ON THE ANALYSIS OF SHELL STRUCTURES
SUBJECTED TO A BLAST ENVIRONMENT:
A FINITE ELEMENT APPROACH
(PART 1)
by
2 ME/1986
Glenn R. Heppler
TECHNISCHE HOGESCHOOL DELFT LUCHTVAART· EN RUIMTEVAARTIECHN1EIC
BIBLIOTHEEK Kluyverweg 1 - DELFT
•
ON THE ANALYSIS OF SHELL STRUCTURES SUBJECTED TO A BLAST ENVIRONMENT:
Submitted May 1985
A FINITE ELEMENT APPROACH (PART 1)
by
-Abstract
The development of a finite element technique designed specifically for the analysis of blast loaded (chemical explosions) or combined
blast and thermally loaded (nuclear explosions) circular cylinders is outlined. Following a brief introduction the formulation of the stiffness matrix is presented which includes discussions of the material and geometric
nonlinearities included in the finite element model. The consistent force vector calculation is illustrated in the next chapter including techniques used to deal with thermal loading and blast wave loading. SUbsequently examples and results are presented which verify the accuracy and versatility of the static formulation of the problem. To lead into the dynamic analysis the consistent mass matrix development is presented along with that for a companion diagonal mass matrix which exhibits dynamically correct behaviour. The requirements of temporal integration schemes are investigated and the necessary and sufficient conditions for a linear single step integration method to be stable when operating on a system of motion equations in which the damping matrix is nonproportional are presented. Finally, examples of dynamic analyses, including a cylinder loaded by a nuclear burst, using the methods outlined herein are presented.
'
.
Acknowledgments
The author would like to thank and acknowledge Professor J. S. Hansen for his initial faith and his continuing encouragement, patience, support and friendship during the course of this work. Dr. J. J. Gottlieb
generously made himself available to clarify, what at the time were, obscure points of blast analysis and in the process provided considerable
encouragement and support. All of his efforts were very much appreciated. The participation of Professor P. C. Hughes on the departmental supervision committee is gratefully acknowledged. Thanks are extended to the numerous students and friends who helped with exchanges of ideas, conversations, and constructive criticisms. Mr. J. Gietz, Mr. W. G. Elliott and Mr. P. Floros deserve special recognition for their efforts in forcing a recalcitrant piece of graphics software to werk as advertised, allowing the author to prepare the figures of chapter 8. Mrs. J. Gilpin, who typed this
manuscript, is thanked for her patience and good humour from the original draft through all the revi sions. Mrs. L. Quintero and Ms. I. Krauze
performed their magic in the preparation of the figures in this volume and are gratefully acknowledged for their efforts.
The most important piece of recognition must surely go to my wife Barbara, whose selfless sacrifices and devotion have made the success of this endeavour possible. It is to her that this tome is dedicated.
• 1.0 2.0 TABLE OF CONTENTS Abstract Acknowledgment Table of Contents List of Tables List of Figures List of Appendices I NTRODUCTI ON 1.1 1.2 1.3 1.4 1.5
The Blast Environment Experimental Background Numerical Treatments
Scope and Aims of the Present Work References
STIFFNESS MATRICES 2.1
2.2
2.3
Element Formulation: Linear 2.1.1 2.1.2 2.1. 3 2.1.4 Int roduct i on Strain-Displacement Relations Constitutive Relations
Strain Energy and Stiffness Matrix Formulation
Element Formulation: Nonlinear Geometry 2.2.1 2.2.2 2.2.3 2.2.4 Int roduct i on Strain-Displacement Relations StrainEnergy and Stiffness Matrix Formul ati on
The Tangent Stiffness Matrix Element Formulation: Elasto-Plasticity 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6
An Introduction to Plasticity Theory Uniaxial Stress-Strain Relation
Effective Stress and Effective Strain Yield Conditions and Hardening Behaviour Stiffness Matrix Formulation
Strain Rate Effects
i i i i i iv viii x xvii 1.1 1.1 1.6 1.9 1.11 1.15 2.1 2.1 2.1 2.2 2.8 2.9 2.13 2.13 2.14 2.16 2.17 2.27 2.27 2.27 2.29 2.32 2.42 2.45
2.5 Nuclear Radiation Effects on Metals 2.52
2.6 Nomenclature 2.56
2.7 References 2.60
3.0 CONSISTENT FORCE VECTOR 3.1
3.1 Int roduct i on 3.1
3.2 Blast Wave Loading 3.1
.
'
3.3 Thermal Loading 3.13
3.4 Nomenclature 3.17
3.5 References 3.20
4.0 STATIC FORMULATION VERIFICATION AND CHARACTERIZATION 4.1
4.1 Introducti on 4.1
4.2 Linear Analysis Examples and Results 4.2
4.3 Linear Analysis Results Summary 4.7
4.4 Nonlinear Geometry Examples and Results 4.7
4.5 Nonlinear Geometry Results Summary 4.13
4.6 Nonlinear Material Examples and Results 4.14
4.7 Nonlinear Material Results Summary 4.19
4.8 References 4.20
5.0 MASS MATRICES 5.1
5.1 Int roduct i on 5.1
5.2 Consistent Mass Matrix Calculation 5.1
5.3 Summary 5.8
5.4 Nomenclature 5.9
5.5 References 5.10
6.0 TEMPORAL INTEGRATION OF THE MOTION EQUATIONS 6.1
6.1 Int roduct i on 6.1
•
6.3 Collocation Schemes
6.3.1 Wilson-9 Results 6.4 Alpha Methods
6.4.1 Newmark-~ Results
6.5 Implicit Operators: Summary of Findings 6.6 Expl i cit Methods
6.7 Temporal Integration of Nonlinear Motion Equations 6.8 Implementation
6.9 Nomenclature 6.10 References
7.0 DYNAMIC FORMULATION RESULTS 7.1 Introduction
7.2 Simply Supported Square Plate 7.3 Clamped Ci rcul ar Plate
7.4 Va ul ted Roof Example 7.5 Surrmary 7.6 References 6.13 6.23 6.24 6.27 6.32 6.33 6.34 6.35 6.38 6.41 7.1 7.1 7.2 7.7 7.9 7.11 7.13 8.0 ANALYSIS OF A CIRCULAR CYLINDER SUBJECTED TO A NUBLEAR BLAST 8.1
8.1 8.2 8.3 8.4 8.5 8.6 Introduction
Weapon and Target Characteristics 8.2.1
8.2.2 Weapon Characteristics Target Characteristics Thermal Load Determination Blast Load Determination 8.4.1
8.4.2 Drag Phase Diffraction Phase The Finite Element Model Cylinder Test Results
8.1 8.1 8.1 8.4 8.6 8.10 8.10 8.17 8.18 8.20
8.6.1 Thenna1 Pre10ad 8.20 8.6.2 Dynamic 81 ast Load Resu1 ts 8.22
8.6.3 Summary and Cone1 usions 8.24
8.7 Nomenc 1 at u re 8.26
8.8 Referenees 8.29
9.0 SUMMARY 9.1
9.1 Referenees 9.3
..
List of Tables Chapter 3
3.1a Base Pressure Coeffieient Versus Reynolds Number 3.1b Separation Angle Versus Reynolds Number
3.2 Drag Coeffieient Versus Reynolds Number 3.3 Referenees for Tables 3.1 to 3.3
Chapter 4
4.1 Thin Cireular Plate Centre Point Deflection Results 4.2 Pineh Test Resul ts
4.3a Shell Roof Test: Oefl eet i on Results at the Nodes
4.3b Shell Roof Test: Axi al Stress Resultant Evaluated at the Nodes 4.3e Shell Roof Test: Moment Resultants Evaluated at the Nodes
Chapter 5
Chapter 8
8.1 Nuclear Burst Characteristics
8.2a Target Cylinder Material Properties 8.2b Target Cyl inder Geometry
8.3a Calculated Temperature Distribution 8.3b Prescribed Temperature Distribution
8.4a Spline Function Coefficients for the Base Pressure Coefficient Interpolation
8.4b Spline Function Coefficients for the Separation Angle Interpolation
8.5 Spline Function Coefficients for the Reflected Overpressure Interpolation
8.6a Di spl acements at Representat ive Nodes for the Thermal Load Results
8.6b Calclated Stresses due to the Applied Thermal Load
8.7a Displacement, Velocity and Acceleration Records of Representative Nodes at t = 600 ~s
8.7b Calculated Stresses at t
=
600 ~.-.cl Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Chapter 3 List of Figures
The Bi-Cubic Finite Element; 80 Degrees of Freedom ( [), V, W, <vx' <vy'> at each node.
Tangent Stiffness Solution Method
Hybrid (Tangent-Initial) Stiffness Solution Method Uniaxial Nonlinear Material Models
Ramberg-Osgood Uniaxial Material Model
Geometrical Representation of the von Mises and Tresca Yield Surfaces in Stress Space
Projections of the von Mises and Tresca Yield Surfaces into the Principal Stress 1-2 Plane and the n-Plane
Strain Rate Dependence of a Linear Hardening Elasto-Plastic Materi al
3.1 The Peak Reflected Overpressure as a Function of the Angular Position Around the Cylinder
3.2a A Circular Cylinder in the Physical Plane of the Parkinson-Jandali Model
3.2b A Representation of a General Body in the Parkinson-Jandali Model Physical and Transform Planes
3.3 Flow Separation Angle for a Circular Cylinder as a Function of Reyno 1 ds Number
3.4 Base Pressure Coefficient for a Circular Cylinder as a Function of Reynolds Number
3.5 Drag Coefficient for a Circular Cylinder as a Function of Reynol ds Number
Chapter 4
4.1 Square Plate Test Model, 1/4 Plate Symmetry Used, Edges May be Simply Supported or Clamped and the Applied Load may be a Centre Poi nt Load or a Unifonn Load.
4.2a Uniform Load on a Square Plate Centre Point Deflection Results 4.2b Point Load on a Square Plate Centre Point Deflection Results 4.3 Ci rcul ar Pl ate Test Geometry. Cl amped or Simply Supported Edge
and a Centre Po i nt Load or a Un iform load
4.4 Lateral Deflection and Slope Results for the Thick Circular Pl ate Test
4.5 Thermal Load on Circular Cylinder Test: Deflection Results 4.6 Thermal Load on a Circular Cylinder Test: Stress Results 4.7 Pinch Test Geometries
4.8a 4.8b 4.9a 4.9b 4.10 4.11 4.12
Deep Vaulted Roof Test: Axial Deflection Results Deep Vaul ted Roof Test: Vertical Deflection Results Deep Vaulted Roof Test: Stress Resul tant Resul ts Deep Vaulted Roof Test: Moment Resul tant Resul ts
Nonl i near Geometry Analysi s of a Simply Supported Square Plate Under a Uni form Load
Nonlinear Geometry Analysis of a Clamped Square Plate Under a Uni form Load
Nonlinear Geometry Analysis of a Clamped Circular Plate Under a Centre Point Load
4.13 Nonlinear Geometry Analysis of a Clamped Circular Plate Under a Un i form Load
4.14 Nonlinear Geometry Analysis of a Shallow Cylindrical Shell Segment Under a Point Load
4.15 Nonlinear Geometry Analysis of a Shallow Cylindrical Shell Segment Under a Uniform Pressure Load
4.16 Nonlinear Geometry Analysis of a Segment of a Sphere Under a Poi nt Load
4.17 Uniaxial Tension Test on an Elasto-Plastic Bar with Unloading and Reloading
4.18 Elasto-Plastic Analysis of a Simply Supported Square Plate Under a Un i form Load
4.19 Elasto-Plastic Analysis of a Cyclically Loaded, Simply Supported Square Pl ate under a Uni form Load
4.20 Elasto-Plastic Analysis of a Simply Supported Circular Plate: Test Geometry
4.21 Elasto-Plastic Analysis of a Simply Supported Circular Plate: Defl ecti on Resul ts
4.22 Elasto-Plastic Analysis of a Shallow Cylindrical Shell Segment Under a Point Load
4.23 Combined Nonlinear Analysis of a Deep Vaulted Roof Under lts Own Weight, Present Results
4.24 Elasto-Plastic Analysis of a Deep Vaulted Roof Under lts Own Weight, Comparison of Results
4.25 Combined Nonlinear Analysis of a Deep Vaulted Roof Under lts Own Weight, Comparison of Results
Chapter 5
5.1 Rigid Body Dynamies Test of the Consistent Mass Matrices 5.2 Rigid Body Dynamies Test of the LlI11ped Mass Matrix
5.3 Canti 1 evered Pl ate Natural Vibration Test Geometry
Chapter 6
6.1 General Collocation method: Stability Boundaries
6.2a General Collocation Method: Spectral Radius Behaviour for a Representative Optimal Parameter Pair
6.2b Wilson-9 Method: Spectral Radius Behaviour, !; = 0 6.2c Wilson-9 Method: Spectral Radius Behaviour, !; = 0.5 6.3a Collocation Method: Algorithmic Damping Ratio Behaviour 6.3b Collocation Method: Relative Period Error Behaviour
6.4b a-Method: Spectral Radius Behaviour, ~
=
0.5 6.4c a-Method: Spectral Radius Behaviour, ~t/T -+- co 6.5 a-Method: Algorithmic Damping Ratio Behaviour 6.6 a-Method: Relative period Error Behaviour 6.7 Newmark-~ Method: Stability Boundaries6.8a Newmark-~ Method: Spectral Radius Behaviour, ~
=
0 6.8b Newmark-~ Method: Spectral Radius Behaviur, ~=
0.5 6.8c Newmark-~ Method: Spectral Radius Behaviour, ~t/T -+- co 6.9 Newmark-~ Method: Al gorithmic Dampi ng Ratio Behaviour6.10 Newmark-~ Method: Relative period Error Behaviour
Chapter 7
Simply Supported Square Plate Under a Heaviside Unifonn Load Example: 7.1 Test Geometry, Quarter Symmetry, 2 x2 Mesh
7.2 Displacement, Velocity and Acceleration Records at the Centre Point
Li near Analysi s
Consistent Mass Matrix
High Dissipation Implicit Integration
7.3 Displacement, Velocity and Acceleration Records at the Centre Point
Li near Analysi s
Consistent Diagonal Mass Matrix
High Dissipation Implicit Integration
7.4 Displacement, Velocity and Acceleration Records at the Centre Point.
Li near Analysi s
Consistent Diagonal Mass Matrix
Newmark-~ Integration
7.5 Displacement, Velocity and Acceleration Records at the Centre Point
Nonlinear Geometry (NLG) Analysis Consistent Diagonal Mass Matrix
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1
•
7.6 Displacement, Velocity and Acceleration Records at the Centre Point
7.7
Elasto-Plastic (Perfectly Plastic) Analysis Consistent Diagonal Mass Matrix
Newmark-~ Integration
Displacement, Velocity and Acceleration Records at the Centre Point
Combined Nonlinear Analysis Consistent Diagonal Mass Matrix Newmark-~ Integration
Clamped Circular Plate Under a Heaviside Uniform Load Example: 7.8 Test Geometry
7.9 Displacement, Velocity and Acceleration Records at the Centre Point
Linear Analysis
Consistent Mass Matrix Newmark-~ Integration One El ement
6t
=
10 s7.10 Displacement, Velocity and Acceleration Records at the Centre Point
Linear Analysis
Consistent Mass Matrix Newmark-~ Integration One El ement
6t
=
4 s7.11 Displacement, Velocity and Acceleration Records at the Centre Point
Li near Analysi s
Consistent Mass Matrix Newmark-~ Integration Three El ements
6t
=
10 s7.12 Displacement, Velocity and Acceleration Records at the Centre Point
Li near Analysi s
Consistent Mass Matrix
High Dissipation Implicit Integration Three El ements
6t
=
10 s7.13 Displacement, Velocity and Acce1eration Records at the Centre Poi nt
Li near Ana1ysi s
Consistent Mass Matrix
Newmark-~ Integration Three E1ements
&
=
4 s7.14 Displacement, Velocity and Acce1eration Records at the Centre Point
Non1 i near Geometry Ana1ysis Consistent Mass Matrix
Newmark-~ Integration One Element
t§.
=
10 s7.15 Displacement, Velocity and Acce1eration Records at the Centre Point
Non1inear Geometry Ana1ysis Consistent Mass Matrix
Newmark-~ Integration Three El ements
6t : 4 s
Vau1ted Roof Under Dead Weight Heaviside Loading Examp1e: 7.16 Test Geometry, Quarter Symmetry, 2><2 Mesh
7.17 Vertica1 Displacement, Velocity and Acce1eration Records at the Centre Point (Point C)
Linear Ana1ysis
Consistent Mass Matrix Newma rk - /3 I nteg rat i on
6t = 7 ms
7.18 Vertica1 Displacement, Velocity and Acce1eration Records at the
7.19
Midspan Edge Point (Point 0)
L i near Ana1ysi s
Consistent Mass Matrix Newma rk -~ I nteg rat i on
6t = 7 ms
Axia1 Displacement, Velocity and Acce1eration Records at the Top of a Diaphragm End Support (Point A)
Li near Ana 1ys is
Cons i stent Mass Mat ri x Newmark-~ I ntegrat i on
6t = 7 ms
xv
..
.t
•
7.20 Axial Displacement, Velocity and Acceleration Records at the Edge of a Diaphragm End Support (Point B)
Li near Ana lysi s
Consistent Mass Matrix
Newmark-~ Integration
l1t = 7 ms
7.21 Axial Displacement Around a Diaphragm End Support (Edge A-B) at Various Times
7.22 Vertical Displacement Around the Midspan {Edge C-D} at Various Times CHAPTER 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8a 8.8b 8.9 8.10a 8. lOb 8.11a 8.11b 8.12
Blast Wave Characteristics General Test Arrangement Target Cylinder Geometry
Finite Element Discretization of the Target Cylinder Cylinder Deformation Due to the Applied Thermal Load
Displacement, Velocity and Acceleration Records, for the Centre Front Node
Displacement Velocity and Acceleration Records
Back Node for the Centre
Cylinder Deformati on at t
=
600 j.IS, Front Vi ewCylinder Deformation at t
=
600 ~s, Rear ViewVariation of the Cylinderls Cross Sectional Shape with Time Cylinder Deformation at t
=
1.65 ms, Front ViewCylinder Deformati on at t
=
1.65 ms, Rear View Cylinder Deformati on at t=
6.3 ms, Front View Cylinder Deformation at t=
6.3 ms, Rear View Cylinder Deformation at t=
10.5 ms, Front Viewt
1
LIST OF APPENDICES
A.O On Satisfying the Rigid Body Motion Requirements A.l A.l Introduction
A.2 Element Formulation A.3 Basis Functions
A.4 Numerical Integration
A.5 NlI1leri ca 1 Tests and Results A.5.l Prescribed Translation A.5.2 Uni form Pressure Test A.5.3 Thermal Test
A.5.4 Pinch Test A.5.5 Shell Roof Test A.6 Discussion A.7 Conclusions A.8 References A.1 A.4 A.6 A.12 A.14 A.l5 A.l5 A.l6 A.17 A.17 A.l9 A.20 A.21
r
I
1.0 INTRODUCTION
1.1 The Blast Environment
The capability of predicting, wi~hout recourse to direct
experiment, the response of a structural system that is subjected to a blast environment is of fundamental interest. Designers of many
structures and vehicles, both civilian and military, have a
considerable interest in designing these systems to withstand the rigours of a blast environment. The structural integrity of the design is of ten of primary interest and because the most vulnerable components are frequently assemblies of plates or shells, it is these structural groups that will be investigated here. The loading
environment associated with an explosive blast is very exacting, and in this regard, the loading environment associated with a nuclear explosion is the most severe. The nuclear burst environment
incorporates many features which are seldom seen with such intensity and synergism. It is exactly this intensity and synergism of the exciting forces which makes the structural response unique in terms of the results that may be expected from more conventional loading
situations. This volume will present a finite element formulation that includes the capability of being able to treat this class of problem. A brief discussion of the burst environment, and its modelling, is given in order to provide an appreciation of the complexity of the loading.
The nature of the environment which results from the detonation of a nuclear weapon depends primarily on: the weapon yield, the altitude at which detonation occurs and the height above the ground. These latter two considerations are usually expressed in terms of a
altitude. The target structurels response to the resulting nuclear burst environment depends on: the structurels proximity to the point of burst, atmospheric conditions, structural geometry and constitutive materials, the surrounding terrain and other slightly less significant parameters [1.1, 1.2].
For the case where the detonation of a nuclear weapon is at a sufficient height above the ground so that the fireball does not come into contact with the ground and the shock front is not influenced by the presence of the ground, a target within the weaponls range of effects will experience: air shock waves, electromagnetic radiation
and nuclear radiation. If the detonation is close to the ground, all the above effects are increased near ground zero due to the
interaction of the ground surface and the blast wave. As a result of the very high pressures at ground zero cratering will be very
extensive, this, in addition to the very high temperatures caused by the close proximity of the fireball, will cause enormous quantities of ground material to be excavated and drawn up into the mushroom cloud where it will be irradiated by the radioactive material present from the bomb. The excavated material , heavily supplemented by debris drawn up into the mushroom cloud by the af ter winds that are present in either case, will later "fall-out" over a very wide area,
increasing the incidence of fall-out radiation over the free air burst situation [1.1].
As noted above the major divisions of nuclear weapons effects are blast, electromagnetic radiation and nuclear radiation. These effects can be subdivided into more specific areas. Blast effects include air blast, ground shock (air induced or direct) and cratering, while
electromagnetic radiation and the target materials. Nuclear radiation is divided into initial effects including gamma and neutron
radiation and residual effects which comprise induced radiation and fall-out. Of these effects it has been found [1.3J that thermal load and airblast constitute the most significant sources of structural excitation at target to weapon separation distances which are meaningful for vulnerability studies of shipboard or aircraft
structures. These two aspects of the 1 oadi ng wi 11 be di scussed separately although they do interact strongly in the final analysis
[1.4-1.7]. The term "burst" will be used to indicate that both the thermal and the aerodynamic loads are being taken into account wnile the term "bl ast" shall be taken to mean that only the aerodynamic loads are included.
In terms of nuclear radiation, alpha particle and beta particle radiation have very short mean free paths and have no significant effect beyond a few meters from ground zero [1. 8J. Consequentl y they can be ignored in relation to other effects while neutron radiation will later be shown to have a negligible effect on the structural response.
The blast wave generated by a nuclear burst can be well represented by a Freidlander wave [1.1, 1.2, 1.9-1.11J which is characterized by an instantaneous initial rise to a peak value,
corresponding to the shock front, followed by an exponential decay of the post-shock flow properties. Although the decay constants and wave lengths are all different, the overpressure, over-density and flow velocity may all be represented in this fashion.
The blast loads that act on the target are due to the blast wave overpressure and the dynamic pressure; with the former of ten
constituting the most significant part of the blast loading on shell structures. The overpressures generated by a nuclear explosion range from several thousands of atmospheres near the point of detonation to fractions of an atmosphere at great distances from the point of
detonation. It is interesting to put these various values into perspective in terms of the damage that can be expected at various overpressure levels. While it is most likely that overpressures on the order of a few thousand atmospheres will destroy just about anything, including many "hardened" structures, it is much less obvious at what level of peak overpressure minimal damage may begin.
Although the magnitude of the incident overpresure will be used as a parameter of comparison, in the following it is important to realize that it is the sum of the blast wave effects that are
important. While it is the overpressure that is most significant in the loading of hollow shell structures the drag forces associated with the dynamic pressure are very significant for solid structures, such as anten na masts, and in cases where overturning of the target is a
possibility, as in the case of vehicles.
A peak overpressure level of less than 0.02 atmospheres (atm) will break glass and knock out windows in most conventional structures [1.12J. Damage to the larynx can be expected to begin at
approximately 1/3 atm [1.13J and most steel framed structures will be damaged beyond use at this same overpressure level [1.12J. Injuries to the gastrointestinal tract may be generated at as little as 2/3 atm peak overpressure [1.13J, while a peak overpressure of 0.8 atm would result in 90% of all persons exposed having their eardrums ruptured
•
lung damage [1.12, 1.13], and various researchers put the onset of severe lung damage at peak overpressure levels from 2 atmospheres [1.12] to 3.5 atmospheres [1.14]. Trucks and similar unhardened vehicles are totally destroyed [1.12] at peak overpressures of 2 atm but only 0.4 atm is required to initiate enough damage so that the vehicle is unusable, at least temporarily. In this latter example the damage may arise due to the overturning of the truck by the dynamic pressure.
In light of these and other considerations outlined in Ref.
[1.3] a peak overpressure in the neighbourhood of 1 atm will be deemed the maximum value that an unhardened structure could be expected to survive. This value is well beyond the guidelines recommended by the Canadian Armed Forces for the blast-resistent design of warship
structures [1.15].
The thermal load arises from the interaction of the
electromagnetic radiation, which is largely in the visible and infrared portions of the spectrum [1.1, 1.11, 1.16, 1.17], with the target materials. The actual thermal load experienced by the target will vary with: distance from the fireball, atmospheric clarity, height of burst and weapon yield [1.1]. The weapon yield determines the amount of energy that is available not only for blast wave
formation but also for thermal radiation. At higher altitudes more of the weapons energy is spent as electromagnetic radiation, due to the lower ambient air density, thus increasing the potential for a
heightened thermal load [1.1, 1.2, 1.9]. The presence of water vapour, dust and aerosols in the atmosphere causes dilution of the thermal load through absorption and scattering, but even at large
elevated temperatures that can be expected in a target, as a result of the thermal radiation, it can be expected that there will be a
significant effect on the yield stress and the ultimate stress parameters as well as on the elastic modulus and the thermal coefficients [1.17J.
1.2 Experimental Background
In the twenty years following the first detonations of nuclear weapons there were over two hundred atmospheric tests of nuclear
explosives conducted by western nations [LIJ. In spite of this large number of tests there is a very noticeable shortage of specific
experimental data in the open literature. Glasstone [1.1J and Brode [1.2J provide the broadest range of material on nuclear weapon effects but offer no specific examples of structural response with which
comparison is feasible.
A relatively large amount of data is available for free-field detonations of large chemical explosions [1.18-1.44J or for
experiments conducted in blast simulators [1.44-1.49J.
Eight major free-field chemical explosive trials are reported: SNOWBALL [1.40J, DIAL PACK [1.19, 1.28-1.32, 1.37, 1.38, 1.41J, DICE THROW [1.19-1.23J, DISTANT PLAIN [1.23], MIXED COMPANY [1.19, 1.35], PRARIE FLAT [1.18, 1.19, 1.25, 1.30, 1.37-1.39, 1.41], SAILOR HAT [1.42], and MISERS BLUFF [1.43J. Two more events have taken place since MISERS BLUFF; they were MILL RACE in 1982 and DIRECT COURSE in 1984, but documentation of the results of the tests conducted at these trials is not widely available yet.
•
overpressure during the positive phase duration of the blast. It should be noted that because the dynamic pressure is proportional to the square of the overpressure (via the Rankine Hugonoit relations) the Friedlander wave is not a good approximation of this flow
parameter. Results for the reactions of some structural assemblies to the blast wave are reported [1.21-1.28, 1.36-1.39] along with the drag coefficient data for circular cylinders [1.18, 1.19, 1.29, 1.30, 1.34, 1.35] and I-beams [1.24]. It is generally the case that the only blast flow parameter measured at the target is the overpressure and therefore reconstruction of the flow field for numeri cal simulations is not feasible. In two notable exceptions [1.58, 1.71] there has been enough data acquired to allow reconstruction of the flow field. Further, ·only net forces acting on the experimental targets are obtainable and not the actual pressure distributions. Rather than detailing the findings of each individual reference it will be sufficient to outline the general findings and results obtained in these tests. At the outset it must be realized that there is a
significant difference between the environments created by the detonation of large quantities of chemical explosive and by the detonation of a nuclear weapon of nominally equivalent yield. Most
importantly, there is no significant thermal load in the case of a chemical explosion and it is not val id to scale data obtained froo a chemical explosion and apply it to a nuclear explosion due to the differences in the explosive charge. Attempts to include the thermal effects of a nuclear detonation, by burning sheets of rocket
propellant or aluminum powder/liquid oxygen mixtures, have been made in both free-field tests and in blast simulators [1.4-1.7, 1.47,
These reports vividly illustrate that, in order for reasonable correlation between a theoretical analysis and experimental results, accurate modelling of the target and the applied loads is essential [1.21, 1.22, 1.25J. Accurate modelling of the loads is complicated by an inability to strictly control the experimental environment. Dust entrained in the post-shock flow [1.18, 1.19J, the presence of a precursor shock or flow turbulence can individually or collectively [1.50J cause significant modifications to the loading of the target. Theoretical models of the the fluid dynamic loading are unable to
include the effects of flow turbulence, target surface roughness, and shading and solidity effects in structural assemblies. Further, they cannot accurately predi ct flow separati on points for a specific test situation. This leads to the necessity of generating approximations of the pressure profiles or drag coefficients based on very tenuous data. Another significant problem associated with comparison between experiment and theoreticalor numeri cal models is that many reports fail to include sufficient data from which a reasonable model can be established.
Smal 1 amounts of graphical data for the response of flat plates and circular cylinders subjected to the thermal radiation pulse from a nuclear weapon are available [1.48], but it is more useful for
qualitative comparison than quantitative. These data are contained in papers that discuss the experimental problems and techniques
associated with a blast simulator [1.47, 1.48J. The results are for al uminum (6061-T6) alloy pl ates 1/8" and 1/4" thick as we" as
ci rcul ar cyl inders with a wall thickness of 0.049" stationed at the 10
•
these specimens; a temperature sufficiently above ambient to induce significant degradation of the mechanical properties.
1.3 Numerical Treatments
Because there is a general unavailability of experimental results for nuclear explosions and since scaling of results from chemical explosions is not strictly valid, there has been considerable effort devoted to developing numeri cal techniques for the prediction of the blast wave [1.1, 1.2, 1.9, 1.33, 1.51-1.56] and thermal load [1.11, 1.16, 1.17, 1.33, 1.57] that arise from a nuclear explosion. Numerous authors have reported techniques for quantifying the physical
parameters associated with the nuclear blast wave [1.1, 1.2, 1.9, 1.33, 1.41, 1.51-1.56] and they have arrived at a set of theoretical and empirical equations and relations that can be judiciously employed to predict some of these parameters. A discussion of their werk is
presented in Ref. 1.3. It is sufficient to no te here that for nuclear explosions, the 1981 DNA 1 kton Nuclear Blast Standard [1.52J is
considered to be the best choice for characterizing this aspect of the explosion environment because it offers the best combination of self consistency and agreement with experiment and theoretical results. The work of Lau and Gottlieb [1.58J and Sadek and Gottlieb [1.59] provides a comparable souree for chemical explosions.
Although these sources are for fixed explosive yields, 1 kton and 1 kilogram respectively, other weapon sizes may be accounted for by applying suitable sealing relations. Hopkinson cube root scaling
[1.60J applies to yields from 1 kton to 10 megatons and agrees wel 1 with experimental data over this range of yields [1.1J. It can be applied to characteristic di stances , times, and impulses [1.1] and
although this scaling method is strictly valid only for model and prototype detonations conducted under identical atmospheric conditions [1.60], it is considered acceptable to ignore variations in the
altitude of the target from that of the detonation if the two are within a few thousand feet of one another [1.1]. Except for the variation of pressure with altitude the cube root scaling law implies that pressure and velocity quantities are unchanged by the scaling and consequently all pressures and velocities will have the same values at the scaled times and distances [1.60]. A more general set of scaling relations which include the effects of differences in atmospheric conditions are provided by Sachs' scaling relations [1.9, 1.60J. It should be recognized that Hopkinson cube root scaling is a special case of Sachs' scaling and that they implicitly assume that the distances of interest are large in relation to the characteristic dimension of the explosive charge, that the effect of gravity and viscosity are small and that the peak pressure experienced is a function of the ambient atmospheric density and speed of sound for a given weapon yield and distance from the point of burst [1.60]. Sachs' scaling relations have also been validated by comparison to experiment [1.60J.
From this brief discussion of the phenomena that may affect the target structure it is evident that their influence, either singly or in combination, can be very dramatic. The air blast will impart
extremely severe dynamic loads to the structure while the thermal load will induce large stresses and strains as well as causing extensive changes in the material properties. The response due to the
combination of these two effects can be greater than the sum of the separate parts since both can be of sufficient intensity to cause
•
extensive nonlinear behaviour in the structural response [1.61]. The complexity of both the structure and the loading environment makes the prospect of employing closed form analysis hopeless and points to the use of numeri cal techniques to determine the response behaviour.
While few finite element treatments of this class of problem have been reported [1.5] there are more applications of the finite
difference method [1.62-1.70] reported. The finite difference applications all employ Kirchhoff thin plate and shell fonnulations with nonlinear geometry terms [1.63, 1.66-1.69], nonlinear material models [1.63, 1.49, 1.65-1.67], strain rate sensitivity [1.63,
1.65-1.67], and temperature dependant material behaviour [1.49, 1.67]. The use of the Kirchhoff thin shell formulation precludes the
possibility of these programmes being used to analyse moderately thick plate or shell geometries. In all cases the loading must be explicitly specified at each time step as input data.
Comparison between the results obtained with these programs and experiment is generally favourable [1.62, 1.49, 1.65-1.66, 1.70] but there is extreme difficulty in reproducing the experimental conditions due to probl ems in accurately measuring the experimental inputs and outputs. None of the experiments used for comparison purposes were actual responses to a nuclear explosion environment.
1. 4 Scope and Aims of the Present Work
In this thesis a finite element fonnulation will be developed to treat the problem of the response of a structure to a nuclear burst environment. The position of the target structure will be restricted to be such that the i nc ident peak overpressure wi 11 not exceed
that most conventional structures would be expected to survive but yet still ensures that the incident shock front can be considered to be planar with regard to its interaction with the target. In addition, the post-shock flow at this overpressure level will be subsonic, thus removing any considerations of bow shocks and associated phenomena. The structure itself will be limited to a circular cylinder that is composed of an isotropic homogeneous material. This choice represents a wide class of generic structures and includes manifestations of all the response interactions that may be expected in shell structures.
Isotropie homogeneous materials have been selected because of the wide amount of data available for materials in this class (metals: steel and aluminum), and the well developed models of their behaviour outside the conventional room temperature 1 inear el astic regime.
Application of the finite element method to problems of this nature requires solution of the semi-discretized motion equation
M~ + C:!. + K ~
= [(
t ) (1.1 )In the above equation M, C, and K are the mass, damping and stiffness matrices while ~, ~,
1,
and [(tl are the acceleration, velocity, displacement and force vectors respectively.The mass matrix is assumed to be constant and positive definite while the damping and stiffness matrices may be specified as functions of velocity or displacement or both and are positive semi-definite for a stable response. The force vector characterizes the loads applied to the structure by the nuclear burst environment.
In the remaining chapters of this volume the specification and development of each term in Eq. will be discussed along with the
•
means of solving the motion equation for the displacement, velocity and acceleration vectors. The specification of the damping
characteristics for a continuum is a very complex issue and general methods of characterizing a damping matrix are not available. Thus this term in the motion equation will be neglected in the sequel. This is considered to be a conservative assumption (in the engineering sense) in that the response calculated in the absence of damping
forces will be greater than the response obtained if damping were present. Further, because the exciting forces are not periodic, resonance will not be a problem and therefore the inclusion of damping is unnecessary to limit this phenomenon.
The development of the stiffness matrix is presented in Chapter 2 where it is shown how nonlinear strains and elasto-plastic isotropic
strain hardening material behaviour can be incorporated into the formulation of a finite element specified in shell coordinates and designed to treat a full range of shell thicknesses ranging from very thin to moderately thick. The total Lagrangian representation is adopted so that all deformations and loads are given with reference to the initial undeformed position of the shell. In the third chapter the consistent force vector is developed from the general form to a form which specifically includes the load sources that will be
encountered under the present type of loading. Numerous examples of the application of the previously developed stiffness matrices and force vectors are presented in the following chapter. These are static problems that illustrate the accuracy and flexibility of the current formulation by comparison to theory, experiment, and other numerical techniques. Having illustrated the superiour convergence and accuracy properties of the present element in the static case the
dynamic formulation is completed with the mass matrices given in
Chapter 5. Both the consistent mass matrix and the companion diagonal mass matrix are shown to be dynamically consistent and accurate. It will be shown in Chapter 6 how the resulting system of motion
equations may be integrated in order to find the displacement, velocity, and acceleration vectors at appropriate times. The
conditions required to achieve the unconditional spectral stability for arbitrarily damped linear structural systems are determined in closed form for several linear single-step implicit temporal
operators.
Comparison to analytical solutions and other numerical results for the dynamic case is made in the seventh chapter. Excellent
agreement is obtained in each instance and the present system is found to be accurate and economic in these problems as well.
To conclude the report an example of a circular cylinder subjected to a nuclear burst environment is given.
•
1. 5 References
1.1 Samuel G1asstone, ed., The Effects of Nuc1ear Weapons, Revised Edition 1964, (United States Atomic Energy Commission, 1962). 1.2 H. L. Brode, "A Review of Nuc1ear Exp10sion Phenomena Pertinent
to Protecti ve Constructi on", The Rand Corporat i on, R-425PR (1964).
1.3 J. S. Hansen, G. R. Hepp1er, "Deve10pment of a Finite Element Capabi1ity for the Ana1ysis of Structura1 Systems Subjected to a Nuc1ear Blast Environment", Phase I, submitted to the Defence Research Establishment Suffield, Department of National Defence, Ralston, Alberta, under Contract No. 03SU.3280007, October 1981. 1.4 Clarence W. Kitchens Jr., Richard E. Lottero, Andrew Mark,
George D. Teel, "Blast Wave Modification during Combined
Thenna1/Blast Simulation Testing", Proc. 7th Int. Sym. Mil. Appl. Blast Simulation, Medicine Hat, Alberta, July 1981, 11, pp. 5.6-1 - 5.6-21.
1.5 Richard J. Pearson, Henry L. Wisniewski, Paul D. Szabodos,
"Synergi sm in Nucl ear Therma 1 IBl ast Loading", Proc. 7th Int. Sym. Mil. Appl. Blast Simu1ation, Medicine Hat, A1berta, July 1981, 11, pp. 5.7-1 - 5.7-18.
1.6 George D. Teel, Fritz H. Oertel , "Testing to Combined Blast and Thennal Effects at BRL", Proc. 7th Int. Sym. Mil. Appl. B1 ast Simu1ation, Medicine Hat, Alberta, July 1981, 11, pp. 6.11 -6.1-21.
1.7 John F. Dishon 111, Richard Miller, "Development of a Thermal
Radiation Simulator for Synergistic Blast and Thermal Radiation
Testing of Full Scale Hardware", Proc. 7th Int. Sym. Mil. Appl.
Blast Simulation, Medicine Hat, Alberta, July 1981, 11, pp. 6.2-1
- 6.2-13.
1.8 W. F. Ranson, J. A. Schaeffel, A. E. Murphree, V. G. Ireland, B.
R. Mull inix, "An Experimental Assessment of Nuclear Weapons
Effects on Structures", U. S. Army Mi ssil e Research, Development
and Engineering Laboratory, Tech. Report RL-76-9, November 1975.
1.9 Harold L. Brode, "A Review of Nuclear Weapons Effects", Annual
Review of Nuclear Science, 18, 1968, pp. 153-202.
1.10 Edward S. Tooley, "A Nuclear Air Burst Analysis Routine", Air
Force Institute of Technology, Wright-Patterson Air Force Base,
Ohi 0, Ma rch 1975.
1.11 Robert G. DeRaad, Harol d E. Mei sterl ing, "A Code for Analysi s of
Nuclear Effects and Systems Survivability, Vol. 3, Blast and
Thermal", Ai r Force Insti tute of Technol ogy, Wright-Patterson Ai r
Force Base, Ohio, May, 1980.
1.12 L. E. Fugelso, L. M. Weiner, T. H. Schiffman, Explosion Effects
Computation Aids, (General American Research Division, 1972).
1. 13 J. T. Ye 1 verton, D. R. Ri chmond, E. R. Fl etcher, "Bi oeffects of
Simulated Muzzle Blasts", Proc. 8th Int. Sym. Mil. Appl. Blast
Simulation, Spiez, Switzerland, June 20-24, 1983, I, pp. VI.6-1 _
VI.6-25.
1.14 A. Jonsson, C. -J. Cl emedson, E. Arvebo, "Experiments with an
Anthopomorphic Dummy for Air Blast Research", Proc. 8th Int. Sym.
,
1.15 N. Bertrand, "Ai r B1 ast and Shock Desi gn of Warshi p Structures", Canadian Armed Forces, April 1979, n. pag.
1.16 David M. Dancer, Dona1 d M. Wil son, "TRIN: Computer Program to Ca1cu1ate Temperature Distributions in Circu1ar and Rectangu1ar Sections Exposed to Therma1 Radiation From Nuc1ear Weapon
Exp1 osi ons", Nava1 Ordance Laboratory TR-72-117, May 1972. 1.17 Joel D. Johnson, "A Sensitivity Study of Therma1 Radiation F1 uence from a Nuc1 ear Ai r Burst", Ai r Force Institute of Techno10gy, Wright-Patterson Air Force Base, Ohio, March 1975. 1.18 S. B. Mellsen, "Drag Measurement on Cy1inders by the Free
F1 ight Method - Operati on PRARIE FLAT", D. R. L S. TN-249, January 1969.
1.19 A. W. M. Gi bb, D. A. Hi 11, "Free F1 i ght Measurement of the Drag Forces on Cy1inders in Event DICE THROW", D.R.LS. TP-453, February 1979.
1.20 F. H. Winfield, "Event DICE THROW Canadian Air Blast Measurements", D.R.LS. TP-451, March 1977.
1.21 C. G. Coffey, G. V. Price, "Blast Response of a UHF Po1e Mast Antenna - Event DICE THROW" D. R. E. S. TP-449, November 1977. 1.22 B. G. Laid1aw, "Blast Response of Lattice Mast - Event DICE
THROW", D.R.LS. TP-452, December 1978.
1.23 G. V. Price, "Numerical Simu1ation of the Air Blast Response of Tapered Cantilevered Beams", D.R.LS. TP-447, November 1977. 1. 24 B. R. Long, "Dynamic Response ofSimp1y Supported Beams Under
Blast Loading", D.R.E.S. TN-195, February 1968.
1.25 B. R. Long, R. Na1yor, "The Behaviour of Two Simp1e Framed Structures Under B1 ast Loadi ng", D. R. E. S. Memorandum No. 110/68.
1. 26 D. S. Wal kinshaw, "On Forced Response of Polygonal Pl ates", D.R.E.S. TN-198, December 1967.
1.27 B. R. Long, B. G. Laidlaw, R. J. Smith, "The Analysis of
Shipboard lattice Antenna Masts Under Air Blast and Underwater Shock Loading", D.R.E.S. TP-431, June 1975.
1.28 Y. S. Kim, P. R. Ukrainetz, "Air Blast Response of a Circular Cantil ever Beam", D. R. E. S. TP-385, October 1971.
1.29 S. B. Mellsen, "Measurement of Drag on Cylinders by the Free Flight Method - Event PRARIE FLAP, D.R.LS. TP-382, December 1971.
1.30 R. Naylor, S. B. Mellsen, "Unsteady Drag From Free Field Blast Waves", D.R.E.S. Memorandum 42/71, January 1973.
1.31 James Gottlieb, David Ritzel, "A Semi-Empirical Equation for the Viscosity of Air", D.R.E.S. TN 454, July 1979.
1.32 Y. S. Kim, P. R. Ukrainetz, "Drag Loading on a Circular Cylinder frOOl an Air Blast Wave", D.R.E.S. TP-392, November 1971.
1.33 N. M. Newmark, D. Haltiwanger, "Air Force Design Manual: Principals and Practices for Design of Hardened Structures", AFSWCR-TDR-62-138, December 1962.
1.34 S. B. Mellsen, "Drag on Cylinders in a 20 to 25 millisecond Blast Wave", D.R.LS. Memorandum No. 114/71, July 1972. 1.35 S. B. Mellsen, "Measurement of Drag on Cylinders by the Free
Flight Method - Event MIXED COMPANY", D.R.E.S. TP-419, March 1974.
1.36 F. Adrian Britt, Robert H. Anderson, "Dynamic Behaviour of
,
Overpressuresll
, The Shock and Vibration Bulletin, No. 37, Part 4,
January 1968, pp. 151-167.
1.37 D. S. Walkinshaw, B. G. Laidlaw, B. R. Long, lIShipboard Lattice Masts Under Blast-loading, Part 1: Strains in Model Masts in Operation PRARIE FLAT and Event DI AL PACK II , D.R.LS. TN-402, May 1973.
1.38 B. R. Long, B. G. Laidlaw, lIShipboard Lattice Antenna Masts Under Blast Loading, Part 11: Comparison of
Experimental Results with Theoretical Predictions by Computer ProgramII, D. R. E. S. TP -418, September 1973.
1.39 Robert Geminder, IIAnalytical and Experimental Results of Lattice Type Structures Subjected to a Blast Loadingll
, The
Shock and Vibration Bulletin, 40, Part 11, December 1969, pp. 101-114.
1.40 Proceedings of the Symposium on Operation SNOWBALL, Vols. land 11, Defense Atomic Support Agency Data Center Special Report 34, August 1965.
1.41 S. B. Mellsen, IICorrelation of Drag Measurements in Operation PRARIE FLAT with Known Steady Flow Valuesll
, D.R.LS.
Memorandum 12/69, April 1969.
1.42 J. M. Dewey, M. A. Ward, IIThe Canadian Participation in SAILOR HAT II , D.R.E.S. Suffield Report No. 211, November 1967.
1.43 Proceedings of the MISERS BLUFF Phase II Results Symposium, Defence Nuclear Agency Report POR-7013, Vols. land 11, March 1979.
1. 44 A. M. Patterson, IINaval Effects Research ProgramII, D. R. L S. Memorandum - 124/68, November 1968.
1.45 Valerie J. Bishop, R. D. Rowe, "The Interaction of a Long
Duration Friedlander Shaped Blast Wave with an Infinitely Long
Right Circular Cylinder", AWRE Report No. 0-38/67.
1.46 J. E. Uppard, "The Drag Coefficient of Bodies of Revolution in Shock Induced Flow", Shock Tube Research: 8th. Int. Shock Tube
Sym., (London: Chapman & Hall, 1971), pp. 9.1 - 9.12.
1.47 Neil Griff, "Simulation of Combined Thermal Radiation - Air
Blast Effects", Proc. 3rd Int. Sym. Mil. Appl. Blast Simulation,
September 1972.
1.48 Neil Griff, J. Proctor, "Facility for Simulation of Thermal
Radiation - Air Blast Interaction Effects", Proc. 3rd Int. Sym.
Mil. Appl. Blast Simulation, September 1972.
1.49 Raffi P. Yeghiayan, Analytical Predictions and Correlation with
Experiments from Thermal /Bl ast Exposure of Ai rcraft Panel s",
(Burlington Mass.: Kamman Avidyne, 1977).
1.50 George W. Ullrich, "Air Blast Over Nonideal Surfaces", Proc.
6th Int. Sym. Mil. Appl. Blast Simulation, Cahors, France, June
25-29, 1979, pp. 1.2.1 - 1.2.29.
1.51 Harold L. Brode, "Numerical Solutions of Spherical Blast
Waves", J. Appl. Physics, 26, No. 6, June 1955, pp. 766-775.
1.52 Charles E. Needham, Joseph E. Crepeau, "The DNA Nuclear Blast
Standard (1 kton)", Systems Science and Software (S-Cubed),
P.O. Box 8243, Albuquerque, New Mexico, 87198, (DNA 5648T),
January 31, 1981.
1.53 Nonnan P. Hobbs, Kenneth R. Wetmore, "PIVUL - A Computer Code
for Rapid Assessment of the Vulnerability of Simple Structures
f
1. 54 H. L. Brode, S. J. Speicher, "Analytic Approximation for
Dynamic Pressures vs. Time", Pacifi c Si erra Research
Corporation, PSR Note 315, May 1980.
1.55 S. J. Speicher, H. L. Brode, "Revised Proceedure for the Analytic Approximation for Dynamic Pressures vs. Time", Pacific Sierra Research Corporation, PSR Note 320, May 1980. 1.56 Charles E. Needham, "Nuclear Blast Standard (1 kton)",
AFWL-TR-73-55 (Rev.), April 1975.
1.57 Donal d M. Wi 1 son, "The Di stri but i on and Hi story of Temperature in Circular Cylinders Exposed to the Thermal Radiation Pulse of a Nuclear Detonation", NOL TR-71-61, June 1971.
1.58 S. C. M. Lau, J. J. Gottl ieb, "Numerical Reconstruction of Part of an Actual Blast-Wave Flow Field to Agree with Available Experimental Data", UTIAS TN-251, August 1984.
1.59 H. S. I. Sadek, J. J. Gottlieb, "Initial Decay of Flow
Properties of Pl anar, Cyl indrical and Spherical Blast Waves", UTIAS TN-244, October 1983.
1.60 Wilfred E. Baker, Peter S. Westine, Fanklin J. Dodge,
Similarity Methods in Engineering Dynamics: Theory and Practice of Scale Modelling, (Rochelle Park, N.J. Hayden Book Co., 1978). 1.61 J. S. Hansen, G. R. Heppler, "Development of a Finite Element
Capabi 1 ity for the Analysi s of Structural Systems Subjected to a Nuclear Blast Environment", Phase 11, submitted to the
Defence Research Establishment Suffield, Department of National Defence, Ralston, Alberta, under Contract No. 03SU.3280007, October 1982.
1.62 N. J. Huffington, J. D. Wortman, "Parametric Inf1uences on the Response of Structural Shel1s", J. Engng. for Ind., November 1975, pp. 1311-1316.
1.63 L. Morino, J. W. Leach, E. A. Witmer, "An Improved Numerical Calculation Technique for Large Elastic Plastic Transient
Defonnations of Thin Shells, Part 1", J. App1. Mech., June 1971, pp. 423-428.
1.64 L. Morino, J. W. Leach, E. A. Witmer, "An Improved Numerical Calculation Technique for Large Elastic Plastic Transient
Defonnations of Thin Shells, Part 1I", J. Appl. Mech., June 1971, pp. 429-436.
1. 65 Norri s J. Huffi ngton Jr., John D. Wortman , "Response of Cyl indrical Shells to Travell ing Load Simulation of X-ray B10woff Impul se", BRL, AD 750 336, 1972.
1. 66 Roger L. Bessey, J. C. Hokan son, "Simultaneous and Runni ng Impul si ve Loading of Cyl indrical Shel1 s", BRL Contractor Report No. 237, June 1975.
1.67 Satyanadham Atluri, Emmett Witmer, John Leech, Luigi Morini, "PETROS3: A Finite Di fference Method and Program for the Calcu1ation of Large Elastic Plastic Dynamically Induced Defonnation of Multilayer Variable Thickness Shells", BRL Contract Report 60, November 1971.
1.68 Luigi Morino, John Leech, Emmett Witmer, "PETROS2: A Finite Difference Program for the Calculation of of Large E1astic Plastic Dynamically Induced Defonnation of General Thin Shell S", BRL Contract Report 12, November 1969.
,
1.69 Joseph M. Santi ago, "Formul ati on of the Large Oefl ecti on Shell Equations for Use in Finite Difference Structural Response Computer Codes", BRL Report 1571, February 1972.
1.70 Norris J. Huffington Jr., "Large Oeflection Elastoplastic Response of Shell Structures", BRL Report 1515, November 1970. 1.71 J. M. Dewey, Proceedings of the Royal Society, A324, 1971, pp.
·
...,
2.0 STIFFNESS MATRICES
2.1.0 ELEMENT FORMULATION: LINEAR 2.1.1 INTRODUCTION
The use of a Mindlin [2.1J or a Reissner [2.2-2.5J type plate theory in finite element formulations for plates and shells is well established and has been shown to give good results for moderately thick configurations
[2.6-2.14J. One of the chief advantages of this type of formulation is that independent displacement and rotation trial functions may be used and that these functions need only be Co continuous in order to meet the
compatibility requirements.
Ahmad et. al. [2.6J introduced one of the earliest applications of this type of plate theory to thick shell structures by incorporating it into the formulation of fully integrated superparametrie bi-quadratic and bi-cubic serendipity elements that where specified in Cartesian coordinates.
Although results where very good for thick plates and shells it was soon realized that, in elements where the transverse shear stresses are included and there is independant specification of displacements and rotations, excessive shear stiffness can result when the thickness is reduced [2.7J. Thi s phenomemon has been termed 111 ocki ngll and is fully explored and
discussed in Refs. 2.8 and 2.10. To obtain satisfactory results in these cases it was necessary to adopt special stategies such as reduced and selective integration [2.7-2.14J which decreased the apparent stiffness of the elements.
Further improvements have been made in the case of plates by using different types and orders of basis functions for the displacement and rotati on degrees of freedom [2.11-2.12J. The IIHeterosi Sll el ement [2.11J
for the transverse deflection and bi-quadratic Lagrange basis functions for the rotation degrees of freedom while McNeill's element [2.12J employs full integration with bi-cubic Lagrange basis functions for the transverse
deflections and bi-quadratic serendipity basis functions for the rotation degrees of freedom.
Leonard and Li [2.15J presented one of the earliest elements designed to deal with strongly curved arbitrary shells by using classical shell theory, to derive stiffness matrices for elements which approximate the shell geometry based on an interpolation of the global Cartesian coordinates of the shell surface. More recently a family of shell coordinate
quadrangular elements designed for deep shell analysis has been reported which uses classical thin shell theory and meets the requirements of compatibility of both displacements and rotations [2.16J.
The purpose of this chapter is to introduce an element which employs a Mindlin type shell theory and which may be used with equal confidence in any combination of thick/thin or deep/shallow shell configurations including the degenerate case of plates. It will be illustrated that complete integration of the element yields consistently accurate results and that reduced
integration is not only unnecessary, but can also lead to unrestrained artificial zero energy modes in situations of minimal restraint. This latter situation is of course unacceptable.
2.1.2 STRAIN-DISPLACEMENT RELATIONS
In order to to treat moderately thick shells it is necessary to include the transverse shear stresses ~yz and ~zx in the strain energy, while at the same time taking azz to be negligible [2.1J. This implies that the usual expressions for the displacements, in terms of the mid-surface deflections,
;
I
are not applicable. The normal deflection and rotation measure that are most convenient to use are a weighted average of the normal deflection over the thickness of the plate and a pair of rotation variables that are,
lIequivalent to but not identical with the components of change of slope of the normal to the undeformed middle surface" [2.2J. This is because it is necessary to allow the normal displacement to vary through the thickness when transverse shears are present [2.2-2.5, 2.18-2.19J and the weighted average interpretation allows this dependance to be included implicitly. With this in mind, the displacements are represented in the form
u
=
ü(x, y) + z<\Ix(x, y) (2.1)v
=
v(x, y) + z<\Iy(x, y) (2.2)w = w (x, y) (2.3)
where for
ü and
v
the superposed bar indicates a displacement at themid-surface and, as indicated above,
w,
<\Ix and <\Iy are the measures of normal displacement and rotation.In order to allow the use of shell coordinates the following linearized strain-displacement relations will be utilized [2.18-2.20J.
E
=
1.. [
~
+ Y-~a
+ W~a
] (2.4) xx a ~x f3 ~y ~z 1 [ Bv +!!. BP + w~
] (2.5) E -~ yy ~y a ~x ~z Yxy=
1..
f3 ~y ~u +1..
a ~v ~x _.}L af3 ~ ~x U ~a (2.6) - af3 ~y(2.7)
(2.8)
where (x, y, z) are orthogonal curvilinear coordinates at the mid-surface of the shell; u, vare the tangential displacements in the x and y directions respectively; w is the nonnal deflection and a and ~ are the Lamé
coefficients defined by [2.20],
~ A B(1 + C z)
- y
(2.9)
(2.10)
The curvatures Cx and Cy ' are defined as the reciprocals of the radii of
curvature, that is Cx
=
1/Rx' Cy=
1/Ry.Substitution of the expressions for u, v and w into the
strain-displacement relations [Eqs. (2.4)-(2.8)] results in strain
expressions in terms of the displacement and rotation components given by
=
.!
[
'Oü +i
'Oa + W 'Oa + Z ( 'O<Vx +~
ca) ]EXX a 'ox ~ öy 'Oz 'ox ~ öy (2.11)
(2.12)
1 [ a OÜ + ov - o~ - oa + Z ( o<)lx + OcJiy Yxy
-a13
oy ~--ox v - -ox U oy a -oy ~-OX oa <)Iy~~)
] (2.13) - <)I x oy-=
~
r
Qov
+ o::i - Vo~
+ Q,I. Z (,1.o~)]
Yyz I-' I-' Ol
DY
Ol I-''I'y - 'l'y Ol (2.14)1 [ OÜ o::i - oa oa
1
Y zx
=
ä a oz + ox - U oz + acJix - Z (cJix oz) _ (2.15)In order that the present results not be restricted to thin, shallow shells the factors I/a and 1/~ are expanded in binomial series,
1 1 2 2
a
=
A (1 - Cxz + Cxz - ••• ) (2.16)(2.17)
2
and are subsequently truncated to terms of O(z). This procedure results in strain-displacement relations of the form
where 1 ~ 1 M A "äx J!Jr 'óy C x 0 0 1 ~B 1 ~ Cy 0 0 Äff ~x 13 ~y B = 1 ~ 1 M 1 ~ 1 'óB 0 0 0 (2.19)
B 'óy - Äff 'óy Ti: ~x - Alr ~
0 0 -C y
B
1 ~ ~y 0 1-c
x 0A
1 ~ ~x 1 0-c
x ~ Cx 'óA 2 A 'óx - AB ~y -C x C 'óB -C 2-i-ax
-f
~y
-C Y BI=
Cx M _ AB ~y.:r
B 'óy Ê- ~ ~B AB 'óx -A
Cx ~ 'óx 0 2 -C 0 C-,t-
~y
Y 2 -C C X 0 X ~ A ~x 1 ~ 1 M A~ ÄIf ~y 1 'óB 1 ~ÄIr 'óx 'Ir 'óy
1 'ó 1 M 1 ~ 1 'óB (2.20) Ir 'óy - Ae" ~y A 'óx - AB ~x 0 -C Y -C X 0 (
3 C x 3 Cy 0 2
~Ê-B ay 2 C a x Tax and