is the incremental nodal displacement vector defined as {A8) = {6}-{S0} (3-46) Equation (3.45) can be solved for. {AS} once the tangent stiffness matrix is evaluated. If the function F as defined in Equations (3.41) and (3.42) is substituted into Equation (3.44) the tangent stiffness matrix may be expressed as It is assumed that the order of differentiation and integration of the integrand may be interchanged. Recalling that, from Equation (3.15), the matrix [B] is independent of {5} whereas [C0]=[2C(<5)] as given in Equation (3.18) and {a},.by the constitutive relations, is a function of {s} which is in turn a function of {5} as given by Equation (3.17), differentiation of Equation (3.47) is evaluated as ^^^^m^ <348) d{8] [ ] V- * 1 oJJ :.d{e}d{6} A matrix {Q} is defined as (3.49) and each term in the matrix may be expressed as = G Q !H!l™> (3 50) ' d r d r where i,j = 1,2,...,7 and k,m = 1,2,3,4 are selected to represent the corresponding transverse degrees of freedomwhil e the corresponding in plane degrees of freedom are zero. Chapter 3. Numerical Formulations 34 Also, from Equation (3.48), using the incremental constitutive relation given in Equation (3.22) Jn] ' (351) d{e) ~[ rJ and using the strain displacement relation of Equation (3.27) Substituting Equations (3.49), (3.51) and (3.52) into Equation (3.48) transforms Equation (3.47) to • The tangent stiffness matrix can now be evaluated for a given displacement {$,} by using Newton- Raphson iteration. The outline for the procedure is as follows: 1. From the previous iteration or load step determine/^,). • 2. Evaluate Equation (3.53) numerically. 3. Solve for [ASl } using Equation (3.45). 4. Use Equation (3.46) to find at the next iteration (i.e. {<5}.+1 = {S}. + {AS, }) 5. Repeat the first four steps until the solution converges to within a specified tolerance. 3.4.10 Numerical Integration In order to solve the equation of motion, the stiffness matrix must be evaluated. Unlike the load vector and mass matrix, the integrand in Equation (3.53) is not an analytical relationship which can be integrated directly. There is no explicit formulation for the stress-strain relationship. The integral must therefore be evaluated numerically. In this formulation Gauss quadrature is employed. Chapter 3. Numerical Formulations 35 The integrand in Equation (3.53) may be expressed in a non-dimensional form as /=J%%, V, where the variables r/, g. are the normal co-ordinates which vary from 0 to 1, and the volume integral is expressed as (354) I £ 17, o^Trfs- = £ tSjC&n, = Z S X W^WJC^VJ , £K) k j i . The integral is transformed to the non-dimensional form where / = / (^,r/,g,) and the variables 77,$" are the transformed normal co-ordinates which vary from -1 to 1. This is then evaluated using Gaussian quadrature as shown where Wj,WJ,Wk are the weighting factors and ^i,r/J,gk are the corresponding sampling points in the x, y, z directions respectively and i, j, are the number of sampling points in each of the three respective directions: • The Gaussian integration scheme can integrate a polynomial of degree (2n-l) exactly when n sampling points are chosen. The displacementfields are linear for u and quadratic for w therefore 3 integration points are required in the radial direction. For elastic analysis, a 2 point Gaussian integration scheme is sufficient to capture the linear strain distribution through the depth of the plate. When the plate material becomes plastic, additional points are required to capture the non• linear stress-strain variation through the thickness. Previous studies [32,33] have shown that in this case, the use of 4 integration points is sufficient for thin rectangular beams. For this study, which used Kirchoff thin plate theory and Bernoulli beam theory, the 4 point Gaussian integration is employed when the material becomes plastic. Chapter 3. Numerical Formulations 36 3.4.11 Temporal Integration The global equations of motion (Equation (3.40)) represent non-linear ordinary differential equations in time. Due to the nonlinearity, direct integration is the most expedient method of solution. In this method the equation of motion is written at a specific instant of time with the time derivatives replaced with afinite differenc e approximation. The direct method chosen for this formulation is the average acceleration method, also known as an implicit Newmark-P method with parameters P=0.25 and y=0.5. This method is second-order accurate and unconditionally stable with no restriction on the size of the time step size except as required for accuracy. Expressing the governing equations of motion in terms offinite differenc e gives (3,55) L J n+l ' L J The termJ({S}n+i) is a non-linear term which represents the internal force vector at the (n+l) time step and may be expanded using a Taylor series expansion to give f({S}J=f{{S}n)+[KT}iAS}^ ' (358) where [Kr] is the global tangent stiffness matrix and . {AS}n.]={S}n.l-{S}n (3.59) is the incremental displacement vector. The acceleration vector may be obtained by rearranging Equation (3.57) to give (3.60) {AJ}-(A^}r^-(l-2^){^ Substitution of this equation into Equation (3.56) gives the velocity vector as Chapter 3. Numerical Formulations 37 (3.61) n+l 6(At)" B [J„ 2>t? Using Equations (3.59),(3.60), and (3.61), the finite difference Equation (3.55) is expressed as (3.62) [K]{ASl.}={p}r where \K\ is an effective stiffness matrix given by (3.63) .3{At)2 l"~J ' /?A/.1 and {P} is the effective load vector given by (3.64) + AA>) I'l When the problem is linear, prior to yielding ofthe material, the tangent stiffness matrix is equal to the elastic stiffness matrix, [KT] - [K\ and the internal force vector may be expressed explicitly in terms ofthe displacement asX{S}n) = [£]{ displacement term {A8)n+i must be replaced by {8}n+i. The resulting effective load vector is expressed as > (3.65) 2X in + j3(At) B(At)\ J„ 2/3 I JnJ + [C] j(At)^" p 3.4.12. Computer Implementation The formulation as presented in Section 3.4 has been implemented in a computer code named NAAPFE which was written in FORTRAN language. It has been run on several computer platforms including PC's, SUN and HP. Chapter 3. Numerical Formulations 38 The NAAPFE code may be used to examine a circular plate problem in terms of its vibrational response to determine its eigenvalues and eigenvectors. Also, its static or transient response may be analysed taking into account linear or non-linear geometric, and material behaviour. The solution procedure is iterative as explained previously. This algorithm proceeds until a norm based on the displacement solution is less than or equal to an allowable tolerance as specified by the user. One of two norms may be chosen; the maximum norm, defined as <5A, (3.66) or the Euclidean norm, defined as where At is the displacement solution for the nodal variable i, SAt is the current correction for that variable, and is the total number of nodal variables. 3.5 NAPSSE 3.5.1. Introduction An existing program, NAPSSE (Non-linear Analysis of Plate Structures by Super. Elements), developed to model mode I response of rectangular orthogonally stiffened or unstiffened plates and beams has been modified to include strain rate sensitivity and to predict mode II failure. The displacement fields in this program were specifically chosen so that only a single super element was required to model each panel bay or stiffener span. That is, all the possible linear and non-linear deformation modes and boundary conditions are represented by the displacement fields in one element. This was accomplished by the addition of carefully selected analytical functions to the regular finite element polynomials for the displacement fields. A summary of the existing program is presented in the following sections. Full details are presented by Koko[31]. Chapter 3. Numerical Formulations 39 . In Subsection 3.5.2 the super element discretization is presented as well as the degrees of freedom for each type of element. Subsection 3.5.3 presents all the displacement functions for each element. Subsection 3.5.4 introduces the strain displacement relations for this formulation and finally Subsection 3.5.5 summarises the implementation of this formulation. 3.5.2. Finite Element Discretization A typical orthogonally stiffened plate is shown in Figure 3.6 with one panel identified as ABCD. This panel may be modelled using one. plate element and two beam elements. The nodes associated with each element and the variables associated with each node are shown in Figure 3.7. Each plate element has 9 actual nodes numbered 1 to 9 as shown. The additional nodes labelled uw to and vi0 to v]5 are actually the amplitudes of the trigonometric functions used to model the in-plane displacements, u and v, and they are lumped at the mid-side and central nodes labelled 5 to 9. In total the plate element has 55 degrees of freedom. Each beam element has 3 actual nodes numbered 1 to 3 as shown in Figure 3.7 and two additional nodes w4 and w5 which again represent, the amplitudes of the trigonometric functions that model the in-plane displacements and are lumped at the midspan node. A beam which spans in a direction parallel to the x axis has three variables at each end node u, w, and wx and four variables at the midspan node u, w, uA and w5 for a total of ten degrees of freedom. If torsion is significant then additional variables are added to model the torsional rotation 6 and the lateral bending displacement v. The additional variables are v,0, 0* at the end nodes and v and 0 at the midspan node for a total of 18 variables. The variables for a beam which spans parallel to the y axis are analogous to those presented for the beam spanning in the x axis. In stiffened plate structures the nodal variables for the beam are assumed to be located at the centroidal plane of the plate to ensure that no additional degrees of freedom are introduced at the beam-plate connection. Continuity between the beam and the plate and between adjacent plates is ensured at the common nodes. ' . ' >' Chapter 3. Numerical Formulations .40 Chapter 3. Numerical Formulations 41 Degrees of Freedom for Plate Element Nodes 1,2,3,4 - u,K w, wx, Wy, Nodes 5,7- u,v,w,wy; plus (u^u,^ and (U^LU respectively Nodes 6,8- u,v,w,w„- plus (vh.M-t) and (v10,v1c) respectively Node 9- tf,iwplus Uu.ViaU^s 1 tf5 //« 3 2 x,u Degrees of Freedom for Beam Element (In x direction) Nodes 1,2 - u,v,w,Wx,6,9x Nodes 3 - u,v,w,e; plus u^,uz Figure 3.7: Degrees of freedomfo r rectangular plate and beam elements. Chapter 3. Numerical Formulations 3.5.3. Displacement Functions The displacement fields for the super elements consist of the usual polynomial functions and additional continuous analytical functions which are smeared together. The analytical functions consist of trigonometric and hyperbolic functions which are chosen to model all possible displacement modes in a stiffened plate structure. The displacement fields for the plates and beams are presented in the following subsections. 3.5.3.1 Plate elements The 55 degree of freedom plate element represented in Figure (3,7) has a displacement field given by (3.68) '10 u = N»u, + sin 2^1, (77), L2 (77), L3 (tj)] > + '13 + sm4^[L (n),L (7 ),L (r/)\ 1 2 7 3 '14 lW15 (3.69) '10 v = ATv, + [LA (4 L2 (£), I3 (£)}sin lm LV12J 13 +[i;(4l2(4Z3^)]sin4^. 14 15 (3.70) w = NJ¥j +.^(^)[//1(r7),^2(r7),i/3 ( 77),//4 ( 77)]' r + ^8 Wx8 ' + *ti)i(ri)w \H^),H2^),H^),H^)\ w Chapter 3. Numerical Formulations 43 where / = 1,2,...,9; j = 1,2,...,16; u, v are the two perpendicular in-plane displacements; w is the lateral displacement; E, = x/a; rj = yfb. and the Lagrange interpolation polynomials, Lh are given by Equations (3.7) and the Hermitian interpolation polynomials, Hu are given by Equations (3.8). represents the first symmetric vibration mode of a clamped beam and is given by a(sinh//£ - sin //£) + (cosh //£ - cos jug) (3.71) where // = 4.7300407448, cos//-cosh// (3:72) « = —rr—— sinh fx - sin ju and cp = a(sinh0.5// - sin0.5//) +(cosh0.5// - cos0.5//) (3.73) •N", and N,v are products of the Lagrange polynomials and H/ are products of the Hermitian polynomials, details of which are given in [31]. w, and v, are the nodal variables in the x and y directions respectively, y/j are the variables associated with the Hermitian polynomials at the corner nodes, namely the lateral displacements, slopes and twists, and are represented by the vector w, w w, w, w~ w~ w~ w~ w~ w, w~ w, • w, w, w, w, 1 (3-74) w yv YV yv n rv r n ,v "l •> \x-> \y> \yy> lf 2x' 2y 2xy> w3> rK3x' iy> 3xy' Ai 4x> 4y> 4xy\ where the subscript denotes differentiation with respect to that variable. Chapter 3. Numerical Formulations 44 3.5.3.2 Beam elements The 18 degree of freedom beam element in the x-direction shown in Figure 3.7 has a displacement field given by (3.75) u2 r + w4 s'mln^ + w5 sin4^ +. w ]x • ^Hte)Mz)Mz)Msl[' w Ix (3.76) w.2x where the primes denote differentiation with respect to x and, e is the eccentricity ofthe stiffeners defined as the distance between centroidal axis of the beam and the mid-plane of the plate. In those cases where torsion and lateral bending are included, the displacement fieldsfo r rotation, 6 and lateral displacement, v are given by 0, (3.77) (3,78) V = [L^),LMLM A beam element in the y-direction has an inplane displacement field,v , that is analogous to that given in Equation (3.75) for the x-direction. Chapter 3. Numerical Formulations 45 3.5.4. Strain Displacement Relations The same assumptions which were made for the development of the axisymmetric formulation are made for the rectangular formulation, namely thin plates and slender beams undergo displacements which are large compared to the plate or beam thickness but small relative to the length. The resulting strain-displacement relations for a plate are du d M> 1 dw (3.79) + — dx Z dx2 2 \dx dv 6 ~w 1 ''dw^ £•„ = —- - Z- dy dy 2\dy J du dv ' d2w dwdw 2si.. = — + — - 2z—— + • dy dx dxdy dxdy where s*, sy, and Sq are the strain components in the xy plane and is the engineering shear strain. For a beam spanning in the x or y direction, only the appropriate normal strain relation, fx, or sy„ is required. The strain displacement matrices may be derived in an similar manner as was done for the axisymmetric formulation. This results in the linear strain displacement matrix [5] for a plate given by ..; (3.80) d d_ : 0 -z- dx dx2 d • d 0 [N] dy d d d2 z dy dx dxdy Chapter 3.' Numerical Formulations 46 and the non-linear strain-displacement matrix [C] for a plate given by i C?N; 3.5.5. Computer Implementation The development of the formulation for the rectangular stiffened plate proceeds in a similar manner as that for the axisymmetric plate as described in Section 3.4. The constitutive relations, temporal integration and stiffness formulation are identical in both formulations. The mass'and damping matrices and the load vector are developed in a similar manner. The numerical integration is appropriate for accurate representation of the problem. Complete details may again be found by referring to [31]. Chapter 4 ANALYTICAL FAILURE FORMULATIONS 4.1 Introduction This chapter outlines the analytical failure models which are used in conjunction with the numerical models presented in Chapter 3 to predict the failure of stiffened or unstiffened metal plates. Although the philosophies of the failure models for the axisymmetric and rectangular finite element formulations are identical, the implementation in the regular finite element program and the super finite element program is different. Several failure models have been presented in the literature as discussed in the literature review. Each has its own limitations and no one model has been shown to.be superior at this time. Given the scope of this research and the philosophy inherent in its application, namely the preliminary design phase, failure models which require a fine discretization of the structure to ensure an accurate representation of the strain and stress fields are not suitable. The finite element discretization of the plate problems in this work has been simplified by the use of a large element grid, Bernoulli beam elements and Kirchoff plate theory. Due to these simplifying assumptions inherent in both finite element programs, the prediction of the strain field in the metal plates is not expected to be accurate at any given location, particularly in the region of the plastic hinge which typically occurs at the plate boundaries. Parameter studies to be discussed later show that the strain as predicted by the finite element formulation.is very dependent on the finite element grid representation of both the circular and rectangular plates. It was found that as the grid size was reduced, the maximum strain at the boundary in the region of the plastic hinge increased without bound. The maximum strain was found therefore to be grid dependent. In addition it was 47 Chapter 4. Analytical Failure Formulations 48 not possible to determine what the true strain should be at the hinge location. In fact, any level of strain can be achieved simply by choosing an appropriate grid spacing. The prediction of failure based on a maximum strain criteria using the finite element strains is therefore not practical and an alternate method is required. Despite the ^determinate nature of the strain distribution, previous studies [32] have shown that the prediction of the overall deformed profile of the blast loaded plates is within engineering design level accuracy. In addition, the prediction of the profile for a given blast intensity is relatively insensitive to the finite element grid discretization. Section 4.2 outlines the theoretical basis for the analytical mode II failure model. Section 4.3 presents the analytical mode HI failure model: Sections 4.4 and 4.5 then provide the details of the implementation of the analytical failure models into each of the numerical formulations, axisymmetric and rectangular respectively. 4.2 Mode II Failure Model 4.2.1 Introduction An analytical failure model has been developed to take into account the limitations described in the previous section. This model uses the deformed profile, as determined by the finite element programs, to calculate the total strain at critical locations in the plate. As discussed, the profile is less sensitive to the grid discretization than the resulting strains. This approach is similar to that developed by Jones for rigid-plastic beams. The Jones model consists of plastic hinges at the beam supports and at midspan while the remainder of the beam is modelled as rigid links. Rigid-plastic theory is used to predict the deformed profile of the beam. An analytical failure model is then used to calculate the maximum strain at the boundary. In the current work, the plate is modelled by the appropriate numerical formulation, axisyrnmetric or rectangular. The boundaries. of the stiffened plates are normally clamped to Chapter 4. Analytical Failure Formulations 49 prevent displacement in any direction. The finite element model typically represents the boundary as rigidly clamped. The use of simple one dimensional beam and two dimensional plate formulations to develop the numerical models results in the inabihty of those formulations to accurately represent a plastic hinge at the boundary. Experimental results have shown that mode TJ and mode m failure of thin metal plates subjected to blast loading typically occurs at the clamped boundaries of the plate. Figure 4.1 shows a typical experimental profile of a thin isotropic plate subjected to blast loading sufficient to cause large permanent deformation; mode I behaviour. The profile shows the presence of plastic hinges with large rotations at the boundaries. Because the finite element model is not able to model the plastic hinge region, the deformed profile near the boundary is not predicted accurately. The overall Figure 4.1: Typical centreline deformation profdes for unstiffened square plates. Chapter 4. Analytical Failure Formulations 50 deformed shape is however more closely predicted and is within engineering design level accuracy. The analytical failure model, therefore, must approximate the actual hinge rotation in order to determine the maximum strain in the plate. This approximation is based on the deformed profile of the plate as predicted by the finite element model. The details of this procedure will be explained in detail in the following sections. .. An analytical model is developed to predict the maximum strain which will occur in the plate without relying on the strains as predicted by the finite element formulation. As reported .earlier, the shear strain is neglected and the Bernouilli-Euler Hypothesis is assumed valid. That, is, normal sections remain plane, undistorted and normal to the axis of the beam in its deformed configuration. The maximum total strain at a point in a thin plate may be expressed as £„u„; = + *4 (4.1) where ea is the membrane strain and Sb is the bending strain at the point. This maximum strain is then compared to a maximum allowable strain. If the maximum has been exceeded then failure is said to have occurred at that point. Section 4.2.2 provides an outline on the formulation to evaluate the membrane strain at the boundary and section 4.2.3 provides the outline on the formulation for evaluation of the bending strain at the boundary. • 4.2.2. Membrane Strain The Failure model developed by Jones assumed that membrane strain was confined to the length of the plastic hinge. The plastic hinges, which occurred at the beam supports and at midspan, were assumed initially to have a length of twice the beam depth. When the midspan deflection ofthe beam reached a value equal to the beam depth, then the axial forces became large enough to cause Chapter 4. Analytical Failure Formulations 51 The membrane strain was calculated simply by examining the deformed shape ofthe beam, which was kinematically admissible based on the assumed hinge locations, and calculating the axial strain based on the change of length of the beam by L'-L : (4.2) £ = L where L 'is the final deformed length of the beam and L is the original undeformed length. The first method used to evaluate the axial strain in this work was based closely on the method developed by Jones. In this case the deformed shape as shown in Figure 4.1 was integrated directly to determine the deformed length Z'. The aire length of the deformed shape is given by where w'(x) is the slope of the deformed profile. From the assumption of small slopes this can be approximated by (44) L'=L+-\ {wfdx 2°Ij '"*'•' - and the axial strain may be evaluated from Equation (4.2) above to give " 2L 0 This approach has been developed for clamped-clamped beams and it must be generalised to account for axisymmetric and rectangular stiffened plates. In the most general case the boundaries of a plate may consist of flexible stiffeners. In that case there may be displacement and rotations of the supports which would effect the axial strain. Because of the formation of plastic hinges at the boundaries, the rotation of the supports does not affect the deflected shape of the plate. This will be discussed in Section 4.6 when the failure model implementation in the numerical formulation for rectangular plates is presented. Examining Equation (4.5) it can be shown that the resulting strain is equal to the average strain over the length ofthe beam. The strain at a point is given by the strain displacement relations, Chapter 4. Analytical Failure Formulations 52 Examining Equation (4.5) it can be shown that the resulting strain is equal to the average strain over the length of the beam. The strain at a point is given by the strain displacement relations, Equations (3.12) for axisymmetric plates and Equations (3.78) for rectangular plates. The strain normal to the boundary, for a beam in the x-direction, is given as £x ~ dx Zdx1 + 2\dx) The axial strain is evaluated at the mid-depth of the plate where z = 0. Therefore the axial strain at a point is given as (47) -^a+H^L\ E* ~ dx + 2Kdx) • If the average strain is sought it may be evaluated by integrating the strains over the length of the beam and dividing by the span length giving If the boundaries are rigidly clamped then the first term in the integrand drops out and what remains is identical to Equation (4.5) above. If this approach is repeated with the axisymmetric plates the same results are obtained. Studies undertaken with both the axisymmetric and rectangular plates show that the average axial strain as calculated using either Equation (3.12) or Equation (3.78) are very insensitive to the grid spacing whereas the strains taken directly from the finite element formulation increase without bound as shown in the following chapters. Therefore this analytical method of calculating the axial strain was adopted in this work. Experimental results as discussed later display an increasing section in the central region of the plates which translates as a rigid body and undergoes very little deformation. This is also duplicated in the finite element results, for both the axisymmetric and rectangular plates. This Chapter 4. Analytical Failure Formulations 53 suggests that to use the average axial strain over the entire span of the plate may be a nonconservative approximation to the strain at the boundary. The model was therefore refined to calculate the strain at the boundary based only on the average strain over the first element adjacent to the boundary. For the case where only one element was used to model an entire bay then the two results would of course be identical. Results of a parameter study, to be presented in the following chapter, of the axial strain based on the first element only compared to that from the first gauss point nearest the boundary show again that the results are much less sensitive to the grid length until the element size is less than approximately one-half the plate thickness. 4.2.3 Bending Strain The failure model developed by Jones calculated the bending strain based on the rotation of the plastic hinge at the boundary. An estimate ofthe length of the plastic hinge was also required. The.finite element model represents the clamped boundary as completely fixed against rotation and translation. Because of the simplified Bernoulli and Kirckhoff plate models, as discussed earlier, the finite element model is not able to represent a plastic hinge at the boundary as shown in Figure (4.1). In examining the profile of the finite element result and the experimental result however, it appears that the maximum slope of the finite element profile is approximately equal to the maximum slope of the experimental profile which occurs at the plastic hinge at the boundary. Therefore, in order a evaluate the bending strain at a plastic hinge at the boundary the analytical failure model assumes that the maximum slope of the deformed profile as predicted by the finite element program is equal to the rotation of the plastic hinge at the boundary. The bending strain of a thin plate may be expressed as ' hK (4.9) where h is the plate thickness and /cis the curvature defined as Chapter 4. Analytical Failure Formulations 54 0 (4.10) K — . £ where 9 is the rotation of the plastic hinge at the boundary and £ is the plastic hinge length. An approximation for the hinge length £ is also required, again because of the finite element formulations' inability to model a plastic hinge. The hinge length was assumed to be a function of the beam depth or plate thickness, h given by £ = ah . (4.11) where the value of a is input by the user. Based on a study by Nonaka[34] on the hinge length of rigid plastic beams of rectangular cross section, a value of ct=2 is recommended. This assumes that the elastic response of the beam is negligible compared to the. plastic response. The. bending strain can now be expressed as 6 (4.12) 2a • and can be evaluated for any plate once the value for 9 has been determined. 4.2.4 Extension to 2D Problems In order to study the failure of metal structures such as stiffened plates, the failure model must be extended to two dimensions. The maximum total normal strain will occur perpendicular to the panel boundaries at the midpoint of the sides of the panel. If the panel boundary consists of a stiffener then there will also be strain in the direction parallel to the boundary. Along the length of the stiffener, plastic hinges will form only at its supports and not at its midpoint. Thus, the strain parallel to the stiffener at the mid-side of the panel will have a negligible bending strain component due to absence of significant rotation. The total strain consists primarily of membrane strain. By monitoring the strains, both parallel and perpendicular to the boundaries at the mid- sides of the panels, the principal strain state of the critical areas of the panels can be monitored. The total strains will be calculated using the theory outlined in the two previous sections. At each time Chapter 4. Analytical Failure Formulations 55 step these strains may be compared to the failure criteria in principal strain space. If all the critical areas in the stiffened panel are monitored in this fashion, its resistance to mode II failure can be assessed. 4.3 Mode III Failure Model Mode III failure, as discussed earlier, is characterised by a transverse shear failure at the boundaries with little permanent deformation in the other regions of the plate. This suggests that the failure occurs at a very early time, compared to the mode I and mode U failures/before the Central region of the plate has had time to respond to the loading. The finite element formulations employed in this work are based on Bernoulli beam elements and Kirchoff plate elements. Shear displacement is therefore not a degree of freedom. The finite element program then does not calculate or monitor shear strains. Therefore mode m failure can not be predicted directly or indirectly from the shear strains. . An attempt was made to develop a model similar to that for mode n failure where the concept of a plastic rotational hinge was employed to estimate the bending strain. For the case of mode HI, Jones[21,22] introduced a sliding hinge at the beam supports. A kinematically admissible displacement fieldwa s postulated and a shear displacement at the hinge was calculated as discussed in Chapter 2. Mode m failure was said to occur when the shear displacement exceeded the beam depth. The finite element formulations are not capable of modelling a shear hinge. The boundary is represented as completely clamped and therefore no transverse displacement occurs at the boundary. As the rotation of the plastic hinge was approximated by the maximum slope of the deformed profile, an attempt to estimate the translation of the sliding hinge was made by selecting the lateral translation of the deformed plate a short distance from the boundary as the hinge Chapter 4. Analytical Failure Formulations 56 displacement. When the displacement at this location reaches the plate thickness then the beam is said to have failed in shear. The shear hinge method was implemented in the finite element formulation and tested but not found to be acceptable. Details ofthe attempts will be presented in the following chapter. An alternate method to predict mode in failure was sought. A simple procedure was employed by Ratzlaff and Kennedy[23] to predict the shear failure of a laterally loaded thin rectangular plate. In their work, a stress based failure criteria was adopted. The maximum shear stress in an elastic rectangular beam is 1.5R (4.13) T = max where R is the support reaction and A is the cross sectional area of the beam. As the central fibres yield, the stress distribution becomes more uniform. When all the fibres in the cross section reach the ultimate shear stress shear failure will occur. The shear stress is therefore assumed to be uniform across the depth of the plate and is calculated as R (4.14) •T=A. . and compared to a maximum ultimate shear stress to determine if mode HI failure has occurred. A very straight forward analytical method may be used to determine the shear stress at the supports at each time step. Equilibrium is used to sum the forces in the transverse direction, including the inertial forces. Thus, the support reactions and the shear stress may be determined. Specific details will be provided in section 4.5. Chapter 4. Analytical Failure Formulations 57 4.4 Mode II and Mode III Interaction Although mode TJ and mode HI failure have thus far been considered separately, experimental evidence indicates that there is some interaction between the two, Menkes and Opat[6] report that as the load increases, modes II and III overlap. Two simple interaction models are developed. The, first is a linear interaction model given as. . "• ' .r . "... (4.15) avg <1 and the second is a quadratic interaction model given as (4.16) < 1 rup •ult Both of these models are introduced into the axisymmetric formulation as presented in the following section. 4.5 Implementation in Axisymmetric Plates 4.5.1 Introduction The axisymmetric formulation as discussed in Section 3.4 is a one dimensional approximation of the circular plate problem. The mode II failure model used with this formulation is therefore also one dimensional. Further, if mode II or mode HI failure is predicted at the boundary, then the entire plate will have become separated from the boundary and the plate will translate away from its support; The following sections provide details oh the implementation of the failure models into the axisymmetric formulation. Sections 4.5.2 and 4.5.3 outline the mode H and mode HI models respectively, while Section 4.5.4 discusses the interaction between the two. Section 4.5.5 presents the post-failure analysis which models the response of the plate after it has been separated from its supports. . , . - Chapter 4. Analytical Failure Formulations 58 4.5.2 Mode II 4.5.2.1 Membrane strain The implementation of the model to determine the membrane strain in the axisymmetric formulation is straight forward Equation (4.8) is evaluated by integrating over the first element adjacent to the boundary using Gauss integration. A three point integration scheme is employed in accordance with the discussion in Section 3.4.8. 4.5.2.2 Bending strain The Hermitian polynomials used as the interpolation functions are smooth continuous cubic functions. In order to evaluate the bending strain, the maximum slope of the deformed profile must be calculated Equation (3.5) expresses the lateral deformation w of the axisymmetric plate. The slope of the deformed profile is then evaluated simply by taking the derivative with respect to the. global co-ordinate r giving (4.17) d w dH^) dH2(t) dH^) dl%{q) 0, ~d~r~ dr dr dr dr 0, evaluating the derivatives and expressing dw/dr as # gives 1 (4.18) The maximum slope is calculated by finding the maximum value of equation (4.18). This maximum will occur where the derivative of the equation is equal to zero. Taking the derivative of equation (4.18) and setting it equal to zero gives d6 . 1 (4.19) [(I2|-6)w, +a(-4 + 6g)0} +(6-.-12|)w2 +a(6^-2)02] = 0 dr a2 Rearranging this equation and solving for the local variable £ gives the location of the maximum slope as Chapter 4. Analytical Failure Formulations 59 ' _ 3w, + 2a<9, -3w2 + a62 (4.20) ' max 6w]+3a0l-6w2+3a02 Substituting this value of £ into Equation (4.18) provides the maximum rotation in the element. The bending strain at the boundary can then be calculated using Equation (4.12). 4.5.3 Mode III In the axisymmetric problem, the shear stress at the boundary is constant along the boundary. Equilibrium requires that the sum of all the forces in the transverse direction be equal to zero which gives R-^q(r,t))dA+^pwdV = 0 ^4'21^ where R is the total reaction at the boundary. , The integral of the pressure over the surface area of the plate is evaluated using Gauss quadrature to allow the most general loading q(r,t) to be specified. That is, q may be a function of time and radius. This integral may be expressed as f q(r,t)dA=27t\q{r,t)rdr (4-22) If the load is not a function of time or the radius then Equation (4.22) reduces to f q(r,l)dA =nr2q ' • . (423) JA The second integral in Equation (4.21) may be expressed in a form similar to the load vector as (424) i^v=phi[NpyA Using the definition of the load vector as given in Equation (3.35) this may be expressed as Chapter 4. Atmlytical Failure Formulations 60 i^fm (4-25> At each time step the integrals in Equation (4.21) are evaluated and the value of R determined. It is assumed that the support reaction is uniformly distributed around the axisymmetric boundary. Therefore, the average shear stress at the boundary is R . (4.26) . Tavg~ iTtrh where h is the plate thickness. At each time step, the shear stress is calculated and compared to the maximum ultimate shear stress [57] to determine if failure has occurred. The ultimate shear stress, however, is a function of the strain rate as was the normal stress. The Cowper-Symonds relationship is assumed to apply to. shear stress as well. This relationship, as given by Equation (3.21) is used to update the ultimate dynamic shear stress at each time step. The calculated shear stress is then compared to the ultimate dynamic shear stress. If the calculated value exceeds the ultimate value then mode III failure is said to have occurred. 4.5.4 Interaction The two interaction models as described by Equations (4.15) and (4.16) are implemented directly into the axisymmetric formulation. The maximum strain e^a, and the average shear Tavg, are evaluated at each time step and compared to their respective limits. When the interaction limit is exceeded failure is said to have occurred. 4.5.5 Post-Failure Analysis In the axisymmetric problem, the strain at the boundary is not a function of 9 and therefore the strain at any point along the boundary is equal. When failure is predicted by mode II, mode HI or interaction between the two, then failure will occur simultaneously along the entire boundary. The Chapter 4. Analytical Failure Formulations 61 plate will therefore be released from its support. In general, this separation will occur while the plate is still undergoing deformation due to the blast loading. It is unlikely that failure will be predicted at exactly the moment that the plate deformation stops. Due to the kinetic energy that the plate has received from the loading, the severed plate will continue to translate through the air as reported in the experimental results. In addition, because the distribution of kinetic energy in the plate is not uniform; the centre of the plate having higher velocity and therefore more kinetic energy then the perimeter, the plate will continue to deform as it translates. This post-failure deformation is significant as discussed in the next chapter. At each time step, the maximum normal strain and the average shear stress are calculated by the analytical failure model. When failure strain is predicted, the boundary conditions ofthe plate are modified. The node at the boundary is originally modelled as completely fixed. After failure it is modelled as completely free. Once failure has occurred the analytical failure model is no longer evaluated. The numerical analysis continues until a steady state behaviour is attained. 4.6 Implementation in Rectangular Plates. 4.6.1 Introduction The prediction of failure in a general stiffened plate is much more difficult then in the axisymmetric plate. The two dimensionality of the problem as well as the presence of flexible supports greatly complicates the analytical failure model. As a result, the failure model is not completely implemented in the finite element formulation for rectangular plates. Despite this shortcoming, the limited study of failure in the rectangular plate problem provides several insights into the behaviour of stiffened and unstiffened plates under blast loading. This information contributes to the understanding of this complex problem. Section 4.6.2 presents the implementation of the mode II model and section 4.6.3 discusses mode III failure. Chapter 4. Analytical Failure Formulations 62 4.6.2 Mode 11 4.6.2.1 Membrane strain Again the implementation of the model for calculating the membrane strain is a straight forward implementation of Equation (4.8). The presence ofthe analytical functions in the shape functions requires the use of a 5 point integration scheme in evaluating the integral. Gauss integration is employed. 4.6.2.2 Bending strain In the super element formulation for the lateral displacement of a plate element as given in Equation (3.69) the presence of hyperbolic and trigonometric functions makes the determination of the maximum slope more difficult. Taking the derivative of Equation (3.69) with respect to the variable x results in the equation for the slope dw/dx given as (4.27) dw dN) w dx dx dx [HArj),H2(Ti),HArr\HM¥> / ^ + w 4>(y) + Mw dx W 6,- -dx-* <- w x.6 The The maximum slope occurs where the derivative with respect to x of equation (4.27) is equal to zero. This gives Chapter 4. Analytical Failure Formulations 63 (4.28) wy$ r + dx w d2\H^),H {^H^lH^)\ 2 wxS • £¥(|Ln' + dx2 The resulting equation is transcendental arid it is not possible to derive an explicit equation for the location ofthe maximum slope as was done for the axisymmetric problem, therefore this equation must be solved by interation. This was accomplished in two steps. This equation is expected to have several roots due to the presence of the hyperbolic and trigonometric functions. The element is divided into 10 equal length segments and Equation (4.28) is evaluated at the endpoint of each segment. If the value of the left side ofthe equation changes sign over the length of one segment then there is at least one root in that interval. Once a segment has been identified as containing a root then the Newton- Raphson method is.used to locate the root in that segment. If Equation (4.28) is rewritten as O(x) = 0 (4.29) then the root of that equation may be found by iteration. The ith iteration ofthe Newton-Raphson method is given as O (*,_,) (4.30) 'i-l 1 o'OO where the prime signifies the derivative with respect to x. This requires the derivative of Equation (4.28) with respect to x or the third derivative of w as given by Equation (3.69). Iteration continues until the solution converges. Once the location of the maximum slope has been Chapter 4. Analytical Failure Formulations 64 determined the magnitude of that slope is determined by substituting the appropriate co-ordinate into Equation (4.27). Figure 4.2: Generalised stiffener displacement. In the general case of a panel boundary at a stiffener, the flexibility of the stiffener may allow it to rotate under the applied load as shown in Figure 4.2. The rotation of the plastic hinge which occurs at that boundary will be affected. The analytical model must therefore take the stiffener rotation into account when estimating the plastic hinge rotation. This is done simply by calculating the difference between the stiffener rotation and the maximum slope of the deformed profile. The rotation of the plastic hinge therefore is calculated as O=Osl-0max (4.31) where 9st is the stiffener rotation and 6L* is the maximum slope perpendicular to the stiffener as calculated by the preceding formulation. Finally, knowing the hinge rotation, the bending strain is evaluated by the use of Equation (4.12). Chapter 4. Analytical Failure Formulations 65 4.6.2.3 Biaxial Strain Limit The normal strain in each principal direction is calculated according to the procedures outlined earlier. Once the biaxial strain state is known, it can be compared to the forming limit curves as developed by the sheet-metal forming industry. Alternatively, because of the shape of the curves, a conservative approach would be to neglect the effect of the minor principal strain and consider only the major principal strain. The effect of the minor strain is to increase the value ofthe limit strainas shown in Figure (2.1). 4.6.3 Mode III In the rectangular plate problem the application of the maximum shear stress criteria is not straight forward. In a typical panel the shear stress is not expected to be uruform along the boundary as in the axisymmetric case. The stress is expected to vary from a very small value in the corners of the panel to a maximum at the mid-side ofthe boundaries. The use of an average value similar to the axisymmetric problem would be unconservative at the critical location; the mid-side. For the general case of a stiffened rectangular plate, the boundaries of a panel would consist of flexible stiffeners. The ability ofthe stiffeners to deflect under load would reduce the reaction at the plate boundary and consequently the shear stress would be reduced. The model developed to predict mode HI failure would require modification. Mode HI failure for rectangular plates has not been implemented in this work. Chapter 5 SQUARE PLATES 5.1 Introduction The failure model implemented in the NAPSSE code as Outlined in Section 4.6 for rectangular plates was evaluated by comparing the numerical results with the experimental results from a series of investigations involving blast loaded square plates. Many experimental and theoretical studies of the large inelastic deformation (mode I failure) of thin square plates have been reported in the literature, however little work has been reported where the load intensity is sufficient to cause complete failure of the test specimen (mode II and mode III failure). An experimental study of blast loaded square plates subjected to intensities sufficient to cause mode I and mode II failure was performed at the University of Capetown by Nurick and Olson as reported in Olson et al.[8] using an experimental'procedure similar to that reported in Nurick et al.[35]. The latter studies included only mode I failure. This chapter provides the details of the numerical and experimental investigations. Section 5.2 discusses the experimental procedure employed by Olson and Nurick at the. University of Capetown and Section 5.3 presents the numerical modelling procedure using NAPSSE. Section 5.4 presents the results of some preliminary investigations on the effects of grid spacing and membrane strain distributions. Finally, Section 5.5 presents the results of a detailed analysis of the square blast loaded plate problem. 66 Chapter 5. Square Plates 67 5.2 Experiments Cold-rolled mild steel plates, 1.6 mm in thickness were cut to make the test specimens. Two series of tests were performed, the first by Nurick et al.[35] in 1985 and the second by Olson and Nurick [8] in 1992. In both series, the specimens were clamped between two 20 mm thick steel plates which had a square opening of 89 x 89 mm in the centre over which the explosive loading was applied. In the first series, each test specimen had a circular shape with a diameter of 200 mm whereas in the second series the specimens were squares 240 x 240 mm. The 20 mm plates were clamped by eight 11 mm diameter high strength bolts on a diameter of 175 mm. This assembly was then attached to a ballistic pendulum which consisted of a short section of I-beam hung from the concrete ceiling of a blasting room by four strands of spring steel wire. The natural period of oscillation ofthe pendulum was 3.20 seconds in comparison to the duration of plastic deformation of the plate which is in the order of 200 usee, and the burn time of the explosive which is in the order of 15 psec. Thus the pendulum motion essentially takes place under no load and is the result of the inertia imparted to the system by the explosion. Since all the plastic deformation is over in essence before the pendulum has moved, the displacement of the pendulum, which is recorded with a pen attached to the rear of the pendulum, records the potential energy ofthe system after the energy used for plastic work has been dissipated The impulse imparted to the system by the explosion may thus be determined from the potential energy of the system. The application of the explosive to the specimen was carried out in a fashion very similar to that used by Nurick et al,[35]. A 12 mm polystyrene pad was attached to the specimen. The explosive was laid out on the pad in two concentric square annuli as shown in Figure 5.1 In the first series, the two anriuli were connected by a single strip of explosive along the centreline whereas in the second series the they were connected by two perpendicular strips. In both cases the detonator was placed in the centre of the plate. The combination of the use of the polystyrene pad and the annular rings of explosive was designed Chapter 5. Square Plates 68 to impart an approximately uniform pressure pulse to the specimen and to prevent spallation of the specimen. Material properties ofthe cold-rolled mild steel plates were determined by performing uniaxial tension tests at different strain rates on samples of the material which resulted in typical stress-strain curves. The Cowper-Symonds relation was then employed to compute the static yield stresses from the test results. The test specimens for the original series of tests in 1985 had a yield stress of 296 MPa and for the second series it was 273 MPa. The average rupture strain was calculated for the second series only from the uniaxial tests and was determined to be 34%. L1 =0.B4L L2=0.49L PLATE SPECIMEN EXPLOSIVE Figure 5.1: Layout of explosives for square plate tests. Chapter 5. Square Plates 69 5.3 Modelling the Experiment The clamping of the specimen by the two 20 mm thick plates was assumed to provide a rigidly clamped boundary to the 89 x 89 mm loaded area ofthe plate. By the use of symmetry, this loaded area has been represented by one quarter of the plate, modelled numerically by five different finite element grids as shown in Figure 5.2. The first,gri d A, used only one element to model the quarter plate. The next three grids each used four elements. The firsto f these, grid B, used four equal sized elements and the second and third, grids C and D, used border elements which had a width of one half and one third of the central element respectively. The final grid, grid E, used nine equal spaced elements. The border elements of grid C which had a width one half the central element were therefore the same width as the border elements of grid E. The explosive charge imparted to the specimen through the polystyrene pad was assumed to produce a uniform pressure loading with a square wave form in time and a duration of 15 us, the approximate burn time of the explosive. The numerical analysis was Table 5.1: Uniform pressure load for given impulse, (square plate) IMPULSE PRESSURE (Ns) (MPa) 5 42.1. 10 84.2 15 126 20 168 25" 210 30 252 35 294 40 337 Chapter 5. Square Plates I B c Ml I D Hi £. E U3 173 Figure 5.2: Finite element grids for preliminary studies. Chapter 5. Square Plates 71 performed for impulses from 5 to 40 Ns in increments of 5 Ns. Table 5.1 shows the uniform pressure corresponding to each impulse. The pressure is equal to the impulse divided by the loaded area of the plate (89mm x 89mm) and the duration of the pressure (15 u.s). The numerical model used the following material properties in its calculations: elastic modulus, E = 197 GPa; density, p = 7830 kg m"3; Poisson's ratio, v = 0.3; tangent modulus, Et= 250 MPa; static yield stress, c0 = 292 MPa; and rupture strain, 8rap = 0.30. This value is lower that reported from the tensile tests results in Section 5.2 to partially account for strain rate effects [36]. The value for o0 was not varied to account for the second series of tests (o0 = 273) except for a few calculations. The effect on the mode I results was not significant and the effect on the mode II results were decreased by only a few percent. Numerical convergence was checked for each grid by using various time steps. It was found that the 1 x 1, 2 x 2 and 3x3 grids required time steps of 1, 0.5 and 0.25 jas respectively. 5.4 Preliminary Studies A series of studies on the effects of the grid spacing on the predicted response of the plate was undertaken. The five different finite element grids as shown in Figure 5.2 were used to model the plate when subjected to impulses of 20, 25 and 40 Ns. In these preliminary studies, the membrane strain is evaluated by calculating the average membrane strain over the entire panel length as described in Subsection 4.2.2. The effect of varying the grid spacing on the deformed centreline profile is shown in Figure 5.3. These profiles result from an impulse of 25 Ns applied as a uniform pressure load to the plate. In all cases mode II failure is predicted. These profiles represent the deformed profile at the time that mode II failure is predicted. Apart from the first case where only one element is used to model a quarter of the plate, the results are very similar. Thus except for the one element grid, the mode II failure model appears to be grid independent, that is the Chapter 5. Square Plates 72 prediction of mode II failure profile is not dependent on the finite element grid which is chosen to model the structure. The data in Table 5.2 summarises some of the results of the preliminary investigation. For each of the impulses investigated, 20, 25 and 40 Ns, and for each of the five grids, the table presents: the midpoint displacement of the plate, the time at failure, the maximum bending strain perpendicular to the boundary at the boundary as predicted by the finite 20 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY Figure 5.3: Effect of grid size on centreline failure profiles. Chapter 5. Square Plates 73 Table 5.2: Preliminary investigation results IMPULSE FINITE MIDPOINT TIME AT BENDING STRAIN (Ns) ELEMENT DISPL. FAILURE NAPSSE MODEL GRID (mm) (usee) (max.) 'A 18.67 105 0.06 0.14 B 21.67 HI 0.13 0.17 20 C 22.16 113 0.19 0.16 D 21.64 109 0.25 0.17 E 21.56 108 . 0.20 0.16 A 16.46 75.5 0.09 0.19 B 14.20 .65 0.17 0.20 25 C 13.65 64.5 0.24 0.20 D 13.20 63 0.32 0.20 E 13.69 62 0.25 0.20 A 10.48 42.5 0.11 0.20 B 7.89 27 0.26 0.23 40 C 6.62 24 0.37 0.24 D 6.10 22.5 .0.46 0.25 E 6.69 24 0.37 0.24 element formulation and the bending strain as predicted by the failure model. In all cases mode II failure was predicted except for the result for the 20 Ns impulse on grid A where mode I failure was predicted. The displacement and strain data represent the results at the time that failure is predicted; no post failure analysis is performed. The results for the midpoint displacement reinforce the conclusion of the profiles in Figure 5.3 for the 25 Ns impulse. Namely that for a variety of impulses the deformed shape at Chapter 5. Square Plates 74 • ' Y failure as represented by the midpoint displacement is essentially independent of the finite element grid. The results from grid A, which only used one element, are significantly different from the other grids on a consistent basis as was shown by the profiles in Figure 5.3. The times at which failure occurs as presented in Table 5.2 again supports the conclusion that the failure model is independent of the finite element grid with the exception of grid A. The maximum bending strain perpendicular to the boundary as calculated directly from the finite element strains shows a clear trend in Table 5.2. As the element nearest the boundary is reduced in size monotonically fromgri d A to grid D the maximum bending strain at the boundary increases monotonically. Also the bending strain as predicted by grids C and E are almost identical. As discussed previously the boundary elements of grid C and E have the same width. The maximum bending strain predicted directly by the finite element formulation is therefore a function of the width of the boundary element. Any magnitude of strain can be obtained simply by selecting the appropriate finite element grid. It is not possible therefore to accurately determine the true bending strain from the finite element formulation when Kirchoff thin plate theory is used. A failure prediction based directly on the finite element strains is therefore not possible because of the dependence of the results on the finite element grid and the inability to determine the true values. An indirect method was developed to calculate the bending strains as described in Section 4.2. As has been shown by this preliminary study, the deformed profiles are relatively insensitive to the grid spacing. Comparisons between the experimental results and numerical results as discussed in reference to Figure 4.1 show that these profiles may be used to determine the bending strain at the plastic hinge. The bending strains as predicted by the resulting failure model are shown in Table 5.2. The resulting strains which are based on the deformed profiles are insensitive to the grid spacing. The failure model is therefore also insensitive to the finite element grid. ' Chapter 5. Square Plates 75 In view of these results, the detailed analysis of the blast loaded rectangular plates presented in the following section is based on grid B, the four element representation of the quarter plate using four equal size elements. The plots in Figures 5.4, 5.5 and 5.6 show the membrane strain resultsVrom the finite element formulation for impulses of 20, 25 and 40 Ns respectively. In general there is a significant variation of the membrane strain within and between elements. The variation within the elements may be attributed to the presence of trigonometric and hyperbolic shape functions. The large discontinuities between the elements is a function ofthe C1 continuity of the out of plane displacements and the C° continuity of the in plane displacements. A general trend can be discerned, however, for each impulse. For the 20 Ns impulse the membrane strain increases from a minimum at the boundary GRID D — GRID E J_ _|_ _L 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 5.4: Effect of grid size on centreline strain distribution. (20 Ns) Chapter 5. Square Plates 76 to a maximum at the centre of the plate. Thus the membrane strains in the region near the boundary are significantly below the average over the entire panel. For the 25 Ns impulse, the membrane strains over the region near the boundary are somewhat uniform but gradually decrease towards the centre of the plate. The average membrane strain as predicted by the failure model is approximately equal to the membrane strains from the finite element formulation in the region near the boundary. In contrast, the 40 Ns impulse shows very large strains in the elements immediately adjacent to the boundary and strains approaching zero in the other elements and the failure model strain is generally below the finite element strains in the boundary region. (0.05) 1 1 r—J 1 1 > 1 L. 1 1 1 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY j Figure 5.5: Effect of grid size on centreline strain distribution. (25 Ns) Chapter 5. Square Plates 77 Figure 5.6: Effect of grid size on centreline strain distribution. (40 Ns) There is a trend, therefore, that with increasing impulse the membrane strain is concentrated within the elements adjacent to the boundary. This will be discussed in more detail in the following section. Figure 5.7 shows this trend for grid B; four equal size elements. As the impulse is increased the membrane strain in the region near the boundary increases and the strain in the element near the centre of the plate decreases to approximately zero. It is interesting to note that the membrane strain at the boundary appears to be almost constant at about 0.25. Experimental evidence shows that mode II failure typically occurs at the plate boundaries. The membrane strain used for the failure model should therefore be based on the strain adjacent to the boundary. Considering the results of this preliminary study it is more Chapter 5. Square Plates 78 0.4 0.3 K- CO 0.2 111 CQ 0.1 LU (0.1) 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 5.7: Effect of impulse on centreline strain distribution. reasonable to calculate an average strain based only on the element adjacent to the boundary. Table 5.3 compares the average strain over the entire plate along the failure line with the average failure over the first element only. Once again since the membrane strain is directly related to the deformed shape of the plate, the membrane strain over all the elements is relatively independent of the finite element grid. When the average is taken over the first element adjacent to the boundary only, the membrane strain is slightly less independent ofthe grid spacing at the higher impulse but still within acceptable limits. Chapter 5. Square Plates Table 5.3: Comparison of membrane strains IMPULSE FINITE MEMBRANE STRAINS AT FAILURE (Ns) ELEMENT GRID ALL ELEMENTS FIRST ELEMENT A 0.11 0.11 B 0.13 0.10 20 C 0.14 0.09 D 0.13 0.08 E 0.14 0.08 A 0.11 0.11 B 0.1.0 0.14 25 C 0.10 0.14 D 0.10 . 0.15 E 0.10 0.12 A 0.11 0.11 B 0.07 0.14 40 C 0.06 0.16 D 0.05 0.19 E 0.06 0.16 Chapter 5. Square Plates 80 5.5 Results And Discussion 5.5.1. Predictions The detailed analysis of the blast loaded square isotropic plates is presented in this section. The plot in Figure 5.8 shows the time history ofthe displacement of the centre of the plate for an impulse of 15 Ns. The displacement increases linearly to a maximum value and then vibrates elastically at a very small amplitude, a behaviour which has been observed experimentally[37,38]. The small elastic vibration compared to the large permanent plastic deformation indicates that most of the kinetic energy received by the plate due to the explosive loading is dissipated by plastic work. In addition, the permanent deflection of the midpoint of the plate is a close approximation to the maximum deflection observed during the time history of the plate. The effect of the strain rate on the displacement of the midpoint is seen to be significant as shown in Figure 5.8. When strain rate effects are included, resulting in a much higher value for the yield stress, the maximum midpoint displacement is reduced from 27.6 mm to 17.8 mm and the time at which these maximums are attained is reduced from about 190 p.s to 120 LIS. In addition to the effect on the midpoint of the plate, the overall deformed profile of the plates is also affected by the strain rate as shown in Figure 5.9. At a time of 86 LIS, which corresponds to the cross shown on Figure 5.8, the midpoint displacement ofthe two profiles, one with the strain rate effect included and one without, are almost identical. The overall deformation profile is significantly different, however, with the profile resulting from the analysis which neglected the strain rate effect showing a larger maximum rotation adjacent to the boundary as well as a longer developed length. This results in a higher predicted maximum strain according to the failure model, in fact, the analysis actually predicted a mode II failure at 86 Lis when strain rate effects are neglected and the midpoint displacement was only 12 mm. In comparison, when the analysis included the strain rate effect, no failure occurred at Chapter 5. Square Plates 81 Chapter 5. Square Plates 82 all even for the maximum midpoint displacement of 17.8 mm. In comparing the permanent displacement profile for the analysis which included strain rate effects to the profile at 86 ps which failed, it can be seen that the mode I permanent profile had a much smaller rotation near the boundary than the mode II failure profile. More discussion of the transient profiles will follow. The predicted trends versus impulse for the strain rate, the time at which first yield occurred and the dynamic yield stress are shown in Figure 5.10. As the impulse increases from 5 to 40 Ns the strain rates increase from 640 to 3160 s"1 and the corresponding dynamic yield stress, as calculated by the Cowper-Symonds relation range from 800 to 900 MPa in comparison to the static yield stress of 292 MPa as reported earlier. In comparison to the load duration time of 15 u.s as determined by the explosive burn time, the time to first yield IMPULSE (Ns) Figure 5.10: Effect of impulse on yield stress, strain rate and time to first yield. Chapter 5. Square Plates 83 decreases monotonically from a value of 9.5 LIS at an impulse of 5 Ns to a value of 3.5 LIS at an impulse of 40 Ns. The profiles shown in Figure 5.11 are the transient deformation profiles of the centreline of a plate subjected to an impulse of 20 Ns. Together they show the time history of the deformation of the plate. Initially the central region translates as a rigid body as is expected from impulsive theory. Only a small region adjacent to the boundaries is plastically deformed. As the deformation continues, the central region contracts until the entire plate is plastically deformed in a characteristic mode I shape at a time of 121.5 p.s. However, the 20 Ns impulse is the first for which mode II failure is predicted. Mode II was predicted at a time of 101 LIS when the midpoint deflection had reached 20.0 mm. Thus mode II failure occurs before the plate has had the opportunity to become fully developed. H3 121.5 A/SEC (Mode I) 20 101 pSEC (Mode II) E E, 15 z 80pSEC o LU 10 I 60pSEC LL LLJ Q 40pSEC 20/JSEC 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY Figure 5.11: Predicted transient centreline deformation profiles. . Chapter 5. Square Plates 84 This trend can be seen over a wide range of impulses as shown in Figure 5.12. The centreline failure profiles for impulses of 10, 15, 20, 25, 30 and 40 Ns as predicted by the numerical analysis are shown. For the 10 and 15 Ns impulses mode I failure is predicted, whereas for the impulses of 20 Ns and larger, mode II failure is predicted. The characteristic shape of the mode I failure can be seen to increase with increasing impulse with a maximum displacement at the midpoint of the plate. Conversely, the midpoint displacement decreases with increasing impulse for the mode II failures. As explained earlier, the mode II profiles for the isotropic square plates represent the deformed shape of the plate at the time that mode II failure is predicted. No post failure analysis has been included for these plates. In reference to the discussion of Figure 5.11, when mode II failure occurs, the plate does not have the time to 25 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY Figure 5.12: Predicted final permanent centreline deformation profiles. Chapter 5. Square Plates 85 reach its fully developed characteristic mode I profile and mode II failure occurs at ever decreasing times as the impulse is increased. Thus the flat central region of the plate increases and the deformation of the plate is concentrated near the boundaries. This results in an increase in the rotation of the plate near the boundaries. These results are highlighted further in Table 5.4. As the impulse for mode II failures is increased from 20 to 40 Ns, the proportion of the total failure strain which is caused by bending increases from 58% to 76 % and the time to failure decreases from 101.0 to 26.5 us. Table 5.4: Strains at mode II failure IMPULSE (Ns) 20' 25 30 35 40 BENDING STRAIN (%) 58 68 75 76 76 MEMBRANE STRAIN (%) 42 32 25 24 24 FAILURE TIME (uSEC) 101 58 38 30.5 26.5 Figure 5.13 plots the total strain distribution perpendicular to the boundary as a function of impulse. For each impulse which causes the failure strain of 0.30 to be reached at the centre of the boundary, the distribution along the boundary is plotted. For the impulse of 20 Ns the strain varies from 0 at the corner to a maximum of 0.30 at the centre. As the impulse is increased, the failure strain of 0.30 is reached simultaneously over an increasing length of the boundary. The local maximum shown near the boundary for the 40 Ns curve has been ignored. This has been interpreted as a numerical modelling error due to the coarse finite element grid's inability to model the highly localised plastic zone. If this error is ignored, then failure would occur over a region which is approximately 70% of the length of the boundary. Chapter 5. Square Plates 86 0.35 DISTANCE FROM BOUNDARY Figure 5.13: Strain perpendicular to boundary at failure. 5;5.2. Comparisons The mode I centreline profile resulting from an impulse of 15 Ns, as predicted from the numerical analysis, is compared to that measured experimentally in Figure 5.14. The general shape is very similar to that obtained from the circular plate as discussed in the previous chapter. The predicted and measured profiles again differ at the boundary, where the plastic hinge is not modelled by the finite element model. The overall profile and the maximum displacement at the midpoint of the plate, as predicted by the numerical model, do closely approximate the experimentally measured results. Once again, the maximum slope of the predicted deformed profile adjacent to the boundary compares very favourably to the actual rotation of the plastic hinge as measured experimentally. Chapter 5. Square Plates 87 20 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY Figure 5.14: Comparison of experimental and predicted centreline profiles. The times at which the model permanent deformation was predicted to occur ranged from 110 to 120 jis. This agrees with other theoretical results[7] but underestimates the experimentally measured times which ranged from 130 to 175 \xs. This discrepancy has not been explained. Figure 5.15 compares the measured and predicted results for the permanent midpoint displacement ofthe square isotropic plates as a function of impulse. The mode I experimental data are from two sources[8,35]. The general trend ofthe mode I prediction curve agrees very well with that of the experimental data. The permanent midpoint displacement as predicted by the numerical model does slightly overpredict the experimental results but in reference to Figure 5.14, it can be seen that the plate deflection is underpredicted for the large majority of the area of the plate and over predicted for a small central area only. Chapter 5. Square Plates 88 Figure 5.15: Midpoint deflection versus impulse. The asterisks represent plate specimens which were completely separated at the boundary and the other solid symbols represent plates for which the specimens were only partially torn. The number attached to the solid symbols indicates the number of torn sides. The mode II curve represents the deflection at the midpoint of the plate when failure according to the maximum strain criteria is first predicted and neglects any post failure deformation. Additional calculations show that the threshold impulse for mode II failure as predicted by the failure model occurs at 19.2 Ns. Chapter 5. Square Plates 89 The mode I data and the partial mode II data show considerable overlap It is thought that small asymmetries in the loading or geometry may cause premature tearing on the side with the highest loading. A similar result was also observed in the axisymmetric plates ofthe previous chapter, but it is much more significant in the square isotropic plates. As discussed in relation to Figure 5.13, the length of the side which reached the failure strain increases rapidly with increasing impulse. The geometry of the square plates may cause it to be more sensitive to nonsymmetric parameters. This results in the impulses causing partially torn specimens falling below the mode II failure prediction and resulting in nonconservative predictions of failure. In contrast, the mode II prediction does provide a lower bound for the impulse which causes the plates to be completely torn out. Once the impulses increase beyond the threshold impulse for mode II failure the experimental data shows a rapid decrease in the permanent deformation with increasing impulse which is mirrored by the numerical prediction as shown by the mode II curve. The prediction of the threshold impulse for mode II failure is very sensitive to the value chosen for the rupture strain. It is not clear what value should be chosen for the high strain rates encountered in these experiments, Some literature suggests that a value as low as 0.20 could be used in comparison to the value of 0.30 used in this analysis as discussed earlier[36], The maximum total strain which occurred for the impulse of 15 Ns was 0.204, therefore choosing a more conservative value for the rupture strain would lower the threshold impulse to under 15 Ns. Chapter 6 CIRCULAR PLATES 6.1 Introduction The failure model as implemented in axisymmetric plates as outlined in Section 4.5, was used to model an experimental investigation similar to that reported in Chapter 5 on square plates and reported by Teeling-Smith and Nurick[7]. This testing program used the same ballistic pendulum and explosive loading technique to- investigate the mode I, II and III failure of clamped circular plates. This chapter presents the details of the numerical and experimental investigations for circular plates. Section 6.2 presents the results of the testing of the finite element formulation for the axisymmetric plates as derived in Section 3.4. Section 6.3 discusses the aspects of the experimental procedure employed by Teeling-Smith and Nurick which were distinct from that employed for the square plates. Section 6.4 presents the numerical modelling procedure using NAAPFE and Section 6.5 presents the results of a detailed analysis of the circular blast loaded plate problem 6.2 Testing of Finite Element Formulation 6.2.1. Introduction In order to verify the axisymmetric finite element formulation as implemented in the NAAPFE code, the numerical results from NAAPFE were compared to known analytical results available in the literature. [39] The comparisons were based on the following: a plate radius 90 Chapter 6. Circular Plates 91 and thickness of, R = 100 mm and h = 10 mm respectively; Poisson's ratio, v = 0.3 and the elastic modulus, E = 197 x 109 N/m2. The results of that comparison are reported in this section. Subsection 6.2.2 compares the static linear elastic results, Subsection 6.2.3 the eigenvalues and Subsection 6.2.4 dynamic elastic results. Subsection 6.2.5 compares the results of the static problem with linear material and non-linear geometry. 6.2.2. Static, Linear Geometry and Elastic Material The equation for the linear elastic lateral deflection, w of a simply supported axisymmetric plate subjected to uniform loading is given by (5 + v) 2(3 + v) (6.1) w(r)= q D4 64D 1 + v 1+v where R is the plate radius, v is Poisson's ratio, q is the uniform pressure, r is the distance from the centre ofthe plate and D is the plate stiffness given by [}= Eh" (6-2) 12(1-v2) The strain energy for the linear axisymmetric plate is given by Equation (3.19). Substituting Equation (6.1) into Equation (3.19) and integrating over the surface area of the plate gives the strain energy for a simply supported axisymmetric plate as 2 4 u = xq R 7+v, (6.3) . ~ 384D 1 + v and the maximum displacement at the centre of the plate is given by qR4 5+v (64) w = ——: max s- A T-\ t 64D 1 + v For the case where the plate has clamped boundaries, the linear elastic displacement is given by Chapter 6. Circular Plates 92 (6.5) w(r) = 4 2R2r2+rA 64D Substituting Equation (6.5) into Equation (3.19) gives the strain energy for the axisymmetric clamped plate as 7cq2R6/ (6.6) U = 384£> and the maximum displacement as (6.7) w •• = max . 64D These analytical solutions are compared to'the numerical results of NAAPFE in Table 6.1. Table 6.1: Comparison of exact static results with NAAPFE Number of Wmax (mm) U(Nm) Elements 1 3.589 252.8424 SIMPLY 2 3.535 254.5501 SUPPORTED 4 3.531 254,6498 10 . ,3.531 254:6562 EXACT :- 3.531 254.6563 1 0.9239 43.5358 :'• 2 0.8668 45:0800 CLAMPED 4 ' , ' 0.8664 ' 45.3432 10 0.8661 45.3496 EXACT 0.866.1 45.3497 Chapter 6. Circular Plates 93 Results are tabulated for different finite element grids of 1, 2, 4 and 10 equal size elements as well as the exact solutions. The results show excellent strain energy convergence from below to the exact solutions. As the number of elements is increased, the model becomes more flexible and the strain energy increases. The average numerical convergence rate is a406, where a is the element length, which compares well to the theoretical rate of a4 as stated in Chapter 3. A comparison of the bending moments calculated numerically by NAAPFE and those calculated analytically was also completed. For the axisymmetric plate, the radial and tangential bending moments per unit length, Mr and Mt respectively, for a linear elastic material and geometry are given as 1'd2w vd\A (6-8) M, = -D \dr r dr j • f\dw d\A (6-9) Mt=-D —— + v 1 V r dr dr Using the equations ofthe lateral displacement w, for the simply supported plate as given in Equation (6.1) the moments given in Equations (6.8) and (6.9) are expressed as 2 (610) M,=^(3+v)(^-, ) :• . 16 . M, = -^[R2(3 + v)-r2(\ + 3v)] (6ll) Table 6.2 summarises the results of the comparisons of the analytical and numerical calculations. The analytical results are simply the solution of Equations (6.10) and (6.11) for the sample problem introduced in Subsection 6.2.1 and the numerical results are the NAAPFE evaluations of Equations (6.8) and (6.9) for the same sample problem. The results show excellent agreement and conversion as the finite element grid is refined from one to ten elements. . Chapter 6. Circular Plates 94 Table 6.2: Variation of element forces with grid type: simply supported NUMBER ELEMENT Mr (kNm/m) . M, (kNm/m) OF ELEMENTS NUMBER NAAPFE EXACT NAAPFE. EXACT ONE 1 280.0 309.4 350.0 353.1 TWO 1 . 379.0 386.7 396.5 397.6 2 173.0 180.5 276.9 278.9 : i 404.1 406.0 408.5. 408.8 FOUR 2. • 352.6 354.5 378.6 379.1 .3 249.4 251.4 319.2 . 319.7 4 . 94.7 96.7 230.1 230.7 . 1 411.2 .411.5 411.9 •411.9 .2 . 402.9 403.2 407.1 407.2 . 3 386.4 386.7 397.6 397.6 4 361.7. 362.0 383.3 383.4 TEN 5 328.7 329.0 364.3 364.4 6 287.4 287.7 . 340.6 340.6 7. 237.9 238.2 312.1 312.2 8 180.2 180.5 278.8 278.9 - 9 114.2 114.5 240.8 240.9 10 39.9 ' 40.2 198.1 198.2 Chapter 6. Circular Plates • ' 95 A similar comparison, of the clamped axisymmetric plate was made by substituting the lateral displacement w>, for the clamped plate given by Equation (6.5) into Equations (6.8) and (6.9) resulting in the expressions a \m„2 , . ,\ „2,2, , .Al (6-12) Mr = -^[R (l + v)-r (3+vj\ (6.13) M, = -^[R2 (1 + v) - r2 (1 + 3v)] Table 6.3 compares the analytical and numerical solutions for the clamped axisymmetric plate . based on these equations. Once again the numerical solutions converge very well to the exact solutions. 6.2.3. Eigenvalues , A comparison of the eigenvalues of the same sample axisymmetric plate was also undertaken. The fundamental frequencies of vibration for the elastic axisymmetric plate are given by -,v. (6.14) Eh3 '2 2 2TTR- \2Y(\-v ) where E, R, and v are the elastic modulus, plate radius and Poisson's ratio respectively, y is the mass per unit area and-X is a parameter which reflects the boundary conditions. Table 6.4 compares the firstfou r fundamental frequencies from the numerical and analytical results for the simply supported and clamped plates. These values also show excellent convergence from above as the.number of elements is increased. As expected, with fewer elements the model is stiffer than the actual structure and therefore the frequencies as calculated numerically are higher than the exact solutions. When the number of elements is increased, the model becomes more flexible and the calculated frequencies approach the exact value. Chapter 6. Circular Plates 96 Table 6.3: Variation of element forces with grid type: clamped NUMBER ELEMENT Mr (lcNm/m) Mt(kNm/m) .. OF ELEMENTS NUMBER NAAPFE EXACT NAAPFE EXACT ONE 1 30.0 59.4 100.0 103.1 TWO 1 129.0 136.7 146.5 147.6 2 :• • -77.0 -69.5 26.9 28.9 1 154.1 156.0 158.5 158.8 FOUR • 2 • 102.6. 104.5 128.6 129.1 •. 3 -0.56 1.37 69.2 69.7 4. -155.6 -153.3 -19.9 -19.3 1 161.2 161.5 161.9 161.9 • 2. ' 152.9 153:3 157.1 157.2 3 136.4 136.7 147.6 147.6 4 : 111.7 112.0 133.3 133.4 TEN'' • • 78.7 79.0 114.3 , 114.4 6 37.4 37.7 90.6 90.6 7 -12.1 -11.8 62.1 62.2 8 -69.8 -69.5 28.8 28.9 9 -135,8 -135.5 -9.2 -9.1 10 -210.1 -209.8 ' -51.9 -51.8 Chapter 6. Circular Plates 97 Table 6.4 Eigenvalues BOUNDARY SOURCE Number of foo foi fo2 f03 CONDITION elements 1 2476 - ' - - SIMPLY NAAPFE 2 2470 9723 27816 32140 SUPPORTED 4 2468 9625 21771 39065 10 2468 9608 21533 38260 EXACT 2469 9608 21525 38218 10.22 39.77 89.10 158.2 1 1194 . 9556 - - CLAMPED NAAPFE 2 1192 7287 20707 45463 4 1192 7187 18059 34606 10 1192 7180 17919 33445 EXACT 1202 7189 17925 33410 Xjj2 4.977 29.76 74.20 138.3 6.2.4. Dynamic Elastic A comparison of the maximum displacements of the sample plate under a static load and a dynamic step load ofthe same magnitude was undertaken for the simply supported and clamped boundary conditions. The results shown in Table 6.5 show a ratio of the dynamic to static displacement of approximately 2 as expected. An exact value of two would only be expected for a single degree of freedom system. Chapter 6. Circular Plates 98 Table 6.5: Dynamic step load BOUNDARY Number of STATIC DYNAMIC RATIO CONDITION elements (mm) (mm) •1 3.589 7.230 . 2.01 SIMPLY 2 3.535 7.178 2.03 SUPPORTED 4 ;. 3.531 7.185 2.04 10 3.531 7.185 2.04 • 1 0.9239 1.846 2.00 CLAMPED 2 - 0.8668 1.8.14 2.09 4 . 0.8664 1.810 2.09 10 0.8661 1.817 2.10 6.2.5. Static, Non-linear Geometry, Linear Material Analytical solutions are also available for the case where the lateral deflections are no longer small and non-linear geometric effects are introduced. An approximate solution for the maximum deflection derived by Timoshenko[39] is available for both the simply supported and clamped axisymmetric plates and given as ' ,qR4 (615) h \h) Eh4 where wa is the centre deflection, q is the uniform load, and R, h, and £ are again the radius, thickness and elastic modulus. A and B are coefficients such that for a simply supported plate A = 1.852 and B = 0.696 while for a clamped plate A = 0.471 and B = 0.171. The same sample plate is used for this comparison except that the uniform load q, is increased to 200 x 10" N/m . to ensure that non-linear geometry is attained. A more exact solution also by Timoshenko is available for the clamped case. Comparison of the numerical and analytical results is shown in Table 6.6 Chapter 6. Circular Plates 99 Table 6.6: Non-linear geometry - linear material BOUNDARY SOURCE Number of w0 CONDITION elements (mm) 1 14.33 SIMPLY NAAPFE 2 . 13.89 SUPPORTED 4 13.88 10 13.88 EQUATION (6.15) 14.48 1 10:22 CLAMPED .• NAAPFE 2 10.58 4 10.61 10 10.61 EQUATION (6.15) 11.03 EXACT 10.57 The comparisons show that for both the simply supported and clamped cases the numerical results converge to a centre deflection which is approximately 4% below the approximate analytical solution given by Equation (6.15). That is, the numerical solution is slightly stiffer than the approximate solution. The numerical solution for the clamped plate is however very close to the more exact solution. 6.3 Experiments A series of experiments was performed by Teeling-Smith and Nurick[7] on blast loaded fully clamped circular mild steel plates. The range of impulses applied to the plates was sufficient to cause mode I, II and III failures. The procedure was very similar to that reported in the Chapter 6. Circular Plates 100 previous chapter on square plates. The test specimens were once again attached to a ballistic pendulum and the explosive applied to a neoprene pad in two concentric rings joined by a single cross leader. The detonator was again placed in the centre of the plate and fixedint o place with a small cube of explosive. The explosive layout is shown in Figure 6.1. The 200 mm square specimens were cut from 1.6mm thick cold rolled mild steel plates and clamped between two 20 mm thick plates with eight 12 mm high strength bolts. The loaded area of the plates where the explosive was placed was a circular area, 100 mm in diameter. The material properties of the specimens was determined in a manner similar to that reported for the square plates in Chapter 5, using a series of quasi-static tensile tests. The 1.6 mm test plates were cut from two different sheets of mild steel for which the average static yield stress was 264 MPa and 277 MPa. Figure 6.1: Layout of explosives for circular plate tests. Chapter 6. Circular Plates 101 6.4 Modelling the Experiment The circular plates were modelled using the NAAPFE code. As discussed earlier, this finite element code uses axisymmetry to model the plate as a one-dimensional problem. A uniform grid of 10 elements was used to represent the plate and the boundary was assumed to be rigidly clamped. The material properties used in the model were as follows: elastic modulus, 3 E= 197 GPa; tangent modulus, Et= 250 MPa; density, p = 7830 kg m"; static yield stress, cry= 270 MPa; ultimate shear stress, xmax = 270 MPa; Poisson's ratio, v = 0.3; and failure strain, sm3X = 0.41. Several analysis by others, most notably that of Jones has used rigid plastic analysis methods to analyse mode I failure of plates. They have assumed that the explosive loading could be assumed to be impulsive. That is, the pressure-time history of the loading need not be known in detail to predict the final deformed shape of the plate but only the total area under the pressure-time history is required. The results from NAAPFE, to be discussed in detail in the following section, indicate that when mode I response occurs, (i.e. large permanent deflection without fracture) the Figure 6.2: Models of pressure loading Chapter 6. Circular Plates 102 deformation is independent ofthe time-pressure history and is a function of the impulse only, as expected. However, this is not the case for the mode II and mode III responses The explosive load was therefore modelled in three different ways as shown in Figure 6.2. All three models assumed that the loading was uniformly distributed over the entire area of the plate and did not vary spatially. The first model assumed a square wave distribution in time whereas the second and third assumed an isosceles triangle variation in time. The first two assumed a load duration of 15 usee; equal to the approximate burn time of the explosive. Bodner and Symonds [40,41] state that the polystyrene buffer pad served to lengthen the pressure pulse acting on the plate and reduce the pressure magnitude, therefore the third loading was assumed to have a duration of two times the explosive burn time or 30 Lisec. The three models are referred to as SI5, T15 and T30 respectively as noted in Figure 6.2. It should be noted that for any given impulse, the peak pressure for the S15 and T30 load models are identical. As discussed in. Section 6.3, the detonator was placed in the centre of the plate. When the detonator fires, the reaction travels along the strips of explosive at a speed of 6500 to 7500 ms"1. and the explosive material is completely burned after approximately 15Ltsec. The burning of the explosive is therefore not uniform or axisymmetric at an instantaneous moment during the short loading phase, although the presence of the polystyrene pad serves to spread out the loading. Several numerical investigations were carried out which attempted to model this behaviour by the use of an axisymmetric pressure wave which travelled out from the centre of the plate. . 6.5 Results and Discussion 6.5.1 Predictions A typical time history plot for the displacement of the centre of the plate subjected to an impulse producing a mode I response is shown in Figure 6.3. The displacement exhibits an approximately linear increase to a maximum value followed by small elastic vibrations in the order of less than one plate thickness similar to the results for the square plates as discussed in Chapter 6. Circular Plates 103 Chapter 5. The peak displacement occurs at a time of between about 130 psec for the two pressure pulses with a 15 usee duration to about 140 psec for the pressure pulse with a 30 usee duration. Thus the load duration in both cases is much shorter than the response time of the plate which supports the use of impulsive theory for mode I analysis. At the end of the loading phase the deflection is only in the order of two deflection thicknesses. The time to first yield of the mild steel plates was in the order of 3 psec which is well before the end of the loading phase and very early in the response history of the plate. Also shown in Figure 6.3 is a time history of the midpoint displacement of the plate when the strain rate effect is not included: As expected, the increased yield stress ofthe strain sensitive plate results in a decreased permanent displacement similar to that reported in the previous chapter for the square plates. The reduction in permanent displacement is of the order of 40%. The effect of strain rate on the material properties and response of the plate is again significant. It is interesting to note that the response ofthe rate sensitive and rate insensitive models are virtually identical until the peak displacement of the rate sensitive plate is reached at a time of about 130 usee, despite the fact that the maximum yield stress is reached at a much earlier time when the strain rate effect is neglected. Table 6.7 shows, as a function ofthe applied impulse, the magnitude of the residual elastic vibration of the plate when plastic deformation has ceased and the steady state response has been reached. The three pressure loads produced similar results and only the results for the square (SI5) pressure load are presented. As the impulse increases from zero to 25 Ns the residual elastic vibration of the plate decreases. This implies that as the impulse increases a larger proportion of the kinetic energy imparted to the plate is dissipated into plastic work. For very low impulses below 5.0 Ns a large portion of the plate remains elastic and energy is available for elastic vibration of the plate. Above an impulse of 25 Ns, mode II or III failure occurs and, as will be discussed shortly, failure occurs at an earlier time in the deformation history of the plate. Consequently less plastic deformation has occurred and more energy is available for elastic vibration. As the impulse is increased above 25 Ns the residual elastic vibration increases. Chapter 6. Circular Plates 104 50 STRAIN RATE IGNORED 40 E E z O h- 30 o STRAIN RATE INCLUDED LU _l LL LU Q 20 _l h- Z LU o 10 100 200 300 400 500 TIME(^SEC) Figure 6.3: Time history of midpoint displacement. The transient deformation profiles of a plate subjected to an impulse of 20.0 Ns are shown in Figure 6.4. Early in the response history of the plate, the uniform pressure loading causes the plate to translate as a rigid body except for the region immediately adjacent to the clamped boundary. As time increases, a decreasing proportion of the central region of the plate continues to translate as a rigid body while the large deformation region near the boundary forms an increasing proportion of the plate. Chapter 6. Circular Plates 105 Table 6.7: Residual elastic vibration IMPULSE VIBRATION FAILURE AMPLITUDE MODE (Ns) (mm) 2.5 • 4.00 I 5.0 1.57 I 7.5 1.10 I 10.0 0.84 I 12.5 0.65 I 15.0 0.52 I 17.5 0.42 I 20 0.36 I 22.5 0.33 I 25.0 0.30 I 27.5 1.40 . II 30.0 1.34 II 32.5 1.41 II 35.0 1.44 II 37.5 1.78 III 40.0 2.04 III 42.5 2.33 . III 45.0 2.48 III 47.5 2.63 III 50.0 2.53 III Chapter 6. Circular Plates 106 It is interesting to note that as the plate profile evolves, the deformation in the region near the boundary remains constant. Only the central region ofthe plate is in motion. For the 20 Ns impulse for example, after 50 Lisec the deformation of the region within the normalised distance of 0.2 from the boundary remains constant. This corresponds to the first two elements of the finite element grid. The deformation ofthe first element remains constant after a time of approximately 30 jisec. The failure model as discussed earlier, calculated an average membrane strain based on the first element only. Therefore the membrane strain will be approximately constant after a time of about 30 p.sec when the deformation of the first element becomes constant. The 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY Figure 6.4: Predicted transient centreline deformation profiles. Chapter 6. Circular Plates • ' . 107 bending strain was determined by the maximum slope of the plate near the boundary which also is determined at a very early time in the response history ofthe plate at approximately 50 psec. The result then is that the maximum total strain for a given impulse on a plate is reached at a very early time. Fig. 6.5 shows a plot of the transient history of the strain at the gauss point nearest the boundary. This plot shows that the maximum radial strain is reached at a time of about 50 usee as expected from examining the profiles. A plot of the strain at all the gauss points, when a plate subjected to an impulse of 15.0 Ns is at its maximum displacement, is shown in Figure 6.6. The vertical dashed lines represent the boundaries of the ten elements. The figure clearly shows the discontinuity of the strains across the element boundaries and the variation of the strains within the elements. It is evident that the strain at any particular gauss point cannot accurately represent the actual strain in the plate. The four gauss points through the depth of the plate are represented by the four different symbols in the figure. The solid symbols represent the exterior gauss points while the square and circular symbols represent the loaded and unloaded faces respectively. The gauss points closest to the boundary show a large variation in strains through the thickness due to the large bending strains, with tension on the loaded face and compression on the unloaded face. The large strain gradient drops off quickly and in the second element all four gauss points through the thickness have essentially the same strain. The bending strain is therefore concentrated in the first element and the strain in the second element is due to uniform membrane strain only. Thus, as the length of the first element is reduced, the bending deformation is concentrated over a shorter distance and consequently the bending strain increases. The average membrane strain over the first element is about 4%. In Figure 6.6 the magnitude of the membrane strain increases as the distance from the boundary increases. This is similar to the results reported for the square plates at impulses which produced mode I failures. Chapter 6. Circular Plates 108 0.25 TOTAL STRAIN 0.2 0.15 0.1 0.05 20 40 60 80 100 120 TIME(^SEC) Figure 6.5: Time history of strains The effect of the length of the finite elements near the boundary on the behaviour of the plate was investigated in detail. A plate subjected to an impulse of 15.0 Ns was modelled with several different finite element grids. The element nearest the boundary was subdivided into an increasing number of elements of uniform length while the other nine elements were not altered. The length of the element nearest the boundary was therefore varied from a length to thickness ratio of 6.25 to 0.26. The peak deflection from these several grids varied from 19.50 mm to 19.74 mm and the time to reach the peak varied from 130.0 to 133.5 usee. Thus the overall deformation of the plates was independent of the grid size over the range studied. Chapter 6. Circular Plates 109 The maximum strain as predicted by the finite element formulation, however, is very dependent on the grid size. Figure 6.7 plots the strain at the gauss point nearest the boundary on the loaded surface of the plate versus the element length. This gauss point is the one closest to the location where tensile tearing of the plate is expected to occur. The maximum strain over the area of the plate occurs at this gauss point. The plot shows that as the element length decreases, the strain increases dramatically. It is apparent that the strain at the gauss 0.25 LOADED FACE • 0.2 - o CO UNLOADED FACE r- o 0.15 Q_ CO CO < 0.1 • 0.05 o • •!!!!!!! < 3- t -0.05 -0.1 _L 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 6.6: Distribution of finite element radial strains. Chapter 6. Circular Plates 110 points in the region of concentrated bending near the boundary can never be grid independent and hence cannot be used for failure prediction. The analytical model presented in Chapter 4 is an attempt to solve this problem. Also plotted in Figure 6.7 is the maximum tensile strain as predicted by the analytical failure model. These results are almost independent of the grid size and therefore indicate that the analytical failure model is much more acceptable for failure prediction. The use of very small elements near the boundary would again result in a dramatic increase in the predicted strain and should be avoided. The membrane strain is obtained by taking the average over the length of the first element near the boundary. Therefore, even though the maximum strain at 1.4 i 1 Q I I I I I I I I I I I I I I 0 1 2 3 4 5 6 7 ELEMENT LENGTH/THICKNESS RATIO Figure 6.7: Total radial strains as a function of element size. Chapter 6. Circular Plates Ill the gauss point near the boundary is very large this a very local effect which is smoothed out by taking the average. The bending strain as predicted by the failure model is a function ofthe maximum slope of the plate. Because the grid size does not significantly affect the overall deformed shape of the plate, the bending strain is also independent of the grid size Figure 6.8 shows a plot ofthe strain ratio {em^lsmp) and the shear stress ratio (T /TU1I ) as given in Equations (4.15) and (4.16) versus time for the square pressure load (S15) at an impulse of 17.5 Ns. This impulse is lower than the critical impulse required to cause mode II failure for all three load models. This plot is a typical representation of the transient value of these ratios at impulses which cause mode I deformations. The strain ratio increases monotonically to a peak value then decreases slightly to a lower value and then oscillates about that value. The peak value occurs at approximately 50 usee, which is much earlier then the time to peak midpoint displacement of approximately 130 |isec. Therefore although the deformation ofthe plate continues to increase after a time of 50 irsec* the strain remains constant. This is a direct result of the transient deformation history of the plate and the boundary element in particular as discussed previously. The typical mode I stress ratio plotted in Figure 6.8 shows a more irregular magnitude versus time. The ratio rises very steeply to a maximum at 15 itsec and drops sharply at the end of the loading phase. Subsequently, it rises again to another lower maximum as the inertia forces become significant. Thereafter, it oscillates irregularly about the zero position. During the loading phase, the stress is a function of both the pressure load and the inertia! load resulting from the acceleration of the plate. After the pressure drops to zero, the stress is a function ofthe inertia forces ,only. As the plate continues to deform because of the momentum it has received after the pressure loading has been removed, its kinetic energy is converted into plastic work. Chapter 6. Circular Plates 112 The plots in Figure 6.9 show the corresponding temporal variation of the two failure model functions. Each failure model shows a similar variation. That is, an initial sharp peak occurs at about 15 u.sec due to the high early shear stress ratio and this is followed by a second higher peak at a time of approximately 40 Ltsec due to an approximately equal contribution from the strain and stress ratios. These times are much less than the time to peak displacement of about 130 usee as noted earlier. The relative sizes of the two peaks in the failure models change as a function of impulse and result in distinctly different responses as will be discussed shortly. STRAIN RATIO -0.3 200 400 TIMEOL/SEC) Figure 6.8: Time history of strain and stress ratios. Chapter 6. Circular Plates 113 LINEAR INTERACTION FAILURE MODEL 0.1 200 400 TIMEOL/SEC) Figure 6.9: Time history of interaction models. As the impulse is increased from zero the plate continues to respond in mode I failure. until a critical impulse is reached after which the plate responds in a mode II failure. This critical impulse is defined as the mode II threshold impulse. The mode II threshold predicted by NAAPFE also appears to be relatively insensitive to the pressure time history of the loading pulse. The linear interaction model for failure predicts a mode II threshold of 20 Ns for the three representations of the pressure pulse: The permanent deflection of the midpoint of the plate at this failure threshold ranges from 16.7 mm to 16.9 mm. The quadratic interaction model predicts a threshold of 27.5 Ns for the S15 and T15 load models and a Chapter 6. Circular Plates 114 threshold of 25 Ns for the T30 model. The corresponding permanent deflections range from 20.3 mm to 20.6 mm. At the mode II threshold, which occurs when the second peak as shown in Figure 6.9 reaches a value of 1, it was observed that both the strain and stress ratio contributions were approximately equal. For the linear interaction model the ratios are approximately equal to 0.5 for all three load representations, whereas for the quadratic interaction model they are approximately equal to 0.70. This indicates that even at the threshold of mode II failure which has been characterised as a tensile failure by several experimentalists, the effect of shear is significant. This is consistent with the recent theoretical predictions of Shen and Jones [26], When xmax was set to a very large value, to eliminate the effect of shear stress on mode II failure as was the case for the square plates, the mode II threshold occurred at an impulse of 32.5 Ns: As the impulse is increased further, the first peak increases in height and the failure function in Figure 6.9 reaches a value of 1 at a very early time The failure is then dominated by the shear ratio and mode III failure is said to have occurred. At the mode II threshold impulse, the failure time ranges from 24 to 38 usee. This compares to the'time for the peak displacement to be reached of 130 to 140 u,sec as discussed earlier. The predicted times to failure are shown plotted versus impulse for the three different pressure load models and the two failure models in Figures 6.10(a) and (b) As the impulse increases above the mode II threshold, the failure time decreases in a monotonia fashion. These plots can be subdivided into two regions; at lower impulses the slope ofthe plot is steeper than at higher impulses. A transition impulse can be identified on each curve which clearly separates the two regions. This impulse is labelled as the threshold for mode III response. Chapter 6. Circular Plates Figure 6.10: Failure time versus impulse. Chapter 6. Circular Plates 116 It is obvious that whereas the mode II threshold was insensitive to the load history the mode III threshold is not. For the-linear interaction model, the mode III threshold varies from 27.5 Ns to 32.5 Ns and for the quadratic interaction it varies from 32.5 Ns to 42.5 Ns. The mode III threshold generally occurs at a time less than 15 psec, the approximate burn time of the explosive. Therefore, failure occurs before the full impulse has been transmitted to the plate. The modelling of the pressure load on the plate therefore becomes more important than the magnitude of the impulse. Any pressure remaining on the plate after failure, if it is uniformly distributed, will only cause rigid body motion of the plate: Transient deflection profiles of plates subjected to impulses of 30 Ns and 40 Ns are shown in Figure 6.11 and Figure 6.12 respectively. These profiles result from the S15 pressure load and the quadratic interaction failure model. As noted earlier, the 30 Ns impulse results in a mode II failure whereas the 40 Ns impulse results in a mode III failure. The first solid line shown in each figure is the profile at failure, which occurs at times of 22.8 and 10.6 usee respectively for the 30 and 40 Ns impulses. ; Figure 6.11 shows that when failure occurs at an early time of 22.8 usee the midpoint displacement of the plate is approximately 4 mm which corresponds to 2.5 plate thicknesses. At this time the plate separates from its boundary and travels through the air. The deformation is seen to continue as a result of stored strain and kinetic energies. The net transverse deformation increases until it reaches a peak at about 120 (isec. The plate then vibrates elastically as it continues to travel at a constant velocity. After failure, the edge of the plate also pulls away from the plate in a radial direction towards the centre of the plate. The maximum normalised distance is 0.1 which corresponds to a real distance of about 5.0 mm or 3 plate thicknesses. The transient profiles ofthe plate subjected to an impulse of 40 Ns as shown in Figure 6.12 are similar to those just discussed in Figure 6.11. At failure, which occurs at an earlier time of 10.6 usee, the deflection of the plate is almost constant along the radius except for a small region adjacent to the boundary. This deflection is equal to 1.8 mm which is approximately one plate thickness. The maximum permanent net deflection is 14.3 mm. The Chapter 6. Circular Plates 117 pulling in of the plate away from the boundary is also evident at this impulse. The maximum distance is approximately 2.5 mm. Comparison of Figure 6.11 and Figure 6.12 shows that the higher impulse produces a flatter permanent profile. The edge of the plate subjected to an impulse of 30 Ns shows a larger permanent rotation then that of the plate subjected to an impulse of 40 Ns. The higher rotation suggests a failure due to tensile tearing or mode II failure at an impulse of 30 Ns, whereas the lower rotation for the 40 Ns impulse suggest a failure where shear is more prominent as in a mode III failure. The determination of the mode III threshold based on the 40 30 E E, z o h 20 - o LU _l LL LU Q 10 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 6.11: Predicted transient centreline deformation profile. (30 Ns) Chapter 6. Circular Plates 118 60 Figure 6.12: Predicted transient centreline deformation profile. (40 Ns) failure profiles is, however, subjective. It is interesting to compare the two profiles at the time of failure for each impulse. The plate subjected to an impulse of 30 Ns fails at a later time allowing the plate to undergo a larger deformation. The profiles are very similar in that they are very flat except for the region adjacent to the boundary. However, because the central region ofthe plate subjected to a 30 Ns impulse has received greater stored strain and kinetic energies at the time of failure than the plate subjected to an impulse of 40 Ns, the final deformed profile of the 30 Ns plate has a much larger net deflection. - Chapter 6. Circular Plates 119 25 Figure 6.13: Midpoint deflection versus impulse. A plot of the deflection versus impulse for the mode II and mode III responses is shown in Figure 6.13 for the quadratic failure model. The upper curves represent the final permanent net midpoint deformations of the severed plate for the various loading models whereas the lower curves represent the net deformation at the time of failure when the plate has just separated from its support. It is evident that the comparisons made between the profiles of the 30 Ns and 40 Ns impulses can be generalised. As the impulse increases, separation of the plate occurs at an earlier time and the kinetic energy distribution in the plate Chapter 6. Circular Plates 120 is more uniform. The increase in net deflection after failure is approximately 15 mm at an impulse of 25 Ns and decreases to a range of between 4 mm to 10 mm for the 50 Ns impulse. Once again this plot shows a sharp transition between the mode II results and the mode III results at the same transition impulses as the earlier plots. In particular the T15 load shows a very steep drop at an impulse of 32.5 Ns and the permanent deflection at impulses greater than 32.5 Ns is only slightly larger than the deflection at failure for the other loading models. In addition, above an impulse of about 42 Ns, the deflection at failure of the SI 5 load is only very slightly larger than that ofthe T30 load and yet the permanent deformation of the square pressure load is significantly larger than that of the triangular load. These results clearly demonstrate the importance of the modelling of the pressure loading for the mode III responses. This contrasts sharply with the results from impulses of 30 Ns or lower which are independent of the modelling of the pressure load as discussed earlier. It is interesting to compare the results of the S15 load with the T30 load! These two models have the same peak pressure for a given impulse. Below an impulse of 30 Ns the failure times are longer then the load duration times for both load models. The failure is therefore a function of the total' impulse and the deformation at failure and the permanent deformations are almost identical. At impulses of 42.5 Ns or higher, the failure time for the triangular loads are approximately 15 usee which is the time at which the peak pressure is reached. Therefore failure occurs at approximately the same pressure and again the failure deformations and permanent deformations are similar. Contrast this with the T15 load which has twice the peak pressure. Failure occurs at an earlier time and the net permanent deflection is much less. The pressure is therefore an important determinant of mode III failure. Finally Figure 6.13 shows the general trend that increasing the impulse above the mode II threshold results in a decrease in the final permanent deformation ofthe plate. Figures 6.14(a) and (b) are plots ofthe stress and strain ratios as a function of impulse for the linear and the quadratic failure models, respectively. As the impulse increases the shear stress ratio increases and the strain ratio decreases. These plots are characterised by an Chapter 6. Circular Plates 111 approximately level plateau at lower impulses, a sharp rise or drop at a transition impulse and another level plateau at higher impulses. For each failure model and pressure load model the transition impulse which marks the sharp changes in value of the ratios is identical to the corresponding transition impulse identified previously in the plots of failure time versus impulse. At impulses above the transition impulse for each case, the shear stress ratio for the linear model is approximately equal to 0.9 and for the quadratic interaction model it is approximately equal to 1.0. Therefore this transition impulse marks the impulse above which the shear stress dominates the failure. This impulse is then defined as the mode III threshold. 1 15 20 25 30 35 40 45 50 55 IMPULSE (Ns) Figure 6.14(a): Stress and strain ratio versus impulse.(Linear model) Chapter 6. Circular Plates 122 25 30 35 40 45 50 55 IMPULSE (Ns) Figure 6.14(b): Stress and strain ratio versus impulse.(Quadratic model) 6.5.2. Comparisons The predicted variation of the kinetic energy with impulse is shown in Figure 6.15 along with the experimental data. The kinetic energy tabulated there is the total, steady state kinetic energy of the severed plate as it travels through the air after all plastic deformation of Chapter 6. Circular Plates 123 12 * 1 I l / $ / t / 10 LINEAR FAILURE MODEL t 1 QUADRATIC FAILURE MODEL 1/ / / EXPERIMENT * / , / / * / / / / / / — / / / / / / * / / * / - // * / / / / 6 —- / / / / * / / / -'A / / / * / / / *^ _ t . 4 - i / / s^T / / _ / / / * / * / . / / * * * * * * i I i I i I i 1 D 10 20 30 40 50 Impulse (Ns) Figure 6.15: Kinetic energy of the severed plate versus impulse. the plate has occurred and only the residual elastic vibration remains. The plotted results are from the S15 load model and both failure models. The results from the other load models were very similar. The predictions show an approximately linear increase in residual kinetic energy with impulse and this is clearly confirmed by the experimental results. The predicted Chapter 6. Circular Plates 124 values are, however significantly higher than those determined experimentally. Part of the discrepancy may be due to the neglect, in the failure: model, of the energy lost in the tearing process. In addition, there is some evidence that some of the experimental kinetic energy is dissipated in a rotational mode which is not accounted for analytically. The plots shown in Figure 6.16(a) to (e) show a comparison of the final permanent profile of an experimental plate and the numerical results of NAAPFE for a variety of impulses. Figure 6.16(a) shows the profiles of a plate subjected to an impulse of 13.3 Ns. This impulse produces a mode I response both numerically and experimentally. The three models of pressure loading produce identical profiles which again confirms that mode I response can be considered a function of the impulse rather than the maximum pressure. The numerical and the experimental profiles at the boundary are in close agreement exceptat the very edge where the finite element program assumes a clamped boundary and zero slope. The experimental profile, however, shows a finite rotation angle at that location which is interpreted as a plastic hinge. As discussed earlier, the simple plate theory formulation of the finite element modelling is unable to represent the plastic hinge. The slope of the experimental profile and that of NAAPFE do however agree very well at the boundary. Therefore, the approximation used to determine the bending strain at the boundary is again well justified, as it. was for the. square plates. The rotation ofthe plastic hinge is approximately equal to the maximum slope of the NAAPFE profile. The midpoint deflection of the plate as predicted by NAAPFE, however, overestimates the experimental result. This apparent discrepancy is discussed further below. Figure 6.16(b) compares the experimental and numerical results at an impulse of 26.2 Ns. At this impulse, the experimental result is a partial (asymmetric) mode II failure. Because the model is axisymmetric it cannot predict a partial failure. The numerical analysis indicates that this impulse is just above the mode II threshold and the profiles shown are those of completely severed plates. Only the net transverse displacement of the plates is plotted here because experimental measurements of the in-plane displacement are not available. However, Chapter 6. Circular Plates 125 the experimental photographs in [7] show qualitatively that the radial displacement at the edge of the plates, which was discussed in reference to Figures 6.11 and 6.12, does occur. Figure 6.16(c) compares the profiles at an impulse of 30.3 Ns. Both the experimental and numerical results at this impulse indicate a mode II failure and the profiles are the net transverse deflections of the severed plate. The profiles at this impulse are in close agreement. Again the effect of the modelling of the load can be seen to be small. Figure 6.16(d) shows the results at an impulse of 39.6 Ns. This impulse is approximately equal to the mode III threshold. The effect of the modelling of the pressure load has now become very significant. The profiles produced by the S15 and T30 load models < appear to provide a good bracket to the experimental profile. Each discrete symbol represents the midpoint displacement of a separate experimental result with the corresponding measured impulse shown in the legend. This shows the variation in experimental results for impulses which are very similar in comparison to the variation in displacement due to the different load models. The variation in experimental results is of the same order as the variation due to the load modelling except for the T15 model which had twice the peak pressure. The final plot showing deformation profiles is shown in Figure 6.16(e) for an impulse of 49.6 Ns. This impulse produces a definite mode III failure response. In this case both the SI5 and T30 load models are slightly above the experimental profile but again within the range of variation of the other experimental midpoint displacements. Chapter 6. Circular Plates 126 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 6.16(a): Comparison of experimental and predicted centreline profiles. (13.3 Ns) Chapter 6. Circular Plates 111 Figure 6.16(b): Comparison of experimental and predicted centreline profiles. (26.2 Ns) Chapter 6. Circular Plates 128 30 DISTANCE FROM BOUNDARY Figure 6.16(c): Comparison of experimental and predicted centreline profiles. (30.3 Ns) Chapter 6. Circular Plates 129 30 EXPERIMENTS 25 •X- 37.75NS O 39.38NS E 39.56NS E A 40.48nS 20 • 41.37nS O h- o LU _J 0) LU 15 - S15 Q I- Z LU < 10 / / CC LU D_ T15 0.2 0.4 0.6 0.8 DISTANCE FROM BOUNDARY Figure 6.16(d): Comparison of experimental and predicted centreline profiles. (39.6 Ns) Chapter 6. Circular Plates 130 Figure 6.16(e): Comparison of experimental and predicted centreline profiles. (49.6 Ns) Chapter 6. Circular Plates 131 Figures 6.17(a) and (b) plot the permanent net midpoint displacement versus impulse for the experiment and each failure model. The experimental results show an increasing deflection for an increasing impulse for mode I behaviour, a maximum deflection at the mode II failure threshold and a decreasing deflection with impulse for impulses greater than the mode II threshold. There does not appear to a distinct mode III threshold in the experimental results. This was also noted by Menkes and Opat[6] in their study on aluminium beams. TESTS OOLD 0 10 20 30 40 50 60 IMPULSE (Ns) Figure 6.17(a): Midpoint deflection versus impulse. (Linear model) Chapter 6. Circular Plates 132 TESTS OOLD 0 10 20 30 40 50 60 IMPULSE (Ns) Figure 6.17(b): Midpoint deflection versus impulse. (Quadratic model) The numerical results show the same general trend, the major difference being the large drop in deflection at the mode II threshold as described earlier. The quadratic interaction failure model appears to correlate more closely overall to the experimental values than the linear interaction model. Comparison ofthe mode I results show that the numerical predictions overestimate the midspan deflection of the results from Teeling-smith and Nurick as discussed in reference to Chapter 6. Circular Plates 133 Figure 6.16(a). Also shown in Figures 6.17(a) and (b) are the results from earlier work by Nurick. These results are from very similar tests performed on different mild steel specimens. The yield stress for the earlier work was higher than those of the Teeling-smith and Nurick results and yet the deflection was also higher contrary to expectations. If the numerical analysis were repeated with this higher yield stress, the experimental and numerical results for that series of tests would be in excellent agreement. However, other work by Nurick [42] has shown the repeatability of mode I deflection with a 90 percent confidence falls within a bandwidth of +/- one plate thickness. As a result, the experimental profiles in Figure 6.16(a) could be larger and that in Figure 6.16(b) could be lower. The experimental data in Figure 6.13 suggests that the deflection decreases uniformly from the value at the mode II threshold. The numerical data as discussed previously can be characterised by two plateaux joined by a transition region. The overall trend however is in agreement with the experimental results. The quadratic interaction failure model again gives results which are generally in good agreement with, the experimental values with the S15 and T30 load models. The T15 load model appears to be too severe. The prediction of the mode II threshold as given by the quadratic interaction model is excellent whereas that given by the linear interaction model is very conservative. Overall, the quadratic failure model gives the best correlation with the experimental results. Chapter 7 STIFFENED SQUARE PLATES 7.1 Introduction The final series of tests consisted of a limited study of blast loaded stiffened square plates carried out in conjunction with this study as reported by Olson et al.[43]. The experiment was identical to that for the isotropic square plates as discussed in Chapter 5, except that the specimens were stiffened plates, fully built-in to their clamped boundaries and machined from solid steel stock.. The specimens consisted of a thin uniform plate with an integral eccentric stiffener of various depths. This chapter discusses the experimental and numerical results and compares them. The loading is such that the impulses are large enough to initiate tearing of the plate. Once again, for this square plate problem no post-failure analysis is included. 7.2 Experiments In the experiments described in the Chapter 5, the plate specimens were clamped between two large clamping blocks which had 89 x 89 mm openings and the assembly was then attached to the ballistic pendulum. In the series of test with the stiffened plates, the clamping blocks were not required as the test specimens were machined from solid steel stock so that the plate and stiffener were integral with each other and the support boundary. The 200 x 200 mm specimens were flame cut from hot rolled steel plates which had a thickness of 12 mm. A flat surface for the machining process was then obtained by grinding one face of the plate 1mm. An 89 x 89 mm section was milled into the central part of the plate with a numerically controlled milling machine leaving a 1.6 mm thick plate with a central 134 ' Chapter 7. Stiffened Square Plates 135 stiffener 3 mm wide. The 6 mm diameter cutter was not able to produce a square joint at the boundary corners or at the stiffener ends and therefore a 3 mm radius curve remained at those locations. This process resulted in a stiffener located on one side of the plate only and a built- in boundary which was asymmetric with regard to the plate thickness as shown in Figure 7.1. Ail the stiffeners were 3 mm wide but had depths of either 9, 5, 4, or 2 mm. Some isotropic plates without a stiffener were also machined. Once each plate had been machined, it was then stress relieved by heat treatment. It was placed in an oven and heated to 650 °C for three hours and then slowly cooled. As was done for the square isotropic plates, the material properties for the stiffened plates were obtained from standard uniaxial tensile tests. In this case, because the heat treating process was completed in three separate batches of specimens, three sets of tensile tests were required. The average yield stress varied from 260 to 302 MPa and the average fracture,strai n 43 43 o o o o CM CO o O si n.6 Figure 7.1: Stiffened plate specimen. Chapter 7. Stiffened Square Plates 136 Table 7.1: Material properties for stiffened plate specimens. Specimen Number Average Static Yield Stress Average Fracture Strain (MPa) (%) 01 - 20 • 260 17.2 21-31 260 18.9 32-44 302, 18.1 varied from 17.2 to 18.9 %. as shown in Table 7.1. Once these fully built-in specimens were attached to the ballistic pendulum, the explosive was attached to the plate in the identical manner as for the square isotropic plates using the 12 mm polystyrene pad and the concentric annuli of explosive as shown in Figure 5.1. 7.3 Modelling the Experiment The experiment was modelled in a method similar to that used for the isotropic square plates. The boundaries of the 89 x 89 mm machined region were assumed to be rigidly clamped including the ends of the stiffener. Only two different finite element grids were used, the 2x2 grid and the 3 x 3 grid. Symmetry was again employed so that only one-quarter of the plate was represented by each grid. A uniformly distributed pressure pulse with a square wave form in time with a duration of 15 LIS was assumed. The material properties for this series of calculations were as follows: elastic modulus, E = 197 GPa; density, p= 7830 kg/m3; Poisson's ratio, v. .= 0.3; tangent modulus, Et = 250 MPa; static yield stress, c0 = 265 MPa; and rupture strain Srup -0.18. The third set of test specimens had a static yield stress, a0 •= 302 MPa and therefore some calculations were Chapter 7. Stiffened Square Plates 137 redone using that value. The true plate thickness and stiffener dimensions varied somewhat due to variations in the machining process including tool sharpness. However a nominal plate thickness of 1.6 mm and a nominal stiffener width of 3 mm was used for all the calculations. A study of the numerical convergence using a variety of time steps was performed and a time step of 0:25 us was found to be adequate for both grids. 7.4 Results and Discussion 7.4.1 Predictions For each stiffener depth an analysis was performed for impulses which ranged from 5 Ns to 30 Ns in increments of 5 Ns. The results ofthe analysis had many similarities with those reported earlier for the isotropic plates. The time history response of the deformation was again linear in the initial stage until a maximum was reached artd then a small amount of residual elastic vibration occurs. The presence of the stiffener causes the residual vibration to be even smaller in magnitude then for the isotropic plates thus the maximum displacement is again considered to the same as the permanent displacement. The significant effect of the strain rate was once again observed. A detailed study of the results of the plates with 3x4 mm stiffeners was performed. Figure 7.2 plots the strain rate, dynamic yield stress and the time of occurrence for first yield versus the impulse as predicted by the numerical analysis. The location of first yield was always at the top and bottom Gauss points closest to the corner of the boundary. As the impulse increased from 5 to 30 Ns the strain rates increased from 850 to 3000 per second and the dynamic yield stresses increased from 750 to 890 MPa. In addition the time required for first yield to occur decreased from 7 to 2.8 us, much shorter than the load duration time of 15 us. Chapter 7. Stiffened Square Plates 138 0 5 10 15 20 25 30 IMPULSE (Ns) Figure 7.2: Effect of impulse on yield stress, strain rate and time to first yield. The profiles shown in Figure 7.3 represent transient deflection profiles for a plate with a 3 x 4 mm stiffener which has been subjected to an impulse of 15 Ns. Initially the maximum displacement occurs at a location between the fixed boundary and the stiffener. This supports the predictions based on impulsive theory that the plate will respond faster than the stiffener in the initial stages. This can be compared to the results from the isotropic plates in Chapter 5 where the profiles were shown to be very flat over the central region of the plate. Figure 7.3 shows that mode II failure is predicted at a time of 62.25 us when the displacement of the centre of the plate at the midpoint ofthe stiffener is 7.5 mm. The maximum displacement is 8.1 mm at a point located approximately halfway between the stiffener and the fixed boundary. If the mode II failure is ignored and the calculations allowed to continue, Figure 7.3 shows the mode I failure or the maximum displacement occurs much later at a time of 130 ps and has a magnitude of 16.1 mm. Chapter 7. Stiffened Square Plates 139 20 . 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY TO STIFFENER CENTRE Figure 7.3: Predicted transient centreline deformation profiles. The profiles shown in Figure 7.4 show the maximum displacement profiles for all the stiffened plates at an impulse of 10 Ns. The effect of increasing stiffener depth can be seen; as the depth increases, the central maximum displacement decreases monotonically a significant amount whereas the rotation near the fixed boundary may increase or decrease a slightly. For the 9 mm stiffener the maximum displacement occurs in the plate between the stiffener and the fixed boundary. For the smaller stiffeners the maximum displacement occurs at the centre of the stiffener. For those impulses that produced a mode I response, the time at which the maximum displacement was reached varied from 115 to 135 \xs for impulses from 5 Ns to mode II initiation. The variation for the 3x9 mm stiffened plate was larger than for the other plates. Chapter 7. Stiffened Square Plates 140 14 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY TO STIFFENER CENTRE Figure 7.4: Predicted final permanent centreline deformation profiles. The profiles in Figure 7.5 are the mode II failure profiles as predicted for the plate with the 3 x 4 mm stiffener when subjected to impulses of 14, 15 20 and 25 Ns. All the initial failures for this plate occurred in the plate at the fixed boundary. At this location the slope of the plate was very high and as a result the bending strain as calculated by the plastic hinge-line model was high as well. The slopes at the stiffener were generally much smaller and the total strain at that location did not therefore govern. Once again, as was the case for the isotropic plates, as the impulse increases for a given stiffener depth, the failure strain is reached simultaneously along an increasing length of the boundary and the times at which the mode II impulse was predicted decreases. Mode II failure times for the 14, 15 20 and 25 Ns impulses were 78.8, 62.2, 33 and 25.2 us respectively. Chapter 7. Stiffened Square Plates 141 16 0 0.2 0.4 0.6 0.8 1 DISTANCE FROM BOUNDARY TO STIFFENER CENTRE Figure 7.5: Predicted failure profile versus impulse. (3x4 mm stiffener) The results of the numerical analysis at failure shown in Table 7.2 are interesting. The threshold impulse at which mode II failure occurs and the magnitude of the bending strain and the membrane strain appear to be independent of the stiffener depth. This is in contrast to the time at which first failure occurs and the mid-point displacement at failure, which both decrease continuously as the stiffener depth is increased, as discussed previously. Table 7.2: Predicted mode II failure results, (ao = 265 MPa) Stiffener Mode II Mid-point Failure Time Strain Size Impulse Displacement (mm) (Ns) (mm) (LLS) Sb 3x0 13.6 11.7 90.2 0.050 0.130 3x2 14.0 10.8 88.0 0.051 0.129 3x4 14.0 8.6 78.7 0.049 0.131 3x5 14.0 8.0 77.2 0.049 0.131 3x9 13.9 6.0 73.7 0.051 0.129 Chapter 7. Stiffened Square Plates 142 7.4.2 Comparisons Figure 7.6 compares the mode I profiles as predicted by the numerical analysis to those determined experimentally. An impulse of 10 Ns was employed for each of the different stiffener depths. The experimental data was obtained for measured impulses which were nominally 10 Ns; the actual measured impulses were 9.44, 10.05, 10.05 and 9.67 Ns for the stiffener depths of 2, 4, 5 and 9 mm respectively. The experimental profiles display abrupt changes or kinks which are results of the digitising process used to measure the profiles and not actually. present in the final deformed shape of the plate and therefore they should be ignored. The experimental result for the plate with the 3 x 9 mm stiffener showed a very small fracture in the region of the plate immediately adjacent to the stiffener whereas the numerical analysis did not predict failure at any location of the plate specimen. As was apparent with the axisymmetric and isotropic square plates, the profiles of the experimental plates show a concentrated plastic deformation occurring at the fixed boundary which is not captured by the numerical analysis due to the clamped boundary conditions. The maximum slope of the deformed profile of the plate as predicted by the numerical analysis does however compare well with the maximum slope of the experimental plate at the boundary which again suggests that the use of the maximum slope to calculate the bending strain at the boundary is acceptable. Aside from the boundary region, the general predicted shape of the deformed plates compare favourably with the experimental profiles. The experimental profiles have a flat region in the centre of the plate adjacent to the stiffener. The smaller stiffeners do not show this result. Instead, deflection of the profiles continue to increase and reaches a maximum at the stiffener. The 3x9 mm stiffener does have a flat region with the maximum deflection attained in the plate at a normalised distance of 0.75 from the boundary. More experimental data is required to determine if the trend towards the maximum displacement in the stiffener is a valid prediction. Chapter 7. Stiffened Square Plates 12 DISTANCE FROM BOUNDARY TO STIFFENER CENTRE Figure 7.6: Comparison of experimental and predicted mode I profiles at 10 Ns. Chapter 7. Stiffened Square Plates 144 The series of plots shown in Figure 7.7 show the permanent midpoint displacement as a function of impulse for both the experimental and numerical results. The plots include both the mode I and mode II results, however the experimental results for the mode II results only include partial failure. Since the numerical analysis does not include post failure analysis the mode II results only show the deformation at the time that first failure occurs. As discussed earlier, the plate would continue to deform after failure occurs because of the inertia of the plate. A distinction has been made in the plots of the experimental data between plates which had a tearing at the boundary (mode IIB) and those which had tearing adjacent to the stiffener (mode IIS). Once again the mode I numerical results for the midpoint displacement are slightly larger than the experimental results. Figure 7.7(a): Permanent midpoint deflection versus impulse. (3x0 mm stiffener) Chapter 7. Stiffened Square Plates 20 EXPERIMENTS MODE I O MODE MB • 35 IMPULSE (Ns) Figure 7.7(b): Permanent midpoint deflection versus impulse. (3x2 mm stiffener) EXPERIMENTS MODE I O MODE MB • 0 5 10 15 20 25 30 IMPULSE (Ns) Figure 7.7(c): Permanent midpoint deflection versus impulse. (3x4 mm stiffener) Chapter 7. Stiffened Square Plates 20 EXPERIMENTS MODE I O - MODEMS * i" — / z o / g H c 0 / Ul - ut 10 01 Q H- Z 0 a / 1 5 MODE II /MODE I / I I I I I 10 15 20 25 30 35 IMPULSE (Ns) Figure 7.7(d): Permanent midpoint deflection versus impulse. (3x5 mm stiffener) 20 EXPERIMENTS MODE 1 O MODE IIS * E15 - E, z o I- o LU / LL10 LU Q 1- Z o CL Q //MODE 1 / 1 1 I I I I 10 15 20 25 30 35 IMPULSE (Ns) Figure 7.7(e): Permanent midpoint deflection versus impulse. (3x9 mm stiffener) Chapter 7. Stiffened Square Plates 147 The mode I curves show a changing shape as the stiffener size increases. For the smaller stiffeners the mode I curves are convex, that is, as the impulse increases the slope of the curve decreases, whereas for the larger stiffeners, the mode I curve is concave, as the impulses increases the slope decreases. The mode II failure as predicted by the numerical analysis always occurred at the fixed boundary where the bending strain caused by the rotation of the plastic hinge at the boundary was large. The bending strain at the stiffener was much lower and the total strain at that location'was, in no occasion, large enough to cause failure. The mode II failures observed experimentally for the plates with the 2 and 4 mm deep stiffeners did occur at the fixed boundary as predicted. The mode II failures for the plates with the 5 and 9 mm stiffeners however, occurred at the stiffeners and not at the fixed boundary. This failure seems to have occurred with very little rotation of the plate and for the plates with the 9 mm stiffeners, the failures occurred at impulses and deformations much lower than predicted numerically. This suggests that these failures may have been dominated by shear effects which have not been included in this study. It has been assumed that the pressure distribution caused by the explosive layout was uniformly distributed over the surface area of the plate. In fact, the detonator and the perpendicular cross leaders combined to produce a concentrated load at the centre of the plate at an early time in the load history of the plate. This high localised loading may combine with the very stiff deeper stiffeners to cause a local shear failure. In addition, the experimental data shows that the plate with the 9 mm deep stiffeners had an average plate thickness of 1.52 mm compared to the nominal average of 1.6 mm. The limited data of this experiment does suggest that the failure model is capable of predicting the mode II Threshold for stiffened plates. More data is required to fully explore this problem. Chapter 8 CONCLUSIONS An approximate analysis procedure has been developed for blast loaded circular and square stiffened and unstiffened plates with loads sufficient to cause mode I, II and III failure. The procedure requires a reduced set of input data enabling it to be useful for the preliminary design of plate structures. The numerical formulation developed for the axisymmetric plates reduced the two dimensional plate problem to a one dimensional beam problem. It employed von Karman theory to model the large displacements. A bi-linear stress strain relation was employed with the vonMises yield criterion and associated flow rule to model the material non-linearities. Virtual work was used to derive the finite element equations. Numerical integration was carried out by Gauss integration and temporal integration was carried out by the Newmark Beta method with Newton-Raphson iteration within each time step. The numerical formulation for the square stiffened and unstiffened plates used super finite elements which employ continuous analytical functions as well as polynomial functions for its displacement fields. The other aspects ofthe formulation were similar to that described previously for the axisymmetric plates. The analytical failure formulation was developed to predict mode II and III failure. Mode II failure due to tensile tearing of the plates at the supports, for both the axisymmetric and square plates, was evaluated by a maximum strain.criterion. Because of the simplifying assumptions of the finite element formulation and the large grid size, which both serve to simplify the failure analysis, the strains from the finite element formulation are not sufficiently accurate at the local level to be used for failure prediction. In fact, the strains are a function of 148 Chapter 8. Conclusions 149 the grid size. The overall deformed shape of the blast loaded plates do however show excellent agreement with the experimental results Consequently the analytical failure model employs the deformed profile to calculate the tensile strain in the plates. The total strain is comprised of membrane and bending strain. The membrane strain is calculated by calculating the average membrane strain over the first element adjacent to the plate boundary. The resulting strain value is relatively grid independent. The bending strain is approximated by considering the maximum slope of the plate in the region of the plate boundary. The bending strain is a function of the curvature of the plate or the rotation of the plastic hinge. The maximum slope of the deformed profile is used to approximate the rotation of the plastic hinge. Mode III failure of the axisymmetric plates is evaluated by considering equilibrium of the plate at each time step ofthe analysis. The pressure loading, if it is still present, and the inertia! force of the plate are evaluated and the reaction force at the boundary is calculated. The shear stress at each time step is then evaluated and compared to the allowable shear stress to determine if failure has occurred. Linear and quadratic interaction relationships between mode II and mode III failure were developed for the axisymmetric plates. The quadratic interaction relationship yielded results which were closer to the experimental data. The linear interaction relationship, which is often employed in engineering design codes, was more conservative in the prediction ofthe mode II threshold impulse and could be employed in the preliminary design phase. Comparisons of the results for the isotropic square plates with experimental data shows very good agreement for mode I loading, with the deformation slightly underpredicted for most of the plate area and slightly overpredicted at the centre. The prediction of the mode II threshold impulse falls within the range of experimental values. The effects of shear on failure is not included for the square plates. The prediction of the mode II threshold is expected to be more conservative with the inclusion of shear effects, however, for a general Chapter 8. Conclusions 150 stiffened plate structure, the flexibility ofthe stiffeners is expected to reduce the effect of shear at the supports as discussed by Duffey [27]. The analysis ofthe circular plates shows an overall trend which is in agreement with the experimental data. Once again the model deformation is slightly overpredicted at the centre of the plate. The prediction ofthe mode II threshold is excellent. The experimental data shows a fairly uniform decrease in the deflection of the centre of the plate whereas the numerical analysis is characterised by two plateaux joined by a transition region. The correct qualitative behaviour is however obtained. That is, at the mode II threshold, the failure is mainly due to tensile tearing and as the impulse increases failure becomes dominated by shear, or mode III failure. Some predictions were made as a result ofthe analysis. The first prediction was that as the impulse increases, the time to failure decreases, until at high impulses, the failure of the plate occurs before the end of the loading phase. No experimental datais available to confirm this result. Another prediction relates to the final deformed shape of the severed plates. The numerical and experimental results agree quite closely. However, the numerical data shows that at failure the deflection ofthe plate is quite small but that the post failure deformation of the plate as it travels through the air is significant, with the result that the final deformed shape of the analysis and the experimental data agree. Once again no experimental data is available for comparison. •A very limited study of the failure of stiffened square plates was also undertaken. Only mode I and mode II analysis was included. Once again the mode I numerical analysis slightly overpredicted the central deflection. The mode II threshold prediction is good except for the larger stiffeners where localised shear effects due to the concentrated load at the detonator location causes a premature mode III type failure. Chapter 8. Conclusions 151 For each plate type; axisymmetric, square isotropic or stiffened, the Overall prediction of the impulse at which failure due to material separation occurs is in the range of experimental values and generally conservative.. The procedure developed in this work has potential for preliminary engineering design-analysis of metal structures. Additional testing on large scale metal panels is required to confirm that the proceedure may be extended to include full size structures. In the future, the model may be extended to include mode III failure and postfailure analysis for square isotropic and stiffened plates. This study has highlighted the need for more investigations into the effects of high strain rates on the material properties such as fracture strain and ultimate shear stress. More detailed analysis of blast loaded plates may be hampered by this lack of information. BIBLIOGRAPHY [I] Jones, N., "A Literature Review on the Dynamic Plastic Response of Structures," Shock Vibr. Dig., Vol. 7, No 8, 1975, pp. 89-105. [2] Jones, N., "Recent Progress in the Dynamic Plastic Behaviour of Structures," Shock Vibr. Dig., Part 1, Vol. 10, No. 9, 1978, pp. 21-33. [3] Jones, N., "Recent Progress in the Dynamic Plastic Behaviour of Structures," Shock Vibr. Dig., Part 2, Vol. 10, No. 10, 1978, pp.13-19. [4] Jones, N., "Recent Progress in the Dynamic Plastic Behaviour of Structures," Shock Vibr. Dig., Part 3, Vol. 13, No. 10, 1981, pp. 3-16. [5] Jones, N., "Recent Progress in the Dynamic Plastic Behaviour of Structures," Shock Vibr. Dig., Part 4, Vol. 17, No. 2, 1985, pp. 35-47. 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