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TRANSIENT ELASTO-PLASTIC RESPONSEOF PULSE LOADED BEAMS
by
SAMER MUHSEN HUSSAIN AL-HABIB B.Sc. (Civil Eng.), M.Sc. (Bridge Eng.)
A thesis submitted for the degree of Doctor of Philosophy
In the Faculty of Engineering of the University of London, and for the Diploma of the membership of the Imperial College
Department of Civil Engineering Imperial College
of Science, Technology and Medicine
November 1989
2
ABSTRACT
The reaction of beams to dynamic transient loads of such severity as to initiate plasticity and cause permanent damage has been studied in detail. An extensive review of the literature has been made to help identify the structural, physical and loading parameters which significantly affect the response, and to establish the adequacy, extent of application and practical utility of the prominent methods of mathematical analysis, covering both continuum and discrete methods. From this, it was concluded that a discrete model of the continuous mass - lumped flexibility type could provide a simple but accurate representation of the dynamic transient elasto-plastic response of beams.
A mathematical model has been developed for a new discrete beam element that brings together simplicity, efficiency and competitive accuracy. Both geometric and material nonlinearities are considered. The constitutive equations employed are based on force-deformation relations - as opposed to stress-strain relations - and yielding of the cross section as a whole. Parameters incorporated in the model are:
- Structural : transverse shear deformations, both elastic and plastic; rotatory, transverse and axial inertias; finite displacements.
- Physical : strain rate sensitivity; strain hardening; moment-shear andmoment-axial force interaction.
- Loading : shape and duration of applied load pulse.
A computer program is developed to encode the full process of numerical solution. This is based upon a central difference time-marching scheme - a direct explicit integration procedure - which proves to be simple, robust and effective for problems involving complex nonlinearity.
The proposed mathematical model is validated against existing experimental and numerical investigations. The solutions furnished are further analysed to contribute new insight into the nonlinear behaviour of beams under dynamic loading. A parametric study is conducted, employing the proposed model as the analytic tool, to explore the degree of influence of various response parameters on the elasto-plastic transient response of pulse loaded beams.
To my parents and brother Amer
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ACKNOWLEDGEMENTS
I began my research studies at King's College London in the Department of Civil Engineering, successively working on the assessment of impact factors for composite bridges and on the collapse behaviour of structures subjected to dynamic loading. I should like to acknowledge the part played in my study of these topics by staff of the Department of Civil Engineering and of the Computer Centre at King's College.
Subsequently, to redirect my research programme to a theme more suited to my particular taste and talent, I decided to transfer my research studentship to the Department of Civil Engineering at Imperial College. For the encouragement of my friends, Dr. A.C. Cassell, Dr. Hussein Jameel, Dr. Ali Al-Hamdani and Dr. Tewabech Zewde in helping me take this critical decision, I am profoundly grateful.
At Imperial College, the responsibility for supervising my research programme was taken by Dr. David Lloyd Smith, Head of the Systems and Mechanics Section. Largely, I owe the successful completion of this work to Dr. Lloyd Smith, who is a scholar with a great deal of generosity, courage and patience. In the regular and frequent discussions we had, he helped me get over the technical difficulties and develop new ideas; his pleasant manners and skills in teaching made it all look very easy. I am grateful.
I would like to extend my thanks to other members of the Imperial College staff : to Mr. Thomas Sippel-Dau, a senior analyst programmer at the Computer Centre, for his timely help with aspects of hardware and software; to Mrs. Kay Crooks, the Librarian in charge of the Departmental library; and to Miss Patricia O'Connell for the remarkable endurance and aptitude she had shown in typing this thesis.
Finally, I would like to express my thanks to Mr. Jawad Ali, a family friend, for his help and concern throughout.
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TABLE OF CONTENTS
Page
TITLE 1
ABSTRACT 2
DEDICATION 3
ACKNOWLEDGEMENTS 4
TABLE OF CONTENTS 5
PART—I THE FIELD OF STUDY AND ITS LITERATURE
CHAPTER 1 INTRODUCTION AND OBJECTIVES 91.1. TRANSIENT ELASTO-PLASTIC RESPONSE 91.2. OBJECTIVES OF THIS RESEARCH 10
CHAPTER 2 PARAMETERS AFFECTING THE RESPONSE 122.1. INTRODUCTION 122.2. MATERIAL PARAMETERS 122.2.1. Strain Rate Sensitivity 122.2.2. Strain Hardening 192.2.3. Material Elasto-Plastic Transition 202.2.4. Material Yield Criterion 24
Effect of Shear Force on Yield in Bending 25Effect of Axial Force on Yield in Bending 28Combined Effects of Shear and Axial Forces 29
2.3. STRUCTURAL PARAMETERS 312.3.1. Transverse Shear 312.3.2. Inertial Forces 352.3.3. Finite Displacements 362.4. LOADING PARAMETERS 382.4.1. Pulse Shape 38
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Page
CHAPTER 3 METHODS OF MATHEMATICAL ANALYSIS 443.1. INTRODUCTION 443.2. CONTINUUM METHODS 443.2.1. Elasto-Plastic Methods 443.2.2. Rigid-Plastic Methods 463.2.2.1. Standard Rigid-Plastic Method 473.2.2.2. Mode Form Approximation 493.2.2.3. Upper and Lower Bounds 513.2.3. Simple Elastic-Plastic Methods 533.3. DISCRETE METHODS 553.3.1. Lumped Parameter Methods 573.3.1.1. Lumped Mass - Continuous Flexibility 573.3.1.2. Lumped Mass - Lumped Flexibility 593.3.1.3. Continuous Mass - Lumped Flexibility 603.3.2. Finite Differences 613.3.3. Finite Elements 62
PART-H CONSTRUCTION OF A MATHEMATICAL MODEL
CHAPTER 4 STRUCTURAL MODELLING IN GEOMETRIC 64NONLINEARITY4.1. INTRODUCTION 644.2. DISCRETE ELEMENT AND NOTATION 644.3. RELATIONS AT ELEMENT LEVEL 704.3.1. Kinematic Relations 70
Relations in Incremental Form 70Relations in Differential Form 70
4.3.2. Kinetic Relations 72Relations in Finite Form 72Relations in Differential Form 77
4.3.3. Inertial Forces 814.4. RELATIONS AT STRUCTURE LEVEL 824.4.1. Kinematic Relations 824.4.2. Kinetic Relations 84
CHAPTER 5 CONSTITUTIVE MODELLING IN MATERIAL 87NONLINEARITY5.1. INTRODUCTION 875.2. ELASTIC CONSTITUTIVE RELATIONS 87
7
5.3. PLASTIC CONSTITUTIVE RELATIONS 905.3.1. Perfect Plasticity 90
Analytic Aspects 90Computational Aspects 93
5.3.2. Strain Rate Sensitivity 975.3.3. Strain Hardening 101
CHAPTER 6 TEMPORAL OP E R A T O R A N D C O M P U T E R 102P R O G R A M6.1. INTRODUCTION 1026.2. TEMPORAL OPERATOR 1026.2.1. Comparisons of Explicit and Implicit Operators 103
Explicit Methods : Advantages 103Explicit Methods : Disadvantages 104Explicit Methods : General Attributes 104Implicit Methods : Advantages 105Implicit Methods : Disadvantages 105Implicit Methods : General Attributes 105Summary 106
6.2.2. Central Finite Difference Explicit Operator 1066.2.2.1. Description and Implementation 1066.2.2.2. Size of Time Step 1076.2.2.3. Numerical Stability 1096.3. COMPUTER PROGRAM 1116.3.1. Structure of Program 1116.3.2. Flow of Computation 112
PA R T-m PRESENTATION AND COMPARISON OF RESULTS
CHAPTER 7 VALIDATION EXAMPLES A N D ANALYSIS OF 114RESULTS7.1. INTRODUCTION 1147.2. SIMPLY SUPPORTED BEAMS 1147.2.1. Comparison with ADINA 1147.2.2. Comparison with Experiments 124
Experimental Setup 124Edge Effects 124Impulse, Pulse Shape and Duration 127Sources of Error 128
Page
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Present Study 1287.3. CLAMPED BEAMS 1437.3.1. Comparison with Experiments and ADINA 1437.3.2. Comparison with Experiments 1617.4. CANTILEVER BEAMS 1697.4.1. Comparison with ADINA 169
CHAPTER 8 PARAMETRIC STUDY 1748.1. INTRODUCTION 1748.2. EFFECT OF TEMPORAL MESH 1748.3. EFFECT OF SPATIAL MESH 1788.4. EFFECT OF MATERIAL YIELD CRITERION 1798.5. EFFECT OF MATERIAL ELASTICITY 1878.6. EFFECT OF PULSE SHAPE 191
PART-IV CLOSURE
CHAPTER 9 CLOSURE 1999.1. REVIEW AND MAIN CONCLUSIONS 1999.2. INDICATIONS FOR FURTHER RESEARCH 204
REFERENCES AND SUBJECT BIBLIOGRAPHY 206
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CHAPTER 1
INTRODUCTION A N D OBJECTIVES
1.1. TRANSIENT ELASTO-PLASTIC RESPONSE
The response of a structure to a dynamic load of short pulse and of such severity as to initiate plasticity and cause permanent damage is usually called transient elasto-plastic response. It is affected by many different parameters that clearly distinguish it from static structural behaviour.
The influencing parameters may be associated with the nature of the material, the structure or its loading. Material parameters involve strain rate sensitivity, strain hardening, elasto-plastic transition and yield criteria. Structural parametersare inclusive of transverse shear deformations, inertial forces and geometry change. Lastly, loading parameters entail the shape and duration of the pulse of the applied load.
The diversity of the impelling parameters coupled with the time dependence of the response, render the analysis of a dynamically loaded structure quite complex. Available methods of analysis can be put into two categories : those by which a structure is analysed as a continuum, and others by which it is analysed in discrete form.
Continuum methods are mathematically rigorous and thus provide valuable insight into dynamic response. However, in order to be of practical assistance, this mathematical complexity is eased by many idealisations related to the material, the structure and its loading. Some of these idealisations or assumptions may be quite restrictive in effect - the assumption of infinitesimal displacements, for example. In methods which involve discrete variables, the idealisations may perhaps be localised - the restriction that plastic hinges may only occur at fixed sites, for example. Such discrete methods incorporate the finite element and finite difference methods which present powerful and versatile analytical tools, and have actually been the focus of research for the past three decades. However, these twomethods demand considerable computational resources and require (human) operator expertise.
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1.2. OBJECTIVES OF THIS RESEARCH
The main theme of this research is the provision of a mathematical means of dynamic analysis - within the category of discrete methods - that is both simple and predictive of real behaviour. Simplicity promotes and enhances understanding, and predictability inspires trust. These two qualities are particularly convenient in the highly complicated domain of nonlinear transient analysis. From this point of view, the effort invested in this research is considered worthwhile.
The structural form considered herein is that of the beam. This choice is believed to be consistent with the fact that beams exhibit most of the majorsources of complexity arising from mixed elasto-plastic behaviour in largedisplacements. An additional factor contributing to the choice is the prevalence - to a certain extent - of experimental evidence.
The specific objectives of this work are summarised in the following statements, in order of execution:
1. The identification of the parameters that significantly affect the transient elasto-plastic response of beams, and the assessment of their influence through a study of the literature.
2. Examination of the prominent mathematical methods of analysis of dynamicelasto-plastic response. This is done within the purpose of identifying thatmethod which, with some development, could have its predictive capability substantially improved, and yet would retain its essential simplicity. The aspects examined are:
- the degree to which these methods can adequately represent the general character of the response;
- specific limitations on applicability; and
- utility for easy and practical implementation.
3. Advancement of a mathematical model for an original beam element that, inboth structural and constitutive aspects, encompasses simplicity, efficiency and competitive accuracy. The intended field of application is the field ofnonlinear transient elasto-plastic dynamics.
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4. The constructing and implementing of a computer program which employs the proposed mathematical model.
5. Validation of the proposed mathematical model through a scheme of comparing results given by the computer program with relevant experimental and numerical studies.
6. A detailed observation of the evolution, predicted by the computer program, of the response variables with the intention of contributing further insight into the dynamic nonlinear response of beams.
7. The carrying out of a numerical parametric study to explore the degree of influence of some of the many structural, material and loading properties that have a bearing on both modelling and solution of problems in dynamic plasticity.
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CHAPTER 2
PARAMETERS AFFECTING THE RESPONSE
2.1. INTRODUCTION
The transient elasto-plastic response of structures is affected by many material, structural and loading parameters. In this chapter the parameters that most affect the dynamic elasto-plastic response of beams are presented by way of a review of the literature.
2.2. MATERIAL PARAMETERS
2.2.1. Strain Rate Sensitivity
Many structural metals under dynamic loading exhibit substantially different stress-strain characteristics than they would under quasi-static loading. Figure 2.1 shows the ranges of strain rates encountered in engineering problems. In general, for metals exhibiting yielding, the yield stress increases continuously with an increasing strain rate. The yield stress is therefore a continuous function of strain rate throughout the plastic response, Figure 2.2. The proportional change in the yield stress of metals ranges from being as large as twice, and sometimes as high as three times, the static yield stress for mild steel to being almost negligible for some high tensile steels. The dependence of yield stress in dynamic response on strain rate is often referred to as viscoplasticity.
In physical terms the effect of strain rate is thought to be associated with the time needed for atomic lattice imperfections, such as dislocations, to form and move across crystal grains, thereby permitting plastic flow to occur (Cottrell 1957). Consequently, the strain-rate dependency of the yield strength is highly sensitive to the metallurgical state of metal, such as, for instance, to variations in alloy content and heat treatment. The implication that this is predominantly a material phenomenon is further supported by experimental results in which approximately the same increase in yield strength is produced by the same strain rate in: shear, torsion, tension and compression, through explosive pressure or through impact
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Liquid behaviour of solids
Plastic behaviour of solids
_____ Elastic behaviourof solids
Inertia forces neglected
Creep
I
Static loading-
— Inertia forces important------------►^-Wave propagation important-**
Deep drawingJ Strip rolling| j Meteor impactj I Ballistic impact
Land vehicle impact Metal cutting
■Rapid loading—► ◄— Hypervelocity —I l
_ i___ i___ i____i___ i____i___ i___ i10'6 10' 5 10"4 10'3 10'2 10' ' 1 10 102 103 104 105 1 06 107 108
| Strain rate (s"1!
ZOD
U11 ?
TO0)O)c C 1 -o i
t5ro'<D■o 1 8
o -_ j o 1 o o.:«l
cc ! § c■ *->
s\ JZ £* i p i J
Figure 2.1. Ranges of strain rate encountered in engineering problems. After Macaulay (1987).
1 4
(a) (b)
(c) (d)
Figure 2,2. Dynamic stress-strain relations:
(a) For elastic3 visco-perfectly plastic material with constant strain rate k.
(b) Same as in (a) but with e = k (e).
(c) For elastic3 work-hardening viscoplastic material with constantk.
(d) Same material as in (c) but with k = e (z ).After Ferzyna (1966).
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(Symonds 1967).
The first clear demonstrations that the mechanical behaviour of metals under dynamic loading conditions could be significantly different from that under static loading were probably those given by J. Hopkinson (1872) and later by B. Hopkinson (1905). Subsequently, numerous studies have been carried out into both the metallurgical and analytical aspects of the phenomenon of strain rate sensitivity in metals. Some of the earlier work concerning the metallurgy of strain rate sensitivity, and reflecting its complexity, is that of Marsh and Campbell (1963) and Campbell (1970, 1973). More recent communications on this matter are those of Clifton (1983) and Blazynski (1987), and a review of additional relevant publications is provided by Nicholas (1982).
The analytic aspects of strain rate sensitivity concern the formulation of constitutive relations and their utilisation, perhaps only in approximate manner, in the establishing of mathematical solutions to dynamical problems.
Perzyna (1962), employing the same general assumptions as those used by Malvern (1951) for one-dimensional problems, devised constitutive relations for multi-axial stress states. The flow law for any generic point on the yield surface is determinable from the associated initial strain rate. However, it is conceivablethat, due to the complicated interaction between rheologic and plastic effects, the constitutive relations may depend upon the complete evaluation of the strain rate history of the material.
The relative merits and disadvantages of the various constitutive relations have not yet been thoroughly examined (Bodner 1984). According to Malvern (1984), the analytical constitutive laws that have actually been applied, or are convenient for application to engineering problems, are those of Perzyna (1966) - the multi-axial overstress model - and of Bodner and Partom (1972); despite the inadequacies of these two models, they represent an improvement over the widespread practice of using a rate-independent plasticity model which includes a "dynamic" yield strength.
A simple empirical form of non-linear viscoplastic constitutive equation was deduced by Cowper and Symonds (1957) from experimental data obtained by Manjoine (1944) on uniaxial tensile and compressive tests on mild steel. This equation has the form,
16
D 1 +Vp
( 2 . 1 )
where
Cp = dynamic y ie ld s t r e s s ;Cg = s t a t i c y ie ld s t r e s s ;e = s t r a in r a te (p er seco n d );D = m a ter ia l co n sta n t w ith s t r a in r a te d im en sion s;P = m a ter ia l c o n sta n t;
The remarkably close fit of equation (2.1) to the uniaxial test data of Manjoine for mild steel and of Hsu and Clifton (1974) for Alpha-titanium is demonstrated in Figure 2.3. Values of material constants D and P appropriate for uniaxial tension and compression tests on typical engineering metals are given in Table 2.1.
M ater ia l D ( s ’ ’ ) P T est data byM ild S te e l 4 0 .4 5 M anjoine (1 9 4 4 ) , and
Aspden and Campbell (1966)S t a in le s s S te e l (304) 100 10 S te ic h e n (1971)Aluminium A llo y (6061-T 6) 6500 4 Bodner and Symonds (1962)A lpha-T itanium (T i-50A ) 120 9 Hsu and C l i f t o n (1974)T itanium A llo y (6A1-4V) 3 . 4x lO G 7 .6 Lindholm and B essey (1969)
Table 2 .1 . M a ter ia l c o n s ta n ts D and P fo r some common m e ta ls .
For the first three entries in Table 2.1, the variations in dynamic yield stressassociated with equation (2.1) are compared in Figure 2.4 for values of strain rate from zero to 100 s“ 1.
Clearly, the non-linear variation of yield stress with strain rate makes forconsiderable mathematical complexity when strain rate sensitivity is incorporated intothe dynamical model. Attempts to account for strain rate sensitivity in anapproximate manner were therefore eagerly pursued.
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Figure 2.2. Nonlinear viscoplastic equation (2.1), (a) fitted to test data for mild steel by Manjoine (1944), and (b) fitted to test data for alpha titanium by Hsu and Clifton (1974).After Symonds and Chon (1974).
Figure 2.4. Variation o f the ratio of dynamic to static yield stresses with strain rate.
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One of the early attempts in approximate corrections for strain rate sensitivity was that by Parkes (1955). The order of strain rates considered were recorded from experiments on mild steel cantilever beams under impact loading, and the corresponding dynamic yield stresses were deduced from comparison with the test data of Manjoine (1944) on mild steel in uniaxial tension and compression. A more representative approximate approach was later introduced by Perrone (1965),it was suggested that the dynamic yield stress is calculated at the strain ratepredicted by the elastic analysis when yield is first reached; thereafter the plastic flow is assumed to be constant throughout. The essence of Perrone's approximation was the fact that in using the exponential equation (2.1) to represent viscoplasticity, almost 80% of the rise in yield stress occurs in the low range of strain rate (up to 10 s- 1 ), as indicated by Figure 2.4. This approach was laterapplied to plates (Perrone 1967), rings and tubes (Perrone 1970) and frames(Perrone 1971).
The immediate appeal of the approximation made by Perrone (1965) is clear from the analytic point of view. However, there are some practical difficulties, such as deciding on what strain rate to choose in cases where secondary (transient) plastic hinges form before the plastic hinges of the dominant mode of deformation. Another difficulty is that reported by Symonds and Jones (1972); it was noted that when the static yield stress is employed in the approximation to define the strain, the dynamic yield stress given by (2.1) is exaggerated, thus causing the correction factor to be overestimated.
This notion of using constant dynamic yield stress over the entire motion, as determined from strain rates representative of the early part of the motion, would apply particularly well to cases where the geometric shape of the displaced configuration remains constant throughout, its magnitude varying with time. When finite displacements must be considered, with a resulting change in the structural response mechanism, application of Perrone's approach must be considered tentative.
Despite these practical difficulties, Perrone's approach remains widely used. In a review by Zukas et al. (1982) it was reported that most analyses of high-rate deformation, such as ballistic impact, have used rate independent plasticity laws that incorporate a dynamic yield strength to account in an average way for rate effects.
2.2.2. Strain Hardening
Strain hardening refers to the increase of yield stress in plastic flow with increasing strain. Strain hardening occurs at room temperature with most metals
2 0
(except pure lead, tin and cadmium which only strain harden below typical room temperature), and under both static and dynamic loading.
In physical terms, strain hardening is due to the generation, movement and interlocking of dislocations during plastic flow, dislocations being small defects in the cyrstal lattices of crystalline solids. The greater the degree of straining, the larger the number of dislocations generated. Owing to their mutual interaction, larger stresses are required to enforce their movement, and hence cause further plastic flow.
In structural problems (Symonds 1967), the need to consider strain hardening may arise, not so much because strain hardening is large, but because some physical feature of the actual response disappears when perfectly plastic behaviour is assumed. For example, in the case of the end impact of a long thin rod, if perfectly plastic behaviour is assumed, then there would be no propagation of plastic stress and strain waves away from the impacted end; an infinitesimal segment of the rod at the struck end therefore acquires infinite strains.
Bodner and Symonds (1962), in a theoretical study of the strain rate sensitive behaviour of cantilever beams subjected to impulsive loading, observed that the incorporation of strain hardening can cause the transfer of energy from one mode of deformation to another mode which is a less efficient absorber of energy. On the other hand Symonds and Jones (1972) in their analysis of the response of mild steel clamped beams loaded impulsively, noted that the correction needed to account for strain hardening in a rigid-perfectly analysis is relatively unimportant compared to effects of strain rate sensitivity and finite displacements, even with a midspan transverse displacement equal to seven times the beam thickness. This sameconclusion was also supported by the theoretical analysis of Jones (1968) on axisymmetric impulsively loaded annular plates.
2.2.3. Material Elasto-Plastic Transition
In elasto-plastic analyses of beams requiring moment-curvature relations, the procedure is often based on the simplifying assumption that the section of the beam is either entirely elastic or entirely plastic. For a beam of rectangular section subjected to pure bending, the variation of the normal stress, as the applied moment is progressively increased, is shown in Figure 2.5. The transition state during which a section is partly elastic and partly plastic, Figure 2.5(d), is disregarded; and further that the section yields only when the fully plastic moment Mp is reached, Figure 2.5(e).
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Figure 2.5. (a) Cross-section o f a rectangular beam, (b) to (e) Variation o f normal stress distribution as the applied moment is progressively increased.
Figure 2.6. Relation between bending moment ratio M/M^ and curvature ratio k/ ky . After Coates et al. (1980).
2 2
The relation between bending moment M and curvature k for a beam ofrectangular cross section is shown in nondimensional form in Figure 2.6. The relation is linear up to point A, which represents the onset of yielding M = My, after which there is the transition state (curve AC). The dotted lines AB-BC form the commonly employed idealisation of the curve AC; here point C corresponds to the fully plastic moment Mp. Theoretically, the full plastic moment Mp is reached only when the axial strain of the extreme fibre in the cross section is infinite.
The ratio of the plastic moment of a section to its yield moment is called the shape factor a = Mp/My-, it is solely a function of the shape of the cross section. The shape factor for a circular cross section is 1.7, for a rectangular 1.5, and for British Universal Beams and American Wide Flange beams 1.15. This is indicated in Figure 2.6 by the reciprocal of a for rectangular and Universal Beams crosssections. Disregarding the transition state amounts to assuming a shape factor of unity.
The effects of the transition state on the dynamic elasto-plastic response have rarely been studied. In a general consideration of the hysteretic (steady-state) behaviour of materials, Iwan (1966) has concluded that only if the level ofexcitation is low does the sharp-cornered elastic-plastic approximation produce conservative estimates of the peak response. Iwan's study was based on an approach which views the actual material as being made of a series of elastic,perfectly plastic elements.
In another study, Raghavan and Rao (1978) investigated the effect of thetransition state through a comparative evaluation of the response as predicted bytwo independent approaches: (1) a stress-strain approach in which a layered model represents the successive penetration of yielding through the depth of the cross section; and (2) a moment-curvature approach which considers instantaneous yielding of the section as a whole (shape factor of unity). The structure in bothapproaches was represented by a finite element discretisation. The main conclusions were:
1. For load durations so short that the load pulse may be replaced by an impulse, the moment-curvature approach gives an upper bound to the actual mean displacements, as shown in Figure 2.7(a).
2. For pulse durations of less than about half the fundamental period, themoment-curvature approach predicts reliably good results, Figure 2.7(b).
23
(a) (b)
3.0-
(c)
M = moment-curvature approach.5 = stress-strain approach.0 = nondimensional time.6 = nondimensional displacement
q = nondimensional loading.X = nondimensional velocity.
(midspan).
Figure 2.7. Histories of nondimensional displacements at midspan:
(a) Impulsively loaded clamped beams, (b) Clamped beams subjected to pulse load, (c) and (d) Clamped beams and clamped square plate r e s p e c t i v e l y s u b j e c t e d to step-rise3 indefinite duration pulses. After Raghavan and Rao (1978).
2 4
3. For longer durations of the pulse, a disregard of the transition state leads to significant underestimation of the peak displacement - an underestimate of approximately 12% is shown in Figure 2.7 (c and d).
2.2.4. Material Yield Criterion
Here the effects of shear and axial forces on yield in bending of beams are reviewed.
There exists a true relation of interaction, in the sense of the yield condition, between the bending moment M and the axial force N at any cross section of a beam, irrespective of general loading and geometry. No such unique interaction relation exists between bending moment M and shear force Q at a cross section; plastification can only be determined through establishment of the correct continuous distributions of stresses in the region of the cross section, and is in any case dependent upon the loading distribution.
The case of a beam under combined shear and bending is specified by one normal stress parallel to the longitudinal axis of the beam and two shearing stresses. If the y axis is taken as the longitudinal beam axis, then these stress components are
°y y = a ’ Txy Tx > Tzy Tz
In the most general case, all three may be arbitrarily distributed throughout the beam - subject to equilibrium and stress boundary conditions. If the beam is symmetric and loaded in one plane only, say the load is parallel to the z axis, then the shear stress rx may reasonably be neglected. Accordingly the yield condition may be written in the form,
a + cl r < a (2 .2 )z y
where a is a constant of a magnitude depending on the yield criterion chosen - a = 2 for Tresca yield criterion and a = V~3 for Mises yield criterion - and Uy is the static yield stress in simple tension.
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Effect of Shear Force on Yield in Bending
The general effect of the presence of shear forces is the reduction in the value of the plastic moment of the cross section. In connection with static limit analysis Drucker (1956) has stated that there is no reason to expect a unique interaction relation between plastic moment and shear force for a given cross section. This is because the stress distribution across a section of an elastic member depends on the whole loading and support conditions of the member. For example, a cantilever of length L carrying a load F at the free end would have the same local situation at the clamped end - shear force F and a bending moment FL - as would a cantilever of length 2L carrying a uniformly distributed load F. However, it is common practice to employ the interaction relationobtained from a specific loading distribution for other cases involving quite different distributions of loading.
An analysis presented by Neal (1968, 1977) utilised states of plane stress for a cantilever of rectangular cross section with breadth b, depth d and length L, carrying a transverse load F at the free end. Equilibrated systems of plane stress were adjusted to be everywhere safe, so that the yield criterion is not violated but is just met at the critical section - the fixed end. Mises yield criterion was used, and nowhere did shear stress rz exceed the yield stress in pure shear 7y = UylJ~2>. The interaction relation suggested was
subject to the requirement that
wherein Qp = bd7y. Condition (2.4) is equivalent to L/d > 0.433.
In a study by Horne (1951), the growth of central yield zone of the cross section in the neighbourhood of the fixed end was considered. The yield criterion used was that of Tresca. The study gave results similar in form to equation (2.3), differing only in the value of numerical constants.
MM,V
2(2 .3 )
2MM,P
(2 .5 )
2 6
Drucker (1956) developed a lower bound solution which was based on more hypothetical stress distributions than in the previous two studies. Again the Tresca yield criterion was used and the interaction relation proposed by Drucker was
MM ( 2 . 6 )
which is represented in Figure 2.8(a). Since M = QL and Mp/Qp = d/2, the effect of increasing load on a particular cantilever is given by the radial line
M__ 2L Q_Mp d Qp ( 2 .7 )
Lines drawn for the two cases L/d = 1 and L/d = 2, Figure 2.8(a), show that M falls below Mp by less than two or three per cent if L/d > 2 - for L/d < 2 the structure can no longer be analysed as a beam. This indicates that the shear effect is entirely negligible for a rectangular beam in static loading.
However, this conclusion does not necessarily apply to conditions of dynamic loading. Under such conditions, applied loads may be considerably greater than those which would cause failure by plastic collapse under statical loading. Consequently, shear forces may become large enough to have a substantial effect on the value of the fully plastic moment, as shown in several different theoretical studies (Symonds 1968, Jones 1976, Nonaka 1977, De Oliveira 1982, Liu and Jones 1988) and experimental investigations (Menkes and Opat 1973, Liu and Jones 1988).
Interaction relations for other beam cross sections have been developed, especially for I-sections where shear effects are larger than for rectangular sections. Notable communications in this respect are those by Neal (1961b, 1961c) and Heyman and Dutton (1954). The relation developed by Neal (1961b) is shown in Figure 2.8(b), with a part enlargement in Figure 2.8(c). The sudden reduction in plastic moment, revealed in the former figure, develops when the applied shear force approaches the plastic shear force capacity QpW of the web area - demonstrated by the experiments of Heyman and Dutton (1954). Also shown in these figures are radial lines for particular ratios L/d, where d, = d - df, and df is the flange thickness. In the static problem, shear effects are negligible for L/d > 4; again this does not imply the same conclusion for dynamic problems.
The simplest form of interaction relation for any shape of cross section is that shown in Figure 2.8(d) - a simple square interaction diagram. In general, when
27
(b) (c)
Figure 2.8. Effect o f shear force on plastic moment of cross-section:
(a) Rectangular cross-section : solid curve for N = 0 (Drucker 1956)3 and dashed curves for 0 < N < 1 (Real 1961a). (b) Typical I-section3 (8WF40)3 (Real 1961b). (c) Fart enlargement of (b). (d) Simple square interaction diagram.
2 8
one needs to consider the effect of shear force upon yield in bending, the following observations of Drucker (1956) should be borne in mind:
1. If a unique interaction diagram between M and Q were to exist, it must be
2. The interaction between M and Q is not uniquely determined by the properties and geometry of the cross section; it depends upon the geometry and loading of the entire beam.
3. All things considered, it does appear that the concept of an interaction curve has enough value to warrant a selection of an approximate relation.
Finally, it may be remarked that few dynamic problems have been considered in which simultaneous plastic deformations in both flexure and shear are allowed to occur in association with an interaction relation other than a simple square.
Effect of Axial Force on Yield in Bending
There is a true interaction relation in what concerns the yield condition relating axial force N and bending moment M. This is the case at any cross section of a beam, irrespective of general loading and geometry. Axial forces and bending moments are both produced by suitable distributions of tensile stress, and their interaction relation can be worked out on the basis of yield induced by uniaxial stress.
In the presence of axial load, either tensile or compressive, the neutral axis of beam no longer divides a section into two equal areas. The plastic moment of a doubly symmetrical section, such as a rectangular or I-section, formed from a homogeneous metal, is always reduced by the presence of axial force. The case of non-symmetric sections bent in the plane of symmetry, such as the T-section, is not so straightforward.
For a rectangular section, the plastic moment varies parabolically with the axial force, for example Horne (1979),
convex.
2MM,
1 N ( 2 . 8)P N.Pj
2 9
where Np is the axial force which induces yield throughout the cross section in the absence of bending moment. This relation is shown as the upper curve (No. 1) in Figure 2.9. The lower curve (No. 2) shows the linear relation between the axial force and the bending moment My at which an extreme fibre first develops the yield stress. The corresponding interaction relations for a typical I-section are also shown in Figure 2.9, curves No. 3 and No. 4.
For a mean axial force equal to 10% of the yield force, N/Np = 0.1, theplastic moment of a rectangular section is reduced by only 1%, and that of a typical I-section by about 2%. This is why, in many practical problems of the small displacement of beams under static conditions, the effect of axial forces on plastic moments is ignored, provided that there are no buckling effects. In other structures, such as arches and rings, axial forces would be effective even for small displacements, as they are required for equilibrium.
In practical problems of dynamic loading causing large displacements, axial forces must often be considered since they may play a principal role in determiningthe load-displacement characteristics of the structure. For example, in the case ofa clamped beam of rectangular cross section subjected to transverse pressure ofhigh initial intensity, it was found that the growth of displacements to magnitudes as large as, or larger than, the beam depth would cause the beam to behave like a plastic string or membrane carrying axial force Np without bending resistance (Symonds and Mentel 1958, Nonaka 1967, Symonds and Jones 1972, Liu and Jones 1988).
In order to simplify theoretical analyses that incorporate effect of axial force on yield in bending, approximate yield criteria are employed. In the case of a rectangular cross section, a square yield criterion - Figure 2.10 - has often been used (Jones 1967, Jones 1971, Symonds and Jones 1972, Liu and Jones 1988).
Combined Effects of Shear and Axial Forces
The combined effects of Q/Qp and N/Np on M/Mp have been considered by Green (1954). He obtained an upper bound solution for a cantilever of rectangular cross section in plane strain subjected to a static end load. Horne (1958) extended Heyman and Dutton's (1954) local analysis, of bending-shear interaction in an I-section, to include the effect of axial force. Neal (1961a, 1961c) developed a lower bound solution for cantilevers of rectangular and I-section. Furthermore,Neal (1961a) suggested that Drucker's (1956) equation (2.6), for bending-shear interaction in the absence of axial force, can be an equally good approximation when N is different from zero if the following substitutions are made:
30
_N_Np
Figure 2.9. Effect of axial force on plastic moment o f rectangular and I-sections. After H o m e (1979).
Figure 2.10. Interaction relations of axial force and bending moment for rectangular cross-sections : (a) Parabolic3 (b) circumscribing squares and ( b r) inscribing square. After Symonds and Jones (1972).
3 1
12
The suggested formula for the effect of shear and axial forces on bending moment for rectangular cross sections is therefore,
M
MP1
(2 .9 )
This is exact for Q/Qp = 0, see equation (2.8). Equation (2.9) is plotted in Figure 2.8(a) for different values of N/Np.
2.3. STRUCTURAL PARAMETERS
2.3.1. Transverse Shear
The transverse displacement of real beams depends on both flexural and shear deformation. Shear deformations are usually neglected in the static and dynamic analyses of beams because, for slender beams, their effect is usually small. However, shear deformations can become important for short beams, higher modes of response (Jones and Wierzbicki 1976, Jones and Guedes Soares 1978), and severe dynamic loading conditions, where the failure may well be a shear failure at the supports (Menkes and Opat 1973, Liu and Jones 1987). Figure 2.11 shows the permanent displacement of a set of clamped aluminium beams in which each successive beam is subjected to a greater intensity of the impulse that is distributed along the whole span. It may be noted that when a shear failure is obtained, it occurs very early in the response, before any significant plastic deformation has developed.
In problems of dynamic plasticity, analytic solutions that account for yielding through bending-shear interaction are often simplified by assuming that the beam
32
Figure 2.11. Permanent displaced configurations of a set of clamped aluminium beams as the applied blast pressure is increased from beam No. 1 to beam No. 11. After Menkes and Opat (1973).
3 3
material is rigid-perfectly plastic and obeys a square yield criterion (Salvadori and Weidlinger 1957, Karunes and Onat 1960, Symonds 1968, Nonaka 1977, Jones and De Oliveira 1982, Jones and Song 1986).
In a recent investigation, Jones and Song (1986) considered the response of simply supported rigid-perfectly plastic beams subjected to partially distributed pressure loading. The material had finite shear strength and obeyed a square yield criterion relating bending and transverse shear force, Figure 2.12. It was shown that the dynamic response is governed by nine different patterns of initial motion: pure bending, pure shear sliding, or a combination of sliding and bending. The curvature depends on the dimensionless parameters v, /xQ and rj, where v - Qp l/2Mp is a dimensionless ratio of shear strength to bending strength, \lq - PQ l 2/2Mp a dimensionless measure of the load intensity pQ compared with the bending strength, and rj = 1^1 is a ratio between the loaded length 1, and the span, as shown in Figures 2.12 and 2.13. For a pressure load whose time distribution is that of arectangular pulse it was concluded that the influence of transverse force Q increases as v decreases. Its influence on the transverse displacement is negligible when the shear strength is high {v > 1.5) and the load is well distributed (0.5 < rj < 1.0). The transverse shear force assumes greater importance when the load becomes more concentrated (rj < 0.5). For rj - 0.1, for example, the effect of the shear force can only be ignored if v > 6 .
In similar analyses of beam, Symonds (1967, 1968) and Nonaka (1977) have concluded that the effect of the transverse shear force is generally unimportant for beams with rectangular and other compact cross sections, since r — 1/d is generally large. For box and I-sections, however, v is smaller (v = l/7d for the American Wide Flange I-section 8WF40), and therefore shear effects can be important for beam spans less than about ten times the depth of cross section.
The discretisation of the beam continuum into finite elements makes consideration of the effect of transverse shear deformation upon dynamic response much less difficult. One such analysis is that undertaken by Wen and Beylerian (1967) in which the elasto-plastic dynamic response of beams was predicted from force-deformation relations. Shear deformations and rotatory inertia were included, and various yield conditions for bending-shear interaction were considered. The study was mainly concerned with the behaviour of beams of I-section. It was concluded that shear effects should be taken into account, even for beams of quite usual proportions. For example, in the case of a simply supported beam of 10 ft span loaded uniformly by an exponentially decaying pressure pulse in which the peak force is equal to the elastic limit load, the permanent shear deformation could account for as much as (35%) of the total permanent displacement. The
3 4
P o f (t )
QNVA
0i
\ * 1
-1 0 1 kw
-1
Figure 2.12. Simply supported beam subjected to partially distributed loading : its configuration3 kinetics and interaction relation.After Jones and Song (1986).
Figure 2.13. Patterns o f initial motion : (a) Pure bending.(b) Pure sliding and combined sliding and bending. After Jones and Song (1986).
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percentage could be much higher for clamped beams, and is dependent on the form of the yield condition. It was indicated that the reliability of the analysis was examined only empirically by the apparent convergence of displacements, moments, and shear forces as the spatial mesh was made finer.
Tuomala, Mikkola and Wolf (1984) employed a finite element stress-strain model to investigate the geometrically and materially nonlinear response of beams and frames in which transverse shear deformations and rotatory inertia were considered. It was concluded that the effect of these two parameters on the deformation of dynamically loaded beams is influenced by the boundary conditions and loading type.
2.3.2. Inertial Forces
In a dynamical problem, the external forces may be considered as comprising the actual applied loads and some additional transient forces called inertia forces. The latter are dependent on the mass of beam and its distribution throughout thespan, and on the acceleration attained; in this sense they are time-dependent ortransient. Inertial forces for the planar motion of beams are therefore associatedwith each of the corresponding components of acceleration - that is, for transverse, rotational and axial motion.
For beams, usually the type of loading of most practical concern is thattransversely applied along the axis of the beam. Therefore, the dominant inertialforces are those due to transverse motion. Transverse inertial forces are the major constituents of a dynamic problem of beams and as such are normally included in continuum and discrete methods of analysis.
Rotatory inertia, is the inertia associated with rotation of the beamcross-section during flexural deformation. In a review of contributions on the transient response of Timoshenko beams, Al-Mousawi (1986) has nominated Bresse (1866) as the first to include a term for the effect of rotatory inertia in theequation of motion for lateral vibration, and also the first to discuss the effect of non-uniform shear stress distribution over a cross section. Al-Mousawi also reported that Timoshenko (1921) was the first to include in the equation for the flexural vibration of elastic beams effects of rotatory inertia and shear deformation, and therefore beams in which both parameters are considered are often calledTimoshenko beams. The effect of rotatory inertia on the elastic flexural vibrationof beams is to lower the natural frequency from that predicted by the simple flexural (Euler-Bernoulli) theory of beams. However, the effect of rotatory inertia
36on the dynamic plastic response of beams is not so clear. According to Jones and De Oliveira (1979) and Tuomala, Mikkola and Wolf (1984), it appears that its effect on midspan displacement is influenced by the kind of boundary conditions of the beam and on the distribution of the applied loading. Jones and De Oliveira have also concluded that rotatory inertia exercises the greatest influence on beams of rectangular cross section, as opposed to those of I-section; the largest reduction in the maximum transverse displacement is 11% approximately.
The other inertial forces are those due to axial motion. In the dynamic transverse loading of beams, axial inertia forces begin to affect the response only when there is a significant geometry change - due to large displacements. The effect of axial inertia on the dynamic response of beams has rarely been mentioned in the literature. Neglecting axial inertial forces would indicate a constant axial force throughout the beam span.
2.3.3. Finite Displacements
For the simplest dynamic modelling of beams, the effects of finite displacements are neglected. This means that the kinetic equations are founded on the undeformed configuration of the structure, and that the response predicted is valid only for infinitesimal displacements. Such problems are invariably easier to solve than when large displacements are allowed.
Finite displacements imply a significant change in the geometry of a structure. This induces in-plane axial forces (membrane forces) which strongly influence the behaviour of axially restrained beams subjected to transverse dynamic loads. When axial forces become significant, interaction of bending and extension plays an important role. When the beam acquires a maximum displacement, of the order of the beam depth or greater, further displacement is then mainly governed by the axial force (Symonds and Mentel 1958, Florence and Firth 1965, Humphreys 1965, Ting 1965, Nonaka 1967, Jones 1971, Symonds and Jones 1972, Jones 1979, Vaziri, Olson and Anderson 1987).
The influence of finite displacements on the dynamic response of rigid plastic, impulsively loaded, fully clamped beams is shown in Figure 2.14, after Jones (1979). It is evident from Figure 2.14 that the classical infinitesimal rigid-plastic theory is inadequate for this particular problem when displacements are larger than the beam thickness.
The difficulty in obtaining exact theoretical solutions, considering geometry
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Wm = maximum -permanent transverse displacement, d = depth of beam.
x = p v p r-/Mp .
p = mass per unit volume o f beam.
VQ = initial velocity.I = half the beam length.
Mp = Oy d2/4.
Figure 2.14. Maximum permanent transverse displacements of a fully clamped beam loaded impulsively : □, Aj Oj 0 experimental results on 6061-T6 aluminium beams3 after Jones (1971).
experimental results on mild steel beams3 after Symonds and Jones (1972). • (a - e) numerical finite difference resultsj after Witmer et al. (1963). (T)infinitesimal analysis (bending only). @ rigid-plastic analysis (RPA) using circumscribing M- N yield criterion3 after Jones (1971).(f)RPA using inscribing M-N yield criterion. (4) RPA using exact yield criterion3 after Jones (1971). (5)as in (2)with strain rate correction. (6)as in (s) with strain rate correction. After Jones (1979).
3 8
change, has led many researchers to develop approximate methods and bounding procedures. Some of these studies are discussed in Chapter 3.
2.4. LOADING PARAMETERS
2.4.1. Pulse Shape
Dynamic loads that could inflict permanent damage on structures are the result of many different causes, such as blasts, collisions, wind, earthquakes ... etc. Transient dynamic loads are often single pulse loads. The actual shape of a pulse is commonly idealised into one of the standard shapes shown in Figure 2.15.
One of the early studies on the dynamic load characteristics, in plastic bending of beams, was by Symonds (1953). The response of rigid-plastic free-free beams, subjected to concentrated force at midspan was examined for different pulse shapes, namely rectangular, half-sine and triangular pulses. It was concluded that, for the same peak force and impulse: a load pulse of half-sine wave shape approximately averages the midspan displacements for rectangular and triangular shapes; consequently, a half-sine pulse shape can be used to average force pulses of mostshapes that are likely to occur in practice. However, if, for analytical convenience,one idealises a pulse shape as rectangular, the overestimate of permanent displacement might be expected to be less than 20 or 30 percent.
An impulsive load is defined as a load of infinite intensity applied during a vanishingly small time interval. This idealisation of an actual dynamic load often makes for convenience, although it may lead to significant errors in some circumstances. For example, for the problem of a simply supported, rigid-plastic beam, subjected to uniformly distributed pressure, Symonds (1968) indicated that,for an exponentially decreasing load pulse, the peak load must be at least ten timesthe static collapse magnitude to give a final displacement within 10% of that of the impulsive load case.
Abrahamson and Lindberg (1971), in a study of pulse loads for pinned and clamped rigid-plastic beams, loaded uniformly, concluded that, for a specified impulse, the peak load is the important factor for long-duration pulses, and that the impulse is the important factor for short-duration pulses. Consequently, pulse shape becomes most significant for mid-duration pulses in which both the peak load and the impulse would have significant affect on the permanent displacement.
Stronge (1974, 1982) studied the transient response of simply supported,
p(t)
(c) LINEAR DECAY
(e) TRIANGULAR
p(t)
t(b) RECTANGULAR
(d) EXPONENTIAL DECAY
( f ) H A L F - S I N E
Figure 2.15. Standard shapes of pulses of applied load. After Youngdahl (1970).
4 0
rigid-plastic beams loaded uniformly by a pressure pulse, without and with aconstant tensile force respectively; displacements were considered infinitesimal. Both studies were aimed at determining the pulse shape that is most effective in inducing permanent displacement at midspan for a specific impulse. Pulses investigated were of the following shapes: rectangular, two rectangular pulses separated by a no load period, step rise exponentially decaying, and various triangular. It was concluded that for a specified impulse,
1. Axial tension has no influence in determining the shape of the most effective pulse.
2. An impulsive pressure causes the largest midspan displacement of arigid-plastic beam.
3. For different pulse shapes of different peak pressures and durations, beam displacement is maximised by applying the largest pressure in the shortest time; therefore, generally speaking, the efficiency of a pulse depends on the peak pressure rather than on details of the pulse shape, Figure 2.16 (a and b).
4. Pulses of approximately equal duration and maximum pressure, but of different shape, resulted in differing final beam displacements; between two triangular pulses, the more compact around t = 0 results in the larger permanent displacement (Youngdahl (1970) describes the compactness of a pulse about t = 0 in terms of a higher effective pressure).
5. The difference in the effect between the two triangular pulse shapes is greatest for low pressures, because then most displacement occurs while the pulse is acting, whereas with high pressures sufficient beam momentum is developed for the displacement to continue after the pressure had ceased.
A study of pulse shape effects, on the dynamic response of simply supported rigid perfectly plastic beams, was recently undertaken by Jones and Song (1986). The beam material has finite shear strength, and obeys a square yield criterion for the interaction between bending moment and transverse shear force. The transverse dynamic load is uniformly and symmetrically distributed over a middle portion of the span. A comparision between the transverse displacement of beams subjected to rectangular, triangular and exponentially decaying pulses was considered, for different spatial load distributions, considering shear slide. It was concluded that triangular and exponential pulses produce less transverse shear sliding and smaller bending deformation than a rectangular pressure pulse that has the same impulse and peak pressure value. The difference is particularly significant for small values
4 1
p( t ) / p
(b)
p = mass per unit length.
It = total applied impulse.N = constant axial force.
I = half beam length.
5 = maximum permanent displ.
t = pulse duration. x = 3N/plZ.5 = X V Pp= 2 Mp/ l Z.p = peak pulse pressure.
Figure 2.16. Maximum permanent displacement at midspan due to various pulses o f applied pressure : (a) Exponentially decaying and rectangular pulses, (b) Two different triangular pulses. After Stronge (1982).
4 2
of peak pressure, but decreases with an increase in pressure.
The variety of shapes of dynamic load pulses presents analytical and computational inconvenience in theoretical analyses, and a considerable problem in experimental simulation of actual loading. This has stimulated interest in studying the influence of shape of transient pulses on the response of structures, and in representing the effects of pulse loading through the least number of parameters.
Youngdahl (1970), studied the plastic response of four different structural systems under the effect of the different (standard) pulse shapes, shown in Figure 2.15. In all cases the material was assumed rigid-perfectly plastic, and thedisplacements infinitesimal. Youngdahl concluded that the response depended strongly on pulse shape for pulses of the same total impulse and the samemaximum load; Figure 2.17(a) shows this for the case of a free-free beam loaded with a central concentrated force.
To eliminate the effect of pulse shape on the response, Youngdahl (1970) introduced the concept of 'effective load' as a correlation parameter, instead of the peak load of the pulse. The use of the effective load and total impulse ascorrelation parameters has been shown to reduce drastically the dependence on pulse shape of the final plastic deformations, because both parameters depend onlyon integrals of the loading and are consequently insensitive to small perturbations inpulse shape. Figure 2.17(b) shows the case of a free-free beam for which the analysis procedure was based on that presented by Symonds (1953). In order to determine the effective load of a pulse on a structure, one needs to know beforehand the time tf at which plastic deformation ends. To determine tf, Youngdahl (1970) had suggested the approximate formula
I * py ( t f - t y ) ( 2 . 10 )
where I is the total impulse, py the static yield load, and ty the time when plastic deformation begins. For a rigid-plastic structure, ty is the time at which the dynamic load first reaches py. Youngdahl emphasised that his conclusions regarding the effect of the pulse shape on rigid-plastic systems might not necessarily obtain when the structural material is elasto-plastic.
4 3
l - half beam span.
p = mass per unit length.
I = total impulse.
Py = 4 M p /l.
W = W0 (tf) pl Py/ I Z.
Figure 2.17. Relation between permanent displacement at midspan
W0 (tj>) and : (a) Peak load (Pmap:) o f pulse3 and (b) effective load J (Pe ) respectively for various pulse shapes. After Youngdahl (1970).
44
METHODS OF MATHEMATICAL ANALYSIS
CHAPTER 3
3.1. INTRODUCTION
Mathematical models of analysis can be classified according to the spatial distribution of the parameters describing the system properties. The two majorclasses are discrete and distributed-parameter models; a minor class is the hybridsystem model, which is partly discrete and partly distributed. Discrete models are generally described by ordinary differential equations, and distributed-parameter models by partial differential equations. Except for some simple classical problems, exact solutions for the response of distributed-parameter systems are difficult, if not impossible to obtain. Accordingly, to solve for the response one has either tomake major simplifying assumptions regarding the structural and material propertiesof the continuum, or to represent the continuum of a distributed-parameter system by a discrete one in a process known as spatial discretisation. In this chapter, continuum and discrete methods of analysis of the dynamic elasto-plastic response of beams are discussed.
3.2. CONTINUUM METHODS
Continuum methods of analysis and associated closed-form solutions are often desirable because they afford valuable insight into the system behaviour. These methods, however, are often too complex to be of practical use in day-to-day engineering problems. Consequently simplifying assumptions are frequently adopted. In the following subsections these assumptions and their effects are discussed in detail through the presentation of existing methods of analysis. It will also be made clear how recent research has been focussed on ways of weakening or completely removing some of the restrictions.
3.2.1. Elasto-Plastic Methods
In methods of analysis of this type, the elasto-plastic constitutive characteristics
4 5
of a structural material are generally assumed to reflect linear elasticity, perfect plasticity and to be insensitive to strain rates. A closed-form solution to the problem of the elasto-plastic response of beams was developed by Duwez, Clark and Bohnenblust (1950); their analysis extends an earlier elastic analysis by Boussinesq (1855) to include the effects of plastic deformations. The approach of Boussinesq was based on the simplified equations commonly used in beam problems, disregarding transverse shear deformations, rotatory inertia, and geometry change,
8 q 8 2z— + pA ------ = 0ay at ^
8m
Q - —
(3.1)
( 3 . 2 )
8 2zM - El ------ ( 3 . 3 )
8y2
where, Q is the shear force, y the distance along beam axis, p the density per unit volume, A the area of cross section, z the transverse displacement, t the time, M the bending moment, and El the flexural rigidity. Boussinesq concluded that for small values of the velocity of impact, the ratio z/t is a function of y 2/t alone. Duwez et al. (1950) made use of this conclusion to reduce the partial differential equations of motion to ordinary differential equations by replacing the variable y, measured from the point of impact, by
1 y 2 /E lr ---------------- , a 2 = — ( 3 . 4 )
4 a 2 t \ f pA
and extended the analysis to include plastic deformations. Solutions obtained from their model involved the following limitations,
1. constant velocity impact only can be dealt with, and unloading cannot be treated,
2. infinite beams only can be solved; no modifications were presented to treat beams of finite length.
A less exact, but more widely applicable elasto-plastic theory, was that
46proposed by Bleich and Salvadori (1955). Here the initial elastic phase wasdescribed in terms of a set of normal modes; this phase ended when the bending moment at some cross section reached the fully plastic moment. In the ensuing elasto-plastic phase, the transverse displacement was represented by a different set of normal modes; one plastic hinge is allowed in the analysis and is assumed toremain stationary, all other parts of the beam behaving elastically. According toSymonds (1967), the method is easy to comprehend and, in principle,straightforward to apply. In practice, however, he noted a considerable amount oflabour in determining the successive sets of normal mode functions and in calculating the permanent angular deformations and lateral displacements in terms of series expansions that either converged slowly or failed to converge. Symonds also indicated that the method was not practicable for the consideration of the spreading plastic zones, that may develop when the applied energy considerably exceeds the elastic strain energy.
The overwhelming complexity of the description of the dynamic behaviour of elasto-plastic beams has precluded the development of a general method of analysis and has lent encouragement to researchers to explore a simpler problem in which a rigid-plastic material is assumed.
3.2.2. Rigid-Plastic Methods
Simplicity in analyses becomes possible only when indealisations are adopted. The complex intermingling of elastic and plastic behaviour, in problems of dynamics, is responsible for the absence of general analytical approaches, and for much of the difficulty of numerical solutions. In treating problems of plastic flow with substantial plastic deformations, one approach to simplification is to disregard elasticity - effectively considering the elastic moduli infinite. In such a rigid-plastic analysis, deformation occurs only in regions where a yield condition is satisfied; everywhere else, motion is governed by rigid-body mechanics. Furthermore,equations governing motion are assumed linear - that is, no change in geometry is considered - and the loading is idealised as impulsive - specifying an initial velocity field.
According to Symonds (1980c), the real value of these idealisations is not that they provide solutions of practical problems, but that they enable a general theory to be developed, even though it might need to be modified or even replaced in certain circumstances. Without the theoretical framework that is based on these idealisations, the general understanding of response patterns would be far more limited, and the nature of the corrections or modifications relatively obscure.
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In the consequent three subsections, the main approaches involving rigid-plastic models in the solution of problems of dynamic plasticity are examined.
3.2.2.1. Standard Rigid-Plastic Method
Rigid plastic methods are essentially based on the concept of plastic hinges.This concept had already been in use in the solution of static problems; its application to dynamic problems was first suggested by Taylor (1948). A discussion of the general characteristics of rigid-plastic solutions of beams of finite length was first given by Lee and Symonds (1952). This was followed by many other beam investigations, such as those by Pian (1952), Symonds (1953, 1954, 1955), Salvadori and DiMaggio (1953), Parkes (1955, 1958), Salvadori and Weidlinger (1957), Symonds and Mentel (1958), Ezra (1958), and Conroy (1964).
In dynamic problems, standard rigid-plastic analyses are based on the law ofconservation of momentum. Using this law, Lee and Symonds (1952) showed that, depending on the magnitude of applied load, plastic hinges and finite plastic regions can be expected to form; generally these plastic hinges, or the boundaries in thecase of plastic regions, are not stationary but would move during the course of theresponse.
The response of rigid-plastic beams is generally described by two phases:
1. Transient phase, with initial velocities determined by the distribution of impulsive forces over the structure. It is in this phase that travelling plastic hinges appear, with the shape of the velocity field over the structure changing with time towards a simpler shape. The response ultimately converges on a shape which is time-independent.
2. Modal phase, which is virtually a one degree of freedom system in which velocity magnitudes decrease linearly with time so that all points on the structure come to rest simultaneously.
In problems of flexural deformations in rigid-plastic beams, the positions and senses of the relative rotation of plastic hinges are assumed apriori, and the magnitudes of bending moments and shearing forces fixed at either the value of zero or the fully plastic section value. For instance, the shear force is zero in the absence of transverse applied forces. This enables the formation of the equations of motion for the rigid segments of the beam between the plastic hinges. Thus, force interactions between the rigid segments depend on the kinematics of the
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beam. However, it is necessary to check that the velocities deduced from the equations of motion do maintain the existence of the assumed plastic hinges. If the relative angular velocity at a particular hinge falls to zero, this hinge must be removed in the force analysis, and a different set of equations of motion obtained. In case of a moving plastic hinge, the equations of motion would contain a term involving the velocity of the hinge along the beam.
The validity of rigid-plastic methods has often been associated with an energy ratio criterion. Analyses were said to be valid if the ratio of plastic work tomaximum elastic strain energy capacity of the structure greatly exceeded unity. However, for load pulses of finite duration, the validity of rigid-plastic methods was also considerably affected by the ratio of pulse duration tf to the fundamentalperiod of vibration of the structure Tf, and thus the energy ratio criterion needed supplementing to take account of this fact. For a single degree of freedommass-spring model, where the spring characteristic was taken as linearly elastic and perfectly plastic, Symonds (1981) and Symonds and Frye (1988) calculated the error in the final displacement of the elasto-plastic system which follows from assuming the elastic component as rigid. It was found that for a rectangular pulse, the error in the final displacement varied for: (1) an energy ratio S = 50 from +4% (fortf/Tf = 0), - 25% (for tf/Tf = 1), to -100% (for tf/Tf = 4), and (2) an energyratio S = 5 from + 20% (for tf/Tf = 0), to - 100% (for tf/Tf = 0.55).
The main restrictions of the standard rigid-plastic approach are:
1. The treatment of the transient phase could be quite difficult and may require some judgement, depending on the structural boundary conditions and type of applied loading. In some cases both plastic hinges and plastic regions appear along the beam, while under other circumstances only plastic hinges or plastic regions are present.
2. The lack of a general criterion for the validity of the analysis.
3. The inaccuracy of the results when the structure is subjected to a pulse loadwhere the peak magnitude does not greatly exceed the magnitude of staticcollapse load, and the pulse duration is long compared to the fundamental period of the corresponding elasto-plastic structure. In such circumstances the final displacement is underestimated by an order of magnitude.
The conditions under which it is permissible to disregard the elastic deformations and adopt a rigid-plastic model are often conditions for which other secondary effects become important. According to experimental evidence, the most
4 9
significant of these secondary effects are the variation of yield stress in a strain rate sensitive material, and the change of mode of behaviour caused by finite displacements in the presence of axial restraints. Attempts have been made to incorporate these two, and other secondary effects, in rigid plastic analyses - for example, the work of Bodner and Symonds (1962), Jones (1971), Symonds and Jones (1972), and Perrone and Bhadra (1979) among others. However, all these studies were subject to some or all of the restrictions mentioned earlier.
3.2.2.2. Mode Form Approximation
This approach basically reduces the standard rigid-plastic approach to that of treating one phase of the response, namely the modal phase. The concept of mode form approximation is founded on the fact that, however the structure is loaded, the complicated initial response changes with time to a much simpler modal form in which most of the energy is dissipated. In this sense, the mode form approximation is equivalent to matching a structure to a model with a single degree of freedom by imposing on the structure a pattern of displacement that can be described at any time by one parameter; and the deformed shape is virtually time-independent.
The mode form approximation was first presented by Martin and Symonds (1966). They showed that the solution of one impulsive loading problem may be approximated by the solution of a problem involving the same structure, but with a different initial velocity field. This field would retain a constant shape over the structure, and velocities at all points would remain in a fixed ratio to each other whilst their magnitudes vary with time. The velocities in an adopted mode form are determined by,
u ( y , t ) - w*( t ) y) ( 3 . 5 )
where « (y) is the shape function or pattern imposed for displacements, velocities and accelerations, and w*(t) is the velocity of the point of main interest. Equation (3.5) represents a complete solution satisfying the dynamic equations and constitutive relations of rigid-plastic behaviour. In an impulsive loading problem the initial velocities are specified over the structure while initial displacements are zero. Here, a mode form solution can not satisfy the initial conditions unless the initial velocity field is of identical shape to that of the adopted assumed form. For this situation an approximate solution may be obtained by a special choice of the initial velocity amplitude so as to minimise the difference Aq between the given initial
velocities u°(y) and those of the approximating mode field w9< (y). Aq can be defined by
5 0
( 3 . 6 )
where m is mass per unit length, and the integration is taken over the entire span of the beam. The value of the initial velocity amplitude of the mode field which minimises Aq is,
w° - [jnv(y) u°(y)dy]/[jiiv>(y) 2dy] ( 3 . 7 )
Equation (3.7) was proposed by Martin and Symonds (1966) as the condition for minimising the difference, in the mean-square mass weighted sense, between the actual velocity field and the approximating mode field, and it is usually referred to as the "minimum Aq" formula.
Since the shape function ^(y) plays a critical role, a method has had to be devised for finding the shape of the most suitable mode, together with a criterion for preferring one mode over another. Compatible with the nature of the mode approximation approach is the notion that the best mode choice in problems of impulsive loading is the one with the longest response time. Martin (1981) presented an iterative method that, in few cycles, determines the shape of this primary mode of the unloaded structure under the assumption of infinitesimal displacements; each iteration was equivalent to the solution of a static limit analysis problem. Criteria for selecting the preferable mode have also been discussed by Symonds (1980d). Martin (1983) considered the convergence to mode form solutions in impulsively loaded, piecewise linear, rigid-plastic structures.
Extensions to the mode form approximation were made to include strain rate sensitivity (Kaliszky 1973) and nonlinear effects due to finite displacements (Kaliszky 1973, Symonds and Chon 1979). Such extensions to meet practical conditions have largely been successful - in most cases, however, at the additional cost of considerable complexity (Symonds 1981).
The general restrictions of the standard rigid-plastic model, mentioned in section 3.2.2.1, are essentially the same for the mode form approximation technique.
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3.2.2.3. Upper and Lower Bounds
Upper and lower bounds on final displacements and response time had been basically developed for impulsively loaded structures of rigid-perfectly plastic material idealisation and under the assumption of infinitesimal displacements. The general idea of taking bounds is to simplify the dynamic analysis of the response by determining only the upper and lower limits on displacements and response time, rather than the whole history. This is done by assuming a mode form structural response, thus reducing the structure to a single degree of freedom system. By its nature, such a system is insensitive to the spatial distribution of the dynamic applied load.
Martin (1964) was the first to present a technique giving the upper bound on the final displacements at any point, and a lower bound on the response time, under the assumptions made above. Martin (1964) based his solution on Drucker's (1959) stability postulate for time independent inelastic materials,
( 3 . 8 )
where apj is any stress state on the yield surface ^(crpj) = 0, <tjj is a stress field on or inside the yield surface, and epj is a plastic strain rate; for any smaller stress magnitude than crpj, y?(qj) < 0 implies tpj = 0. The work was also based on the equation of virtual work rate,
•k *T i l l , ds -1 l CTit € . . dv =
J 1Js v v
*pu.u. l l dv ( 3 . 9 )
where Tj are tractions acting on the surface s, Uj are accelerations resulting from Tj, ay are stresses in the body, u* any assumed kinematically admissible velocity field with corresponding stress and strain-rate fields q j and e*j, and i = 1, 2, 3 to indicate three-dimensional applicability and j = 1, 2, .... r indicates the number of generalised stresses and strains.
To obtain an upper bound on displacements an artificial and constant system of surface loads is introduced. To obtain a lower bound on the response duration, an artificial system of constant velocities and strain rates is introduced that would satisfy compatibility equations and the specified fixing conditions on the surface. In practice, the crudeness of the upper bound on displacements stems from the fact
5 2
that only the total kinetic energy is specified, the bound must hold for all distributions of initial velocity with the same initial kinetic energy. Martin (1964) reported that the upper bound in one case overestimated the actual displacement by 100%, and that it may even be less accurate in more complex configurations. The lower bound on response time performs better, since it is almost always close to the exact solution.
Morales and Nevill (1970) presented a technique for obtaining lower bounds on the displacements of dynamically loaded rigid-plastic continua, complementing the work of Martin (1964). Their solution was also based on Drucker's stability postulate equation (3.8), using the virtual work rate equation (3.9) and assuming a kinematically admissible velocity field. In practice, their lower bounds on displacements had the same degree of crudeness observed in the upper bounds mentioned above.
An investigation was made by Stronge (1985) into the manner by which structural characteristics and load distribution influence the accuracy of the upper and lower bounds on final displacements, as determined by the techniques of Martin (1964) and Morales and Nevill (1970). It was concluded that, generally, the range between the upper and lower bounds on final displacements is small for one-dimensional deformation (such as the flexural deformation of beams) in response to uniformly distributed loads, and larger for two-dimensional deformations (such as the bending and stretching deformation of plates) and more concentrated loads.
To complete the list, Lee (1972) presented a technique for obtaining an upper bound on response time. The work was based on the mode response concept and the equation of virtual work rate, equation (3.9). In the mode response concept, all velocity fields having the same kinetic energy, the dissipation rate is a minimum; this property implies that the principal mode motion is the most persistent of all responses starting from the same initial kinetic energy, whether in mode form or otherwise.
Several studies had been made, since the early work of Martin (1964), to include effects of strain-rate sensitivity, elastic deformations, finite displacements and shear deformations. These were incorporated individually, and mostly to determine an upper bound on final displacement. Their inclusion, essentially, implies added analytic complexity in return for a more representative modelling.
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3.2.3. Simple Elastic-Plastic Methods
The development of the Simple Elastic-Plastic (SEP) approach was prompted by the need to overcome the general restrictions common in the earlier approaches, namely standard rigid-plastic analysis and the mode form approximation. The SEP approach was first proposed by Symonds and Raphanel (1979). In essence, it "sandwiches" the mode form approximation between artificially separated elastic phases.
In the SEP solution, the response is treated as either entirely elastic, or entirely plastic, and consists of three distinct stages: the first initially elastic, a subsequent stage of rigid-perfectly plastic deformation and a concluding elastic vibration. A brief description of each stage is given below,
1. The solution of the initial elastic stage may be obtained using a discrete model with one or more degrees of freedom; this is permissible since this stage is completely separated from the following rigid-plastic stage. When to terminate the initial elastic stage is a decision still surrounded by debate; Symonds and co-workers had suggested a couple of resolutions which will be discussed later.
2. In this second stage, the response is rigid-perfectly plastic and displacementsproceed in a mode form. This mode form has its velocity amplitude determined from the velocity field at the end of the preceding elastic phase by applying the "minimum formula, equation (3.7). The stage ends when all velocities are reduced to zero simultaneously and consequently thedisplacements are at a maximum.
3. The final stage in the response is that of elastic vibrations. It is included in the analysis to provide a means of determining the amplitude of permanent displacements. Some ideas put forward by Symonds and co-workers are subsequently discussed.
The SEP approach emphasises simplicity, and allows the assumption of rigid-plastic behaviour to be relaxed by including the initial elastic response. This facilitates the treatment of general finite pulse shapes, and the inclusion of strain rate sensitivity. Strain rates of the elastic stage are used in a constitutive law of viscoplastic behaviour in order to determine a corresponding global yield condition; this is used throughout the perfectly-plastic stage as the appropriate dynamic yield stress.
In cases where the structural behaviour may change from flexural to membrane
5 4
action, as in clamped beams undergoing large displacements, the rigid-plastic stage would consist of two distinct consecutive mode forms. Each mode has its velocity amplitude determined from the velocity field at the end of the preceding phase by applying the "minimum Aq" formula which is sensitive to the velocity distribution over the structure.
The simple elastic-plastic approach had been used to predict the response of clamped beams (Symonds 1980a, 1980b), cantilever beam with tip mass (Symonds and Fleming 1984), and portal frames (Symonds and Raphanel 1979, 1984,Symonds, Kolsky and Mosquera 1984, Symonds and Mosquera 1985, Mosquera, Kolsky and Symonds 1985, Mosquera, Symonds and Kolsky 1985).
Problems associated with the practical application of the SEP approach are:
1. When the initial elastic stage is terminated, it has displacements and velocity fields different in shape from those of the succeeding rigid-plastic stage, hence necessitating the use of a device to minimise the loss in kinetic energy when changing from one stage to another.
2. The time at which the initial elastic stage ceases is not clearly defined. It has been mentioned (Symonds and Raphanel 1984, Symonds 1980a, 1980b) that this time (say t , ) can be determined by invoking a global yield condition defined from the positions of the plastic hinges participating in the subsequent rigid-plastic stage - for example, the global yield moment Mqj is,
MCl
n- JM (t, ) n ~ l 1 Mr ( 3 . 1 0 )
where the M j(t,) are the bending moments computed at assumed plastic hinge sections from the elastic solution at time t v i = 1, 2, ... n, and Mp is the yield moment at these yield sections. An alternative approach that seemed preferable (Symonds, Kolsky and Mosquera 1984), especially for long pulses, starts by treating t , as an arbitrary independent variable, and for any choice of t 1 the full SEP solution is carried through. When the plastic work isplotted against several choices of time 11, maximum plastic work is seen to occur at some particular value of t ,; this value is then employed in the analysis; the approach generally yields conservative results.
5 5
3. Determination of the permanent displacements. Symonds (1981) assumed thatthe amplitude of the elastic vibrations subsequent to the rigid-plastic stage isequal to that of the initial elastic displacement, and is thus subtracted fromthe maximum displacement to give the permanent plastic displacement. This approach seemed appropriate for a single degree of freedom mass-spring system, but proved unsatisfactory for portal frames. An alternative approach was suggested by Symonds, Kolsky and Mosquera (1984), whereby the amplitude of the final elastic vibration is determined from the fundamental mode appearing at the end of the rigid-plastic stage. The shape in this final elastic vibration could differ from that of the initial elastic motion. As before, the amplitude of the final elastic vibration is subtracted from the maximum displacement to obtain the permanent displacement.
According to Symonds and co-workers, SEP solutions for frames offer amaximum disagreement of about 20% from those obtained from experiment and from the finite element code ABAQUS (Hibbit 1984). The good agreement, for the particular examples used, was seen as a corroboration of the concepts presented in the SEP approach, in particular of the simplification obtained by separating the response into elastic and rigid-plastic stages. It was concluded that much further research is needed - on the one hand to make the process more convenient, and on the other to establish its limitations and possible extensions.
Apart from the above evaluation, it should be remembered that the SEP approach is an extension of the mode form approximation model: obtained by adding initial and terminal elastic stages. The mode form approximation is in itself equivalent to matching the structure to a single degree of freedom model by imposing a pattern of displacement that does not change with time. Therefore, the SEP approach can provide no information about the distribution of internal plastic deformations away from the locality of the plastic hinges defining the mode; especially with high energy loading that would result in large displacements, such as the case of a cantilever beam with a tip mass subjected to a short pulse. This was examined recently by Reid and Gui (1987) who concluded that a more accurate understanding would be gained from an adaptation of the travelling hinge concept that incorporates the effect of elastic vibrations throughout the analysis.
3.3. DISCRETE METHODS
The difficulties in obtaining closed form solutions to cases of structural continuity are due to the inherent difficulty of solving partial differential equations and in satisfying boundary conditions, in particular for cases of two and three
56
spatial dimensions. These difficulties can be circumvented by eliminating the spatial dependence from the equations describing the response through discretisation in space (Meirovitch 1980).
There are two major classes of discretisation methods, the first consists of lumping the system properties, and the second is based on the expansion of the solution in a finite series of given function, either over the whole structure or over finite elements of the structure. The first class encompasses the methods of lumped parameters and finite differences; the second class encompasses methods of Rayleigh-Ritz type and weighted residuals methods, including that due to Galerkin. It can be said that the first class is more intuitive in character, while the second is more analytical. A third class of discrete methods is that involving boundary elements or the method of boundary integral equations, in which only the boundary of the continuum need be discretised. Considerable advantage can be gained by this reduction and simplification of the discretisation; however, for nonlinear problems boundary element methods lose much of their appeal as volume integrals appear which require additional discretisation of the continuum.
Within the two major classes of discrete methods, there are three basic approaches for modelling that are probably of most interest in the dynamic analysis of nonlinear elasto-plastic structures. They are lumped parameter, finite difference, and finite element models. Historically, the lumped parameter model involving lumped mass and lumped flexibility, has been called the finite difference method; this method has grown immensely in sophistication over the years to rival the finite element method in versatility, and indeed, despite separate historical developments, some degree of convergence has been reached lately between the two, Belytschko (1983). The original finite difference method, and other methods of lumped mass and/or flexibility, will be labelled here as lumped parameter methods.
Finite difference methods start with a statement of the problem as a system of differential equations, together with appropriate boundary and initial conditions, and proceed to replace the derivatives with a discrete analog. Finite element methods may be based on a differential equation system or upon a variational statement of the problem; the relevant field variables - for instance, displacements - are represented piecewise within each finite element by a combination of prescribed continuous functions.
Time integration may be performed either by modal analysis or by direct integration. Modal analysis requires transformation into the frequency domain, which are only valid in a linear regime. Direct integration can be used, in explicit and implicit versions, for linear and nonlinear systems.
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In the subsections that follow, the lumped parameter methods are discussed first, followed by brief discussions of the finite difference and finite element methods.
3.3.1. Lumped Parameter Methods
3.3.1.1. Lumped Mass - Continuous Flexibility
This model is formed by lumping the distributed mass of the structure into a number of concentrated masses attached at discrete points to a massless structure in which the flexibility is continuously distributed. The applied load is normally also lumped at the mass points, Figure 3.1(a) and (b).
This model was used by Berg and DaDeppo (1960), to study the transient elasto-plastic response of multistorey frames loaded laterally at the joints between the floors and columns. The mass of the structure was concentrated at the joints. Viscous damping and shear and axial forces were considered in the analysis. In the equations of motion, the continuous flexibility of the model was determined using the method of the influence coefficients. When plasticity is reached at any joint, the frame was assumed to behave as though the member were hinged at thatlocation, with constant moments applied on either side of the hinge. Plasticity waspredicted using the Predictor-Corrector method proposed by Milne (1949). ThePredictor is essentially an elastic solution which satisfies dynamics, but generally violates material laws at certain points in the structure, in this case by calling for a bending moment larger than the yield moment. The Corrector solution consists of a self-equilibrated structure with plastic hinges at appropriate points such that a superposition of this solution with the Predictor solution would result in a system satisfying both dynamic and material laws.
Heidebrecht, Fleming and Lee (1963) used a lumped mass continuous flexibility model to study the transient elastic-perfectly plastic response of single- and multi-span beams. Plastic hinges occurred at points of concentrated masses where the applied load was lumped. The method of analysis used throughout the response was the conjugate beam method, normally employed in elastic analyses, but which had been extended by Lee (1958) to include the formation of plastichinges.
Wen and Toridis (1964) extended the method proposed by Heidebracht et al. to include constitutive behaviour other than elastic-perfectly plastic.
5 8
rmJ2
L U M P E D —-J LUMPE LOADING j ^ M A S S
7 7i C O N T I N U O U S F L E X I B I L I T Y(b)
RIGID ANDMASSLESS PANEL —
h *2 h-1 h h+11,, _ . ^ i__i__i i_* UL — — ~ ~wPmm \* y
r<z;
Figure 3.1. Original structure and lumped parameter models.(a) Original structure, (b) Lumped mass - continuous flexibility model, (c) Lumped mass - lumped flexibility model, (d) Continuous mass-lumped flexibility model. After Wen and Toridis (1964).
5 9
In all of the above mentioned analyses, the following parameters were neglected : shear deformations, axial deformations, rotatory inertia, geometrychange, force interaction in the plastic response, and strain rate dependence of the yield stress. Also, a common feature is the absence of experimental or numerical verification of the solutions.
3.3.1.2. Lumped Mass - Lumped Flexibility
This model consists of a series of rigid and massless panels connected by flexible joints where the mass is lumped. A lumped mass at a joint is the sum of the distributed mass from a length tributary to the joint. The deformations at a flexible joint represent all deformation changes in the original structure over a length tributary to the joint. For simplicity of solution, the applied load is normally lumped at the joints, Figure 3.1(c). This model is the most widely used amongst the three models associated with the lumped parameter method, most probably due to the simplicity inherent in both formulating the equations of motion and in solving them numerically.
In a study by Baron, Bleich and Weidlinger (1961), such a model was utilised for representing the transient elasto-plastic response of beams. The equations of motion were formulated using Hamilton's principle to avoid the problems of establishing vectorial equations of equilibrium. When plasticity developed at any joint j a corrective displacement was introduced to reduce the bending moment to the moment capacity of the cross section. This corrective displacement was determined from,
ycn - e . \L.J jn (3 .1 1 )
where, y£ is the displacement at joint n due to a corrective displacement at joint j, 0j is a generalised coordinate proportional to the angle of the plastic rotation at joint j, and ^jn is the displacement at the points n of a fictional elastic beam if a hinge were introduced at a point x = xj along beam axis, and the two portions of the structure on either side of the point j were rotated by a certain angle. If plasticity were to occur at several points, then the corrected displacement at any joint n would be obtained from the summation,
y = T 6 . $ . n j j * jn (3 .1 2 )
60
The expression for the overall displacement at n is,
yn - ynel + y„ " ynei + 2 «j ^jn <3-13>j
where y n e j *s t ^e e l a s t i c d i s p l a c e m e n t at j o i n t n .Response parameters neglected in the analysis of Baron et al. were, shear forces and deformations, axial forces and deformations, rotatory inertia, geometry change, and strain rate sensitivity. Numerical results for simply supported beams loaded by an exponentially decaying pulse were compared with results of single degree of freedom elasto-plastic and rigid-plastic models; no experimental validation was attempted.
A subsequent method of analysis based on the lumped mass - lumped flexibility model was that developed by Witmer, Balmer, Leech and Pian (1963) who investigated the transient elasto-plastic response of two-dimensional and axisymmetric structures. The continuum of the structure was divided into a system of lumped masses and massless links. The cross sectional area of each link wasassumed to consist of a number of discrete, evenly spaced layers that had equal cross sectional areas, and could carry normal stresses. These layers were separated by material that could not sustain normal stresses and had infinite shear rigidity; in this way stress and strain in the structure could be defined by individual normal stresses at the separate layers. The applied load was lumped at mass points. The equations of motion were set up using direct dynamic equilibrium of forces acting on an element, and were integrated temporally using the central finite difference explicit operator. Parameters neglected in the analysis were, shear deformations, rotatory inertia, and effect of force interaction in the plastic response.Comparisons with experimental results were made for the different structures studied. In their conclusion, Witmer et al. mentioned that, to permit reasonably reliable predictions of large transient and permanent displacements, their methodrequired use of a finely graded mesh along the span and across the depth of the structure. They also concluded that it was desirable to use their model as alearning tool to guide the formulation of simple, less time-consuming prediction methods retaining the essential features of the dynamic problem.
3.3.1.3. Continuous Mass - Lumped Flexibility
This model consists of a series of rigid panels having the same mass distribution as in the continuum, and linked together by flexible joints.
6 1
Deformations at a flexible joint represent those of the continuum over a length tributary to that joint. The applied load can be considered as lumped or distributed over the rigid panels, Figure 3.1(d). Due to the retention of the mass distribution, the equations of motion of individual elements are coupled via the accelerations; it makes the model computationally harder to solve than the lumped mass - lumped flexibility model, however, this can be overcome by the use of an appropriate integration scheme.
Amongst the early studies in which this model was used, to predict the transient elasto-plastic response of beams, is that by Wen and Toridis (1964). The response was considered to be mainly flexural, and the analysis lacked manyimportant response parameters. However, a more elaborate study was later made by Wen and Beylerian (1967) using Timoshenko beam elements of a lumped mass - lumped flexibility type, but with each mass having rotatory inertia. Other response parameters considered were shear forces and deformations and the effect of moment-shear interaction on plastic yielding. The applied load was lumped at mass points. The equations of motion were set up using direct equilibrium, andplasticity treated by the Predictor-Corrector approach. Response parameters neglected in this model were axial forces and deformations, finite displacements, and strain rate sensitivity. Also in the treatment of the elastic stage, the constitutive equations had a diagonal stiffness matrix, implying that bending moments depended only on flexural rotation, and shear forces only on sheardeformations. No experimental or numerical verifications were attempted; instead, the reliability of the model was judged by the convergence of the results as the discretisation mesh was made finer, and whether results made good physical sense.
The continuous mass - lumped flexibility approach, to the present author'sknowledge, has not been improved upon since. This may well have been due to the computational power and improved continuum modelling associated with the then emerging finite element method.
3.3.2. Finite Differences
This method may still be regarded as an advanced form of the lumped parameter method. Its advanced features include the discrete representation of two- and three-dimensional continua, and constitutive equations in terms of stresses and strains which can account for large plastic deformations, rupture and thermal properties. Other features, according to Goicolea Ruigomez (1985), also include high node velocities in the range of Mach 1, the ability to represent dynamical problems involving interacting continua (such as high velocity impact between two
6 2
solids), mixed mesh geometries and irregular zoning (topologically at least).
In general, the discretising of a structural continuum would lead to a semi-discrete set of differential equations. Finite difference operators provide local approximation for a system of differential equations; consequentely it is not possible to perform a one-step global solution, and recourse must be made to relaxation and iteration techniques. For explicit time-marching models, the semi-discrete equations of motion become fully discrete algebraic equations, implying that only local approximations to the original partial differential equations are performed within each time step, no iterations being required. This fact was exploited in the development of earlier (1950's) finite difference codes to solve problems of transient elasto-plastic response in nuclear and defence applications. However, the incapability of the finite difference method to generate a one-step global solution, together with the restriction of the difference scheme to regular topological zoning, led to its eclipse in the 1960's by the finite element method; the latter seemed advantageous for linear systems.
Interest in nonlinear and wave propagation regimes in the late 1960's and in the 1970's led to the understanding that, for such regimes, both finite element and finite difference methods require the performance of some scheme of iteration. Indeed both methods seem to be converging towards each other as the attractive features of each are successively implemented in the other. For example,relaxation techniques are used in finite element methods to avoid the assembling of large global coefficient matrices in nonlinear problems; and in the finite difference method, irregular zoning is used to improve the refinement of the continuum discretisation.
3.3.3. Finite Elements
Early work on the finite element method was pursued by three separate groups of researchers : applied mathematicians, engineers and physicists. The conceptappears to have been developed independently for the serving of different initial purposes in each of these groups. In engineering, the first application of the finite element method to structural continua was by Clough (1960), although the theoretical bases for the method had already been set by Courant (1943), a mathematician; and applications to structural analysis had been proposed earlier by Argyris and Kelsey (1954).
The finite element method has the essential features of a Rayleigh-Ritz method with enhanced versatility. In the classical Rayleigh-Ritz method, the
63
solution is approximated by a finite series of admissible functions that are defined over the entire domain. For systems with complex geometry, the task of producing admissible functions is very difficult indeed. In other cases, even when admissiblefunctions can be produced, they may be very complicated and difficult to workwith. The basic idea behind the finite element method is that, instead of defining the admissible functions over the entire domain, they are defined only over relatively small subdomains called finite elements. This not only permits the application to complicated geometries, but also permits the use of very simpleadmissible functions. Indeed, for the most part, the admissible functions are low-degree polynomials (known as interpolation or shape functions); they arecomputationally attractive, since integrals involving such polynomials can beevaluated in closed form, thus eliminating errors that may result from numericalintegration.
It is interesting to note the ability of the finite element method to give a one-step solution to the linear problem. Consequently, it has almost become the standard tool of structural analysis. In fact, when the structural geometry isrelatively simple, and there are no sudden changes in the system properties (suchas stiffness and mass distribution), other methods become more convenient to use, such as the classical Rayleigh-Ritz method for linear analysis, and the lumped parameter and finite difference methods for both linear and nonlinear analyses.
For analyses of nonlinear response, the advantage of the finite element method to give a one-step solution over the finite difference method disappears, because in both methods some form of iteration has to be performed. For large systems, to avoid the assemblage of global coefficient matrices, relaxation techniques are used in the finite element method.
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CHAPTER 4
STRUCTURAL MODELLING IN GEOMETRIC NONLINEARITY
4.1. INTRODUCTION
From the literature review presented in the preceding two chapters, it is seen that a discrete model within the continuous mass - lumped flexibility concept can provide an adequate representation of the dynamic transient elasto-plastic response of beams. It is therefore thought appropriate to develop a mathematical model for a new discrete beam element that will simultaneously confer simplicity, efficiency and competitive accuracy upon the response analysis.
In this chapter, a detailed description is presented of the structural aspects of the developed element. Initially, this element is specified and associated notation defined; it is then pursued through the exhibition of the kinetic-kinematic relations of both element and structural levels.
4.2. DISCRETE ELEMENT AND NOTATION
The discrete element developed in this study is a two-dimensional cantilever beam element. The element is prismatic and rigid, with a mass distributedcontinuously throughout the span; all deformations are assumed concentrated, or lumped, at the free end. No assumptions are made concerning the size of the rigid body displacements of the element, but element deformations are assumed small.
Generally, beam elements in a discretised plane skeletal structure can support six end forces, Figure 4.1(a); of these forces, only three can be independent. For a cantilever configuration, a possible choice of the independent forces is that illustrated in Figure 4.1(b). Correspondingly, the element can sustain three independent deformations, Figure 4.1(c).
For planar motion of the cantilever element developed in this study, the three independent deformations are : flexural rotation, transverse shear deformation, and
65
(c ) Independent deformations.
Figure 4.1. Element forces and deformations.
element j- j
element
element j +-j
Figure 4.2. Typical discrete element.
6 6
axial deformation. Each type of deformation is represented jointly by a spring and a friction dissipator - the former simulates elastic behaviour and the latter plastic behaviour. This is shown for a single element in Figure 4.2.
Inertial forces associated with the rigid body displacements of the beam element are all accounted for; these may be referred to as rotatory inertia, transverse inertia, and axial inertia.
The lumped deformations are required to simulate the material characteristics of homogeneity, isotropic linear elasticity, and either perfect plasticity or viscoplasticity.
A typical element in a displaced and deformed configuration is shown in Figure 4.3. The notation used in this figure and in others of subsequent sections of this chapter are defined below.
1. General:
d( ) derivative of a quantity.( ) second derivative of a quantity.A ( ) increment of a quantity.[ ] array or matrix.[ ]T transpose of matrix.
bold typeface indicates an array or matrix.
2. Kinematics:
ArjJ1 increment in element nodal rotation and displacements between times t-1and t, m = 1, 2 is the number of element end, and n = 1,2,3 indicates rigid body rotation, horizontal displacement, and vertical displacement respectively.
Axn increment in element independent deformations between times t-1 and t,n = 1,2,3 indicates flexural rotation, transverse deformation and axial deformation (shortening) respectively.
element rigid body rotation measured from the horizontal axis y at time t.
V5 o similar to <pA but measured at time t-1.
Figure 4.3. Typical discrete element in a displaced and deformed configuration.
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\p element rigid body rotation measured from the horizontal axis y afterallowing for element transverse deformation and axial shortening at time t.
e = - \J/.
L, length of element after allowing for axial shortening at time t.
L,J length of line joining the two ends of the element after allowing fortransverse and axial deformations.
L 0 original length of element at t 0 = 0.
Ay0 horizontal projection of the element at time t 0 = 0 .
Az0 vertical projection of the element at time t 0 = 0 .
structure nodal displacements, i indicates number of node, and n = 1,2,3 indicates rotation, horizontal displacement and vertical displacement respectively.
3. Kinetics:
Rg1 element nodal moment and forces, m = 1, 2 indicates number ofelement end, n = 1,2,3 indicates moment, horizontal force, and vertical force respectively.
XJJ1 element end forces, m = 1,2 indicates number of element end,n = 1,2,3 indicates bending moment, shear force, and axial force respectively. X*, X \, X^ are nominated as element independent forces.
F£ element concentrated applied load, c indicates centre of element andn = 1,2 indicates moment and transverse follower force respectively.
element uniformly distributed applied follower pressure.
U£ element inertial forces, c indicates centre of element, and n = 1,2,3indicates inertial moment due to rotatory inertia, horizontal inertial force, and vertical inertial force respectively.
69
p^ structure nodal applied loads, i indicates number of node, and n = 1,2,3indicates moment, horizontal load and vertical load respectively.
4. Others:
s Sine <p,.
c Cosine </? 1.
s' Sine \p.
c' Cosine \p.
B strain-displacement matrix.
W inertia matrix.
H load matrix.
J geometric nonlinearity matrix.
C connectivity matrix.
m mass matrix.
X element independent forces = [X 2 X 2 X 2 ]T
m total mass of element,
d depth of element cross section.
Icxx mass moment of inertia of the prismatic element with respect to acentroidal axis xc normal to the plane yz of motion.
kcxx radius of gyration of the mass of element with respect to a centroidalaxis xc normal to the plane yz; k ^ x = J Icxx/m. For a rectangular prism such as the element used here, k<?xx = (1/12)(L2 + d 2).
7 0
4.3. RELATIONS AT ELEMENT LEVEL
4.3.1. Kinematic Relations
Relations in Incremental Form
Element independent deformations can be expressed in terms of element nodal displacements by the following incremental relations (Figure 4.3).
Ax
w h e r e ,
i i
Ax =
Ax
l; - { W o + [■Ar -
- Ar i (4 .1 )
s in - v ] (4 .2 )
+ L cos - (4 .3 )
(Az + [Ar2L o L 3 - w i r f (4 .4 )
\[/ = a r c t a nAz + fAr2 - Ar1]0 l 3_______ 3J
Ay + |Ar2 - Ar1] o l 2 2 J
<P1 Ar1 +0? i o
Relations in Differential Form
(4 .5 )
(4 .6 )
Because element independent deformations are assumed small, the above relations should be written in differential form before being employed in the analysis,
dx = d r 2 - d r 1i i i (4 .7 )
dx - dL’ s i n - f ] - LJ c o s j ^ - d^] (4 .8 )
dx^ = dL c o s ^ - L| s i n £ d i - d^j (4 .9 )
7 1
where,
dL = [O —c ' - s ']s ' dr (4 .1 0 )
d\p = 0 s ' / L ' - c ' / L ' 0 - s ' / L ' c ' / L ' d r ( 4 . 1 1 )
do? = drr i i (4 .1 2 )
and where dr d r 1 1 1 2 2 2dr dr dr dr dr2 3 1 2 3
TFor c o n s i s t e n c e w i t h the
small deformations assumption, the angle 0 = p 1 - is assumed small - that is, sin0 = 6 and cos0 = 1. Taking this into consideration, equations (4.7), (4.8), and (4.9) may be rewritten
dx = d r 2 - d r 1 (4 .1 3 )1 i 1
dx = L 'd r 1 - s in p [d r 2 - d r 1] + co s <p [d r 2 - d r 1] (4 .1 4 )2 1 1 * i l 2 2J r i l 3 3 J
dx = - L '0 d r 1 + co s y? [d r 2 - d r 1] + s in p [d r 2 - d r 1] (4 .1 5 )3 1 *1 l 2 2J r 1 l 3 3J
Substituting relations (4.10), (4.11), and (4.12) in equations (4.13), (4.14), and (4.15), and presenting the results in matrix form, yields
■ *dx -1 1i
dx - L 1 s - c . - s c2 1
dx - L ' 0 - c - s c s3 1
dr
dr
dr
(4 .1 6 )
dr
dr
dr
7 2
The above equations can be briefly written as
dx = B (r) dr (4 .1 7 )
Equations (4.16) and (4.17) embody the relation between element independent deformations and element nodal displacements for a regime of small deformations. The relating matrix B may be called the strain-displacement matrix.
4.3.2. Kinetic Relations
Relations in Finite Form
A free body diagram of a typical element is shown in Figure 4.4(a). Itdisplays element end forces, applied loading, and inertial forces. Employingequilibrium conditions for a typical element in its local axes, it is feasible toestablish a relationship between element forces (X ], and X p at the rigid end,and independent element forces (X |, X | and X^) at the deformable end of the element,
X1 + x 2 L’ c o s e + x 2 + x2 L’ sin# - U + U fL /2-| sin1 2 1 1 3 1 1 2 L 0 J 1
d .2U 3 [ V 2 ] C ° S * 1 + F 1 + F 2 [ V 2 ] + F 2 [ V 2 ] (4 .1 8 )
x1 + x2 + Uc sin CP + Fc + Fd L - U° cos w = 0 ” - ” r ’ ** 2 0 (4 .1 9 )
X1 + X2 - UC cos p - U C sin^3 = 0 (4 .2 0 )
Taking element end forces to one side of the equations and constructing relations (4.18), (4.19) and (4.20) in matrix fashion
7 3
//
//
//
Figure 4.4. Kinetics o f a typical element, (a) End forces3 applied loading and inertial forces, (b) Forces at element nodes.
7 4
x11 -1 -L' -L' 6 1 i
x1 -12
x 1 -13
„ .2X1 1
2X 12
x 2 13
The forces acting at element nodes are shown in Figure 4.4(b). Establishing the equilibrium equations and setting them in matrix form yields
7 5
i
2
R1
3
■1 . . 1 . x1
1 11
- s c | . x 11 21
c s | . x11 31
. 1 1 x21
11 „,2. 1 . - s c X1 21 .,2. 1 . c s X3
(4.22)
Later in the analysis, it will be required to deal with nodal forces R which are applied to a structural node rather than an element node. Substituting the negative of equations (4.21) in equations (4.22), it is possible to relate structure nodal force R to element, independent forces X, inertial forces U0 and applied loads F. This produces the subsequent relations.
’R11 -1 -L' -L' e
R12 s - c
R13 • -c - s
R21 1 •
R22 - s c
R23 c s
+
76
- [ V 2] ( V 2] u
-1 -L /2o ■ L2/2o
L so
-c -L c0(4 .2 3 )
Relations (4.23) can be written briefly in the following manner
R = B*(r) X + W(rJ) Uc + HCr1) F (4 .2 4 )
In comparing the coefficient matrix B*(r) with matrix B(r), of equations (4.17), it can be established that
B*(r) = BT (r)
This equality obtains despite the fact that the kinematic relations (4.17) and kinetic relations (4.24) are independent. Equations (4.23) may be rewritten
R’ - BT( r ) X (4 .2 5 )
w h ere,
7 7
R' = R - W(r1) Uc - H ( r 1) Fi i
By taking the inner product of the quantities involved in equations (4.17) and (4.25), the following relation is realised
s t a t i c a l l y e q u i l i b r a t e dI I I
(d x ) T X ( 4 . 2 6 )I I
_ \_k i n e m a t i c a l l y c o m p a t i b l e
Relations (4.26) give the variational representation of the kinetic-kinematic duality known as d'Alembert's Principle.
(d r ) T R'
Relations in Differential Form
The differential form of equations (4.23) may thus be written
dR1
dR12
dR13
-1 -L' - L '0
s - c
- c - s
dR2i 1
dR2 - s c2
dR2 c s3
dX'
dX‘
dX'+
c ’ s ' I . - c ' - s ' 1 1
c . . 1 .11
s . . 1 .1
d r 11
d r 12
d r 13\11
d r 21
1- c . . 1 .
1d r 22
1- s . . 1 . d r 23
. 6c ' 0 s ' l . - B e ' -Os' dr1I 1
Is . . I . dr1
I 2I
-c . . I . dr1I 3II
dr2I iI
-s . . I . dr2I 2I
c . . I . dr23
+
+
1 - [ V 2] s [Lo/ 2 3 0 dUC ~[L0/2) C
1 dUC2
1 dUC3+ uc2
•
7 9
- ( V 21 dr -1 - M - [ h» dF
L so dF
-c -L co dF
c
d r 1
2
S+ (4 .2 7 )
Equations (4.27) can be composed in the following concise manner:
dR = B * ( r ) dX + W(r’ ) dU° + H ( r ’ ) dF + J ( r ) dr (4 .2 8 )
where,
J =
('V 2] [- CU3 - O • K
tX 3 + F 3 + V , ] + S M
fx2 + Fc + L Fdl - c |x2ll 2 2 0 2 J l 3J
K i - •(*;]
[X>] + C (X 3 ]
S' K + 0X32] - C ' ( V + - S ' [ X 2 + 0 X g]
(4.29)
COo
8 1
4.3.3. Inertial Forces
The beam element developed in this study has a continuous rigid mass with deformations concentrated at one end. The various springs that are used to model element flexibilities are massless, consequently their deformations do not generate inertial forces. The only types of movement that are dual to corresponding inertial forces are the element rigid body rotation and displacements (r], r^, r^). Inertial forces associated with a typical element are shown in Figure 4.4(a). They can be expressed in terms of the rigid body displacements in the subsequent matrix form
„c *Ui
„cU = m2..CU3
k2cxx
1K - M [sin f o ) ^
1 K + i L o/ 2 ) i COS * o ] ^
(4 .3 0 )
or briefly as
UC = m r ( 4 . 3 1 )
Equations (4.31) have the following differential form
dUc - (dm)r + m dr ( 4 . 3 2 )
In this study, the mass of the element is required to be constant throughout the response; therefore equations (4.32) develop into
dUc - m dr ( 4 . 3 3 )
or in expanded form
8 2
dU° k 2 d r 11 c x x i
dUC2 = m 1 - [ v 2Hsin * . KdU°3 1 d**3 + [L0/2HCOS
4.4. RELATIONS AT STRUCTURE LEVEL
4.4.1. Kinematic Relations
In a preceding section, 4.2.1, relations had been established between element independent deformations (xn, n = 1, 2, 3) and element nodal displacements (r^, r^, n = 1, 2, 3). In this section, relations will be established between element nodal displacements and structure nodal displacements (q^, n = 1, 2, 3). For a typical node joining two beam elements (j—1 and j) within a beam structure, the set q of degrees of freedom is shown in Figure 4.5. By imposing compatibility, the following relations emerge:
1----------1" 3CN r-
U 1 • •
2r2 , J - i • 1 •
2r 3 , J - l 1
1r 1 . J 1 •
ir 2, J • 1
ir 3 > J • 1
(4.35)
Equations (4.35) may be written in brief as
r = C q (4.36)
8 3
Figure 4.5. Generalised coordinates of a typical joining two elements.
Figure 4,6. Nodal forces of a typical inner node elements.
inner node
joining two
8 4
and in differential form as
dr = C dq ( 4 . 3 7 )
where C is the connectivity matrix. Relations (4.37) are valid for any structural node, and in the presence of kinematic boundary conditions, the corresponding degrees of freedom are restricted.
4.4.2. Kinetic Relations
Relationships have been formed earlier, section 4.3.2, amongst structure nodal forces (Rj\, R^, n = 1, 2, 3) associated with a beam element and the element's independent forces (X^, n = 1, 2, 3), inertial forces (U&, n = 1, 2, 3) and applied loads (F^, F^, F^). In this section, connections will be established between a set of nodal loads applied at any structural node (p^, n = 1, 2, 3) and structure nodal forces of the elements concurrent at that node. For such a characteristic node, Figure 4.6, the imposition of equilibrium conditions yields
iP, 1 . .111 1iP
11 . 1 . 12 1iP„
11 1 . . 13
R1 . 1 . J
R1 . 2»J
R1 .3 * J
Equations (4.38) may be presented in brief as
Tp - C R
( 4 . 3 8 )
( 4 . 39 )
and in differential form as
8 5
dp - CTdR ( 4 . 4 0 )
where CT is the transpose of the connectivity matrix established in equations (4.35). Equations (4.40) are valid for any node on the structure; in the presence of kinetic boundary conditions, the corresponding force is constrained.
In substituting the expression for R, as given by equations (4.27), into relations (4.40), it becomes feasible to present the nodal equations of equilibrium in terms of element forces and element applied loads. For a representative inner node in a beam structure, the nodal equations of equilibrium may thus be depicted accordingly
mj l1} [d K ) j + W K]j (4.42)
8 6
M K l j + mj M j ( 4 . 4 3 )
Relations (4.41), (4.42) and (4.43) form a semi-discrete set of differential equations of motion for any node i located within a beam structure. Upon employing a suitable temporal operator, these equations may be turned into a fully discrete set of simultaneous linear algebraic equations. One such operator is that associated with central finite differences (explicit integration regime). This operator is chosen for the present study. Reasons for its selection and a general description of its attributes are presented in Chapter 6.
8 7
CONSTITUTIVE MODELLING IN MATERIAL NONLINEARITY
CHAPTER 5
5.1. INTRODUCTION
In this chapter, the constitutive aspects of the developed mathematical model are presented. A brief description of the elastic relations is initially set out. This is followed by the portrayal of the plastic constitutive laws which are exhibited in three separate sections with regard to perfect plasticity, strain rate sensitivity and strain hardening. For the cantilever beam element developed in this work the deformations, both elastic and plastic, are assumed to be concentrated or lumped at the free end.
5.2. ELASTIC CONSTITUTIVE RELATIONS
For the cantilever beam element developed in this study, the lumped elastic deformations are representative of flexural rotation, transverse shear deformation and axial deformation. Disregarding any of these actions is tantamount to assuming that the corresponding stiffnesses are infinitely large.
The adopted elastic constitutive equations in their flexibility form are
Lq LoX — X1 El 2EI i
Lo L3 k' L .,2X = ------ -- + ---- X2 2EI 3EI GA 2
2X • - X3 EA 3
8 8
or briefly
x = Kf X (5 .2 )
where L0 is the original length of element, E is Young's modulus of elasticity, I is the second moment of area of cross section, k' is the cross section shape factor for shear, G is the shear modulus of elasticity where 2G = E/(l + y), and A is the area of element cross section.
The coefficient k' is a dimensionless quantity dependent on the shape of the cross section. It is introduced to account for the fact that the shear stress andstrain are not uniformly distributed over the cross section. According to Timoshenko (1940), k' is a numerical factor with which the average shearing stress must be multiplied in order to obtain the shearing stress at the centroid of the cross section. Amongst several writers, Cowper (1966) has pointed out that thecustomary values of k', given for instance by Timoshenko (1940), lead to unsatisfactory results when Timoshenko's beam equations are used to calculate responses in the high frequency spectrum of vibrating beams. It is also pointed out that the distribution of shear strain over a cross section depends on the mode of vibration of the beam, and hence it varies with the frequency. The unsatisfactory results, it is claimed, arise from the use of static stress distributions. Cowper's approach was to derive the Timoshenko beam equations by integration of theequations of three-dimensional elasticity theory. From such dynamicalconsiderations a formula for k' was obtained and numerical values were calculated for a number of cross sections. These, together with the corresponding valuesgiven by Timoshenko, are reproduced in Table 5.1.
Cowper (1 9 6 6 ): For any v
v - 0 .0
v = 0 .3
R e c t a n g l e C ir c le= 12 + U p
1 0 ( 1 + v)1.200
1.177
_ 7 + 6 v6(1 + v)1 .1661 .128
Timoshenko (1 9 4 0 ): 1 .500 1.333
T able 5 . 1 . Shear f a c to r k' fo r r e c ta n g u la r and c ir c u la r c r o ss s e c t io n s .
Cowper further pointed out that the nature of his approximations regarding the shear stresses would suggest that the corresponding values of k' are most satisfactory for static and low frequency deformation of beams. However, the results given by
8 9
that author agree well with those of other researchers who have estimated valuesfor k' for high frequency vibration modes. It seems that the work of Cowperoffers an adequate source for the numerical estimation of values for k \ and onewhich is therefore adopted in the present study.
By inverting equations (5.1), the stiffness form of the elastic constitutiveequations, as used in the present study, is obtained
9X1
?X =2
,2X3
4EI 1 + a / 4 6EI 1
L0 1 + a Lo1 + a;
6EI „ 2
1 12EI l
l + O ! L 3 1 + a
EA
'X i
X 2
X 3
(5 .3 )
where a = 12EI k'/(L^ GA). Equation (5.3) may be written briefly
X = K s x (5.4)
As the shear rigidity of a cross section becomes infinitely large, the value of o tends to zero and the stiffness matrix Ks reduces to
X' =
4EI
6EI
6E I
12EI (5 .5 )
EA
The matrix is the elastic stiffness matrix where only flexural and axialdeformations are accounted for. The differential form of equations (5.4) is
9 0
d X = K s dx ( 5 .6 )
for a constant matrix Ks
5.3. PLASTIC CONSTITUTIVE RELATIONS
5.3.1. Perfect Plasticity
Analytic Aspects
In the cases of single- or multi-stress states, there will exist some function A - ACrj or A ( r it r 2, . .. ., r n) where Tn is the nth stress component - which characterises the yield condition. This function will here be called the plastic potential function. For normalised stresses, the constitutive state of stress can be described by,
A or,, r 2 , . . .., r n ) < 0 Elastic stateA (F\ 1 r 2 , . . • ■. r n ) - 0 Plastic stateA o v r 2 . .. • •. r n > > 0 Insupportable state
A geometric interpretation of this yield function ACI , T2) for a biaxial stress state ( r if r 2) is shown in Figure 5.1. The state of stress at a particular point in the structure is then represented by a stress point with coordinates T] and orequivalently by the vector r 1 extending from the origin to that point. The stress points for which A (T ,, T2) = 0 represent a curve which is generally closed and contains the origin - the stress free state. It will be assumed that the yield curve contains a convex set of points. If the stress point is in the interior of this set, there is no plastic flow; if the stress point remains on the yield curve plastic flow may occur; and a stress point outside the set contained by the yield curve cannot be supported by a perfectly plastic material.
The plastic constitutive law may be stated in the same geometrical terms. It may be done by superposing a set of plastic strain rate axes 7 ^ and 7 2p on Figure 5.1 in coincidence with the stress axes T, and r 2. The plastic strain rate can then be represented by a vector, such as 7 with components (7 ]p> 7 ^ ) , which originates from the corresponding current stress point T1 with components (F]» r p . With the geometric system thus defined, the plastic flow rule proposed by Mises (1928) may be expressed algebraically by
9 1
Figure 5.1. Yield euvve and flow Tuie.
cm a
9 2
3A7 ip ar,
7 2p S3A_a r 2
( 5 .7 )
where g is an arbitrary non-negative scalar. The rule exemplifies a normal flow rule because the partial derivatives (3A /3rit 3A /3r2) form the components of a vector which is normal to the yield surface A = 0 at the current stress point ( r i? r 2). The plastic strain rate vector 7 ^ with components given by equations (5.7) is therefore always normal to the yield surface at the current stress point, and it is always outward pointing in virtue of the non-negative nature of the scalar g.
If the yield curve is continuous with non-zero curvature at all points, the flow rule establishes a unique correspondence between the state of stress and strain rate; in perfect plasticity this is subject to the limitation that the strain rate magnitude is indeterminate in the scalar g.
In the case of a yield curve with a straight side, a strain rate vector normal to that side does not uniquely define the stress point; and in the case of a yield curve with a corner, the direction of strain rate can not be uniquely defined at the corner. At a corner point, difficulties arise because of the intersection of two or more surfaces. For such a stress state, the normal to the yield surface becomes indeterminate. However the ambiguity is largely removed if the criterion for yielding at such a point is applied for each function separately. This is plausible because each yield function represents a separate yield criterion. Algebraically the normal at a corner may be regarded as a linear combination, with positive coefficients, of the normals to the intersecting surfaces. Thus, for a biaxial stress state, if A, (r 1, T2) = 0 and A2 ( r it T2) = 0 are the equations of two sides intersecting to form a corner, the strain rate vector has components
3A, 3A 27 i = g i ----- + g 2 ----- ( 5 -8 )3r , 3 r ,
3A, 3A 27 2 = g i ----- + S 2 -----3 r 2 3 r 2
where g n and g 2 are non-negative but otherwise arbitrary scalars.
In the present study, the concern is with small deformations. A basic assumption of the classical plasticity theory - Hill (1950) - is that during an infinitesimal increment of stress, changes of strain are assumed to be divisable into elastic and plastic parts in an additive decomposition manner
9 3
dy = d7 e + cfyp (5 .9 )
Other decomposition proposals exist in the literature; however the approach presented by equation (5.9) is adopted in this study.
Computational Aspects
In the present study, forces and deformations are used throughout the analysis instead of stresses and strains. The relevant internal forces are bending moment X n, shear force X 2 and axial force X 3, their corresponding deformations being x p x 2 and x 3 respectively.
Computationally, yield is forseen by a Predictor-Corrector approach. The Predictor solution is essentially an elastic solution which satisfies kinetics and kinematics, but violates constitutive laws at certain points in the structure. TheCorrector solution consists of a self-equilibrated structure with plastic hinges at appropriate points such that a superposition of both solutions would result in asystem satisfying kinetics, kinematics and the constitutive laws.
For constitutive models where the yield is governed by a single internal force, say X v in the two-dimensional space of X 1 and X 2, the yield curve acquires the configuration of two infinite straight lines parallel to the X 2-axis at values of X, = ± X ,p , where X ,p is the plastic capacity of the homogeneous section with respect to X v This is shown for normalised forces X {, X ’ in Figure 5.2.Generally, a force increment vector dX’ is made up of two components dXJ and dX2, the latter component being always elastic, as is assumed a priori. If the constitutive laws are violated by the increment dX \ the former force component dXJ would be composed of two parts dXJg and dXJp; let them be called elasticand plastic respectively, Figure 5.2. The elastic parts dX2e and dX{e only of each force component are fed back into the equations of motion, and a plastic hinge is inserted into the structure at the particular site where plastic activation has occurred. Thus, this correction now ensures the satisfaction of kinetics, kinematics and also constitutive laws.
For constitutive models where the yield is governed jointly by both internal forces - X, and X 2, the yield curve in the two-dimensional space of X, and X 2
attains the shape of a plane closed curve. This is shown for normalised forces in Figure 5.3. The force increment vector dX' (vector AC) which violates the yield condition is shown in Figure 5.3 to originate from point A on the yield curve. In practice this is not always the case; in the present study, when the increment vector commences from point A' rather than from A, the solution is advanced to
94
Figure 5.2. Yield surface governed by a single force.
95
Figure 5.3. Yield surface governed by two forces.
96
point A by interpolating the solutions at A' and C. The error thus introduced is kept acceptably small because: (1 ) very small time steps are used in thistime-marching analysis, and (2) the single time step which marks the opportunity for such error occurs only in the transition from an elastic state to a plastic one, or vice-versa.
Initially in the analysis, the force increment vector AC is decomposed intoelastic and plastic parts - vectors AB and BC respectively. The former should be tangential to the yield curve at point A, and the latter perpendicular at the same point. However, since vector AC is very small, the vector AB is approximately tangential to the yield curve at point A. Such a process may be viewed as piecewise linearisation of the yield curve; however, this is done only for a smallpart of the yield curve and where required, in contrast to other approaches in which linearisation is done a priori. Vector BC is perpendicular to vector AB and hence (approximately) to the yield curve.
Computationally, the decomposition process of vector AC is done by regarding this vector as the diameter of a circle. If AC is not perpendicular to the yield curve, then its circle would intersect the curve at a second point B which would represent the corrected state of stress. Such a construction thereby identifies astress-state which now satisfies the yield conditions.
This manner of practical division of the increment vector is attractive and simple; the computer programming side, however, is not necessarily so elementary when one considers the fact that, in a real situation, one may be dealing with all four quadrants of the yield curve.
The corrective increment dX£ and error increment dXp may be decomposedinto their constituent components with respect to X{ and X^; thus dX£ isdecomposed into dXJe and dX^e as suggested in Figure 5.3. These corrective components are fed back to the equation of motion to determine the correct accelerations and a plastic hinge inserted at the activated section of the structure.
In the present study, if a third internal force X 3 is included, it is assumednot to interact with the other two X\ and X*. In such a case, the plane curve inFigure 5.3 would represent a cross-section through a cylindrical yield surface in three dimensions, as for example in Figure 8.6(b) in Chapter 8.
The yield criterion adopted in this study is that of Tresca. A beam undercombined shear and bending, and loaded in one plane only, would have the following yield condition
9 7
a 2 + At 2 < a* (5 .1 0 )
where a is the normal stress parallel to the longitudinal beam axis, r is the shearing stress perpendicular to the same axis, and Oy is the static yield stress in simple tension.
5.3.2. Strain Rate Sensitivity
The perfectly plastic behaviour of solids postulated earlier is probably a fiction as the maximum stress which can be carried by a solid is almost unvaryingly associated with the rate at which this is applied. In general, for metals exhibiting yielding, the yield stress increases continuously with an increasing strain rate throughout the plastic response. For the additive decomposition concept of the strain rate increment vector
d7 = d7e + d7p (5 .11 )
it is assumed that the plastic part of the strain rate represents combined viscous and plastic effects. The effect of strain on the material yielding is determined from the elastic strain rate vector 7 e identified with the stress point at the relevant instant in time. In the present study, the term elastic-viscoplastic indicatesmaterials showing viscous properties in the plastic region only. Viscoplastic materials are assumed non-strain hardening.
For a yield surface dominated by a single internal force, the change in the shape of the surface resembles that shown in Figure 5.4. The dashed lines represent yield surfaces for various strain rates occurring during the response time. Increasing strain rate, as can be seen, causes the yield surface to expand; upon unloading, the surface would contract, due to the decreasing strain rate, until it coincides with the initial (static) yield curve at zero rate of strain. Analytically, there exists a difficulty in interpreting this contraction process. For example, let it be assumed that the stress point A at time t 3 lies on the yield surface as shown in Figure 5.4, and that thereafter the strain rate starts to decrease such that, at a later time t 4, the yield surface has contracted to that indicated by the number 2 in the figure. The stress point, however, is that denoted by the symbol B which lies in between the yield surfaces designated by the numbers 3 and 2. The dwelling problem is that the whole of the force increment vector dX', or AB in the figure, lies outside the yield surface. The solution developed in the present study is quite
9 8
Figure 5.4. Strain rate dependent yield surface dominated by a single force.
9 9
simple and seems capable of producing good numerical results, as will be demonstrated in Chapter 7. First of all, vector dX' is put into its constitutent force components dX{ and dXj, see the inset of Figure 5.4. Secondly, these two vectors are in turn decomposed. The vector dXJ is decomposed into two vectors, dXJe which allows the yield condition to be satisfied, and dXJp (of opposite direction) which represents an error. The vector dX^ is essentially totally elastic. It follows from this interpretation that the new stress point at time t 4 is point C lying on the yield surface indicated by the number 2. This solution adheres to the rules of perfect plasticity advocated in the preceding section (5.3.1).
For a yield surface governed by two forces, the change in its shape during the deformation process is influenced by the two corresponding deformation rates. Normally, these strain rates have different magnitudes. Consequently, the viscoplastic yield surface will change both its size and shape during the response history, Figure 5.5. The analysis of this problem has not been tackled in the present work.
The effect of strain rate on the dynamic yield stress of discrete beams has been accounted for in this research in three different manners:
1. The dynamic yield stress is determined by upgrading the static yield stress in accordance with the initial strain rate at midspan - being the strain rate at which incipient yield at midspan first occurs. Thereafter, this dynamic yield stress is kept constant in time and used for the whole beam span. After its initial evaluation, the dynamic yield stress is both space and time independent.
2. The dynamic yield stress is worked out for each element in accordance with the initial strain rate at that element, and is subsequently kept constant. The yield stress is thus space dependent but time independent.
3. The dynamic yield stress is calculated at each element for each time step all through the response. For this analysis, the stress is both space and time dependent.
The constitutive equation (2.1) is employed throughout this study for the evaluation of the dynamic yield stress. It is basically an empirical formula of nonlinear nature, deduced from experimental data.
1 0 0
Figure 5.5. Strain rate dependent yield surface dominated by two forces.
Figure 5.6. Stress-strain relationship in a strain hardening material3 with yielding dominated by a single stress component.
1 0 1
5.3.3. Strain Hardening
In the present study, a module has been included in the numerical analysis to deal, in a limited way, with the strain hardening aspect of material behaviour. For yielding dominated by a single stress component, Figure 5.6, the stress-strain characteristic is approximated by linear segments OA, AB, BC, ... with effective moduli E, E p1? E p2, .... respectively in tension, and similarly for compression. If the material in unstressed from point C, the path CF of elastic recovery is taken, the modulus being E - that of the initial elastic path OA. Yielding in compression is then assumed to occur at point F for which the yield strength is the same as that associated with point D.
The response of a simply supported beam made of a strain hardening and strain rate sensitive material (2024-0 aluminium alloy) - reported in the work of Balmer and Witmer (1964) - was predicted numerically in the present study. An insignificant difference in the midspan transverse displacement was obtained between results for the elastic, strain hardening material and those for an "equivalent" elastic, perfectly plastic material in which the yield strength is set at an average value in the range of yield strength of the strain hardening material. It is not feasible to decide whether this indifference is due to: insufficient experimental data on the material behaviour under dynamic loading, fortuitous numerical results for this particular material, or the generally inconsequential effect of strain hardening on the dynamic response. Accordingly, no results are presented here.
It was mentioned in the previous section (5.3.2) that, almost invariably, the yield stress for solids is associated with the rate at which strain is induced. Experimental studies in which both strain hardening and strain rate sensitivity are measured and their interaction observed are almost non-existent; and similarly, there are difficulties in establishing constitutive laws which properly incorporate the interaction between these phenomena. Such circumstances cast a shadow of doubt on the ability to account for strain hardening in a general study of transient elasto-viscoplastic response of structures under conditions of geometric nonlinearity.
CHAPTER 6
TEMPORAL OPERATOR AND COMPUTER PROGRAM
6.1. INTRODUCTION
In the first part of this chapter, the general features are portrayed of the two major classes of temporal integration of the semidiscretised equations of motion. This is then followed by a brief description of the selected operator, with emphasis on the aspects that warranted its choice. In the closing part of this chapter, an indication is made of the developed computer code, its functions and the various associated facets.
6.2. TEMPORAL OPERATOR
The most important factors in the efficiency of dynamic calculations are generally associated with the solution procedure employed. The choice of solution technique is critical. In some circumstances, a particular device may lead to an inefficient and overly expensive solution; in other circumstances, the same device may give rise to such an accumulation of errors that the solution becomes meaningless. With an inefficient technique, the analyst is often tempted to reduce solution costs by surrendering control of accuracy. Closed form solutions, which are of value in solving equations for simple models, are generally of little value in solving equations for complex multi-degree of freedom models. For these, recourse is generally made to numerical techniques appropriate for computer processing.
The strategy considered here for the solution of general systems of ordinary differential equations is that of the direct integration procedure. In this procedure, the equations of motion (with damping neglected)
m x + K x = R (6.1)
are integrated using a numerical step-by-step process. The term "direct" indicates that, prior to the numerical solution, no transformation of the equations into a
1 0 3
different form is carried out, such as occurs in the mode superposition procedure.
Direct integration methods, according to Bathe and Wilson (1976), are based on two ideas:
1. The equations of motion (6.1) are satisfied only at discrete times t0 , t 1? ...., tn, tn+1, ••••» *f *n interval (tQ, tf) of solution.
2. The variation of displacements, velocities and accelerations within each interval or time increment Atn = tn+1 - tn is assumed.
In this work, the time increment or time step size Atn is assumed fixed throughout the entire interval of solution.
Within the direct integration procedure, there are two basic classes of methods : explicit and implicit. In explicit methods the displacements at time tn+1 aredetermined in terms of accelerations and displacements at the immediately preceding time tn ; and if a diagonal mass matrix is used, no simultaneous system of equations need be solved. In implicit methods, on the other hand, the equations fordisplacements at time tn+1 also involve the accelerations at that time; hence thedetermination of displacements at tn+1 involves the solution of a simultaneous system of equations. It is the solution of this system of equations that usually consumes most of the time associated with the solution of the problem at hand.Thus, explicit methods are considerably more efficient per time step than implicitmethods. However, the time step size in explicit methods is restricted bynumerical stability requirements which may result in a time increment much smaller than that needed for the requisite accuracy; for implicit methods, on the otherhand, the time step size is restricted only by accuracy requirements.
The choice of a temporal operator for the solution of nonlinear dynamic problems may prove more demanding than that for a linear problem; grossinaccuracies or high computer costs can arise from an inadvertent selection of operator. In the next section the advantages and disadvantages of explicit and implicit operators are presented, with emphasis on nonlinear modelling.
6.2.1. Comparisons of Explicit and Implicit Operators
Explicit Methods: Advantages
1. A relatively small number of computational operations per time step.
1 0 4
2. Simplicity in algorithm logic and structure. Hence, complex nonlinearity, be it constitutive, geometric or boundary type, can be treated easily with virtually no added cost from the linear case.
3. Requires relatively little "in-core" computer storage.
4. Reliable in terms of accuracy and completion of the computation.
5. Absence of an iteration procedure within each time step brings considerable simplification in use. The only decision required is the time step size, and if this is efficiently chosen by the computer program an explicit method gives rise to a fully automatic procedure.
6. Uncoupling of the equations of motion. No global stiffness matrices need be assembled.
Explicit Methods : Disadvantages
1. Conditionally stable, hence a large number of time steps may be required.The size of time step is bounded by the smallest traversal time of the longitudinal wave across an element. This limitation is consistent with the local uncoupled integration of the equations of motion : a stress wavetravelling across an element within one time step would affect the surrounding elements, hence the behaviour of that element would no longer be independent from the rest of the model.
2. Limitations concerning mass representations and starting conditions.
Explicit Methods : General Attributes
1. The integration of the equations of motion requires little computer effort, thus the cost of any computation varies directly with the number of elements, the complexity of the element and the complexity of the constitutive models used.
2. Cost of a large number of time steps is compensated by the very low cost of a single explicit step.
3. Spatial discretisation errors are always the dominant type, because the truncation errors due to time integration in the stable domain are usually far less (of the order of At2).
1 0 5
4. As a rough guideline, explicit methods are best suited for wave propagationproblems where there is a high frequency content.
Implicit Methods : Advantages
1. Much larger time step can be used because of enhanced stability.
2. Higher degree of generality of application.
3. Accurate and economic solutions can be obtained when applied to the appropriate type of problem in which the time step size is not restricted - for example, by nonlinearities or wave propagation.
Implicit Methods : Disadvantages
1. Greater complexity and size in the software, particularly if Newton-Raphson solution techniques are used.
2. Less reliability.
3. Greater computer-core storage requirements.
Implicit Methods : General Attributes
1. The solution of the linear or nonlinear algebraic system of semidiscretised ordinary differential equations is performed basically by two means:
- Newton-Raphson methods, which for linear systems become direct elimination (triangulation and back substitution); for nonlinear systems, the matrix often needs to be retriangulated if reasonable convergence is to be achieved.
Iterative methods, the simplest of which are the Jacobi and Gauss-Seidel methods.
2. If high frequency components of the solution are relevant then it is necessary to use time step sizes of the order required for stability in explicit methods.
3. In nonlinear systems, the achieving of dynamic equilibrium, which is indespensible for controlling the accuracy of the solution, involves elaborate methods of solution. The user is faced with a variety of choices such as:
106
choice of a method for iteration within a time step;
choice of time step length according to the type of problem and the convergence of the iteration procedure within the time step;
choice of a tolerance in the verification of dynamic equilibrium.
4. In dynamic nonlinear problems, implicit methods generally require a significant amount of stiffness updating and force equilibrium iterations; and for the same solution steps, more matrix manipulations are required. These additional computational costs necessitate the use of a time step size at least a few times larger than the stability limit of an explicit operator in order for an implicit operator to be cost-effective.
Summary
Both the explicit and implicit methods have problem classes for which they are most suitable. At times, the simultaneous applications of the two methods to a single spatial mesh provides the optimal solution strategy. Currently, implicit methods are most commonly used in finite element programs, and explicit methods are widely used in nonlinear computer codes - as for many classes of such problems they provide an efficient, easily programmed transient solution techniques.
6.2.2. Central Finite Difference Explicit Operator
6.2.2.1. Description and Implementation
In this operator, it is assumed that the accelerations at time t are
1xt = --- (xt+At " 2xt + xt-At) (6-2)At2
and velocities at the same time are
1x t “ — ( Xt+At - Xt-A t) ( 6 -3 )2At
The solution for the displacements is obtained by considering the equations of motion (6.1 ) at time t
1 0 7
m x t + K x t = Rt ( 6 . 4 )
Substituting the expressions for xt and xt into relations (6.4) yields
1 „ , 2 x \ 1 ,(— ) m X t+At ~ R t ~ K - (----) m x t - (--- ) m
l A t 2 J L A t 2 J L A t 2 J Xt-A t ( 6 . 5 )
from which the solution for displacements xt+^t at time t+At is obtained. From the above relations, it can thus be noted that the solution for the displacements Xt+At *s based on using the equilibrium conditions at time t and involves xt_^t. As a consequence of this feature, a special starting procedure is required. At time zero, Xq and Xq are known and x0 can be calculated from equations (6.4); relations (6.2) and (6.3) can then be used to obtain the displacements x_^t.
6 .2.2.2. Size of Time Step
The size of time step (At) for an explicit integration procedure must be less than Ay/c, where Ay is the shortest distance between element nodes in a one-dimensional structural mesh, and c is the longitudinal wave speed, such that
cAt--- < 1A y
( 6 . 6 )
This can be stated in terms of the rate of information flow in the discrete model : a condition necessary for stability is that the rate of numerical information flow must be greater than or equal to the rate of information flow in the continuous problem.
In an analysis by Bathe and Wilson (1976), it was established that in linear analysis the central finite difference operator is stable when
2At < ------- ( 6 . 7 )^max
where is the highest frequency of vibration in a discretised system. For aone-dimensional, two-node element
2 c^max ^ ~ 7~Ay
( 6 . 8)
1 0 8
which, when substituted in relation (6.7), yields relation (6.6).
For nonlinear analyses, the above results can, with caution, also be applied, since for each time step the nonlinear response may be thought of as a set of linear analyses : that is, unless the system stiffens or softens during the response, in which cases adjustment of the size of time step must be performed such that criterion (6.6) is satisfied at all times. This is because, in nonlinear dynamics, the response is highly path-dependent, and any error admitted in the incremental solution at a particular time directly affects the solution at a subsequent time. Hence, stringent attention to the time step size is required.
According to Stagliano and Mente (1979), for thin beams (thickness less than one half the length) modelled with a lumped mass matrix, c^ ax can be approximated by
2wmax ” ~
EP
( 6 . 9 )
where 1 is the smallest element length, E is the Young's modulus of elasticity and p is the material density. The value J E/p is the approximate velocity of the longitudinal wave through the beam material. Thus for thin beams, the spatial mesh size will determine the maximum allowable size of time step. If, however, the beam is thicker than half the length, the highest natural frequency of the discretised structure will become dependent on the bending stiffness of the beam, and not the longitudinal wave speed. The approximation of coj^x for a lumped mass system becomes
~ 4d wmax ~ l2EP
( 6 . 10)
where d is the thickness of the beam. As a more conservative choice for nonlinear analyses, the same authors considered the time step size as
1 . 6At < ------- ( 6 . 11 )° W x
Relations (6.9), (6.10) and (6.11) are used throughout this study in determining the size of the time step of integration.
1 0 9
6.2.2.3. Numerical Stability
Integration schemes that require the use of a time step size smaller than a certain size, such as that of the central finite difference, are termed conditionally stable. An intuitive explanation of the stability restriction was provided by Mullen and Belytschko (1983); they considered the information flow through aone-dimensional mesh for both explicit and implicit methods:
1. In the central finite difference procedure (explicit technique) only thedisplacements at nodes i - 1 to i + 1 would affect the node i at the nexttime step; this limited flow of information is shown in Figure 6.1(a). The stability limit can be thought of as a requirement that the physical information flow rate does not exceed the computational information flow rate Ay/At.
2. The information flow in an implicitly integrated mesh is shown in Figure6.1(b). The horizontal information flow paths in the implicit method resultsfrom the coupling of the acceleration and displacements at the next time step. This coupling requires the solution of a system of simultaneous equations, butalso produces an infinite information flow rate, hence the unconditionalstability of these methods.
In an unstable solution, errors resulting from the numerical integration or round-off in the computer would, in a linear system, grow exponentially and trigger "overflow" traps. In nonlinear calculations, however, a numerical instability is not always as wild or as striking. This is because nonlinear processes are often capable of dissipating a large amount of energy, so the instability could be limited to asmall part of the mesh, a process sometimes called an "arrested instability".
Another aspect related to the stability of solution is the refinement of thespatial mesh in regions of interest. In static elasticity, this practice has fewdrawbacks. However, the situation is different in wave propagation problems, and the following spurious effects may be encountered:
1. The coarse mesh eliminates any frequency content beyond its resolution.
2. The coarse mesh reflects back a significant part of the high frequency waves generated in the region of interest; this leads to an overestimate of the high frequency response characteristics.
For explicit operators, numerical stability is assured if the time step size is small enough to allow the accurate integration of the response in the highest
1 1 0
(a ) Explicit integration information flow.
f+3Af --- ►
t+2At
/+Af -«--- H H ----►- -«--- ►
x x+Ax x+2Ax x*3Ax
(b) Implicit integration information flow.
Figure 6.1. Patterns of information flow in the spaee-time domain o f explicit and implicit operators. After Mullen and Belytschko (1983).
frequency component, and the worthiness of the solution to a dynamic problem is secured through the maximum time step criterion.
I l l
6.3. COMPUTER PROGRAM
As a complementary part of this research work, a computer program has been developed to encode the functions required for obtaining a numerical solution. The specific problems encompassed are those of the transient elasto-plastic response of pulse loaded beams.
The developed code is named TEPA2D, an acronym for Transient Elasto-Plastic Analysis in 2 Dimensions. The computer language used throughout the code is FORTRAN 77. In the main equation solving routine, double precision is requested. The program was developed, debugged and tested on a VAX minicomputer - VMS operating system - and examples were run on VAX workstations. The graphics post-processing package used was UNIRAS, which is apowerful suite of graphics programs that includes, amongst others, the UNIGRAPH program for the presentation of scientific data in graphical form.
In the following sections, some aspects of the developed code are presented inbrief.
6.3.1. Structure of Program
The integrated body of the program is made up of several subroutines, eachof which is assigned a specific function. The main four incorporating routines are
1. MESHES: The spatial mesh is generated here; this is done through two subroutines, AMESHG for automatic mesh generation and UMESHG for user mesh generation.
2. APPLOD: The applied loads are treated here. This is managed via twosubroutines: FORCES for identification of the type of applied load, and its assignment to a specific part of the element assembly; and HISTRY, for identification of the history of the load.
3. BOMAGE : For recognition and allocation of boundary conditions, material properties and some initial geometrical features.
1 1 2
4. EXECUT: The semidiscretised ordinary differential equations of motion aresolved here by utilisation of the temporal operator that of the central finitedifference. It is divided into six subroutines which are:
- INICON, for starting up the integration process by dealing with theprescribed initial conditions;
- INNER, for the solution of the equations of motion of an inner node;
- BOUND, for the solution of the equations of motion of a boundarynode;
- GAUSS, embodies the Gaussian elimination algorithm for solving a set of linear simultaneous equations;
- PLAST, a corrective procedure to cater for the development ofplasticity;
- ENERGY, for checking the energy balance of the system.
The structure of the overall code follows a modular form to facilitate the coding, debugging, testing and further development.
6.3.2. Flow of Computation
The flow of the computation logic of the main solving routine (EXECUT) is summarised in a diagrammatic form in Figure 6.2.
The output generated in the above mentioned routine can be composed in a manner destined for a graphics post-processor or for a readable printout; the output in both cases can be controlled to produce data for a specified number of node(s) at all or every specific number of time steps.
1 1 3
Figure 6.2. Flow o f computation logic in the main solution routine.
1 1 4
CHAPTER 7
VALIDATION EXAMPLES AND ANALYSIS OF RESULTS
7.1. INTRODUCTION
The purpose of this chapter is to present a number of examples with two main objectives in mind. Firstly, to validate and establish the limits of the proposed mathematical model; and secondly, to provide a detailed account of the response of beams with different boundary conditions under pulse loading. The validation examples chosen include experimental tests and numerical solutions reported in the literature.
Generally there are two types of dynamic response. Wave propagation is a type of response in which the behaviour at the wavefront is of engineering importance, such as in shock response from explosions or impact, and including problems where wave focussing, reflection and diffraction play a major role. Inertia-dominated response is that in which low frequencies preponderate. In the present work, the type of problems solved lie in-between these two. Firstly, there are examples in which wave propagation is important only in the early stage of the response where the applied load is of high intensity and of a duration typically measured in microseconds, such as for the examples presented in sections 7.2.2, 7.3.1 and 7.3.2. Secondly, there are examples of inertia-dominated response in which the applied load is of low intensity and of indefinite duration. Examples of this type are presented in sections 7.2.1 and 7.4.1.
7.2. SIMPLY SUPPORTED BEAMS
7.2.1. Comparison with ADINA
The example solved is a simply supported beam upon which a uniformly distributed pressure is suddenly applied in the form of a step-rise pulse of indefinite duration, Figure 7.1. The intensity of the applied pressure is equal to 15% of the static collapse, load and the material of the beam is elastic-perfectly plastic.
DISP
LACE
MENT
AT M
IDSP
AN (i
nche
s)
BEAM THICKNESS * 1 in
0.75 p - - o
timeSTEP PRESSURE
E * 3 * I04 kip/in2 v = 0.3
<7y * 50 kip/ in2p * 0 .733 x I03lbf sec2/ in 4p = STATIC COLLAPSE LOAD o
Figure 7.1. Beam geometry3 material properties and applied loading. After Bathe (1975). (1 in = 2.54 cm3 1 Ibf = 4.45 N).
0TIME (m illiseconds)
1 2 3 4 5 6 7 8 9 10 11 12
Figure 7.2. History of transverse displacement at midspan. (1 in - 2. 54 cm ) .
116
This example was presented and solved by Bathe (1975) using the Finite Element code ADINA. The aim was to indicate the importance of error control through equilibrium iterations in a dynamic analysis which uses implicit integration schemes. This example has since been used in the literature either for the same purpose (Geradin, Hogge and Idelsohn 1983) or as a validation example (Mofflin, Olson and Anderson 1985, Lee, Alwis, Swaddwudhipong and Mairantz 1988). In his solution, Bathe (1975) represented one quarter of the beam by six 8-node isoparametric elements, and employed an implicit integration scheme (Newmark Method (3 = y = £) with equilibrium iterations and a time step of0.5 x 10“ 4s (1 /1 00th of the fundamental period for linear undamped natural vibrations).
In the present study, this beam has been analysed in terms of eighteen 2-node continuous mass - lumped flexibility beam elements. The number of elements was determined experimentally, initially starting from four elements and increasing the number until convergence was achieved for the fully elastic solution; this process is explained further in Chapter 8, section 8.3. An explicit integration, time-marching scheme (central finite differences) was used with a time step of 2.75 x 10~6s. It may be re-stated here that explicit schemes do not require equilibrium iterations, but do require a small size time step for their stability. The size of the time step was determined using the criteria presented in Chapter 6, section 6.2.2.2. Sheardeformations were considered in the analysis with a shear factor calculated ask' = 1.177 using the formula recorded in Chapter 5, section 5.2. Geometric nonlinearity and rotatory inertia were also considered in the analysis.
The elastic and elasto-plastic solutions of the present study both compare well with those of the FE Code ADINA, as seen in Figure 7.2 where the time scale on the horizontal axis is non-dimensionalised through division by the time step used in the ADINA solution. For the full elasto-plastic response, the CPU time required was 2 minutes on a VAX/8600 minicomputer including output generation. There was slight inward movement of the roller support due to geometric nonlinearity. If such movement were restrained, the effect would have been to decrease the midspan displacement by approximately 4%.
The analysis of this beam example is carried further by an enquiry into the behaviour of other response parameters. In Figure 7.3, the history of the profile of transverse displacements is shown, in time sequence, in graphs (a) to (g). In graph (a) it is seen that initially the displacement at midspan lags behind those at quarter the span from each support; this is shown as negative bending moment (at approximately 0.35 ms) in Figure 7.5. The history of the development of plastic hinges in flexure is shown in Figure 7.4. It can be seen that the first plastic
( a ) cu rv e s 1 <o 0 from. 0.0 to 0.35 milliseconds
(a) curves 15 to 28 from 0.88 to 1.61 milliseconds
( b ) cvervas 7 to 14 fro m 0.40 to 0.80 milliseconds
(d) curves 29 to 42 from 1.67 to 2.42 m illiseconds
1 * I ' 1 ' I6 8 10 12
NUMBER OF NODEFigure 7.3 (continued on next page)
NUMBER OF NODE
117
DISP
LACE
MENT
(inc
hes)
DISP
LACE
MENT
(inc
hes)
( 0 ) curves 43 to 56 from 2.46 to 3.32 milliseconds
( g ) cu rv e s 71 to 8 4 fr o m 4 .1 0 to 4 .8 5 m illise c o n d s
DISP
LACE
MENT
(inc
hes)
( f ) curves 57 to 70 from 3.29 to 4.04 milliseconds
Continuation of Figure 7.3. History o f the profile of transverse displacements shown in time sequence in graphs (a) to (g). (1 in = 2.54 cm).
113
DISTANCE ALONG BEAM SPAN {inches)0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30J____._I___i____I___._I____.__I___.____I__,_I______i I .___I i_I____.___I_____. I_.___I___.___I___.___L_i___L_ 5
4
3
2
1
~i—i—i—i—i—i—|—i—i—i—i—i—|—i—i—i—|—i—i—i—i—i—|—i—i—i—i—i—i—«—i—«—i—i—i—«—r 00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
NUMBER OF NODE
Figure 7.4. History o f the development o f plastic hinges. (1 in = 2.54 cm).
1UU
8 0 -*8 9 0£ o oQ)to
f 70 4 o75 6 0 lo
■8£oo0)to
g
5 0 -4 0 -3 0 -20 -
10 -
P resen t S tu d y , 18 E lem en ts O P la s tic H inge in F lexure
e
o8O
8I I
1 I 1
: I TIME
(milli
secon
ds)
1 2 0
hinge forms at midspan and thereafter it quickly spreads to adjacent elements forming a plastic region of three elements. This is shown as three discrete hinges due to the discrete deformation idealisation adopted for the beam element. The plastic region changes length with time, alternating between one, three, and for a fraction of a millisecond five elements before it finally ceases when the midspan displacement reaches its peak at approximately 4.25 ms; thereafter, elastic vibration ensues. It is worth mentioning here that in the present analysis plasticity at a section occurs suddenly, the whole section becoming plastic when M = Mp = <jy bd2/4, and likewise it suddenly returns to elastic response when M < Mp. This fact could account for the discrepancy between the presentsolution and that of ADINA; which is similar to the findings of Raghavan and Rao (1978) mentioned in Chapter 2, section 2.2.3. The slight asymmetry in the response reflects the asymmetry in the supports and in the beam elements.
The history of bending moment at midspan is shown in Figure 7.5. It is seen that for the first 0.15 ms there is negligible bending moment as most of the span sufficiently remote from the supports translates as a rigid body. This is followed by a negative bending moment (for 0.12 ms) due to a higher transverse inertia force at midspan, and thereafter the bending moment builds up elastically in an almost linear manner until the formation of a plastic hinge is signalled when the bending moment attains the value Mp = 50,000 lbf.in, Figure 7.5. There follows a very high frequency oscillation for a duration of 3 ms, afterwhich plastic hinges cease and a lower frequency oscillation occurs around a mean value of 37.5 kip.in; the period of this latter oscillation is 5 ms. Superimposed upon this oscillation is a secondary oscillation of much higher frequency (« 182,000/s) which could be due to the propagation and interaction of elastic flexural waves locked in the midspan element by plastic deformations.
The history of the shear force at one of the supports is shown in Figure 7.6 which indicates a rapid rise to maximum value of nearly 0.17 of the shear capacity of cross section. From the time 1.5 ms, at which the first plastic hinge forms, there appears a high frequency oscillation which again is superimposed on one of lower frequency (200/s or 5 ms period). This lower frequency oscillation occurs around a mean value of 5 kip which is equal to the support reaction force in a static analysis.
In Figure 7.7 the history of the bending moment profile is shown in time sequence in graphs (a) to (d). Similarly, in Figure 7.8 the history of shear force profile is shown. In graph (c), the shear force profile resembles that in a static analysis, but for increasing time, and without damping, the profile becomes decidedly chaotic.
SHEA
R FO
RCE (lbf)
BEND
ING
MOM
ENT (lb
f.i
TIM E (m i l l i s e c o n d s )4 5 6 7 8 9■ • * •_ 10 11 12 i i i
■'— i— 1— i— 1— i— ■— i— r_ “i— '— i— 1— i— 1— i— >-0 20 40 60 80 100 120 140 160 180 200 220 240TIM E (s e c o n d s ) / O.S IO-4 (seco7ids)
Figure 7.5. History of the bending moment at midspan. (1 in = 2.54 om, 1 lbj> = 4.45 N).
TIM E (m i l l i s e c o n d s)0 1 2 3 4 5 6 7 8 9 10 11 12
TIM E (s e c o n d s ) / 0.5a: 10-4 (seconds)
Figure 7.6. History of the shear force at the beam support. (1 Ibf = 4.45 N).
BEND
ING
MOM.
/ B
ENDI
NG M
OM. C
APAC
ITY
BEND
ING
MOME
NT (l
brii
(a) curves 1 <o 9 from 0.0 to 0.62 milliseconds (b) curves 10 to 14 from 0.58 to 0.80 milliseconds
( c ) c u r v e s 15 to 2 8 f r o m 0 .8 6 to 1 .61 m illis e c o n d s ( d ) c u rv e s 2 9 to 4 2 f r o m 1 .8 7 to 2 .4 2 m illis e c o n d s
Figure 7.7. History o f the profile o f bending m o m e n t s h o w n in time sequence in graphs (a) to (d). (1 in = 2.54, 1 Ibf = 4.45 N ) .
SHEA
R IV
RCX
(lb,)
SHEA
R FO
RCE
(lb,)
i
0 2 4 a 8 10 12 14 18 18NUMBER OF NODE
(b) curves 5 to 14 from 0.28 to 0.50 millistcands
( o ) cu rv es 16 to 28 fr o m 0 .68 to 1.81 m illiseco n d s (d) cu rv es 2 8 lo 4 2 fr o m 1 .67 lo 2 .42 m illiaoconda
Figure 7.8. History o f the profile o f shear force3 shown in time sequence in graphs (a) to (d). (1 in = 2.54 cm3 1 lbf = 4.45 N ) .
123
1 2 4
7.2.2. Comparison With Experiments
A set of detailed and carefully performed experiments on impulsively loaded structures were conducted in the early 1960's by Clark, E.N. and co-workers (Clark, Schmitt and Ellington 1962, Clark, Schmitt, Ellington, Engle and Nicolaides 1965). Their aim was to provide reliable experimental data for the validation of theoretical models for various structural elements of aircraft frames. Beams, loaded explosively, were tested to provide data on well-defined simple models under accurately known inputs. Simple and clamped supports were chosen to examine the influence of predominant bending and predominant stretching behaviour respectively. To further focus on these effects shear deformations were minimised by considering specimen beams in which the ratio of span to depth was large.
Experimental Setup
Simply supported beams were supported on 3 in diameter fixed steel rods 10 in apart, and with similar rods directly above on top of the beam, fashioned into a roller assembly which pivoted about the fixed rod, Figure 7.9(a). This arrangement permitted the beam to deform without introducing tension at the supported ends, and yet it prevented vertical motion of the beam (rebound). The top rods were made from aluminium to keep mass and moment of inertia low. Framing camera pictures showed that roller did not start to move until the beam had undergone almost maximum displacement. All beams were 12 in long and either in or in in thickness, supported on uprights 10 in apart, resulting in 1 in overhand at each end. The beams were loaded with a sheet of High Explosive (H.E.) (trade name DuPont EL 506 D) centered over a layer of 0.055 in thick polyethylene, Figure 7.9(a) and (b). This layer was necessary to attenuate and lengthen the pressure pulse and prevent spalling of the beam material. A fanshaped piece of the same H.E. was taped to the sample to act as a lead and wasdetonated by an electric blasting cap.
Edge Effects
The explosive laid along the central portion of the beam has two exposed ends. It was therefore necessary that experiments be conducted to determine the magnitude of any edge effects. This was accomplished by arranging 16 individually moveable metal time pieces $ in thick, in long and 1 .2 in wide in the manner of a simple time piece of i in x 4 in x 1.2 in dimensions. The arrangement wasloaded with a 2 in long piece of explosive centrally located. This arrangement andthe results of seven sets of measurements are shown in Figure 7.10. The edge effects are necessarily due to:
1 2 5
r
E = 10.6 x 10* lbf/in2V = 0.367Gy = 43^800 lbf/in2
p = 0.00025 lbf.s2/ i n h
Strain Rate Sensitivity Constantsj
D = 6500/s
P = 4.0Depth (d) = 0.253 inBreadth (b) = 1.187 inImpulse 0.4851 Ibf.s
7 ^ 7
1 ; - 1
4-2 .228 '—P
-10.25*----------------------- f( c )
Figure 7.9. (a) Experimental setup used by Clark et al. (1965).
(b) Configuration o f beam used in experiments. (c) Configuration of idealised beam used in the present study. (1 in = 2.54 cm, H b f = 4.45 N).
126p o l y e t h y l e n e
L A Y E RTIMEPIECESEGMENTS
Figure 7.10. Influence o f edge effect on the spatial distribution of the pressure load. After Clark et al. (1965). (1 in = 2.54 cm).
(a) (b)
Figure 7.11. (a) Idealised pressure pulse, (b) Shape of pressure pulse o f an explosive charge as it varies with distance r from detonation point3 a = air3 g = gas - pressure measured in atmospheres (0 = atmospheric pressure). After Granstrom (1956). (1 ft = 0.30 m ) .
1 2 7
1. The expansion of the H.E. from the edge faster than if confined by more explosive.
2. The layer of polyethylene, underlying the H.E. sheet and extending beyond it, weakens the spatial concentration of impulse. However it is noted that the total imparted impulse is almost exactly that obtained by considering only the portion of the specimen covered by the H.E. layer and neglecting edge effects, as the impulse deficiency inboard is compensated by the impulse increment outboard of that edge.
Impulse, Pulse Shape and Duration
A definitive evaluation of dynamic response using theoretical means requires accurate detailed knowledge - amongst other parameters - of the dynamic forcing function; this function is usually the factor of greatest uncertainty. In theexperiments of Clark et al. contact-detonation-induced impulse was chosen in preference to air blast or magnetically induced forces because of the short duration loading, typically of 10-20 /is. This short duration meant that structural motion and changes in orientation of the structure with respect to the blast field occur to a significant degree only after the blast field has subsided; thus, modifications to the loading caused by motion of the structure are thereby largely pre-empted. Clark et al. indicated that the impulse delivered per unit weight of the H.E. was 18.6 x 104 dyne.s/gm H.E. (1 dyne = 2.248 x 10“ e lbf = 1 x 10- 5N).However, in a subsequent study by Witmer, Clark and Balmer (1967), it wasmentioned that, because of improved measurement and experimental techniques,calibration experiments conducted by Stanford Research Institute (SRI) on H.E. lent more credibility to the values: 20.2 x 104 dyne.s/gm H.E. and 21.6 x 104
dyne.s/gm H.E. These two values were attained using Aluminium alloy (6061-T6) time pieces, and for the former a polyurethane foam buffer was used whilst for the latter a buffer of neoprene rubber. The numbers cited above are respectively 8.6% and 16.0% higher than those deduced by Clark et al. (1962).
The shape and duration of the pressure pulse delivered by the detonation of the H.E. were not indicated by Clark et al. In a more recent work, Stagliano andMente (1979) used Clark's beam experiments as a simple verification of theapplicability of the FE code ADINA to blast-wave loading structural response.They proposed a pulse shape of triangular configuration consisting of 2 /is linear rise and 10 /is linear decay, Figure 7.11(a). The time distribution of a pressure pulse associated with an explosive charge, and measured at three stations remotefrom the detonation point (Granstrom 1956) is shown in Figure 7.11(b). It is seen that blast overpressure is followed by a suction effect of much smaller magnitude,
1 2 8
an effect which is normally ignored for short duration loading. The time span of the forcing function in Figure 7.11(a) was derived from the detonation wave velocity through the explosive material.
Sources of Error
Deficiencies and origins of inaccuracies in the experiments performed by Clark et al. may generally be summarised in the following comments:
1. Dynamic twisting, as the H.E. detonation front travelled across the width ofthe beam, resulted in different midspan displacements for front and rear edges.
2. Dynamic material property did not sufficiently cover the full range of ratesensitivity and strain hardening attained in the actual experiments.
3. Uncertainty of the forcing function was evident, despite considerable effortsdevoted to impulse calibration.
4. The usual scatter occurred in static stress-strain properties due to uncertaintyin measurements and material variations in specimen lots.
Present Study
The simply supported test beam employed for the comparison with the present study results is Beam No. 121. It has a depth of in and made of non-strain hardening 6061-T 6 aluminium alloy. In order to make use of the experimental data provided, a number of idealisations related to geometry and loading were made. In the experiments, the beam material was allowed to move inboard of the stationary supports to prevent any stretching due to geometric nonlinearity. In the present study, the above arrangement was replaced by a beam of 10.25 in span, supported on a fixed pin at one end and a roller at the other. The new length of the beam for numerical modelling was thus set as the average of the lengths of the experimental beam in its undeformed and deformed configurations. This idealisation was necessitated by the difficulty of modelling a temporally changing beam length. Due to the edge effects discussed earlier and shown in Figure 7.10, the spatial distribution of the applied pressure was extended along the beam span by a total of 10-12%, whilst keeping the total impulse constant. The above mentioned idealisations are shown in Figure 7.9(c).
As for the impulse per unit weight of H.E., the average of the two figures given by the SRI was considered; this meant an increase by 12.5% on the value
1 2 9
given by Clark et al. The pulse shape and duration implemented are those shown in Figure 7.11(a).
In the present study, the beam has been discretised into 46 elements with an integration time step size of 0.33 x 10- 6s. Typical CPU time required for any of the runs shown in Figure 7.12 was 25 minutes on a VAX/8600 minicomputer. In the analysis, shear deformations (k' = 1.173 for v = 0.367), rotatory inertia and geometric nonlinearity were considered.
Material viscoplasticity in the form of strain rate sensitivity was accounted for by employing the empirical nonlinear viscoplastic constitutive relation proposed by Cowper and Symonds (1957), (Chapter 2, section 2.2); for Aluminium alloy (6061-T 6), the appropriate constants for this equation are D = 6500/s and P = 4.0. Strain rate sensitivity effect on the dynamic yield stress has been considered in accordance with the three different manners described in Chapter 5, section 5.3.2. The strain rate insensitive response was also considered.
The response for each of these four cases, together with the experimentalresponse recorded by Clark et al. (1965), are shown in Figure 7.12. It is seen that strain rate sensitivity is quite significant for Aluminium alloy (6061-T 6), even though it is generally considered as strain rate insensitive. The solution closest to the experimental results is that where both space and time dependency of the yield stress were considered. It is also seen that time independency results in a stiffer material, leading to lower peak displacements and to a higher frequency of terminal elastic vibrations. The lowest peak displacement at midspan occurs for the model in which strain rate sensitivity is space dependent but time independent. In all of the four solutions shown in Figure 7.12, terminal elastic vibrations were not ingood agreement with the experiment. This is due to:
1. Material damping, which is not considered in the present study.
2. Loss of energy in partial plasticity of beam cross sections.
3. Loss of energy at beam supports due to friction.
In Figure 7.13(a) to (d) is shown the history of the development of plastic hinges in flexure (positive bending moment) and contraflexure (negative bending moment) for the aforementioned four cases of strain rate sensitivity. The case in whichstrain rate sensitivity is space and time independent is chosen for further analysis ofthe beam response. It is typical of analyses reported in much of the publishedliterature, and it also bears close resemblance in hinge evolution to that of space
TIME (milliseconds)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
TIME (microseconds)Figure 7.12. History o f transverse displacement at midspan. (1 in - 2.54 cm).
TIME
(mic
rose
con
ds)
TIM
E (m
tcro
asci
‘10a0
(a) DISTANCE ALONG BEAM SPAN (in c h e s )2 3 4 5 6 7 8
IP r e s e n t S tu d y , 4 6 E le m e n ts , S tr a in B a te I n s e n s i t iv e* NtgaMvm Plamtio Hinge in riewurm O Pweitiee P iettic Hinge in Pleetee
2 -
1 -
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‘ . r16 2 0 2 6 3 0
NUMBER OF NODE
°88«00@LiliAAi ^ 4
10I
3 6T4 0 4 6
>1030
(b)
iDISTANCE ALONG BEAM SPAN (in c h es)
2 3 4 6 6 7 8 10
*1030
2 -
1 -
DISTANCE ALONC BEAM SPA N (in c h e s )2 3 4 6 6 7 8
J_I___I__i__ I__.__ I__ ___ I__i___I__ ___LP r e s e n t S tu d y , 4 8 E le m e n ts , I n i t ia l S t r a in B a te a t e a c h E le m e n t a N e g a tiv e P la s t ic H in g e i n F le x u re O P o s i t i v e P la s t ic H in g e i n F le x u re
oiloo8°
Oilo
8° oM m
ooo
—I—T
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■ T10 15 2 0 2 5 3 0
NUMBER OF NODE35 4 0
101
I"s g g y g o■ i • i ■ i ■ i ■ i T
4 5
*1030
(d)i
DISTANCE ALONG BEAM SPAN (in c h e s )3 4 5 6 7 8 10
Figure 7.13. History o f plastic hinge development for four oases of strain rate sensitivity. (1 in = 2.54 cm). 131
1 3 2
and time dependency.
The initial history (0 - 250 /is) of plastic hinge development in Figure 7.13(b) is shown magnified in Figure 7.14. Plasticity commenced at the centre at 18 /is in the form of single positive hinge which quickly spreads to an adjacent element for few microseconds, then concentrates back at the centre until the time of 50 /is when all plasticity ceases for an interval of 16 /is. Thereafter, negative plastic hinges appear at 0.195 1 distance from each support (1 = total length of beam); these start to move inwards but are arrested at a distance 0.24 1 from each support when all plasticity ceases for a period of 21 /is. Subsequently, negative plastic hinges develop at 0.13 1 from each support which then form a negative plastic region extending over a maximum of three elements at any one time for nearly 52 /is, and for the first 12 /is of this period a positive plastic hinge reappears atmidspan. Once more all plasticity ceases for 37 /is, followed by the activation ofa central positive plastic region. The alternating nature of the plastic response (positive then negative hinges) indicates flexural wave propagation through the beam material; this will be discussed later in the analysis.
The history of transverse displacements along the beam span is shown in Figure 7.15 in time sequence from graph (a) to (g). In graph (a) it can be seen that, for the whole time period covered in the graph, the boundary conditions of the beam have not been identified by the flexure wave which is travelling towards the supports from the region of load application. From graphs (a), (b) and (c) it can be seen that higher modes of deformation prevail until about 0.65 ms; thereafter, the fundamental mode of deformation becomes dominant, with elastic recovery beginning at about 1.6 ms as shown in graphs (f) and (g). Within eachgraph, curves close to each other indicate slow rate of change.
In Figure 7.16(a) is shown the history of bending moment at midspan, andFigure 7.16(b) shows the initial 1 ms of the history. The response presents similar characteristics to that in Figure 7.5 (section 7.2.1.), except for a short period (0.08 ms) of negative moment (-0 .2 Mp) during the plastic phase.
In Figure 7.17 the profile of the bending moment history is shown in timesequence in graphs (a) to (n); the maximum and minimum values on the vertical or z axis represent Mp and -Mp respectively. Graphs (a) to (b) show the response while the pressure pulse is still active. It is seen that the central region of the beam initially moves as a rigid body, until a bending wave has passedthrough. In impulsive loading, the lack of local equilibrium manifests itself by specific particles moving and adjusting themselves to the instantaneous stress distribution. This ability to adjust is propagated at characteristic speeds of wave
TIME
0250
200 H
•8eg 1 5 0to ob
100 -
DISTANCE ALONG BEAM SPAN (inches)2 3 4 5 6 7 8
P resen t S tu d y , 4 6 E lem en ts P la s tic H inge in F lexure : S 0a N egative O P o sitive
■I t
I 1
§
9_L
10±
@ In itia l S tra in R ate a t M idspan
l oA
i l A
5 0 -
0 l 1 ™ i ■ i • i ■ i • i ■ i10 15 2 0 2 5 3 0 3 5 4 0NUMBER OF NODE
0 4 5
F-igure 7,14. Initial 250 ys o f the history o f plastic hinge development. Yield stress determined from initial strain rate at midspan. (1 in = 2.54 cm).
133
DISP
LACE
MENT
(inc
hes)
DISP
LACE
MENT
(inc
hes)
(a) curves 17 to 32 from 0.40 to 0.775 milliseconds (d) curves 33 to 48 from 0.00 to 1.175 milliseconds
Figure 7.15 (continued on next page)
134
DISP
LACE
MENT
(in
ches
) DI
SPLA
CEME
NT (i
nche
s)
(e) curves 49 to 64 from 1.12 to 1.57 milliseconds
1.00 -
0 .5 0 -
0.00 T 1 r2 0 2 5 3 0
NUMBER OF NODE
(g) c u r v e s 8 1 to 9 0 f r o m 2 .0 0 to 2 .3 7 m il l i s e c o n d s
i— .— i— ■ i— '— r15 20 25 30
NUMBER OF NODE
DISP
LACE
MENT
(in
ches
)
(f) curves 65 to 80 from 1.60 to 1.97 milliseconds
NUMBER OF NODE
Continuation o f Figure 7.15. History o f the profile o f transverse displaoements shown in time sequence in graphs (a) to (g). (1 in = 2.54 cm).
136
TIME (m illiseconds )0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
(a)
TIME (m illiseconds)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(b)
Figure 7.16. History of bending moment at midspan : (a) For the first 6 ms. (b) For the first 1 ms. (1 in = 2.54 in,1 Ibf = 4.45 N).
BE
ND
ING
MO
M.
/ B
EN
DIN
G M
OM
. C
AP
AC
ITY
B
EN
DIN
G M
OM
. /
BE
ND
ING
MO
M.
CA
PA
CIT
Y
BEND
ING M
OMEN
T (lbf
in)
BEND
ING M
OMEN
T (Ib^
in)
( q ) curves 9 to 12 from 0.014 to 0.019 m illiseconds
10 15 20 25 30 35 40 45NUMBER OF NODE
Figure 7.17 (continued on next page) 137
BEND
ING
MOME
NT (
Zb,
.in)
BEND
ING
MOME
NT (l
brin
)
10 16 SO 2 5 3 0NUMBER OF NODE
3 6 4 0 4 5 10 15 2 0 2 5 3 0 3 5 4 0NUMBER OF NODE
4 5
10 15 20 25 30NUMBER OF NODE
3 5 4 0 4 5 10 1 5 2 0 2 5 3 0NUM BER OF NODE
3 5 4 0 4 5
Figure 7.17 (continued on next page) 138
BEND
ING
MOME
NT (l
b,.in)
BE
NDIN
G MO
MENT
(Ib^i
n)
Hi curves 37 to 44 from 0.150 to 0.325 milliseconds ( j ) curves 45 to 52 from 0.350 to 0.525 milliseconds
1000.0 -
500.0 -
0.0
-500.0 -
-1000.0 -i ■ ■ ■ . i . ■ — T' « ■ i | i i i i | i i ■ i |
15 2 0 2 5 3 0 3 5 4 0 4 5NUMBER OF NODE NUMBER OF NODE
10 15 20 25 30NUMBER OF NODE
3 5 4 0 4 5 1 0 1 5 2 0 2 5 3 0NUM BER OF NODE
3 5 4 0 4 5
Figure 7.17 (continued on next page) 139
BEND
ING
MOME
NT (l
brin
)
(m) curves 69 to 70 from 0.950 to 1.125 milliseconds (n) curves 77 to 84 from 1.150 to 1.350 milliseconds1200.0 -
1000.0 -
000.0 -
600.0 -
400.0 -
200.0 -
0.0 I ' 1 ' 1 I • ' ■ • I2 0 2 5 3 0
NUMBER OF NODE
Figure 7.17. History o f the profile o f bending moment shown in time sequence in graphs (a) to (n). (1 in = 2.54 cm).
1 4 1
propagation. In graphs (a) to (d) it is seen that the beam response is initially independent of the beam boundary conditions. Also evident is the characteristic dispersion of flexural waves as the speed of each component wave in a particular flexural wave packet is dependent on its wavelength.
Kolsky (1963) has mentioned that, in a study on cylindrical bars employing the exact theory (Poisson's ratio of 0.29), it was calculated that a flexural pulse propagating along the bar would have particular Fourier components (whose wavelengths are about three times that of the bar radius) travelling ahead of components of other wavelengths and which would therefore appear at the head of the pulse. For bars with other than circular cross sections, however, the exact treatment becomes exceedingly complex, and in only a few cases have solutions been attempted.
Taking the above mentioned case of cylindrical bars as a rough guide, it can be seen in Figure 7.17, graph (a) to (d) that waves with shorter wavelengths travel ahead. In graph (d), it is seen that the first plastic hinge is positive and has developed at the beam centre. In graph (e) it is seen that a flexural wave has reached the beam support and is reflected back towards the centre, and in graph (f) it has interacted with the outgoing wave and negative plastic regions have formed away from beam centre. As the outgoing wave of greater wavelength is still pushing towards the supports, the negative plastic region for the left half is moving from between nodes 9-12 (see graph (f)) to between nodes 5-7 in graph (h), the central hinge is disappearing in graph (g) and some wave turbulence is developing due to wave interaction. The above demonstrates the effect that elastic flexural wave propagation can have on the formation and distribution of plastic hinges in a metal beam. In graphs (i) to (n), it is seen that the wave of larger wavelength reflecting from the supports towards the centre, and this, coupled with the discontinuity in geometry already inflicted by the presence of plastic hinges, causes overall turbulence in wave propagation. However, the general trend in the bending moment diagram is converging upon the fundamental form that would have prevailed in a similar static analysis.
The history of the shear force at one of the beam supports is shown in Figure 7.18(a). The initial 1 ms response is magnified in Figure 7.18(b). It can be seen that for the first 30 /*s there is no reaction at the beam supports until a shear wave has travelled from the point of load application. Thereafter, the shear force oscillates with high frequency between the values of 0.2 and - 0.2 of the section plastic shear capacity; this however is superimposed on a much lower frequency oscillation around a mean shear force value of nil.
SHEA
R FO
RCE
(lb,)
SHEA
R FO
RCE
(lb,)
1 4 2
TIME (m illiseconds)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 8.0
' * ' * 1.000.800.600.400.20
0.00
- 0.20
-0.40-0.60-0.80- 1.000.0 1000.0 2000.0 3000.0 4000.0 5000.0 6000.0
TIME (m icrosecon ds)
(a)
TIME (m illiseco n d s)0.0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9 1.0
6000.0
4000.0
2000.0
0.0
- 2000.0
-4000.0
-6000.0o!o 200.0 400.0 600.0 800.0 1000.0*
TIME (m icrosecon ds)(b)
Figure 7.18. History of shear force at beam support : (a) For the first 6 ms. (b) For the first 1 ms. (1 Ibf = 4.45 N).
6000.0
4000.0 -
2000.0 -
0.0
■ 2000.0 -
-4000.0 -
-6000.0 -— P resen t S tu d y , 46 E lem en ts
SHEA
R FO
RCE
/ SH
EAR
FORC
E CA
PACIT
Y SH
EAR
FORC
E /
SHEA
R FO
RCE
CAPA
CITY
1 4 3
In Figure 7.19, the profile of the shear force history is shown in time sequence in graphs (a) to (f), the applied pressure pulse being active during the time period covered by graphs (a) and (b). It can be seen in graphs (a) to (d) that the propagation of shear waves is from the region of applied load towards the beam supports. Dispersion in this type of lateral vibration is less evident than in flexural waves. Graphs (e) to (h) show the interaction between reflected and outgoing waves. Graphs (i) to (k) show that the shear force is statistically insignificant.
7.3. CLAMPED BEAMS
7.3.1. Comparison with Experiments and ADINA
In order to examine the dominantly stretching behaviour of beams, Clark,Schmitt, Ellington, Engle and Nicolaides (1965) tested a fully clamped beam of rectangular section and \ in depth (Beam N o .l l l ) . It was made from non-strain hardening 6061-T 6 aluminium alloy. Experimental conditions were similar to those for their test on the simply supported beam mentioned in section 7.2.2. Theclamped ends were achieved by bolting a 1 in thick serrated steel plate on top of each beam end to heavy steel stands; these were designed to withstand considerably more force than the tested beams could apply without significant deformation. To further ensure against slippage, the beam - which had 10 in supported length - was extended 6in beyond each support. In Figure 7.20 (a) is shown the beamconfiguration, loading geometry and material properties.
A numerical analysis of the response of Beam N o .l l l was carried out byStagliano and Mente (1979) using the FE code ADINA. Ten three-dimensional beam elements were used to model half the beam span, and the geometric nonlinearity apparent from the experiments necessitated the use of the updated Lagrangian formulation.
For the numerical analysis of the present study, the spatial distribution of the applied pressure along the beam span was extended by 10 - 1 2 %, while keeping the total impulse constant, Figure 7.20. This is done to cater for the edge effects mentioned in section 7.2.2. The impulse per unit weight of H .E., the pulse shape and pulse duration were as discussed in section 7.2.2. The whole beam was discretised spatially into 82 elements, and the integration time step size was 0.1 x 10_ 6s. The typical CPU time for a 1.3 ms response time was 32 minutes on a VAX/8600 minicomputer. The number of elements used might seem at first excessive. It is certainly true that a solution using 40 elements only and an
SHEA
R FO
RCE
(lbf)
SHEA
R FO
RCE
(lb,)
(a) curves 1 to 4 from 0.00 io 0.0058 milliseconds (b) curves 5 io 8 from 0.0073 to 0.012 milliseconds
0 5 10 15 2 0 2 5 3 0 3 5 4 0 4 6NUMBER OF NODE
0 5 10 1 5 2 0 2 5 3 0 3 5 4 0 4 5NUM BER OF NODE
0 5 10 15 2 0 2 5 3 0 3 5 4 0 4 5NUMBER OF NODE
0 5 10 15 20 25 30 35 40 45NUMBER OF NODE
Figure 7.19 (continued on next page)
SHEA
R FO
RCE
(lb,)
SHEA
R FO
RCE
(lb,)
(e) curves 17 io 21 from 0.030 to 0.050 milliseconds (f) curves 22 to 26 from 0.055 to 0.075 milliseconds
10 15 20 25 30NUM BER OF NODE
35 40 45 10 15 20 25 30NUM BER OF NODE
35 40
Figure 7.19 (continued on next page).
45
145
SHEA
R FO
RCE
(lb,)
SHEA
R FO
RCE
(lb,)
(i) curves 37 to 44 from 0.150 to 0.325 milliseconds (g ) curves 45 to 52 from 0.350 to 0.525 milliseconds
6000.0 -4000.0 -2000.0 -
0.0- 2000.0 -
-4000.0-6000.0 -
I 1 ' 1 1 I ' 1 1 ' I ' ' 1 ' I15 2 0 2 5 3 0
NUMBER OF NODE
l ' ' ’ ' I ' • ' ' l15 2 0 2 5 3 0
N UM BER OF NODE
(k) c u r v e s 5 3 to 0 0 f r o m 0 .5 5 0 to 0 .7 2 5 m il l i s e c o n d s
Figure 7.19. History o f the profile o f shear force shown in time sequence in graphs (a) to (k). (1 Ibf = 4.45 N).
XW
WV
WW
J
s\\\
\\\\V
vV
^(a)
BUFFER XHE LAYER
(b)
I
4— 1.988"— +- \ I10.0*
IDEALISED HE LAYER"SI------- -=h__________
2-195* - +
100
DETONATIONPOINT
lb f/in ‘E = 10.7 x 1 0 ~ &
V = 0.367Oy = 41, 200 lbj>/in2
p = 0.000251 lbf.s2/ i n h
Strain Rate Sensitivity Constants,
D = 6500/s (or 7000/s)
S'P = 4.0 (or 2.
ss Depth (d) = 0.242 in
sBreadth (b) = 1.195 in
sss
Impulse = 0.6472 Ibf.s
Figure 7.20. Beam geometry, loading and material properties, (a) Configuration o f beam used in the experimental study o f Clark et al. (1965). (b) Configuration o f idealised beam used in the present study. (1 in = 2.54 cm, 1 Ib-f = 4.45 N).
1 4 8
integration time step size of 0.5 x 10“ 6s would yield as good a response with regard to the displacement at midspan and with a CPU time of 3.2 minutes - only 10% of that required above. However, other response parameters, such as plastic hinge development, flexure wave propagation and axial force build-up, would suffer a certain degree of chaos that does not manifest itself clearly when the sole indicator considered is the displacement at midspan. Shear deformations (k' = 1.173 for v - 0.367), rotatory inertia and geometric nonlinearity were considered. Interaction between bending moment and axial force was implemented numerically in the manner described in Chapter 5, section 5.3.1; the interaction equation used was the exact parabolic equation (2 .8) for rectangular sections (Chapter 2, section 2.2.4). Strain rate sensitivity was considered in three different fashions:
1. The dynamic yield stress is determined from the initial strain rate at midspanusing equation (2.1) with equation constants D = 7000/s and P = 2.13. These two values were determined by Clark et al. (1965) from uniaxial tension tests under strain rates of up to 15 in/in/s; however strain rates of up to3000 in/in/s were recorded in actual tests on beams. The resulting dynamic bending moment capacity of the cross section was calculated from equation (2.1) as 1.23 times the static value, and the dynamic axial force capacity was 1.09 times the corresponding static value.
2. The same as in (1) above, but with D = 6500/s and P = 4.0. These twovalues are occasionally mentioned in the literature with regard to 6061-T6 aluminium alloy (Bodner and Symonds 1962, Ting 1964, Symonds 1965, Reidand Reddy 1979). There is, however, some uncertainty about these values because of conflicting test data (Jones 1979). Nevertheless, there is clear experimental evidence that, for commercial-purity aluminium, there is asignificant increase in yield stress with strain rate, although it is reduced withincreased alloy content (Harding 1987). The resulting dynamic bendingmoment capacity of the cross section was 1.48 times the static value and the dynamic axial force capacity was 1.23 times the static value.
3. Strain rate insensitive yield stress.
It is seen here that, in accounting for strain rate sensitivity, only one manner - time and space independent - of the various manner described in Chapter 5,section 5.3.2, is employed. This is because, for a strain rate sensitive material inwhich the yield surface is governed by two or more forces, the yield surface would be continuously changing its shape and size throughout the response. At the present time, it is not yet feasible to model such a complex behaviour.
1 4 9
The history of transverse displacement at midspan is shown in Figure 7.21(a); solutions obtained in the present study, for the three ways of accounting for strain rate sensitivity mentioned above, are compared to the experimental response of Clark et al. (1965). It is seen that the solution with D = 6500/s and P = 4.0 gives the lowest midspan peak displacement - about 87% of the experimental value, a percentage equal to that of the equivalent solution for the simply supported beam of section 7.2.2. (see Figure 7.12). The effect on the response of the interaction between axial force and bending moment is discussed in detail in Chapter 8, section 8.4. In Figure 7.2.1(b) is shown the responses produced by Stagliano and Mente (1979) using the finite element code ADINA with different solution strategies. Ten three-dimensional beam elements were used to model half the beam span. Also shown is their solution for the response utilising the finite difference code DEPROB - part of the NOVA package developed by Lee and Mente (1976) - with a spatial discretisation of thirty masses per half span.
The case in which the dynamic yield stress is determined from the initial strain rate at midspan, with equation constants D = 6500/s and P = 4, is chosen for further analysis of the beam response.
The history of plastic hinge development along beam span is shown in Figure 7.22, the initial 250 /is being magnified in Figure 7.23. It can be seen from Figure 7.22 that plastic hinges are widely spread along the span; for example, positive plastic hinges form in chronological order at the central region, fixed ends, regions near the quarter-points of the span from each support and finally oncemore at the fixed ends. The latter hinges occur in the rebound stage when theaxial force in the beam changes, as a result of permanent plastic extension and geometric nonlinearity, from tensile to compressive. Negative plastic hinges are initially distributed between 0.35 1 and 0.12 1 of the span 1 from each support, and later at the clamped ends. It is noted that, although the short pulse load wasapplied at the beam's centre, plastic hinges ceased to exist there at a very early stage (after 70 /is) in the response. Hinges in the regions 0.12 1 - 0.35 1 from each support and at the fixed ends seem to have taken up most of the energy dissipated in plastic deformations. This is thought to be due to the reduction with time of the plastic moment of resistance of the cross section as a result of the axial force build-up in time. Consequently, as flexure waves propagate through the beam span, more energy is dissipated in plastic deformations than in a beam devoid of such build-up. It is shown in Figure 7.25 that, as early as 80 /is, the axialtension generated by geometric nonlinearity has reached 0.46 of the axial forceplastic capacity of the cross section, thus reducing the plastic moment of resistance to 78% of its full value and initiating plastic hinges along the beam.
Midspan Deflection (In)
DIS
PL
AC
EM
EN
T A
T M
IDS
PA
N (
inc
he
s)
150
T IM E (m illis e c o n d s )0.0 0.2 0.4 0.6 0.8 1.0 1.2
(a) Solutions from the present study.
1.0 u
0 .8
0.6
0.4
0.2
0.2 0.4
ADINATemporalOperator K/E Int.Pts. At
O Nevmark 1/1 5x3 2A Newmark 3/2 5x3 2□ Cen.Diff. - 5x3 2+ Cen.Diff. - 7x5 2
---DEPROB Cen.Diff.- - EXPERIMENT
1
30x12
1
0.75
1.0 1.2 1.4 1.60.8
Time (msec)
(b) Solutions from F E code ADINA and FD code DEPROBj K = cycles between updates o f material stiffnessj E = cycles between
equilibrium iterations and At = size of time step. After Staglia.no et al. (1979).
Figure 7.21. History of transverse displacement at midspan. (1 in = 2.54 c m ) .
DIS
PL
AC
EM
EN
T A
T M
IDS
PA
N (
i
TIME
0
1000 -
DISTANCE ALONG BEAM SPAN ( in ch es ) 3 4 5 6 7 8 10
P resen t S tu d y , 8 2 E lem en ts, In itia l S tra in R ate a t M idspana N egative P lastic Hinge in Flexure O P ositive P lastic Hinge in Flexure
■8gooCOo&-u
8 0 0 -
6 0 0
4 0 0 -
200 -
£ com pressive aocial force com pressive ax ia l forceU
1 0 15 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0NUMBER OF NODE
Figure 7.22. History o f plastic hinge development. (1 in = 2.54 cm).
1000
8 0 0
6 0 0
4 0 0
200
0
(_n
TIME
(micr
osec
onds
)
02 5 0
200 -
•3gI 1 5 0CO0CJ1
§I100 -
5 0 -I
0
DISTANCE ALONG BEAM SPAN (inches) 3 4 5 6 7__.___ i___ .___ i i ___ .___ i 8
u
P resen t S tu dy , 8 2 E lem ents In itia l S tra in Rate a t M idspan& N egative P lastic H inge in Flexure O P o sitive P lastic H inge in Flexure
*iiill,
o
o
tilloo§ -a°°8° Q j ^ ° ° A<
lit.
I A * *K 00 0 A **
i i
A*
9i 102 5 0
- 200
- 1 50
I- 100
0I I I | I I I I | I I I I | I I l"l | I I I I | I I I I | I I ll[ Tl I I | I I I I | I I1 I I | I I I I | I I I I'] I I I I | I I I I {"IT I I | I I I I | I
5 10 15 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0NUMBER OF NODE
- 5 0
0
Figure 7.23. Initial 250 ps o f the history o f plastic hinge development. (1 in = 2.54 e m ) .
tnfo
TIME
(micr
oseco
nds)
1 5 3
F ro m F ig u re 7 .2 3 it can b e s e e n th a t th e g en era l tren d fo r n e g a tiv e h in g e s is to m o v e a w a y fro m th e cen tre to w a rd s th e f ix e d e n d s , sta rtin g fro m 0 .3 4 1 from e a ch en d to 0 .1 2 1 w ith o c c a s io n a l p e r io d s o f arrest a n d b ack w ard m o v e m e n t, startin g in t im e fro m 14 /is u n til 1 2 0 /is. P o sitiv e p la stic h in g e s fo rm in itia lly(12 /is) a t th e ed g e s o f th e lo a d e d reg io n fo r a fe w m ic r o se c o n d s b e fo reco n c e n tr a tin g in a cen tra l r eg io n o f 2 - 4 e le m e n ts fro m 2 0 - 7 0 /is; th erea fter th ey start m o v in g to w a rd s th e su p p o rts w ith th e sa m e v e lo c ity o f p ro p a g a tio n as th e n e g a tiv e h in g e s , b u t 0 .3 7 5 1 rearw ard o f th e m , u n til th e e la p s e o f 1 4 0 /is . D u r in g th e p e r io d 7 0 - 9 5 /is , p o s itiv e p la stic h in g e s fo rm at th e c la m p e d en d s; th is is co n tra ry to w h a t w o u ld b e e x p e c te d in a s im ila r sta tic a n a ly sis . A p er io d o fce ssa tio n o f a ll p la stic d e fo rm a tio n s fo llo w s fro m 1 4 0 - 1 6 0 /is , a fterw h ich n eg a tiv ep lastic h in g e s fo r m at th e f ix e d e n d s fro m 1 6 0 - 4 0 0 /is . T h e n p o s it iv e p lastic reg io n s fo rm a t 0 .1 8 1 - 0 .2 4 1 fro m e a c h en d at 2 3 0 /is a n d m o v e in w ard s w ith t im e , r ea c h in g 0 .3 5 1 a t 4 7 0 /is (F ig u re 7 .2 2 ) . T h is c o m p le x p a ttern o f h in ged e v e lo p m e n t a n d m o tio n m ay b e a ttr ib u ted to th ree fa cto rs:
1 . th e e x is t e n c e an d tem p o ra l v a r ia tio n o f th e a x ia l fo r c e w h ich resu lts in th eo sc il la t io n o f th e p la stic b e n d in g m o m e n t ca p a c ity ;
2 . th e a c c o u n tin g fo r flex u ra l w a v e p r o p a g a tio n in th e a n a ly sis ;
3 . th e c h a r a c ter is t ic o f c la m p ed e n d s in re fle c t in g w a v es n e g a tiv e ly .
In F ig u re 7 .2 4 is sh o w n th e h is to r ie s o f b en d in g m o m e n t at m id sp a n an d at o n e o f th e f ix e d en d s . It is n o te d th a t a f lex u re w a v e first r e a c h e s m id sp a n ata b o u t 6 .5 /is a n d th e fix ed en d s a t a b o u t 3 2 /is (a lso s e e F ig u re 7 .2 7 ( a ) ) . V ery h igh fr e q u e n c y o sc illa tio n s d e v e lo p in b o th h isto r ies fo llo w in g th e fo rm a tio n o f p lastic h in g e s ; th e se a re su p e r im p o se d o n a m o tio n w ith co m p a ra tiv e ly lo w freq u e n cy c o m p o n e n ts . A t th e f ix e d e n d s in th e p e r io d fr o m 1 6 0 /is to 4 0 0 /is, th ere e x is ts a n e g a tiv e p la stic h in g e c o n tin u o u s ly e v o lv in g d e sp ite th e v a r ia tio n o fth e b e n d in g m o m e n t in t im e fr o m - 0 .7 2 to - 0 .9 2 to - 0 .3 6 tim es th e o rig in alb en d in g m o m e n t ca p a c ity o f th e cro ss s e c t io n .
In F ig u re 7 .2 5 , th ere is sh o w n th e h isto r ies o f a x ia l fo r c e at m id sp a n an d at th e su p p o rts . It is s e e n th a t an a x ia l fo r c e first d e v e lo p s a t m id sp a n aro u n d 6 /is an d a t th e f ix e d e n d s rou n d 2 0 /is (a lso s e e F ig u re 7 .2 9 ( a ) ) . N o tin g th a t w a v es o f a x ia l fo r c e o r ig in a te from th e e d g e o f th e lo a d ed r eg io n (a b o u t 3 .9 1 in fro m th efix ed e n d s ) , th e sp ee d o f p ro p a g a tio n o f th e e la st ic w a v es m a y b e estim a ted as « 1 9 5 ,1 9 0 in /s , co m p a r in g w ith th e th e o r e t ic a l v a lu e o f s Z E /p = 2 0 4 ,4 7 0 in /s . T h e n ear c o in c id e n c e o f th e tw o a x ia l fo r c e s (a t m id sp a n a n d a t th e su p p o rts) in d ica te s th e u n ifo r m ity o f b u ild -u p o f th e a x ia l fo r c e a lo n g th e b e a m sp a n , e x c e p t fo r th e
BEND
ING
MOME
NT (l
b..i
TIME (m illise c o n d s)0.0 0.2 0 .4 0.6 0.8 1.0 1.2
Figure 7.24. Histories of bending moment at midspan and fixed end. (1 in - 2.54 em)3 Ibf = 4.45 N).
BEND
ING
MOM.
/ B
ENDI
NG M
OM. C
APAC
ITY
FORC
E (lb
,)1 4 0 0 0 . 012000.010000.0
8 0 0 0 .06 0 0 0 .04 0 0 0 . 02000.0
0.0- 2000.0 - 4 0 0 0 . 0 - 6 0 0 0 . 0 - 8 0 0 0 . 0
- 10000.0 - 12000.0 - 1 4 0 0 0 . 0
0 .0 2 0 0 . 0 4 0 0 . 0 6 0 0 . 0 8 0 0 . 0 1 0 0 0 .0 1 2 0 0 .0TIME (m icro seco n d s)
TIME (m illise c o n d s)0.0 0.2 0 .4 0 .6 0.8 1.0 1.2
Figure 7.25. History o f shear foroe at fixed end and histories o f axial force at midspan and fixed end. (1 lb^ = 4.45 N).
155
156
in itia l sh ort p e r io d o f a p p ro x im a te ly 4 0 /is . T h e p r ed o m in a n tly te n s ile a x ia l fo rce c h a n g e s to c o m p r ess iv e in th e p e r io d b e tw e e n 6 0 0 - 1 1 2 0 /is. T h is c o in c id e s w ith th e sta g e o f s o - c a l le d " elastic recov ery " ; h o w e v e r , p o s it iv e p la stic h in g es ap p ear at th e f ix e d en d s d u rin g th is s ta g e b e tw e e n 6 8 0 - 7 8 0 /is. A g a in , th e v ery h igh fr e q u e n c y o sc illa tio n is n o te d . A lso in F ig u re 7 .2 5 is sh o w n th e h isto ry o f sh ear fo r c e at o n e o f th e f ix e d e n d s , sh ea r fo r c e d e v e lo p in g first a t a b o u t 31 /is. S in ce F ig u re 7 .2 5 in c lu d es a w a v e o f sh ea r fo r c e tra v e llin g to w a rd s th e fix ed en d from th e e d g e o f th e lo a d ed r e g io n , it is p o ss ib le to e s tim a te th e s p e e d o f p rop a gation o f th e w a v e as « 1 3 8 ,0 0 0 in /s . T h is co m p a re s w ith a th e o r e tic a l va lu e o f y ~ G /p = 1 2 4 ,8 6 9 in /s fo r P o isso n 's ra tio v o f 0 .3 6 7 . In th e first 7 0 /is , th e va lu e o f th e sh ear fo r c e rea ch es a m a x im u m an d a m in im u m o f 84% and -7 5 % r e sp e c t iv e ly o f th e sh ea r ca p a c ity o f th e cro ss s e c t io n . T h e r e a fte r , th e p ercen ta g e d e c r e a se s fo r m o st o f th e r e sp o n se t im e , o sc illa tin g b e tw e e n ± 17% o f th e sec tio n sh ea r c a p a c ity arou n d a m ea n v a lu e o f n il. A su p er im p o sed h ig h freq u en cyo sc illa t io n w ill b e o b se r v e d to start a t a b o u t 7 0 /-is. T h is is th e t im e at w h ich a p la stic h in g e first d e v e lo p s at th e f ix e d e n d .
In F igu re 7 .2 6 , grap h s (a ) to (g ) , is sh o w n th e p r o f ile o f tran sversed isp la c e m e n t h is to ry . It ca n b e s e e n fr o m gra p h s (a ) a n d (b ) th a t h ig h er m o d es o f d e fo r m a tio n p rev a il u n til 3 0 0 /is (75% o f th e t im e to a c h ie v e p e a k d isp la cem en t at m id sp a n ). T h is c a n a lso b e co n je c tu r e d fro m th e c o m p le x p ic tu re o f p la stic h in ge d e v e lo p m e n t sh o w n in F ig u re 7 .2 2 . E la s tic rec o v er y starts a t a b o u t 4 0 0 /is, as sh o w n in grap h (c ) on w a rd s. T h e p e r io d o f c o m p r e ss iv e a x ia l fo rce(6 0 0 - 1 1 2 0 /is) is c o v e r e d in gra p h s ( e ) , (f ) and (g ) . In g rap h ( f ) , partia lb u ck lin g o f th e b e a m is q u ite e v id e n t.
In F ig u re 7 .2 7 , grap h s (a ) to (h ) , is sh o w n th e p r o file o f th e b en d in g m o m en t h isto ry . T h e h o r iz o n ta l a x is starts fro m n o d e 2 an d en d s a t n o d e 8 1 ; th is is du e to th e m o d e llin g o f th e f ix e d en d su p p o rts o n th e a d o p te d b e a m e le m e n t. In g rap h (a ) th e d isp e rs io n ch a rac ter istic o f th e flex u ra l w a v es is e v id e n c e d b y th ein c r e a se in th e n u m b e r o f o sc illa tio n s in th e w a v e p a ck e t as th e h ig h freq u en cyc o m p o n e n ts m o v e o u t a h ea d o f th e m a in d istu rb a n ce . A lso n o te d is th e reversa l in th e flex u ra l w a v e as it r e fle c ts fro m th e f ix e d en d ; th is d id n o t o ccu r at th e s im p le su p p o rt as sh o w n in F ig u re 7 .1 7 .
In graph ( e ) , a t a b o u t 3 0 0 /is , th e m a x im u m b en d in g m o m e n t is s e e n to reach ± 92% o f its n o rm a l p la stic ca p a c ity w h ile th e c o rre sp o n d in g a x ia l fo r c e is at 26% o f its n o rm a l p la stic c a p a c ity . A t th is t im e , th e tra n sv erse d isp la c e m e n t at m id sp an is 2 .1 5 larger th a n th e b e a m d e p th . It h as b e en su g g ested (S y m o n d s and M en te l 1 9 5 8 , N o n a k a 1 9 6 7 , S y m o n d s an d J o n e s 1 9 7 2 , L iu a n d J o n e s 1 9 8 8 , S ch u b ak , A n d erso n an d O ls o n 1 9 8 9 ) th at a fu lly c la m p e d b eam m a y e n te r a "p lastic string"
DISP
LACE
MENT
(in
ches
) DI
SPLA
CEME
NT (i
nche
s)
(a) curves 1 to 16 from 0.0 to 0.150 milliseconds (b) curves 17 to 32 from 0.160 to 0.310 milliseconds
( o ) c u rv e s 3 3 to 4 6 f r o m 0 .3 2 0 to 0 .4 7 0 m il lis e c o n d s ( d ) c u r v e s 4 9 to 6 4 f r o m 0 .4 6 0 to 0 .6 3 0 m il l is e c o n d s
Figure 7.26 (continued on next page). 157
DISP
LACE
MENT
(inc
hes)
DISP
LACE
MENT
(inc
hes)
(e) curves 85 to 80 from 0.640 to 0.790 millisecontis
DISP
LACE
MENT
(inc
hes)
curves 81 to 96 from 0.800 to 0.050 milliseconds
NUMBER OF NODE
Figure 7.26. History o f the profile o f transverse displacements at midspan shown in time sequence in graphs (a) to (g). (1 in = 2.54 cm).
BEND
ING
MOME
NT (l
brin
) BE
NDIN
G MO
MENT
{Ib^
in]
(a) curves 1 to 4 from 0.0 to 0.0378 milliseconds (b) curves 5 to B from 0.0429 to 0.0752 milliseconds
2 9 10 2 3 3 0 3 7 4 4 51 5 8 0 5 7 2 79NUMBER OF NODE
2 9 18 2 3 3 0 3 7 4 4 51 5 8 8 5 7 2 7 9NUMBER OF NODE
2 9 10 23 30 37 4 4 51 5 0 05 7 2 79NUMBER OF NODE
2 9 18 2 3 3 0 3 7 4 4 51 5 8 6 5 7 2 7 9NUMBER OF NODE
Figure 7.27 (eontinued on next page). Ln<X>
BEND
ING
MOME
NT (l
brin
) BE
NDIN
G MO
MENT
(lbr
in)
(e) curves 17 to 32 from 0.160 to 0.310 milliseconds (f) curves 33 to 43 from 0.32 to 0.47 milliseconds1000.00 -
6 0 0 .0 0 -
0.00
- 5 0 0 .0 0 -
- 1000.00i i i 1111111111 [ m u 1111111 i j 111111 |'i 1111111
2 9 16 2 3 3 0 3 7 4 4 51NUMBER OF NODE
I...I... 1111 ll l| III3 0 37 4 4 5 1 68
NUMBER OF NODE
( g ) c u r v e s 4 9 to 6 4 f r o m 0 .4 6 to 0 .6 3 m il l is e c o n d s c u r v e s 6 5 to 8 0 f r o m 0 .6 4 to 0 .7 9 m illis e c o n d s
1000.00 -
5 0 0 .0 0
0.00
- 5 0 0 . 0 0 -
-1000.00 - 1111111111111| 111111| 111111| 11 i m p 11111| 111111| m 1111 n I'l'i'n i
9 16 2 3 3 0 3 7 4 4 51 5 8 6 5NUMBER OF NODE
7 2
1000.00
5 0 0 .0 0 -
0.00
- 5 0 0 .0 0 -
-1000.00 -i3 0 3 7 4 4 51
NUMBER OF NODE
Figure 7.27. History o f the profile of bending moment shown in time sequence in graphs (a) to (h). (1 in = 2.54 c m 1 Ibf = 4.45 N).
160
161
s ta te w h e n th e m a x im u m tra n sv erse d isp la c e m e n t e x c e e d s a p p r o x im a te ly th e b eam th ick n e ss . In th is e x a m p le at le a st , th e r e is n o c lea r ly d e fin e d p la stic str in g state as c a n b e d e d u ced fro m th e p r ev io u sly m a d e s ta te m e n t a n d fro m th e h isto ry o f a x ia l fo r c e s sh o w n in F ig u re 7 .2 5 . T h e still p a r t ly -p la s t ic reb o u n d sta te is to be o b serv ed in grap h (f) et seq.t w ith grap h (h ) g iv in g an a lm o st m irror im a g e o f w h a t is sh o w n in grap h ( f ) .
In F ig u re 7 .2 8 , grap h s (a ) to (h ) , is sh o w n th e p r o file o f sh ea r fo r c e h istory . It is s e e n in grap h (a ) th a t , aw a y fro m th e b e a m su p p o rts , th e m a x im u m shear fo r c e o ccu rs v ery ea r ly a t a ro u n d 1 2 j/s, a t im e co m p a ra b le w ith th e d u ration o f th e a p p lie d p ressu re p u lse . T h e m a x im u m sh ea r fo r c e s at th e f ix e d en d s o ccu rs at a b o u t 5 5 jxs, grap h (b ) , w h e n th e tra n sv erse d isp la c e m e n t a t m id sp a n a p p ro x im a te ly eq u a ls th e b e a m d e p th . F ro m g rap h (c ) o n w a rd s, th e sh ea r fo r c e b e co m es in s ig n if ic a n t; a n d , fro m grap h s (f) a n d (g ) , th e reb ou n d sta te is o n c e m ore r e p r e se n te d .
In F ig u re 7 .2 9 , grap h s (a ) to ( f ) , is sh o w n th e p r o file o f a x ia l fo r c e h istory . G ra p h (a ) in d ica te s th a t , a fter th e a x ia l w a v e r e a c h e s th e b e a m su p p o r ts , th e axial fo r c e m a g n itu d e b e c o m e s (s ta tist ica lly ) u n ifo rm th ro u g h o u t th e b e a m . Su ch u n ifo r m ity is e s ta b lish ed v ery ea r ly in th e r e sp o n se at a rou n d 4 0 ns (d esp ite th e fa c t th a t ax ia l in ertia w as c o n s id e red in th e a n a ly s is ) , a fa c t w h ich co rro b o r a te s th e a ssu m p tio n o f u n ifo rm ity o f te n m a d e in th e lite ra tu re (S ym o n d s a n d M e n te l 1 9 5 8 , S y m o n d s an d J o n e s 1 9 7 2 , L iu a n d J o n e s 1 9 8 8 ) . D u r in g th e reb o u n d s ta te , w h ich c o m m e n c e s in grap h (d ) , th e p rev a ilin g a x ia l te n s ile fo rce c h a n g e s to co m p r essiv e , g ra p h s (e ) an d ( f) .
7 .3 .2 . C o m p a r iso n w ith E x p e r im e n ts
B e a m N o .95 is o n e in a s e t o f c la m p e d b e a m s, te sted u n d er im p u ls iv e load in g c o n d it io n s b y C lark , S c h m itt , E llin g to n , E n g le a n d N ico la id es (1 9 6 5 ) . It is a lm ost id e n tic a l to th e c la m p e d b e a m , B e a m N o . l l l , d iscu ssed in th e p r ev io u s s e c t io n , th e d if fe r e n c e s b e in g in th e b e a m d ep th (^ in ) , a n d th e m a g n itu d e o f th e ap p lied im p u lse (56% th a t o f B e a m N o . l l l ) , F ig u re 7 .3 0 ( a ) and (b ). T h is te s t b ea m is o fte n e m p lo y e d as a v a lid a tio n e x a m p le (W itm e r , B a lm er , L e e c h a n d P ia n 1 9 63 , W u a n d W itm er 1 9 7 1 , B e ly tsch k o an d M a rch erta s 1 9 7 4 , H u g h e s , L iu an d L evit 1 9 8 0 ) .
In th e p r esen t stu d y , B e a m N o .95 is d iscre tised in to 8 4 e le m e n ts , w ith th e c e n tr a l 18 e le m e n ts su b je c te d to an im p u ls iv e p ressu re lo a d in g . T h is arran gem en t g iv e s a lo n g itu d in a l sp rea d o f 1 0 .9 % o f th e a p p lie d p ressu re , w h ich is req u ired to
SHEA
R FO
RCE
(lb,)
SHEA
R FO
RCE
(lb,)
2 9 16 2 3 3 0 3 7 4 4 61 5 0 0 5 7 2 79NUMBER OF NODE
2 9 16 2 3 30 3 7 4 4 51 5 0 65 7 2 79NUMBER OF NODE
9 18 2 3 3 0 3 7 4 4 51 5 8 6 5 7 2NUMBER OF NODE
7 9 9 10 2 3 3 0 3 7 4 4 51 5 0 0 5 7 2 7 9NUMBER OF NODE
Figure 7.28 (continued on next page). 162
SHEA
R FO
RCE
(Zb,
) SH
EAR
FORC
E (lb
,)
curves 17 to 32 from 0.160 to 0.310 milliseconds curves 33 to 48 from 0.320 to 0.470 milliseconds
4 0 0 0 .0 0 -
2000.00 -
0.00
- 2000.00 -
- 4 0 0 0 . 0 0 -
I..I....I... I3 0 3 7 4 4 51
NUMBER OF NODE.."I1..1... I"1
3 0 3 7 4 4 5 1NUMBER OF NODE
c u r v e s 4 0 to 6 4 f r o m 0 .4 8 0 to 0 .6 3 0 m il l is e c o n d s
4 0 0 0 .0 0 -
2000.00 -
0.00
- 2000.00 -
- 4 0 0 0 .0 0 -
c u r v e s 6 5 to BO f r o m 0 .6 4 0 to 0 .7 9 0 m illis e c o n d s
4 0 0 0 .0 0 -
2000.00 -
0.00
S -2000.00 -5- 4 0 0 0 .0 0 -
................................................................9 16 2 3 3 0 3 7 4 4 51 5 0 6 5 7 2 7 9
NUMBER OF NODE
Figure 7.28. History o f the -profile of shear force shown in time sequence in graphs (a) to (h). (1 Ibf = 4.45 N).
163
AXIA
L FO
RCE
(lb,)
AXIA
L FO
RCE
(Ibj)
( cl) curves 1 to 8 from 0.0 to 0.075 milliseconds
2 9 16 2 3 3 0 3 7 44 51 5 0 6 5 7 2 79NUMBER OF NODE
( a ) c u r v e s 17 to 3 2 f r o m 0 .1 6 0 to 0 .3 1 0 m il l is e c o n d s (d) c u r v e s 3 3 to 4B f r o m 0 .3 2 to 0 .4 7 m il l is e c o n d s
10000.00 -
0.00
-10000.00 -
TIB 2 3 3 0 3 7 4 4 51
NUMBER OF NODE
npfmi|im w p5 8 6 5 7 2 7 9
10000.00
0.00
-10000.00 -
I ......... I ............I ............. T3 0 3 7 4 4 51
NUMBER OF NODE
Figure 7.29(continued on next page). 164
AXIA
L FO
RCE
(lb,)
(e) curves 49 to 04 from 0.48 to 0.03 milliseconds
10000.00 -
0.00
-10000.00 -i
NUM BER OF NODE
(f) curves 65 to 00 from 0.64 to 0.70 milliseconds
10000.00 -
0.00
-10000.00 -
NUMBER OF NODE
Figure 7.29. History o f the profile o f axial force shown in time sequence in graphs (a) to (f). (1 Ibf = 4.45 N).
165
v\\\
\Nxy
xvvv
(a)
BUFFER
4
HE LAYER
773. .______•1.932"-+
DETONATIONPOINT
m "
(b) DEALISED HE LAYER
r -||_______
2-143 —sj*-
i 10.0* 4
E = 10.8 x 1 0 6 lbf/in2 V = 0.367 Oy = 41 j 600 lbf/in2
p = 0.0002525 Ibf.s2/in'*
Strain Rate Sensitivity Constants3
D = 6500/s (or 7000/s)
P = 4 . 0 (or 2.13)
Depth (d) = 0.124 in
Breadth (b) = 1.195 in
Impulse = 0.3637 lbj?.s
Figure 7.30. Beam geometry3 loading and material properties, (a) Configuration used in the experiments o f Clark et al. (1965). (b) Configuration of the idealised beam used in the present study. (1 in = 2.54 em3 1 lb^ = 4.45 N).
167a c c o u n t fo r th e ed g e e f fe c ts m e n tio n e d in se c t io n 7 .2 .2 . S h ea r d e fo rm a tio n s(k ' = 1 .1 7 3 fo r v = 0 .3 6 7 ) , ro ta to ry in ertia an d g e o m e tr ic n o n lin ea r ity are c o n s id e r e d in th e an a lysis . T h e in te r a c tio n b e tw e e n b e n d in g m o m e n t an d ax ia l fo r c e is im p le m e n te d n u m er ica lly in th e w ay d escr ib ed in th e p r ev io u s se c t io n ,7 .3 .1 . ; stra in ra te sen s itiv ity is c o n s id e r e d in a c c o rd a n ce w ith th e th ree d iffe ren t m a n n ers d e sc r ib ed in th at sa m e s e c t io n . T h e p u lse sh a p e a n d d u ra tion o f th e a p p lied p ressu re are as sh o w n p r ev io u sly in F ig u re 7 .1 1 ( a ) . W ith a t im e step o f0 .2 x 1 0 ” Gs , an d for a r e sp o n se t im e o f 1 .5 m s, th e co m p u te r C P U tim e req uired is 1 9 m in u te s o n a V A X /8 6 0 0 m in ic o m p u te r .
T h e h is to r ie s o f m id sp a n d isp la c e m e n t, fo r th e th r e e d iffe r e n t a cco u n ts o f stra in ra te s e n s it iv ity , an d th e e x p e r im e n ta l record a re sh o w n in F ig u re 7 .3 1 ( a ) . It is s e e n th a t th e m a g n itu d es o f th e p e a k m id sp a n d isp la c e m e n ts in a n y o f th e re sp o n se s sh o w n a re a t lea st 5 .6 t im e s th e b eam d e p th . C o m p a r in g th ed isp la c e m e n t h is to ry o f B e a m N o .9 5 w ith th at o f B e a m N o . l l l - F ig u re 7 .2 1 (a ) - it is n o te d th a t th e so lu tio n (in w h ich th e c o n sta n ts D = 6 5 0 0 /s a n d P = 4 .0 w ere u sed ) g a v e a b e tter p r ed ic tio n fo r th e p e a k d isp la c e m e n t th an th e eq u iv a len t so lu tio n fo r B e a m N o . l l l . T h is is b e c a u se , in th e fo r m e r , o w in g to its lesser stren g th in f le x u r e (25% th a t fo r B e a m N o . l l l ) , th e te n s ile a x ia l fo r c e b u ild s up fa ster a n d h e n c e a ssu m e its in flu e n tia l ro le ea r lier o n . T h e d isp la c e m e n t atm id sp a n fo r B e a m N o .95 eq u a ls th e b e a m d e p th a t aro u n d 6 .2 5 % th e t im e to th e p e a k d isp la c e m e n t; th is is n e a r ly h a lf th e c o rre sp o n d in g v a lu e fo r B e a m N o . l l l . A lso n o te d in F ig u re 7 .3 1 ( a ) is th e irregu lar p a ttern o f d isp la c e m e n t h isto ry a fter th e in itia l 3 0 0 /is . T h is m a y b e a ttr ib u ted to th e in te n se h in g e a c tiv ity an d th e in te r a c tio n b e tw e e n b en d in g m o m e n t a n d a x ia l fo r c e .
T h e so lu tio n o f tw o d iffe r e n t f in ite e le m e n t sc h e m e s are sh o w n in F igu re 7 .3 1 ( b ) . T h e first is b y B e ly tsch k o a n d M a rch erta s (1 9 7 4 ) in w h ich 2 0 triangu lar e le m e n ts w e r e e m p lo y e d to m o d e l \ th e sp a n , a n d an e x p lic it in teg ra tio n sch e m e w as u t ilise d . T ran sv erse sh ea r d e fo r m a tio n s w ere n o t c o n s id e r e d , an d th e b eam m a ter ia l w as id ea lised as stra in ra te in se n s it iv e . T h e se c o n d so lu tio n is b y H u g h es , L iu an d L e v it (1 9 8 0 ) . H e r e , 1 0 rec tan g u la r e le m e n ts w ere u sed in m o d e llin g th e sp a n , w ith 4 - p o in t G au ssian q u ad ratu re a cro ss th e b ea m th ic k n e ss , th e w h o le m o d e l b e in g su b je c te d to an im p lic it in teg ra tio n sc h e m e (th e N ew m a rk sc h e m e w ith (3 =
7 = £) w ith a tim e s tep s iz e o f 1 x 1 0 - e s; tra n sverse sh ea r d e fo r m a tio n s and k in em a tic h a rd en in g w ere c o n s id e r e d , a n d th e b eam m a ter ia l a ssu m ed stra in rate in d e p e n d e n t. In b o th o f th e a b o v e so lu tio n s , th e a p p lied lo a d w as id ea lised as im p u ls iv e b y g iv in g th e lo a d ed e le m e n ts a n in itia l v e lo c ity o f 5 2 0 0 in /s .
In F ig u re 7 .3 1 ( c ) a re sh o w n tw o fu rth er n u m er ica l so lu tio n s . T h e first is afin ite e le m e n t so lu tio n b y W u a n d W itm er (1 9 7 1 ) . In th is c a se , 1 0 b eam e le m e n ts
1 6 8
0.00 0.25T IM E (m illis e c o n d s )
0.50 0 .75 1.00 1.25 1.501.1Present S tu dy , 84 Elem ents,S tra in P ate S e n s itiv ity According to,
1 In itia l S tra in Rate a t M idspan, D*=7000 s"\ P = 2.132 -a * In itia l S tra in Rate a t M idspan, Z)=8500 s '1, P = 4.00
•x S tra in R ate Insen sitiveE xperim en t (a fter Clark et at. 1065)
0.0
*x* • •. ....—-O---------0^ -------
0.0 250 .0 500.0 750 .0 1000.0 1250.077A/2S1 (m ic ro seco n d s )
1500.0
(a) Solutions from the present study.
(b) Solutions of two finite element schemes compared to experiments. After Hughes et al. (1980).
(c) Finite difference solution of Witmer et al. (1963) and finite element solution of Hu et al. (1971).
Figure 7.31. History of transverse displacement at midspan. (1 in = 2.54 c m ) .
169
w ere u sed to m o d e l j th e sp a n , an d a n e x p lic it in teg ra tio n s c h e m e (cen tra l fin ite d iffe r e n c e s ) w as e m p lo y e d w ith a t im e s te p s iz e o f 1 x 1 0 - e s; n o tra n sv erse sh ear d e fo r m a tio n s w ere c o n s id e r e d , an d stra in ra te in d e p e n d e n t m a ter ia l b eh a v iou r a ssu m ed . T h e se c o n d is a fin ite d if fe r e n c e sc h e m e by W itm er , B a lm e r , L e e c h and P ia n (1 9 6 3 ) . F o r th is so lu tio n 3 0 m a sse s w ere u sed to m o d e l £ th e sp a n , and 6 f la n g e s w ere u sed to rep resen t th e b e a m d e p th . E x p lic it in teg ra tio n w as em p lo y ed (cen tra l f in ite d if fe r e n c e s ) w ith a t im e s te p s iz e o f 0 .6 6 7 x 1 0 " 6s; tra n sv erse sh ear d e fo r m a tio n s , ro ta to ry in ertia an d stra in ra te sen s itiv ity w ere n o t a cco u n ted for . A g a in , in b o th o f th e a b o v e sc h e m e s th e lo a d in p u t w as id e a lise d as im p u ls iv e by g iv in g an in itia l v e lo c ity o f 5 0 0 0 in /s to th e lo a d ed e le m e n ts .
In F ig u re 7 .3 2 is sh o w n th e h is to ry o f p la stic h in g e d e v e lo p m e n t a lo n g th e b e a m sp a n . It is n o te d th a t th e r e sp o n se is m o re in ten se th an fo r B e a m N o . l l l (F ig u re 7 .2 2 ) . T h e p a ttern o f h in g e d e v e lo p m e n t is a lso s im ila r e x c e p t fo r:
1 . A n ex tra p o s it iv e p la stic reg io n o f v a r ied len g th d e v e lo p in g b e tw e e n 0 .1 4 1 and 0 .0 6 1 fro m e a c h su p p o rt at 6 0 - 1 0 0 //s.
2 . T h e fo rm a tio n o f a n e g a tiv e p la stic r e g io n o f 2 to 3 e le m e n ts in len g th atm id sp a n b e tw e e n 1 2 0 - 2 0 0 /is and 3 2 0 - 4 6 0 /is.
3 . T h e r e la tiv e ly sh o rter p er io d o f d u ra tio n o f p o s itiv e h in g e s a t th e fix ed en d sin th e reb o u n d s ta g e w h en th e b e a m is u n d er c o m p r e ss io n .
In F ig u re 7 .3 3 is sh o w n m a g n ified th e in itia l 2 0 0 /is o f F ig u re 7 .3 2 . It can b es e e n th at th e g e n e r a l p ic tu re is s im ila r to th a t o f F ig u re 7 .2 3 fo r B e a m N o . l l l ,b u t w ith sp a tia lly m o r e e x te n d e d p la stic z o n e s ; it m ay a lso b e s e e n th a t th e m ostin te n se h in g e fo r m a tio n a c tiv ity o ccu rs b e tw e e n 6 0 - 1 0 0 /is , w h e n a t 7 6 /as th ere areas m a n y as e ig h t d iffe r e n t lo c a t io n s o f s im u lta n eo u s p la stic d e fo r m a tio n . In d eed ,F ig u re 7 .3 3 d o e s s e e m ex tra o rd in a ry ; an d w ith th e in ab ility o f e x is t in g ex p e r im e n ta l te c h n iq u e s to d e te c t an d m o n ito r p la stic h in g e s o f v ery sh o rt d u ra tio n ( 2 - 4 /is ) , it w o u ld b e d iff ic u lt to p r o v id e e x p e r im e n ta l co rro b o ra tio n o f th e d iffu se d istrib u tiono f p la stic a c tiv a t io n so p ro v id ed .
7 .4 . C A N T IL E V E R B E A M S
7 .4 .1 . C o m p a r iso n w ith A D I N A
T h e b e a m e x a m p le co n s id e red h e r e is an en tir e ly e la st ic ca n tile v e r load ed u n ifo rm ly a lo n g th e sp a n b y a s t e p -r is e p u lse o f in fin ite d u ra tio n ; th e b eam
TIME
(micr
oseco
mP re se n t S tu d y , 84 E lem en ts a Negative Plastic Hinge in Flexure
O Positive Plastic Hinge in Flexure
%
700600500400300200
100
800
0
: 800 ; 700 : 600 : 500 : 400 : 300: 200
: loo
1 o0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84
NUMBER OF NODE
In itia l S tr a in R a te a t M idspan , D= 6500 s “\ P = 4
Figure 7.32. History o f plastic hinge development.
TIME
(micr
osec
onds
)
200
180
160
140
120100
80
60
40
20
0
P resen t S tu d y , 84 E lem en ts a Negative Plastic Hinge in FlexureO Positive Plastic Hinge in Flexure
r r rp i i | i m | i i i ; i i i | i i i | i i i | i i i | i 1 1 | i i i | » i i p i i | i i i | i i i | i i i ; i i rp i i"| n i | i i i | i i i | i i i"f~ 0
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84NUMBER OF NODE
Figure 7.33. Initial 200 ys of the history of plastic hinge development.
TIME
(micr
osec
onds
)
1 7 2
c o n fig u r a tio n , lo a d in g an d m a ter ia l p r o p er tie s are sh o w n in F ig u re 7 .3 4 . T h is e x a m p le w as p r esen te d and so lv e d b y B a th e (1 9 7 5 ) u sin g th e f in ite e le m e n t c o d e A D I N A ; it serv ed as an e x e r c is e to h ig h lig h t th e im p o r ta n c e o f eq u ilib riu m ite ra tio n s w h e n im p lic it in teg ra tio n sc h e m e s a re u sed to p r ed ic t a g e o m e tr ic a lly n o n lin e a r d y n a m ic re sp o n se . T h e sa m e e x a m p le h a s a lso b e e n e m p lo y e d by G e r a d in , H o g g e an d Id e lso h n (1 9 8 3 ) fo r th e sa m e p u r p o se , a n d b y Y a n g an d S aigal (1 9 8 4 ) a s a v a lid a tio n e x a m p le . In h is a n a ly sis , B a th e (1 9 7 5 ) d iscre tised th e b eam in to f iv e 8 - n o d e p la n e stress iso p a ra m etr ic f in ite e le m e n ts , a n d e m p lo y e d an im p lic it in teg ra tio n sc h e m e (th e N ew m a rk s c h e m e w ith (S = y = £) w ith a tim e s tep s iz e o f 45 x 1 0 “ 6s (1 /1 2 6 th o f th e fu n d a m en ta l p e r io d fo r lin ea r u n d a m p ed e la st ic v ib ra tio n s) an d a m in im u m o f o n e eq u ilib r iu m itera tio n p er tim e s te p . W ith th e d e v e lo p m e n t o f large d isp la c e m e n ts , th e a p p lied d istr ib u ted p ressu re a lw ays rem a in s n o rm a l to th e u p p er su r fa c e o f th e b e a m (a fo llo w e r fo r c e ) .
In th e p r esen t stu d y , th e b e a m h as b e e n d iscre tised in to th irty 2 - n o d e c o n t in u o u s m a ss an d lu m p ed f le x ib ility b eam e le m e n ts . E x p lic it in teg ra tio n (cen tra l f in ite d if fe r e n c e s ) w as e m p lo y e d w ith a t im e s tep s iz e o f 0 .4 x 1 0 “ 6s an d w ith n o eq u ilib r iu m itera tio n s . T h e v ery sm a ll t im e s tep s iz e u sed h e r e is d u e to th e fa ct th a t e x p lic it in teg ra tio n s c h e m e s a re m o re su ita b le fo r v ery sh o rt d u ra tion lo a d in g (ty p ica lly m ea su re d in m ic r o se c o n d s) , su ch as c o n v e n tio n a l b last lo a d in g , w h e re w ave p r o p a g a tio n m o d e llin g is o f im p o r ta n c e an d th e re sp o n se is n o n lin ea r (sec tio n s7 .2 .2 , 7 .3 .1 a n d 7 .3 .2 ) . T h e to ta l c o m p u ter C P U tim e req u ired w as 2 7 m in u tes o n th e V A X /8 6 0 0 m in ic o m p u te r fo r a re sp o n se tim e o f 1 2 m s. In th e p resen t so lu t io n , sh ea r d e fo rm a tio n s (k* = 1 .1 8 3 fo r v = 0 .2 ) , ro ta to ry in er tia , g eo m e tr ic n o n lin e a r ity , a n d d isp la c e m e n t d e p e n d e n t fo llo w e r lo a d in g w ere co n s id e red .
T h e h is to ry o f tra n sv erse d isp la c e m e n t a t th e fr e e en d o f th e b e a m is sh o w n in F ig u re 7 .3 5 fo r th e p r esen t stu d y a n d A D I N A so lu tio n s . It ca n b e s e e n th at th e p e a k d isp la c e m e n t is a lm o st 6 t im e s th e b eam d e p th a n d 0 .6 tim es th e b eam le n g th . B a th e (1 9 7 5 ) m e n tio n e d th a t a m a jo r ch a ra c ter is tic o f th e c a n tilev er b eam r e sp o n se w as th e s t iffe n in g w ith in cre a s in g d isp la c e m e n t; i f tw o su c c e ss iv e c o m p le te c y c le s o f th e re sp o n se a re c o m p a r e d th is s t iffe n in g e f f e c t resu lts in a su b stan tia l d e c r e a se in a m p litu d e an d a sh o r te n in g o f th e e f fe c t iv e p e r io d o f o sc il la t io n , F igu re 7 .3 5 , a n d it m a y fu rth er resu lt in c o n v e r g e n c e d iff ic u lt ie s w h en eq u ilib riu m ite ra tio n s a re p e r fo r m e d . F o r th e so lu tio n o f th e p r e se n t stu d y th e p er io d o fo sc il la t io n r e m a in e d sta b le d u e to th e sm a ll t im e s te p u sed . H o w e v e r , th e a m p litu d e in c r e a se d w ith in c r e a s in g n u m b er o f cy c le s ; th is m a y b e a ttr ib u ted to a p p r o x im a tio n s m a d e in th e k in em a tic an d k in e tic re la tio n s at th e e le m e n t le v e l w ith regard to g eo m e tr ic n o n lin e a r ity .
DISP
LACE
MENT
AT TH
E TIP
(inc
hes)
1 7 3
p Ib/inf
Td±
A
L = 10 in
d = 1 inE = 1 .2x 10 * v =0.2
! bf / i n 2 i iP Ib /in
b = 1 in p = I 0 ' 6 lbf 2 4 sec / i nc. o a
♦>.t
Figure 7.34. Beam geometry3 material properties and applied loading. After Bathe (1975). (1 in = 2.54 cm3 1 Ibf = 4.45 N).
TIME (m illisecon ds )0 1 2 3 4 5 6 7 8 9 10 11 12
Figure 7.55. History of transverse displacement at free end. (1 in = 2.54 cm).
1 7 4
CHAPTER 8
PARAMETRIC STUDY
8 .1 . IN T R O D U C T IO N
T h e in te n tio n in th is ch a p ter is to e x a m in e so m e o f th e m a n y criteria th at h a v e a b ear in g o n b o th m o d e llin g an d so lu tio n in p ro b lem s o f d y n a m ic p la stic ity . T h e first tw o p a ra m eters in v estig a te d c o n c e r n th e d iscre t isa tio n o f th e stru ctural m o d e l. T h e se c o n d tw o p a ra m eters a re a sso c ia ted w ith th e c o n stitu tiv e m o d e llin g o f th e stru ctu ra l m a ter ia l. In th e fin a l p a ra m eter c o n s id e r e d , th e attr ib u tes o f th e a p p lie d lo a d p u lse a re ex p lo r e d .
8 .2 . E F F E C T O F T E M P O R A L M E S H
In th e d y n a m ic a n a lysis o f b e a m s su b je c te d to tra n sien t lo a d in g o f v ery sh ort d u ra tio n , th e s iz e o f t im e s tep , u sed in th e e x p lic it t im e -m a r c h in g in teg ra tio n o f th e eq u a tio n o f m o tio n , h as to b e sm a ller th an th e tim e req u ired fo r a lo n g itu d in a l w a v e to tra n sv erse th e len g th o f a s in g le e le m e n t . A c c o u n tin g fo r th e a b o v e , it is b e st to c h o o s e a t im e s tep s iz e n e a r e s t th e largest p e rm iss ib le (cr itica l) to avoid a c c u m u la tio n o f r o u n d - o f f errors ca u sed b y an in cre a sed n u m b er o f n u m erica l o p e r a tio n s .
H a v in g sa id th a t , it s e e m s th a t th e r e are ca ses w h e re th e ex ist in g criteria for d e te r m in in g th e cr itica l s iz e o f t im e s te p a re n o t a d eq u a te . O n e su ch ca se is th at o f a d y n a m ic r e sp o n se w h ich in c lu d e s stru ctural in sta b ility in th e fo rm o f sn a p -b u c k lin g . In a r e c e n t c o m m u n ic a t io n b y S y m o n d s an d Y u (1 9 8 5 ) , a p rob lem o f e la s to -p la s t ic b e a m d y n a m ics w as p r esen te d - s p e c if ic a lly , a p in -e n d e d b eam su b je c te d to a lo a d w h ich is sp a tia lly u n ifo rm ly d istr ib u ted a n d tem p o ra lly h as th e fo rm o f a sh o r t rec tan g u la r p u lse , F ig u re 8 .1 . It w as fo u n d fro m a n u m erica l a n a ly sis o f th is p r o b le m , u sin g th e f in ite e le m e n t c o d e A B A Q U S , th a t a fter a p eak tra n sv erse d isp la c e m e n t in th e d ir e c tio n o f load a p p lica tio n (p o s it iv e ) th e b eam reb o u n d ed an d a tta in ed a p e rm a n en t resid u a l d isp la cem en t in th e o p p o site d irec tio n (n e g a t iv e ); th is is sh o w n in F ig u re 8 .2 as cu rv e n u m b er 1 . F u r th erm o re , th e sa m e a u th o rs p r e se n te d h is to r ies o f tra n sv erse d isp la cem en ts a t m id sp a n as p red ic ted by
xx
:b
h
1 7 5
(a) ^ t t p t t t tI------ t ----- 1------ l ----- 1
PPo
( d )
1—b—1 * (b)
E = 80 x 1 09 N / m Z V = 0.367 Oy= 0.3 x 109 N/ m 2 p = 2700 k g / m3 Depth (d) = 4277? Breadth (b) = 2i9?m7 Length (21) = 200mm p = I P . 2 N/mm
tQ = 0.5 millisecond
Figure 8.1. Beam configurations material properties3 interaction relation between bending moment and axial force3 stress-strain relation and shape o f applied pulse. After Symonds and Yu (1985).
TIME IN MSECS
Figure 8.2. History o f transverse displacement at midspan as predicted by eight different finite element and finite difference computer codes. After Symonds and Yu (1985).
176
se v e n o th e r f in ite e le m e n t a n d fin ite d iffe r e n c e c o d e s , n a m e ly : A N S Y S , D Y C A S T , M A R C , M E N T O R , R E P S IL , W H A M S a n d W R E C K E R , F ig u re 8 .2 (id en tifica tio n n u m b ers d o n o t re la te to th e a b o v e o r d e r ). It ca n b e s e e n th at a ll c o d e s ag ree on th e m a g n itu d e o f th e first p e a k d isp la c e m e n t (1 0 .5 m m ), th e t im e o f its o ccu rren ce (0 .5 5 m s) an d th e su b seq u en t re sp o n se u p to a p p ro x im a te ly 0 .8 m s. T h erea fter , th e r e is w id esp rea d d isa g r e e m e n t. T h e re sp o n se o f o n e o f th e c o d e s was rep r e se n te d b y tw o cu rv es m ark ed 3 b a n d 3 c ; th e data fo r th e se are id en tica l e x c e p t th a t th e fo r c e p u lse w as term in a ted b y ram p s o f 0 .2 m s a n d 0 .1 m s d u ra tio n r e sp e c tiv e ly w h ilst k e ep in g th e im p u lse co n sta n t.
In a su b seq u en t c o m m u n ic a tio n b y S y m o n d s , M cN am a ra a n d G e n n a (1 9 8 6) a p p ea rs th e rem ark : " T h e b e a m u n d er th e s ta ted lo a d in g u n d e rg o e s a d e form ation th a t , fo llo w in g e la st ic an d s m a ll-d e f le c t io n e la s to -p la s t ic b e n d in g , b e c o m e s o n e o f p r im a rily p la stic e x te n s io n . T h e r e c o v er y p r o c ess th en in v o lv es a sh a llo w arch w ith a v e lo c ity f ie ld ten d in g to fla tten th e a rch . T h is in d u ces c o m p r e ss iv e s tresses , and in th e r ig h t c ircu m sta n ces lea d s to th e in sta b ilitie s a n d c o m p u ta tio n a l d ifficu lties m e n tio n e d a b o v e ” . F u rth er stu d ies w ere rep o r ted b y G e n n a a n d S y m o n d s (1 9 8 7 ) a n d Y a n k e lev sk y (1 9 8 8 ) .
C u rren tly , th e c o r r e c t so lu tio n fo r th is p ro b lem is still u n k n ow n ; no e x p e r im e n ta l v e r if ic a tio n h as b e e n p r o v id e d . H o w e v e r , e x p e r im e n ts b y R oss, S tr ick la n d an d S ierak o w sk i (1 9 7 7 ) h a v e sh o w n su p p o rtin g e v id e n c e fo r th e rea lisa tion o f th e ty p e o f a b n o rm a l b eh a v io u r p r ed ic ted b y th e fin ite e le m e n t c o d e A B A Q U S .
T h ro u g h a p p lica tio n o f th e b e a m m o d e l d e v e lo p e d in th e p r e se n t stu d y , th e e f fe c t o f t im e step s iz e o n th e p ro b lem p r e se n te d b y S y m o n d s a n d Y u is ex a m in ed . T h e h is to ry o f th e tra n sv erse d isp la c e m e n t at m id sp a n is d isp la y ed in F ig u re 8 .3 (a) a n d (b ) . F o r th e re sp o n se sh o w n in g rap h (a ) , th e b eam w as d iscre tised in to 10 b e a m e le m e n ts , an d s iz e s o f t im e step w ere v a r ied fro m 0 .2 /is to 2 .5 /is; the cr itica l s iz e , as p r ed ic ted fro m eq u a tio n (6 .1 0 ) , is 2 .9 9 /is. In a ll o f th e so lu t io n s , th e m a g n itu d e an d tim e o f o c c u r r e n c e o f th e first p o s it iv e peak d isp la c e m e n t w as c o n s is te n t an d in a g r e e m e n t w ith th e resu lts o f th e various n u m er ica l c o d e s sh o w n in F ig u re 8 .2 . T h e p o s it iv e p ea k d isp la c e m e n t w as fo llo w ed b y a n e g a tiv e p e a k , e x c e p t fo r th e so lu tio n w h e re th e s tep s iz e w as 2 .5 /is (84% th e cr itica l s iz e ) w h ich b e c a m e u n stab le b e fo r e rea ch in g th e n e g a tiv e p ea k . A fter th e n e g a tiv e p e a k d isp la c e m e n t, th e so lu tio n o f th e sm a lle s t t im e s tep s iz e (0 .2 /is) led to a p o s it iv e resid u a l d isp la c e m e n t, w h ilst th a t o f th e te n t im e s larger tim e step (2 .0 /is) led to a n e g a tiv e resid u a l d isp la c e m e n t. F o r th e re sp o n se sh o w n in graph (b ) th e n u m b er o f d iscre t isa tio n e le m e n ts w as d o u b led to 2 0 , a n d t im e step sizes v a ried fro m 0 .1 /is to 1 .2 5 /is; th e cr itica l s iz e is ca lcu la ted at 1 .4 9 /is. A ga in , in a ll so lu tio n s th e first p o s it iv e p e a k d isp la c e m e n t w as c o n s is te n t an d in a greem en t
DIS
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EN
T A
T M
IDS
PA
N (
cm
)
1.21.00.8
0.6
0.40.2
0.0- 0.2
- 0.4- 0.6
- 0.8
- 1.0- 1.2
TIM E (m illis ec o n d s ) T IM E (m illis ec o n d s )0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.2
O 1.0g
0.8
5 0.6& 0.4^ 0.2 ^ 0.0 ^ -0.2 p- ° . 4
3-o.e §5 - 0.8 ^ -1.0
- 1.20.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
TIM E (m illis ec o n d s ) T IM E (m illis ec o n d s )
(a) (b)
Figure 8.3. History o f transverse displacement at midspan for various spatial and temporal meshes.
1 7 8
w ith th o se sh o w n in F igu re 8 .2 . H o w e v e r , th is w as fo llo w e d b y a p o s it iv e trough in stea d o f a n e g a tiv e p eak . T h e so lu tio n o f th e 1 .2 5 /is s te p s iz e (84% o f th ecr itica l s iz e ) b e c a m e u n stab le b e fo r e rea c h in g th e first p o s it iv e tro u g h . It is n o ted th a t th e re sp o n ses in grap h s (a ) a n d (b ) are id en tica l up to th e t im e 0 .8 m s, and are sim ilar to th o se in F igu re 8 .2 fo r th e sa m e tim e p e r io d .
It ca n b e s e e n in F ig u re 8 .3 th a t , as th e sp a tia l m esh b e c a m e fin er , th e resid u a l d isp la c e m e n t a tta in ed at m id sp a n b e c a m e p o s it iv e , reg a rd less o f th e s iz e o f th e in teg ra io n t im e s te p . F o r th e c o a rser m e sh , h o w e v e r , th e h is to r ies o f m id sp an d isp la c e m e n t a fter a p p ro x im a te ly 0 .8 m s b e c a m e a fu n c tio n o f t im e s tep s iz e , and led in so m e c a se s to th e a b n o rm al r e sp o n se d escr ib ed b y S y m o n d s and Y u (1 9 8 5 ) .F ro m th is it m a y b e said th a t th e m o d e llin g o f p la stic ity c o u p le d w ith th e se ttin go f th e t im e s te p s iz e an d th e s iz e o f th e sp a tia l m esh p la y a n im p o rta n t ro le inth is c h a n g in g b e h a v io u r . T h is is s o , b e c a u se in n u m er ica l s t e p - b y - s t e p so lu tio n s o f e la s to -p la s t ic d y n a m ics , if th e stru ctu ra l m a ter ia l y ie ld s at th e b e g in n in g o f a tim e s tep p la stic d e fo r m a tio n s c o n t in u e at a co n sta n t ra te d u rin g th e tim e co v e r e d in th a t s te p . H e n c e id e a lly , as th e sp a tia l a n d tem p o ra l m e sh e s b e c o m e fin er , p la stic ity is m o d e lle d m o re rea listica lly a n d co n s is ten t so lu tio n s m a y b e fu rn ish ed . It is s e e n in g rap h (b ) th a t, in th e t im e p er io d o f th e reb o u n d v ib ra tio n s, th ere is so m e d e g r e e o f c h a o s , th o u g h n o t d iv e r g e n c e ; th is is d u e to th e e f fe c t o f t im e step s iz e o n th e m o d e llin g o f th e in ter a c tio n b e tw e e n b en d in g m o m e n t an d a x ia l fo r c e .
8 .3 . E F F E C T O F S P A T IA L M E S H
It is d e sira b le to e m p lo y th e la rg est sp a tia l m esh fe a s ib le , c o n s is te n t w ith th e p r ed ic tio n a ccu ra cy so u g h t. O th e r fa c to rs b e in g eq u a l, th is is d esired in o rd er to m in im ise c o m p u ta tio n t im e an d n u m er ica l r o u n d - o f f errors. T h e d y n a m ic resp on se o f stru ctu res is d e p e n d e n t, a m o n g st o th e r fa c to rs , u p o n :
1 . T h e in te n s ity , d istr ib u tion a n d t im e h isto ry o f th e fo rc in g fu n c tio n .
2 . T h e req u ired accu ra cy .
A s su c h , th e e s ta b lish m en t o f a fo rm u la tio n to p red ic t a d eq u a te ly th e s iz e o f th e sp a tia l m e s h , p a rticu lar ly in a d y n a m ic a n a ly sis , h as n o t b e e n rea lised so far. C o n se q u e n tly , c h o ic e o f th e sp a tia l m e sh s iz e is a m a tter le ft to th e a n a ly st's ju d g e m e n t, o r is e sta b lish ed via su c c e s s iv e tria ls.
In p r o b le m s o f e la s to -p la s t ic d y n a m ic s , th is u su a lly in v o lv e s sta rtin g w ith an in itia l co a r se m e s h , tak in g th e e la s t o - p la s t ic so lu tio n to c o m p le t io n and re fin in g th e
1 7 9
m e sh fo r a n e w tr ia l i f n e cessa r y . A s lig h tly d iffe r e n t p r o c e d u r e , o n e th a t m a y b e m o r e r ev ea lin g an d le ss t im e -c o n s u m in g , is to ta k e o n ly th e e la st ic so lu tio n to c o m p le t io n in e a c h tr ia l. T h is is sh o w n in F ig u re 8 .4 (a ) a n d (b ) fo r th e d y n am ic r e sp o n se o f th e s im p ly su p p o rted b eam d iscu ssed in se c t io n 7 .2 .1 . In grap h (a ) is sh o w n th e h is to ry o f tra n sv erse d isp la c e m e n t a t m id sp a n fo r sp a tia l m esh s iz e s o f 4 , 6 , 1 2 an d 18 e le m e n ts , u sin g th e b e a m m o d e l d e v e lo p e d in th is s tu d y , to g e th er w ith th e re sp o n se p r ed ic ted via th e f in ite e le m e n t c o d e A D I N A (in w h ich six 8 - n o d e d iso p a ra m etr ic b e a m e le m e n ts w e r e u sed fo r m o d e llin g o n e qu arter th e b e a m ). T h e c o n v e r g e n c e o f th e so lu tio n is ev id e n t as th e sp a tia l m esh is m ad e f in e r . T h e r e f in e m e n t o f th e m e sh s iz e is ta k en fu r th er , a n d in graph (b ) is sh o w n th e re sp o n se p r ed ic tio n s u sin g 1 8 , 2 4 , 3 0 an d 3 6 e le m e n ts . In graph (b ) it is s e e n th a t th ere is in s ig n ific a n t v a r ia tio n in th e so lu tio n . F ro m th e a b o v e tw os c h e m e s , it w as c o n se q u e n tly p o ss ib le to c h o o s e a m esh s iz e o f 18 e le m e n ts .
In F ig u re 8 .5 (a ) an d (b ) s im ila r s c h e m e s w ere fo llo w e d fo r th e e la s to -p la s t ic s o lu t io n . In g rap h (a ) th e c o n v e r g e n c e to w a rd s th e A D I N A so lu tio n is n o ta b le . In g ra p h (b ) , h o w e v e r , th e r e is s lig h t d iv e r g e n c e fro m th e A D I N A so lu tio n - th a t for 1 8 e le m e n ts b e in g c lo se s t w h ile th a t fo r 3 6 e le m e n ts is m o st r e m o te . T h is can b e a ttr ib u ted to th e c h a n g e in a sp ec t ra tio ( le n g th to d e p th ) in a s in g le e le m e n t (1 .6 6 7 v s . 0 .8 3 3 fo r a m e sh s iz e o f 18 a n d 3 6 e le m e n ts r e sp e c tiv e ly ) w h ich in th e p resen t stu d y a d v erse ly a f fe c ts th e p r ed ic tio n o f sh ea r d e fo rm a tio n a t th e e le m e n t le v e l.
8 .4 . E F F E C T O F M A T E R IA L Y IE L D C R IT E R IO N
T h e a im in th is se c t io n is to stu d y th e e f fe c t o f th e y ie ld cr iter io n in ap r o b le m o f d y n a m ic p la stic r esp o n se w ith p r ed o m in a n t in te r a c tio n b e tw e e n b en d in g m o m e n t an d g e n e r a te d a x ia l fo r c e . T h e e f f e c t o f th e y ie ld cr iter io n o n p la stic ity in sh ea r is n o t c o n s id e r e d b eca u se o f th e u n a v a ila b ility o f e x p e r im e n ta l data n e e d e d fo r v er if ic a tio n .
A s m e n tio n e d in C h a p ter 2 , s e c t io n 2 .2 .4 th ere is a u n iq u e ly d e term in a b lep la stic in ter a c tio n r e la tio n b e tw een th e a x ia l fo r c e N a n d b e n d in g m o m e n t M at a n y cro ss s e c t io n in a p articu lar b e a m . F o r d o u b ly sy m m e tr ica l cro ss s e c tio n s , th e p la stic m o m e n t is r ed u ce d by th e p r e s e n c e o f a x ia l fo r c e . F o r a rectan gu lar s e c t io n in p a rticu lar , th e p la stic m o m e n t u n d er th e e f fe c t o f a x ia l fo r c e variesp a ra b o lica lly w ith th e a x ia l fo r c e (M /M p = 1 - ( N /N p ) 2); th is is d isp lay ed inF ig u re 8 .6 (a ) an d id e n tif ie d as " exact r e la tio n " .
In p r o b lem s o f d y n a m ic p la stic ity w h e r e g eo m e tr ic n o n lin e a r ity is co n s id e red , e m p lo y in g th e p a ra b o lic in ter a c tio n r e la tio n b e g e ts co n s id e ra b le c o m p lic a t io n in th e
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TIME (m illiseconds)0 2 4 6 8 10 12
TIME (m illiseconds)0 2 4 6 8 10 12
Figure 8.4. History o f transverse displacement at midspan for various spatial meshes considering only the elastic response. (1 in = 2.54 cm).
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0 50 100 150 200T I M E ( s e c o n d s ) / 0.5a; 10-4 ( s e c o n d s )
0 50 100 150 200T I M E ( s e c o n d s ) / 0.5a; 10”4 (seconds)
12------.------1------.------1------.------1------.------1------.------1------.----- 'tn 0.7 i
o• g o . e -
* * * * * * * ___ 1___ 1___ 1___
AV* v \ Z V / '
/ ^
£ 0 . 5 -Cq3 0.4 -
_ j ^ 0.3 - /
/ — • P r e s e n t S t u d y , 18 E l e m e n t s / — P r e s e n t S t u d y , 12 E l e m e n t s
[ — P r e s e n t S t u d y , 6 E l e m e n t s I • • P r e s e n t S t u d y , 4 E l e m e n t s
/ — A D I N A , 24 E l e m e n t s
I 0 '2 '
3 o.i -
/ — P r e s e n t S t u d y , 36 E l e m e n t s j — P r e s e n t S t u d y , 30 E l e m e n t s / — P r e s e n t S t u d y , 24 E l e m e n t s
/ • • P r e s e n t S t u d y , 18 E l e m e n t s j — A D I N A , 24 E l e m e n t s
- \£ .---- -------- 1------- -------- 1------- -------- 1---- —*------- 1------- ■— S 0.0 -J* 1 1 1 1 1 1 J
Figure 8.5. History o f transverse displacement at midspan for various spatial meshes considering the elasto-plastic response. (1 in = 2.54 cm).
CO
N / N r
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(b)
Figure 8.6. Material yield criteria.
182
1 8 3
so lu tio n p r o c e d u r e , in particu lar in a n a ly tic so lu tio n s . T o s im p lify th e so lu tion p r o c e ss , J o n e s (1 9 6 7 ) p r o p o sed to r e p la c e th e tw o p a ra b o lic a rcs b y a sq u are y ie ld cu rv e w h ich co u ld b e e ith e r c ircu m scr ib in g o r in scrib in g b y th e a c tu a l y ie ld cu rv e , th e fo rm er d e f in e d b y M /M p = ± 1 an d N /N p = ± 1 , an d th e la tter by M /M p = ± 0 .6 1 8 a n d N /N p = ± 0 .6 1 8 , F ig u re 8 .6 (a ) .
In a c o m m u n ic a t io n by S y m o n d s a n d J o n e s (1 9 7 2 ) , th is s im p lif ica tio n w as e m p lo y e d in stu d y in g th e d yn a m ic p la stic r e sp o n se o f c la m p ed b e a m s. T h e m a teria l w as id ea lised as r ig id -p la s t ic w ith a c o r r e c t io n fo r stra in ra te sen s it iv ity b ased on in itia l stra in ra te . In a d d itio n , f in ite d isp la cem en ts w ere a lso co n s id e red .C o m p a r iso n s w ere m a d e w ith sev era l th e o r e t ic a l so lu tio n s , a n d it w as s e e n th a t th e p a ra b o lic in te r a c tio n so lu tio n an d th e m e th o d o f th e "square y ie ld cr iter ion " a g reed q u ite c lo se ly w ith e a c h o th e r , p ro v id ed th a t th e "p lastic strin g" p h a se in th e la tter a p p ro a ch w as c o n s id e r e d o n ly w h en th e tra n sv erse d isp la c e m e n t e x c e e d e d th e b eam d e p th . T h is w as fo u n d to b e v a lid w h e th e r th e r e w as c o r r e c t io n fo r strain rate sen s it iv ity or n o t . It w as ten ta tiv e ly su g g es ted th a t th e c ircu m sc r ib in g an d in scrib in g sq u are y ie ld cr iter ia p rov id ed lo w e r a n d u p p er b o u n d s r e sp e c t iv e ly o n th e m ax im u m d isp la c e m e n ts . T h is su g g estio n w as fu r th er em p h a sised b y L iu a n d J o n e s (1 9 8 8 ) . In th e ir th e o r e t ic a l stu d y o f r ig id -p la s t ic c la m p e d b eam s stru ck tra n sv erse ly b y am a ss , an d p a rticu lar ly fo r th e ca se o f im p a c t at m id sp a n , p r ed ic tio n s o f th e m a x im u m d isp la c e m e n t b ased o n th e c ircu m scr ib in g a n d in scr ib in g y ie ld criteria w e r e fo u n d to b o u n d th o s e p r ed ic ted b y N o n a k a (1 9 6 7 ) b y a p a ra b o lic y ie ld cu rv e . In a c o rre sp o n d in g e x p e r im e n ta l s tu d y , L iu an d J o n e s (1 9 8 7 ) rep o r ted th a t th e ex p e r im e n ta l resu lts a g reed ra th er w e ll w ith p red ic tio n s fro m th e ir a b o v e m en tio n ed stu d y if an in scr ib ed sq u are y ie ld c r ite r io n w ere u sed to g e th e r w ith a d y n a m ic y ie ldstress b a sed o n in itia l stra in ra te . H o w e v e r , r eserv a tio n s w ere m a d e w ith regard toth e b a la n c in g e f fe c t s o f th e la tter tw o p a ra m eters .
In th e a n a ly sis o f th e p r e se n t stu d y , n o t o n ly p la stic d e fo rm a tio n s are c o n s id e r e d , but a lso e la st ic d e fo rm a tio n s - f le x u r a l, sh ear o r a x ia l - th ro u g h o u t th ee n tir e r e sp o n se . C o n s id er th e tw o -d im e n s io n a l an d n o rm a lised stress resu ltan t sp a ceo f X ; = M /M p v s . X 3 = N /N p . F o r a co m b in a tio n o f XJ an d X 3 w h ichco rre sp o n d s to a p o in t o n th e cu rren t y ie ld su r fa c e , th e n e x t in c r e m e n t (d X { , d X 3) in th e stress r esu lta n ts w ill ca u se in c r e m e n ts ( d x { , d x ’ ) in th e co rresp o n d in g lyn o rm a lise d flex u ra l a n d ax ia l stra in resu lta n ts x j = x , / x t p a n d x J = x 3/ x 3p . In th e se e x p r e s s io n s , th e n o rm a lisin g p a ra m eters a re x ^ = M p L 0 /E I andx 3p = N p L 0 /E A . I f th e stra in resu lta n t sp a ce is su p er im p o sed o n th e stressresu lta n t s p a c e , th e tw o d im en sio n a l v e c to r rep resen tin g th e stra in resu ltan tin c r e m e n t w ill b e d irec ted ou tw ard s fr o m th e y ie ld su r fa ce . T h is v e c to r m ay be r e so lv e d in to p la stic an d e la stic c o m p o n e n ts w h ich , b e ca u se o f th e n o rm a lisa tio n , a re d irec ted p e rp en d icu la r ly and ta n g e n tia lly to th e y ie ld su r fa c e r e sp ec tiv e ly . In
184
the present study, the structural material had finite strength in shear - equal to Qp = crybd/2. Consequently, the parabolic yield criterion had the shape of a finite parabolic cylinder, Figure 8.6(b), and the square yield criteria those of two finite square tubes. Other parameters considered in the analysis were rotatory inertia and finite displacements.
Two clamped beam examples are presented here to evaluate the effect of the three yield criteria mentioned above on the history of transverse displacements at midspan. The midspan displacement of the beam discussed in section 7.3.1 - Beam No. I l l - is shown in Figure 8.7(a) and (b). The case in which the beam material was considered strain rate sensitive is shown in Figure 8.7(a). It is seen that, quantitatively and qualitatively, the nearest solution to the experimental result is that of the parabolic yield criterion. A lower amplitude was recorded for the inscribing square solution, and the circumscribing square solution is seen to be unacceptable - its peak displacement standing at only 67% of the experimental value, and in part of the recovery stage the beam was displaced in the negative direction. In Figure 8.7(b) the response is shown for the case of a strain rate insensitive material. In general, response features are similar to those observed in the previous case, except that here the displacements are larger in magnitude with the parabolic yield criterion giving displacements in excess of the experimental result. Also, there was no negative displacement in the 'recovery' phase for the circumscribing square criterion. The response of the beam discussed in section 7.3.2 - Beam No. 95 - is shown in Figure 8.8(a) and (b) for the two respective cases of strain rate sensitive and insensitive beam material. It can be said that, generally, similar observations to those made in the previous example also hold here. For this beam, the peak displacement recorded in the experiment is 5.75 times the beam depth compared to a value of 3.0 in the previous example. The responses shown in Figure 8.7 and 8.8 suggest that aluminium 6061-T6 is a strain rate sensitive material.
Finally, the conclusions that can be made from this investigation are:
1. In an elasto-plastic dynamic (transient) response of clamped beams of rectangular cross section, inscribing and circumscribing square yield criteria do not essentially provide upper and lower bounds on maximum displacements.
2. If it is necessary to use a square yield criterion in order to simplify the analysis, then it is best to make use of the inscribing square yield criterion, whether the material is strain rate sensitive or not.
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Present Study, 82 Elements, S train Rate Sensitive According to In itial S train Rate a t Midspan Z>=6500 s~\ P =4.0,•0 Inscribing SquareCircumscribing Square -N- Exact Relation— Experim ent (after Clark et al. 1065)
I 1 ' 1 1 I..... I ...... I"1 11 ' I i i i i i
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» ■ * ‘ 1 ■ * i J.Present Study, 82 Elements, S train Rate Insensitive, Bending Moment—Axial Force Interaction Based on:•© Inscribing Square •+* Circumscribing Square Exact RelationExperiment (after Clark et al. 1965)
x X x*~~-x — *■— .©...o.. .
i » i i i r
Figure 8.7. History o f transverse displacement at midspan o f Beam Ho. 111. (1 in = 2.54 e m ) .
185
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0.00 0.25TIME (milliseconds)0.50 0.75 1.00 1.25 1.50_L ■ ■ ■ 1 ‘ * . i .... i ... .
P r e s e n t S tu d y , 84 E le m e n ts , S t r a i n R a te S e n s i t i v e 1.0 H A c c o r d in g to I n i t i a l S t r a i n R a te a t M id s p a n ,D=6500 5f‘. P = 4.00,•0 I n s c r ib in g S q u a r eC ir c u m s c r ib in g S q u a r e 0 0 -| E x a c t R e la t i o n— E x p e r im e n t { a f t e r C la rk e t a l . 1965)
0.2 -
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Co0)-ci0
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0.8 -
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P r e s e n t S tu d y , 84 E le m e n ts , S t r a i n R a te I n s e n s i t i v e , B e n d in g M o m e n t—A x ia l F o rc e I n te r a c t io n B a s e d on: •0 I n s c r ib in g S q u a r e -*• C ir c u m s c r ib in g S q u a r e -H- E x a c t R e la t io nE x p e r im e n t ( a f t e r C la rk e t a l . 1865)
' .../toe; 0 - •
3 0.2 -
I ' ' ' ' I250.0 500.0 750.0 1000.0 1250.0 1500.0
TIME (m icro seco n d s)0.0 » i i 11 i i i i i i 11 i i i
0.0 250.0 500.0 750.0 1000.0 1250.0 1500.0TIME (m icro seco n d s)
Figure 8.8. History o f transverse displacement at midspan of Beam Ho. 95. (1 in = 2.54 cm).
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8.5. EFFECT OF MATERIAL ELASTICITY
In problems of dynamic plasticity, the incorporating of elastic deformations in analytic methods of study requires the solution of partial differential equations with moving boundaries; and if finite displacements are included these equations become nonlinear. This apparent difficulty has prompted the development of procedures in which material elasticity is neglected. However, there remained the question of establishing the range of validity of such methods. Two main criteria have been cited in this regard:
1. The duration of dynamic loading being necessarily short compared with the fundamental period of natural elastic vibrations.
2. The energy ratio, defined as the external applied energy divided by the maximum amount of strain energy a structure can absorb in a wholly elastic manner, largely exceeds unity.
There have been different interpretations of the aforementioned definition of the energy ratio, and discrepant numerical estimations of the term "largely exceeds unity". One expression for the energy ratio was that proposed by Seiler, Cotter and Symonds (1956).
where, S is the energy ratio as interpreted by Seiler et al., m the mass per unit length of beam, 1 is half the beam span, V0 the initial velocity, Mp the plastic moment, El the flexural rigidity. The term mlV^/2 represents the initial kinetic energy, and the term M^l/EI represents an upper bound for the elastic strain energy which can be stored in the beam. In a sample problem solved by Seiler et al. of a uniform free beam subjected to impulsive loading, the spatial distribution of initial velocity had the pattern of a half sine wave. It was established that the rigid-plastic solution can be expected to furnish plausible results only for values of S of the order of 10 or more.
The conditions in which it is permissible to disregard elastic deformations are often those in which other effects become influential, such as the increase of yield stress at high rates of strain and the change of deformation mode caused by finite displacements - particularly in the presence of axial constraints.
( 8 . 1 )
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In what follows, some of the validation examples presented in Chapter 7 are employed to shed some light on the earlier acknowledged criteria for the soundness of discounting material elasticity. In the present study, the neglect of material elasticity can be accomplished approximately by increasing the magnitude of Young's modulus of elasticity (E) ten fold; no further increase in magnitude is feasible as this would require using an impracticably small size for the time step. Themagnitude of the shear modulus of elasticity is accordingly increased. Strain rate sensitivity is, where relevant, considered and the associated dynamic yield stress is based upon the initial strain rate magnitude at midspan. Finite displacements and rotatory inertia are also accounted for. The energy ratio definition adopted is that expressed in equation (8.1 ).
In the following examples, the applied pressure load has a very short duration (12 /*s). The histories of the displacements at midspan of the fully clampedbeams, Beam No. I l l and Beam No. 95, are respectively shown in Figure 8.9(a) and (b). For both examples, it was not possible to use the exact parabolic yield criterion relating bending moment and axial force, this inconvenience is brought about by the small size of the integration time step necessitated by the increase inthe value of E by ten fold. Consequently, only the inscribing and circumscribingsquare yield criteria are used. In Figure 8.9(a) it is seen that, for the inscribing square solution, the maximum and permanent displacements at midspan are respectively 82% and 110 % of those of the elasto-plastic solution (incorporating the exact yield criterion). The energy ratio for Beam No. I l l is S - 22. For Beam No. 95, however, the energy ratio is S - 27 and the corresponding displacements were 83% and 97% respectively.
For the solution of the simply supported beam, Beam No. 121, strain ratesensitivity is considered by upgrading the value of the dynamic yield stress at each element in each time step. With the occurrence of only small axial forces, nointeraction with the bending moment plastic capacity was necessary. For this beam, the energy ratio is S - 10. From Figure 8.10, it may be noted that when Young's modulus E is set at ten times its normal value the peak and permanent displacements at midspan are respectively 91% and 121% of the corresponding displacements of the normal elasto-plastic solution.
Next, consideration is made of a case in which the two criteria, mentioned in the beginning of this section, are violated. The simply supported beam example presented in section 7.2.1. of this chapter is subjected to a dynamic pressure loading which is of infinite duration and has a magnitude equal to onlythree-quarters of the static collapse load. The history of transverse displacement at midspan is shown in Figure 8.11. It is seen that there is gross disagreement
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» 1 ■ ■ 1 ■ ■ 1 ■ 1 ■ ■ ■ ‘ 1 ■ ■ ■ * * » ■ ■ ‘ 1 ■ ■ ■ ■ 1 ■ ■ . P r e s e n t S tu d y , 82 E le m e n ts , S t r a i n R a te S e n s i t i v e
A c c o r d in g to I n i t i a l S t r a i n R a te a t M id s p a n 27=8500 F=4.
f f ' i ' | ' i ' i ............................... | i i i i ................................. i i i i ! i i
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0.9 -P r e s e n t S tu d y , 84 E le m e n ts , S t r a i n R a te S e n s i t i v e
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0.2 - I »0 E T en T im e s L a r g e r , I n s c r ib in g S q u a r eI -*• E T en T im e s L a r g e r , C ir c u m s c r ib in g S q u a r e0.1 - 1 E N o r m a l, E x a c t R e la t io n (m+n*= lT
1 E x p e r im e n t (a f t e r C la rk e t a l . 196o)0.0 J
0.0 250.0 500.0 750.0 1000.0 1250.0 1500.0TIME (m icro seco n d s)
Figure 8.9. History of transverse displacement at midspan of fully damped beams, (a) Beam No. 111.(b) Beam No. 95. (1 in = 2.54 cm).
189
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P r e s e n t S t u d y , 46 E l e m e n t s:S t r a i n R a t e S e n s i t i v e (D=6500 s'1, P = 4), S t r a i n R a t e a t e a c h E l e m e n t U p g r a d e d e a c h T im e S t e p .
..a-—^ ^ __^ — t,— a-....,*';—*
•a »e Ten Tim es Larger E N orm alE xperim en t (a fte r Clark e t al. 1965) T
Figure 8.10. History o f transverse displacement at midspan o f a simply supported beam. Beam No. 121. (1 in = 2.54 cm).
Figure 8.11. History of transverse displacementat midspan of a simply supported beam.(1 in = 2.54 cm).
191
between the solution in which E is increased ten fold, and those corresponding to the normal elasto-plastic solution of the present study and the finite element code ADINA.
Founding an opinion on the results of the above examples leads one to conclude that, in problems of dynamic plasticity of beams, the neglecting of material elasticity is credible in the following cases:
1. For a simply supported beam loaded by a very short duration pulse, and having an energy ratio S > 1 0 .
2. For a fully clamped beam subjected to a very short duration pulse and having an energy ratio S > 2 2 .
3. For any beam, when the duration of the applied load is short compared with the fundamental period of elastic vibrations and the associated energy is much higher than the maximum elastic strain energy.
8.6. EFFECT OF PULSE SHAPE
Transient dynamic loads generally constitute a single pulse. In most practical situations, this single pulse is characterised by a rapid rise of pressure and a subsequent gradual decay.
The outcome of various investigations into the effect of pulse shape on the dynamic plastic response of beams is outlined in Chapter 2, section 2.4.1. In most of the examinations previously carried out, beams have been idealised as rigid-plastic, and in few as a single degree of freedom elasto-plastic system.
In the present work, an attempt is being made to study the effect of pulse shape (for pulses of very short duration) on the elasto-plastic dynamic response of beams. The simply supported beam, Beam No. 121 (see section 7.2.2) is used as a means towards this end. It is exposed to high pressure load at the central portion of the span. The present analysis involved the following responseparameters : transverse shear force and deformations, rotatory inertia, finite displacements and strain rate sensitivity (dynamic yield stress upgraded at each time step and for each element). Variations in the shape of pulses employed are shown in Figure 8.12(a), (b) and (c). They include rectangular pulses, and an assortment of triangular pulses - namely, step rise-linear decay, linear rise-linear decay and linear rise-step decay. Peak forces, for all shapes, are normalised with respect to
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8 12 16 20 TIME (jis)
i
Figure 8.12. Pulse shapes considered.
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that of the 2 fis linear rise-1 0 fts linear decay triangular pulse, used originally in section 7.2.2.
Histories of transverse displacements at midspan, for three different pulses of similar total impulse and duration (4 /*s) - Figure 8.12(a) - are shown in Figure 8.13(a). It is observed that, the linear rise-linear decay and rectangular pulses give comparable estimates of the maximum and permanent displacements, despite the former pulse having a peak force twice that of the rectangular one. The step rise triangular pulse furnished an approximately 1 2 % smaller maximum displacement.Next, pulses of similar shape and impulse - to the above mentioned pulses - areconsidered, but with durations five times longer (at 20 /zs); the results are displayed in Figure 8.13(b). It is noted that, almost all the predictions, including theexperimental, are in close agreement, despite differences in the time of occurrence and magnitude of the peak pressure. Comparing the results in Figures 8.13(a) and (b), it can be said that (except for the 4 [is duration step rise triangular pulse):
1. Peak displacements are substantially identical.
2. Permanent displacements vary by less than 10% from the lowest valuerecorded.
3. For similar impulse and shape of pulse, the shorter the duration of the pulse,the larger is the permanent displacement produced.
Next, two different triangular pulses are considered; they have similar impulse,peak force and duration, but contrast in the time at which the peak force occurs.One pulse has the configuration of step rise-linear decay, the other of linear rise-step decay. Pulses of 4 /ts and 20 fis are examined simultaneously, their respective normalised peak forces being 3 and 0.6, Figure 8.12(b). Histories of transverse displacement at midspan (with strain rate sensitivity maintained) are exhibited in Figure 8.14(a). It is observed that, for the 4 ps duration pulses, the difference between the two pulse geometries is remarkable, with the linear rise pulse producing 29% and 46% larger peak and permanent displacements respectively than for the step rise pulse. This finding is in contrast with the conjecture ofStronge (1982), presented in section 2.4.1 of Chapter 2, with regard to the response of rigid-plastic beams : between two triangular pulses, the most compact around time nil produces the larger midspan displacement in a simply supported beam. However, as the duration of the two pulses is prolonged to 20 ps, the difference between the corresponding responses is very considerably reduced. Itmay be observed from Figure 8.13(a) and (b) that the response for the linear rise-linear decay triangular pulse roughly averages the responses shown in Figure
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. S t r a i n R a t e S e n s i t i v e (D-6500 s~\ P = 4) S t r a i n R a t e a t e a c h E l e m e n t U p g r a d e d e a c h
•0 S t e p R i s e T r i a n g u l a r P u l s e , 20f i s D u r a t i o ni--- 1--- ---- 1--- ---- 1— --- 1--- ---- 1--- -—
Figure 8.13. History of transverse displacement at midspan for the various pulse shapes illustrated inFigure 8.12(a). (1 in = 2.54 em).
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E x p e r im e n t ( a f t e r C la rk e t a l . 1965)-*• L in e a r R i s e , 20f i s D u r a t i o n _ .Step R is e , 20p s D u r a t io n r T ria n g u la r •€>■ L in e a r R i s e , 4 f i s D u r a t io n c n,./ooc Sfep R is e , 4/xs D u r a t io n ■ /u-t-ses
T0.0 1000.0T--- 1--- 1 --- 1--- 1--- 1---1—
2000.0 3000.0 4000.0 5000.0 6000.0 TIME (m icrosecon ds)
Figure 8.14. History of transverse displacements at midspan for the various pulse shapes illustrated inFigure 8.12 (b). (1 in = 2.54 cm).
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8.14(a) for the linear rise-step decay and step rise-linear decay pulses of the same total impulse and duration. The cases mentioned in Figure 8.14(a) were reconsidered, but with strain rate sensitivity abandoned. From the results shown in Figure 8.14(b), one may surmise that the same general observations made earlier also apply here, except for an enhanced difference in displacement.
The final examination undertaken is of a group of triangular pulses that have equal impulse and are of the linear rise-linear decay form. The rise time is set at a constant value of 2 fts, and the subsequent decay time varied from 2, 6, 10, 14 to 18 j/s, Figure 8.12(c). Generally, this temporal form of pulse provides asuitable idealisation for blast loading. Contemplating the history of transverse displacement at midspan shown in Figure 8.15, it is noted that there is a strikingly close fit of all the responses, including the experimental one, up to the time of the first peak displacement. Subsequently, slight variations are observed in permanent displacements. The responses due to the 16 and 20 fts pulses are indistinguishable throughout the considered history.
From the preceding inspections of the elasto-plastic behaviour of simply supported beams exposed to very short transient pulse loads of equal impulse, the following remarks may be registered:
1. Abandoning strain rate sensitivity in a susceptible material enhances the differences in displacements for various pulse shapes.
2. For equal peak forces, a pulse of rectangular form delivers a larger midspan displacement than a triangular pulse of linear rise-linear decay form.
3. Amongst the three different forms of triangular pulse with equal peak force - namely, step rise-linear decay, linear rise-linear decay and linear rise-step decay - the latter produces the largest permanent displacement.
4. For similar peak force and duration, a triangular pulse of linear rise-linear decay form averages the responses due to triangular pulses of linear rise-step decay and step rise-linear decay type.
5. Triangular pulses of linear rise-linear decay form with the rise time set at a shorter duration than the decay time, provide a proper representation of a pressure pulse generated by blast type loading.
6 . For pulses of equal total impulse and of duration up to 10% of the fundamental period of the beam, the effect of the pulse shape on the response
DISP
LACE
MENT
AT M
ID SP
AN (
inche
s)
TIME (m icroseconds)Figure 8.15. History o f transverse displacement at midspan for the various pulse shapes illustrated in the inset (also illustrated in Figure 8.12(c)).(1 in = 2.54 cm).
is marked : for a duration between 10% and 50% of the fundamental period, the effect of pulse shape on the response is much less significant.
For pulses of similar shape, beam displacement is maximised by applying the largest pressure in the shortest time.
199CHAPTER 9
CLOSURE
9.1. REVIEW AND MAIN CONCLUSIONS
In this section, the aim is to provide a concise review of the studies undertaken in this work and to point up the main conclusions.
Chapter 1 furnishes a brief description of the general field of the present study and a statement of the main theme and objectives.
The parameters that most affect the dynamic elasto-plastic response of beams are identified and their effects reviewed in Chapter 2. Such an approach formed an important initial step; it introduced the author to the general field of dynamic plasticity and helped in focussing on and ultimately identifying the parameters that needed to be incorporated into the proposed mathematical tool. In the domain of nonlinear transient response, the following parameters have a significant effect, but are not usually incorporated into the simplest models : strain rate sensitivity, axial and transverse shear forces in the plastic yield criterion, transverse shear deformation, rotatory inertia, finite displacements and the shape and duration of the applied load pulse.
The prominent methods of mathematical analysis are examined in Chapter 3 with regard to their adequacy, extent of application and practical utility.Mathematical models of analysis are classified according to the spatial distribution of the parameters describing the system properties, the two major classes being those associated with continuum and discrete models. Most difficulty in obtaining a solution to a continuum problem resides in the solving of partial differentialequations and the satisfying of appropriate boundary conditions, particularly in two and three spatial dimensions. In engineering applications, where an approximate solution is usually acceptable, this difficulty can be circumvented through discretisation of the spatial domain of the problem. Discrete methods can be put into two major classes : the first consists of lumping the system properties atdiscrete points in the domain; and the second is based on the expansion of thesolution in a finite series of given functions, either over the whole structure or
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over finite elements of the structure. One of the models belonging to the class having lumped properties is that of continuous mass - lumped flexibility. This model consists of a series of rigid panels which retain the continuous mass distribution, but are linked by flexible joints. The deformations at any flexible joint represent those of the continuum over a length tributary to that joint.
From the knowledge acquired through the work presented in Chapters 2 and 3, it was decided that a discrete model of the continuous mass - lumped flexibility type could provide a simple but accurate representation of the dynamic transient elasto-plastic response of beams.
In Chapter 4, a discrete beam element is proposed and a detailed account is made of the kinetic-kinematic relations at both element and structural levels. The discrete element developed is a two-dimensional cantilever beam element. It is prismatic and rigid with its mass distributed uniformly throughout the span; all deformations are assumed lumped at the free end. No restrictions are made on the size of the rigid body displacements of the element, but element deformations are assumed small. Inertial forces associated with the rigid body displacements are all accounted for; these may be referred to as rotatory inertia, transverse inertia and axial inertia. For planar motion, the three independent deformations are flexural rotation, transverse shear deformation and axial deformation. Each type of deformation is represented jointly by a spring and a friction dissipator - the former simulates elastic behaviour and the latter plastic behaviour. The lumped deformations are required to simulate the material characteristics of homogeneity, isotropic linear elasticity and either perfect plasticity or viscoplasticity.
The constitutive relations incorporated in the developed beam element are presented in Chapter 5. Initially, a brief description of the elastic relations is made. This is followed by the introduction of constitutive laws for perfectplasticity and viscoplasticity (strain rate sensitivity); a brief mention is also made of strain hardening. The normal or associated flow rule has been adopted for all forms of plastic behaviour, and yield of the structural material is forecast by a Predictor-Corrector procedure. In perfect plasticity, the constitutive models treated are those where the yield may be governed by one or two internal forces. In the latter case, the yield curve has the shape of a plane closed curve in the two dimensional space of the associated internal forces. Computationally, thedecomposition procedure of the strain increment vector into elastic and plastic parts is based on the local piecewise linearisation of the yield curve and by regarding this vector as a diameter of a circle with the elastic and plastic parts represented by the half-circle chords. This procedure is performed whenever necessary along the time-marching transient response, and may take place in any of the four
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quadrants of the yield curve.
The maximum stress which can be carried by a solid is almost invariably associated with the rate at which this is applied. In this study, the effect of strain on the material yielding is determined from the elastic strain rate vectorcorresponding with the current stress state, and strain sensitive materials are assumed not to strain harden. During unloading, a decreasing strain rate causes the yield surface to contract. Analytically, there exists a difficulty in interpreting this contraction process since a stress point may lie between two locations of theyield surface that are one time step apart. A scheme is proposed in the presentstudy to solve this problem for yield curves dominated by single internal force; this scheme proved to be simple and capable of producing good numerical results. Furthermore, it has enabled account to be taken of the variation of the dynamic yield stress at each element of the discrete structure for each time step along the time-advancing numerical solution.
The spatial discretisation of a continuum reduces the equations of motion from a set of partial differential equations into a set of ordinary differential equations with respect to time. Upon the employment of a suitable temporal operator, these latter equations may be turned into a fully discrete set of simultaneous linear algebraic equations. The temporal operator employed in this study, together with the developed computer program, are presented in Chapter 6. The chosenoperator is that of a central finite difference time-marching scheme, which is a direct explicit integration procedure. Explicit procedures are often best suited for wave propagation problems in which there is a high frequency content. Both algorithmic logic and structure of such procedures are simple, providing an excellent medium through which complex nonlinearity can be treated - be it constitutive, geometric or boundary type - with virtually no additional cost compared with the linear case. The absence of any iteration procedure associated with explicit operators brings significant simplification in use, since the only decision required is that related to the size of the integration time step. The time step size must be smaller than the smallest traversal time of a stress wave across an element, which leads to the demand for a large number of time steps for adequately representing the complete transient solution. However, for problems of concern in this study, the time step size is of necessity small, whether the operator is of an explicit or implicit nature. This arises from the need to model accurately applied load pulses of short duration.
In Chapter 7, a number of examples were presented with two objectives in mind. Firstly, to validate and establish the limits of the proposed mathematical model; and secondly, to contribute further insight into the nonlinear behaviour of
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beams under dynamic loading. Generally, there are two types of dynamic response. Wave propagation is a type of response in which the behaviour at the wavefront is of engineering importance, such as in shock response from explosions or impact. Inertia-dominated response is that in which low frequencies preponderate. In thepresent work, the type of problems solved lie between these two. There areexamples in which wave propagation is important only in the early stage of theresponse, where the applied load is of high intensity and of a duration typically measured in microseconds; also, there are examples of inertia-dominated response in which the applied load is of low intensity and of indefinite duration. While the adopted explicit integration procedure is most economical for responses of short duration, for those of long duration it is often considered that an implicit scheme would be more advantageous. Although this may be substantiated for linear systems, the advantages of the implicit scheme for long duration are very much reduced when nonlinear systems are considered. For both types of examples, comparisons were made with other experimental and/or numerical studies. It canbe said that the solutions obtained using the proposed mathematical model compared very well with the above studies and confirmed the suitability of the model for the prediction of the nonlinear elasto-plastic response of beams under dynamic transient loading.
The solutions of the examples were further considered with respect to the evolution of different response variables. As a result, several conclusions may be stated:
- For a yield surface dominated by a single internal force, the effect of strain rate sensitivity is best predicted by modelling the dynamic yield stress as a space and time dependent variable which is evaluated within each element for each time step throughout the response.
Strain rate sensitivity proved significant for Aluminium alloy (6061-T6), even though it is generally considered as strain rate insensitive. The solution closest to the results obtained by experiments is that in which both space and time dependency of the yield stress are considered.
Flexural wave propagation through the beam material gives rise to the alternating formation and cessation of plastic hinges in flexure.
- Flexural wave propagation significantly influences the deformation patternduring the initial phase of the response. In this phase, higher modes may dominate the fundamental. These modes may well lead to the crash of thenumerical analysis if the structure is not discretised sufficiently finely.
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For fully clamped beams subjected to a short pulse load applied at the beam centre, plastic hinges cease to exist there at a very early stage in the response. Instead, hinges formed at and around the quarter points of the span and at the clamped ends seem to have taken up most of the energy dissipated in plastic deformation. This is due to the reduction in the plastic moment of resistance of the cross section as a result of the build-up, with time, of the axial force. Consequently, as flexural waves propagate throughthe beam span, more energy is dissipated in plastic deformations than in a beam devoid of such a build-up.
- There is a complex and diffuse pattern of the distribution of plastic activation in the geometrically nonlinear response of fully clamped beams. This is attributable to three factors. Firstly, the accounting for flexural wave propagation in the analysis; secondly, the generation and temporal variation of axial force which results in the oscillation of the plastic bending moment capacity; and thirdly, the characteristic of clamped ends in reflecting waves negatively.
- In general, for fully clamped beams, the following may be stated:
(a) Even with a midspan displacement in excess of twice the beam depth, a"plastic string" state exists only transiently.
(b) After the axial waves reach the beam supports, the magnitude of theaxial force becomes (statistically) uniform throughout the beam in theremaining response.
A parametric study is presented in Chapter 8. This study is conducted, employing the developed mathematical model as the analytic tool, to explore the degree of influence of some of the many criteria that have a bearing on both modelling and solution in problems of dynamic plasticity. In particular, a study is made of the effects on the dynamic response of the size of temporal and spatial meshes, material yield criteria and material elasticity, and shape of applied pulse. Detailed conclusions with respect to each parameter have been given in Chapter 8 . However, certain essential points may be distilled in the following manner.
In problems of dynamic plasticity, the size of the time step employed in explicit time-marching schemes determines whether a numerical solution will crash or run to completion. However, in cases where snap-buckling cannot be ruled out completely, it transpires that the existing criteria for determining the critical size of the time step are not adequate for establishing conformity of
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the results. In such cases, the sizes of both temporal and spatial meshes play significant roles in furnishing consistent solutions.
Inscribing and circumscribing square yield criteria, representing the effect of axial force on the plastic bending moment of the cross section, do not essentially provide upper and lower bounds on maximum displacements in beams. However, if it is necessary to use a square yield criterion to simplify the analysis, then the closer correlation with experimental results is provided by the inscribing version; this is true whether the material is strain rate sensitive or not.
- The neglect of material elasticity in problems of dynamic plasticity becomescredible for beams loaded with a very short duration pulse provided that the energy ratio S > 1 0 for simply supported beams, or S > 2 2 for beams with fully clamped supports.
- Let the maximum displacements induced in a beam by two different loadpulses, each of short duration and exhibiting the same total impulse, be A1
and A 2 respectively. Let these values be obtained for a beam of strainsensitive material in which both space and time dependency of the yield stress are considered. Then the absolute value | A, - A 2 \ of the difference inmaximum displacements is increased by the neglect of strain rate sensitivity.
The effect of the pulse shape on the response is marked for pulses of equal total impulse and of duration up to 10% of the fundamental period of thebeam. For a duration between 10% and 50%, however, the effect is muchless significant.
The essence of this work, as mentioned in Chapter 1, is the provision of a mathematical tool that is simple and predictive for the analysis of the dynamic nonlinear elasto-plastic response of beams. This main aim has been accomplished, and the considerable aid to understanding and interpretation of complex structuraland constitutive phenomena provided by the inherent simplicity of the model hasbeen demonstrated.
9.2. INDICATIONS FOR FURTHER RESEARCH
In the course of the present study, some aspects have emerged which warrant further research. These are, briefly,
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A consistent constitutive model is required to represent the influence of strain rates on a yield surface governed by two internal forces, together with an implementation of this model through numerical analysis.
Experimental studies are needed to determine the interactive effect, if any, between strain hardening and strain rate sensitivity.
Experimental techniques should be developed for the detection and monitoring of intense and diffuse plastic activity in structures subjected to dynamic transient load to aid the verification of numerical studies.
Models are required for structural members of non-uniform cross sections.
An expansion is envisaged of the capability of the mathematical model, developed in this work, to analyse frame structures of two and three spatial dimensions.
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REFERENCES AND SUBJECT BIBLIOGRAPHY
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