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Nonlinear rigid-plastic analysis of stiffened plates under blast loads Schubak, Robert Brian 1991

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NONLINEAR RIGID-PLASTIC  ANALYSIS  STIFFENED PLATES U N D E R BLAST  LOADS  by ROBERT BRIAN SCHUBAK B . A . S c , The University of British Columbia, 1984 M . A . S c , The University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  April 1991 © Robert Brian Schubak, 1991.  OF  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and  study. I further agree that permission for extensive  copying of this thesis for scholarly purposes may department or  by  his or  her  representatives.  be  granted by the head of  It is understood  that  copying  publication of this thesis for financial gain shall not be allowed without my permission.  Department The University of British Columbia Vancouver, Canada Date  DE-6  (2/88)  my or  written  Abstract T h e l a r g e d u c t i l e d e f o r m a t i o n r e s p o n s e of s t i f f e n e d p l a t e s s u b j e c t e d t o b l a s t l o a d s is i n v e s t i g a t e d a n d s i m p l i f i e d m e t h o d s of a n a l y s i s o f s u c h r e s p o n s e are d e v e l o p e d . S i m p l i f i c a t i o n is d e r i v e d f r o m m o d e l l i n g s t i f f e n e d p l a t e s as s i n g l y s y m m e t r i c b e a m s or as g r i l l a g e s t h e r e o f . T h e s e b e a m s are f u r t h e r a s s u m e d t o b e h a v e i n a r i g i d , p e r f e c t l y p l a s t i c m a n n e r a n d to have piecewise linear b e n d i n g moment-axial force c a p a c i t y i n t e r a c t i o n relations, o t h e r w i s e k n o w n as y i e l d c u r v e s . A b l a s t l o a d e d , one-way s t i f f e n e d p l a t e is m o d e l l e d as a s i n g l y s y m m e t r i c b e a m c o m p r i s e d o f o n e stiffener a n d i t s t r i b u t a r y p l a t i n g , a n d s u b j e c t e d t o a u n i f o r m l y d i s t r i b u t e d l i n e l o a d . F o r a s t i f f e n e d p l a t e h a v i n g edges f u l l y r e s t r a i n e d a g a i n s t r o t a t i o n s a n d t r a n s l a t i o n s , b o t h t r a n s v e r s e a n d i n - p l a n e , use of t h e p i e c e w i s e l i n e a r y i e l d c u r v e d i v i d e s t h e r e s p o n s e of t h e b e a m m o d e l i n t o t w o d i s t i n c t p h a s e s : a n i n i t i a l s m a l l d i s p l a c e m e n t p h a s e w h e r e i n t h e b e a m r e s p o n d s as a p l a s t i c h i n g e m e c h a n i s m , a n d a f i n a l l a r g e d i s p l a c e m e n t p h a s e w h e r e i n t h e b e a m r e s p o n d s as a p l a s t i c s t r i n g . If t h e l i n e l o a d is r e s t r i c t e d t o b e a b l a s t - t y p e p u l s e , s u c h r e s p o n s e is g o v e r n e d b y l i n e a r d i f f e r e n t i a l e q u a t i o n s a n d so m a y be s o l v e d i n c l o s e d f o r m . E x a m p l e s of a one-way s t i f f e n e d p l a t e s u b j e c t e d t o v a r i o u s blastt y p e pulses demonstrate good agreement between the present rigid-plastic f o r m u l a t i o n and elastic-plastic b e a m finite element and finite strip solutions. T h e r e s p o n s e of a one-way s t i f f e n e d p l a t e is a l t e r n a t i v e l y a n a l y s e d b y a p p r o x i m a t i n g i t as a s e q u e n c e of i n s t a n t a n e o u s m o d e r e s p o n s e s . A n i n s t a n t a n e o u s m o d e is a n a l o g o u s t o a n o r m a l m o d e o f l i n e a r v i b r a t i o n , b u t b e c a u s e of s y s t e m n o n l i n e a r i t y e x i s t s for o n l y t h e i n s t a n t a n d d e f o r m e d c o n f i g u r a t i o n c o n s i d e r e d . T h e i n s t a n t a n e o u s m o d e shapes are  determined by an extremum principle which maximizes the rate of change of the stiffened plate's kinetic energy. T h i s approximate rigid-plastic response is not solved i n closed form but rather by a semi-analytical time-stepping algorithm. Instantaneous mode solutions compare very well w i t h the closed-form results. T h e instantaneous mode analysis is extended to the case of two-way stiffened plates, which are modelled by grillages of singly symmetric beams. For two examples of blast loaded two-way stiffened plates, instantaneous mode solutions are compared to results from super finite element analyses.  In one of these examples the comparison between  analyses is extremely good; i n the other, although the magnitudes of displacement response differ between the analyses, the predicted durations and mechanisms of response are i n agreement. Incomplete fixity of a stiffened plate's edges is accounted for i n the beam and grillage models by way of rigid-plastic links connecting the beams to their rigid supports. Like the beams, these links are assumed to have piecewise linear yield curves, but w i t h reduced bending moment and axial force capacities. T h e instantaneous mode solution is modified accordingly, and its results again compare well w i t h those of beam finite element analyses. Modifications to the closed-form and instantaneous mode solutions to account for strain rate sensitivity of the panel material are presented. In the closed-form solution, such modification takes the form of an effective dynamic yield stress to be used throughout the rigid-plastic analysis. In the time-stepping instantaneous mode solution, a dynamic yield stress is calculated at each time step and used w i t h i n that time step only. W i t h these modifications i n place, the responses of rate-sensitive one-way stiffened plates predicted by the present analyses once again compare well w i t h finite element and finite strip solutions.  iii  Table of Contents  Abstract  ii  List of Tables  ix  List of Figures  x  Notation  xv  Acknowledgement  xviii  1 Introduction  1  1.1  Blast Loads  1  1.2  Structural Response  2  1.2.1  Experimental Studies  4  1.2.2  Methods of Analysis  5  1.2.3  Rigid-Plastic Analysis  6  1.2.4  Mode Approximations  9  1.3  Purpose and Scope of the Present Study  2 Beams and One-Way Stiffened Plates  10 11  2.1  Introduction  11  2.2  Interaction Relation—Stress Resultants  14  2.3  Hinge Mechanism Response  19  iv  2.3.1  Travelling Hinge Mechanism  19  2.3.2  Midspan Hinge Mechanism  23  2.4  String Response  24  2.5  Admissibility of the Hinge Mechanisms  26  2.6  Example Applications  30  2.6.1  Rectangular Pulses  30  2.6.2  H O B Tests  34  2.6.3  Isoresponse Curves  43  2.6.4  Equivalent Rectangular Pulses  47  3 Instantaneous Mode Response 3.1  3.2  3.3  53  Introduction  53  3.1.1  Mode Approximation Technique  54  3.1.2  Instantaneous Mode Approximations  55  Instantaneous Mode Solution for Beams  57  3.2.1  Bending Response  57  3.2.2  String Response  61  3.2.3  Solution Algorithm . . .  62  %  Examples  66  3.3.1  Rectangular Pulses  66  3.3.2  H O B #315 Test  70  3.3.3  Sensitivity to the Time Interval At  70  4 Two-Way Stiffened Plates  76  4.1  Introduction  76  4.2  Problem Formulation  79  4.3  Instantaneous Mode Solution  81 v  4.4  4.3.1  Selection of the Mode Shape  82  4.3.2  Application of the Extremum Principle  86  4.3.3  Solution Algorithm: RIPTAB2  89  Examples  94  4.4.1  DRES1B Panel  94  4.4.2  Square Stiffened Plate  100  5 Partial End Fixity  105  5.1  Introduction  5.2  Modifications to RIPTAB2  108  5.2.1  Kinematics  108  5.2.2  Interaction Relations  110  5.2.3  Integrating the End Slippage Rates  114  5.3  . .  105  Examples  116  5.3.1  Rectangular Pulses  116  5.3.2  H O B #327  126  6 Strain Rate Effects  130  6.1  Introduction  130  6.2  Rate-Sensitive Section Behaviour  131  6.3  Rate-Sensitive Hinge Behaviour  137  6.4  Modifications to RIP T A B 2  139  6.5  Modifications to R I P T A B  140  6.5.1  Impulse-Momentum Method  142  6.5.2  Symonds-Jones. Method  144  6.6  Example Applications  146  6.6.1  147  H O B Tests vi  6.6.2 7  Event MISTY PICTURE  154  Conclusion  160  7.1  Summary  160  7.2  Validity and Limitations of the Analyses  164  7.2.1  Problem Geometry  164  7.2.2  Material Properties  165  7.2.3  Beam Response  167  Bibliography  170  A Associated Flow Rule  176  B Response to Ideal Impulses  179  B . l Hinge Mechanism Response B.2 String Response  179 181  B. 3 Dynamic Admissibility  183  C Response to General Blast-Type Pulses C. l  Linear Representation of the Pulse  C.2 Hinge Mechanism Response  187 188  C.2.1  Travelling Hinge Mechanism  188  C.2.2  Midspan Hinge Mechanism  189  C. 3 String Response  190  D Convergence of Impulsive Responses D. l  187  Convexity Condition of Plasticity  191 191  D.2 Convergence of Solutions  192  vii  E  Lee's Extremum Principle  194  E.l  Derivation of the Extremum Principle  194  E.2  Properties of the Extrema  196  viii  List of Tables  5.1  Sequence of occurrence of the stress states for a beam with partial end fixity. 114  5.2  Stress resultants and end slippage coefficients for a beam with partial end fixity  B.l  115  Permanent midspan displacement of a singly symmetric beam subjected to an ideal impulse  183  ix  List of Figures  1.1  Free-field blast wave pressure as a function of time at a particular distance f r o m the blast source  3  1.2  T y p i c a l blast-type pulse  3  1.3  U n i a x i a l stress-strain curve for a rigid, perfectly plastic material  7  2.1  One-way stiffened plate subjected to uniformly distributed pressure pulse.  12  2.2  F u l l y clamped beam subjected to uniformly distributed pulse loading. . .  12  2.3  Free body diagram of a segment of the deformed beam  14  2.4  Y i e l d curve and associated flow rule for a singly symmetric I-beam.  2.5  Linear yield curve approximation  2.6  A s y m m e t r i c sandwich beam section corresponding to the linear interaction  . . .  15 16  relation  18  2.7  Travelling hinge mechanism: (a) displacement field; (b) velocity field. . .  19  2.8  M i d s p a n hinge mechanism: (a) displacement field; (b) velocity field. . . .  23  2.9  D R E S stiffened plate geometry  31  2.10 Rectangular pressure pulse  31  2.11 M i d s p a n displacement response of the D R E S panel to rectangular pulses of 2 msec duration  33  2.12 Measured pressure pulse and approximations for H O B #315  35  2.13 M i d s p a n displacement response of the D R E S panel subjected to the H O B #315 pressure pulse.  36  x  2.14 Measured pressures and approximation for H O B #327  38  2.15 M i d s p a n displacement response of the D R E S panel subjected to the H O B #327 pressure pulse  39  2.16 Measured pressures and approximation for H O B #338  41  2.17 M i d s p a n displacement response of the D R E S M panel subjected to the H O B #338 pressure pulse  42  2.18 Rectangular and triangular pressure pulses  44  2.19 Isoresponse curves for the D R E S panel subjected to rectangular pressure pulses. . . . . . . . . . .  45  2.20 Isoresponse curves for the D R E S panel subjected to triangular a n d rectangular pressure pulses.  46  2.21 Trilinear and equivalent rectangular representations of the H O B #315 pulse. 48 2.22 M i d s p a n displacement response of the D R E S panel to the trilinear and equivalent rectangular representations of the H O B #315 pulse  49  2.23 Isoresponse curves for the D R E S panel subjected to triangular and "equivalent" rectangular pressure pulses 3.1  51  M i d s p a n displacement response of the D R E S panel under rectangular pulses, as calculated by the I M S w i t h hinge mechanism string response, the I M S w i t h a cosine mode string response, and R I P T A B  67  3.2  Position of the travelling hinges under rectangular pulses  68  3.3  M i d s p a n displacement response of the D R E S panel under H O B #315 pulse, as calculated by the I M S w i t h hinge mechanism string response,  3.4  the I M S w i t h a cosine mode string response, and R I P T A B  71  Position of the travelling hinges under H O B #315 pulse  72  xi  3.5  Midspan displacement response of the DRES panel under HOB #315 pulse, as calculated by the IMS with hinge mechanism string response, for different time steps At  3.6  74  Position of the travelling hinges under HOB #315 pulse for different time steps At  75  4.1  Two-way stiffened plate subjected to a uniformly distributed pressure pulse. 77  4.2  Effective beam sections and distribution of the masses in the grillage model of a stiffened plate  80  4.3  Travelling hinge-line mechanism for a beam grillage  83  4.4  DRES IB stiffened plate geometry  95  4.5  Displacement response of the DRES1B panel under the HOB #315 pressure pulse  96  4.6  Displacement profiles of the longitudinal stiffener ABCD  97  4.7  Displacement profiles of the transverse stiffeners BE and CF  98  4.8  Square stiffened plate  101  4.9  Displacement response of the square stiffened plate  102  4.10 Maximum displacement profiles of the square panel's stiffeners  103  5.1  Partially clamped beam subjected to a uniformly distributed load pulse. . 107  5.2  Travelling hinge response of a beam with partial end fixity  109  5.3  Yield curves for a beam and its support connections  Ill  5.4  Linearized three-parameter yield curves  Ill  5.5  Four cases of the linear yield curve approximation  5.6  Midspan displacement response of the rotationally free, axially restrained DRES panel under rectangular pulses of 2 msec duration  xii  112 117  5.7  Travelling hinge position in the rotationally free, axially restrained DRES panel under rectangular pulses of 2 msec duration  5.8 Yield curves for the cases A = ±0.538, p. = 0 and v = 1  118 119  5.9 Variation of the permanent midpanel displacement of the DRES panel with end fixities p and v: loaded from the plating side (A = 0.538) by a rectangular pulse with p = 129 psi m  121  5.10 Variation of the permanent midpanel displacement of the DRES panel with end fixities p and v. loaded from the plating side (A = 0.538) by a rectangular pulse with p = 258 psi. m  122  5.11 Variation of the permanent midpanel displacement of the DRES panel with end fixities p, and v. loaded from the stiffener side (A = —0.538) by a rectangular pulse with p = 129 psi m  123  5.12 Midspan displacement response of the simply supported DRES panel subjected to the HOB #327 pressure pulse  127  5.13 In-plane end slippage of the simply supported DRES panel subjected to the HOB #327 pressure pulse  128  6.1 Yield stress-strain rate curves for mild steel 6.2  131  Strain rate and stress distributions for a fully plastic sandwich section in hogging near the supports during (a) bending response, (b) string response. 132  6.3  Strain rate and stress distributions for a fully plastic sandwich section in sagging near midspan during (a) bending response, (b) string response. . 135  6.4 Plastic hinge slip-line field for a rectangular section  137  6.5  138  Plastic hinge slip-line field for an unsymmetric sandwich section  6.6 Typical dynamic response of simple rate-sensitive structures subjected to blast loads  141  xiii  6.7  Midspan displacement response of the rate-sensitive DRES panel subjected to the HOB #315 pressure pulse  6.8  148  Yield stress versus midspan displacement of the rate-sensitive DRES panel subjected to the HOB #315 pressure pulse  6.9  150  Midspan displacement response of the rate-sensitive DRES panel subjected to the HOB #315 pressure pulse  151  6.10 Midspan displacement response of the rate-sensitive DRESM panel subjected to the HOB #338 pressure pulse  153  6.11 Measured pressure pulse and triangular approximation for event MISTY PICTURE  155  6.12 Measured pressure pulse and equivalent rectangular representation for event MISTY PICTURE  156  6.13 Midspan displacement response of the DRES panel to the MISTY PICT U R E pressure pulse  .  6.14 ADINA finite element model of the DRES panel A.l  157 158  (a) Singly symmetric beam section; (b) Fully plastic stress distibution corresponding to a plastic hinge; (c) Generalized strain rates  177  A. 2 Relationship between the yield curve and plastic deformations of the beam section  178  B. l Values of A and pi for which the travelling hinge mechanism is statically admissible under impulsive loading  185  B. 2 Distribution of the bending moment in an impulsively loaded beam. . . .  186  C. l  Linearized representation of a general blast-type pressure pulse  188  D. l  Geometric representation of the convexity condition of plasticity.  192  xiv  Notation A  midpanel acceleration (instantaneous mode)  D  energy dissipated in plastic deformations  E  external work due to applied loads  I  nominal impulse  J  Lee's kinetic energy functional  K  kinetic energy  L  stiffened plate half-span  M  bending moment  Mo  ultimate moment capacity of a beam  N  axial force  o  axial force capacity of a beam  U  in-plane end slippage  V  midpanel velocity (instantaneous mode)  w  transverse midpanel displacement  Wj  permanent transverse midpanel displacement  X,Y  locations of travelling hinges  a  stiffener spacing  e  hinge extension  m  mass per unit length of effective beam in one--way stiffened plate  N  m ,m x  m  p  y  mass per unit length of stiffeners in two-way stiffened plate mass per unit area of plating in two-way stiffened plate XV  n ,n  numbers of stiffeners running in coordinate directions, x and y  p  dynamic pressure load  x  p  y  0  q  static collapse pressure dynamic line load  q  static collapse line load  r  material strain rate sensitivity parameter  t  time  tf  time at which panel comes to rest  t  duration of blast pulse  t  time at which string response begins  t  time at which travelling hinge passes through section at x  0  p  s  x  t  time at which travelling hinges meet at midspan  w  transverse displacement  x,y  orthogonal in-plane coordinates  z  transverse coordinate  z  distance between neutral axis and centroidal axis  a, /3  scalar time functions for instantaneous mode response  e  strain  x  e  0  material strain rate sensitivity parameter  6  hinge rotation  K  beam curvature  A  beam section asymmetry parameter  p  support moment capacity reduction factor  v  support axial capacity reduction factor  p  mass density of panel material  xvi  dynamic yield stress  <?d  static yield stress effective dynamic yield stress instantaneous mode shape functions A  kinetic energy of the difference between two fields  At  time step duration  AW  increment of midpanel displacement within time step  0  rotational end slippage  T!,T 2  in-plane end slippage parameters  XVll  Acknowledgement I am indebted to the many faculty, staff and students who generously provided their valued advice and guidance throughout the course of my study and research. In particular, the efforts of Dr. M.D. Olson and Dr. D.L. Anderson are greatly appreciated. The financial support of the Canadian Department of National Defence (through a contract with the Defence Research Establishment Suffield) is gratefully acknowledged. Finally, I would like to thank my family and friends for their encouragement and unfailing support of my academic endeavours. This work is dedicated to my father, Robert Sr., and to my late mother, Rita.  xviii  Chapter 1 Introduction The possibility of being subjected to loads which arise from a nearby explosion exists for almost any structure. This possibility is usually so remote that the structure is designed without regard to such blast loads. Certain industrial and military structures, however, are recognized as being prone to these loads and must be made blast-resistant. Ship structures may often be placed in this category. Typical ship structures are constructed of ductile metal such as mild steel, with plating welded to transverse stiffeners. Blastresistant design of such stiffened plates requires a thorough understanding of the nature of blast loads and of the stiffened plates' response thereto.  1.1  Blast Loads  An explosion will cause a pressure wave, usually called a blast wave, to propagate through the air. The effect of a free-field blast wave as it moves through a point is characterized by the following: an abrupt—generally considered to be instantaneous—rise in the pressure above the ambient atmospheric level to some peak value; an overpressure or. positive phase during which the pressure decays from the peak value to the ambient level; and an underpressure or negative phase during which the pressure drops below the ambient level. The pressure-time history for a free-field blast wave at some distance from the  1  Chapter 1.  Introduction  2  source is shown in Figure 1.1. If the blast wave arises from a conventional explosion and is not too distant from its source, the peak overpressure is much larger than the peak underpressure so that, in practice, the negative phase of the blast wave may be neglected. When the free-field blast wave encounters a structure it is reflected and refracted by that structure. As a result, the pressures acting on the structure are higher than those measured in the free field. If the stiffened plate under consideration is isolated from other structures (in terms of blast wave reflections) and is not so large that some points on it are significantly closer to the blast source than others, the pressures acting on the stiffened plate may be taken to be spatially invariant. Furthermore, these pressures will still be characterized by an essentially instantaneous rise to a peak value followed by a monotonic decay to the ambient atmospheric pressure. Such a pressure load is commonly referred to as a blast-type pulse and is shown in Figure 1.2, where p(t) is the overpressure acting on the structure and the time origin t = 0 is defined as the moment of incidence of the blast wave on the structure. Blast-type pulses arising from conventional explosions can load structures to peak intensities of several Megapascals (MPa) over durations typically ranging from microseconds to milliseconds. In extreme cases, blast-type pulses may be idealized as impulses having infinite intensity and infinitesimal duration. In the present study, details of the blast loads themselves will not be investigated or discussed beyond the general observations recounted above. The reader is instead referred to the published works of Henrych [13] and of Newmark and Hansen [46].  1.2  Structural Response  The response of structures subjected to blast and other large dynamic loads has received much attention in the literature. Exhaustive reviews of the work conducted in this field have been published by Jones [26,27,28,29,30] and by Ari-Gur et al [1]. Their efforts  Chapter 1. Introduction  Pressure  Time Figure 1.1: Free-field blast wave pressure as a function of time at a particular dist from the blast source.  Figure 1.2: Typical blast-type pulse.  Chapter 1.  Introduction  4  will not be repeated herein. Instead, a small number of works will be cited in order to introduce the reader to the principle response phenomena and to the main concepts which will be used herein to model these phenomena. Detailed discussions of these concepts and phenomena, presented in later chapters, will be accompanied by more thorough reviews of the literature.  1.2.1  E x p e r i m e n t a l Studies  Experimental studies of the response of structures to blast loads have provided a qualitatively good, if quantitatively incomplete, understanding of the problem. Tests of impulsively loaded clamped beams by Humphreys [23] and by Menkes and Opat [44] have identified three different damage modes. Mode I is simply large inelastic deformation of the beam. Under the impulsive load, plastically deforming sections which are commonly idealized as plastic hinges form at the supports and move inward to the midspan as the beam is deflected. A second set of plastic hinges remains at the supports and allows large hogging rotations to occur. Higher impulsive loads bring the onset of Mode II: tearing (tensile failure) in the beam's outer fibres at the supports. The large plastic rotations at the supports lead to large strains in the outer fibres which presumably exceed an ultimate rupture strain [25]. With still higher impulsive loads comes Mode III damage: transverse shear failure at the supports. A pure, well defined shear failure occurs so soon after the application of the load that very little deformation of the severed beam takes place. Shear failure is also generally considered to be strain controlled [25]. Tests of impulsively loaded clamped plates of circular plan conducted by TeelingSmith and Nurick [70] and of square plan conducted by Ross et al [53] show very similar damage modes to those for beams. Mode I response in plates is characterized by hinge lines which move inward to the midpanel point. Mode II tearing and Mode III shear  Chapter 1. Introduction  5  failures occur in lines at the clamped edges of a plate under progressively higher impulses. Houlston, Slater and Ritzel conducted a series of experiments [19,59,60] in which full scale one-way stiffened plates with fully clamped edges were subjected to blast loads. Ductile inelastic (Mode I) response of the panels was observed and recorded. Postshot surveys of the panels indicated permanent deformations which were consistent with those of blast loaded beams and plates, and although no observations were made of the transient displacement profiles of the panels, it is likely that moving hinge lines characterized their Mode I response. Notably, the relative displacements between the plating and the stiffeners were small compared to the overall displacements when the loading was very intense (several times greater than the static collapse pressure). The applied blast loads were not intense enough to cause Mode II or Mode III damage to the panels, but it is reasonable to assume that such response is possible under sufficient loading.  1.2.2  M e t h o d s of Analysis  The blast response of structures, even of simple elements like beams and plates, is seen to be complex. Even if tearing and shear failures (Modes II and III) of the structure are neglected, i.e., the structure is assumed to be infinitely ductile, analysis remains difficult. The observed inelastic deformations indicate that plasticity and perhaps strain hardening and strain rate sensitivity of the material need be modelled. Large deflections of plates and axially restrained beams induce membrane stresses in the structure, so that geometric nonlinearities must also be taken into account. The finite element method (FEM) is well-suited to the analysis of nonlinear problems such as the blast response of structures. In fact, general F E M packages which can solve these problems already exist. They are, however, extremely expensive to use and may  Chapter 1. Introduction  6  require tedious data manipulation. General F E M packages are therefore poorly suited to preliminary design of blast loaded structures. Furthermore, as Jones has astutely pointed out [30], the loads to which the structure will be subjected are seldom known to a level of accuracy commensurate with the F E M model. Simplified methods, usually meaning approximate methods, of analysis are therefore sufficient, indeed preferable, for use in the preliminary design and occasionally the final design of a blast-resistant structure. Theoretical techniques, typically based upon sweeping and occasionally restrictive assumptions and simplifications, have been and remain the subject of much investigation. As well as being well suited to preliminary design and analysis, simplified theoretical techniques may clarify the response, providing the analyst with more insight into the problem than would a F E M solution. When properly formulated, the simplifying assumptions focus on the governing response phenomena and neglect the superfluous. Such is the case when the material behaviour of a structure under very intense blast loading is modelled using the rigid-plastic idealization.  1.2.3  Rigid-Plastic Analysis  Blast loads can input a tremendous amount of energy to a structure. This external work becomes kinetic energy which in turn is either stored within the structure as elastic strain energy or dissipated in plastic deformation of the structure. The external work is often much greater than the maximum amount of strain energy which can be absorbed by the structure in a wholly elastic manner. In such cases, material plasticity dominates the response and the structure's elasticity may be neglected. A first order theory will thus idealize the material as being rigid, perfectly plastic—a uniaxially loaded fibre of the material will not deform until the yield stress cr is reached, after which unconfined 0  flow of the fibre occurs. The stress-strain relation for a rigid, perfectly plastic material  Chapter 1. Introduction  7  Stress ' cr . 0  1  *- Strain  Figure 1.3: Uniaxial stress-strain curve for a rigid, perfectly plastic material. is shown in Figure 1.3. The validity of the rigid-plastic material idealization is subject to two conditions. In comparing theoretical and experimental results for impulsively loaded cantilever beams, Bodner and Symonds [3] found that elasticity could be neglected if the external work done by the impulse was at least three times greater than the maximum elastic strain energy. This energy ratio criterion is generally held to be valid for a wide range of structures. A second condition is that the load pulse duration should be small in comparison to the fundamental period of elastic vibration of the structure. Symonds and Frye [66] have determined that neglecting elasticity can lead to large errors when the load pulse has a nonzero rise time and a long duration. Natural modes of elastic vibration are excited in resonance by such pulses and so elastic response is significant. This effect should be much less pronounced for zero rise time blast-type pulses, however, and is generally considered to be secondary to the energy ratio criterion. Elementary rigid-plastic theory combines rigid, perfectly plastic material behaviour with infinitesimal displacement theory. Infinitesimal displacements correspond to bendingonly response of beams and plates. Symonds [62] applied elementary rigid-plastic theory to the analysis of simply supported and clamped beams under blast-type pulses. The  Chapter 1.  Introduction  8  response of simply supported and clamped plates to rectangular pulses was derived using elementary rigid-plastic theory by Hopkins and Prager [18] and by Florence [7], respectively. Cox and Morland used the elementary theory to derive the response of simply supported square plates to rectangular pulses. In all of these analyses, travelling hinge or hinge line mechanisms resembling those observed in the previously discussed experiments were predicted by the rigid-plastic theory. Displacements were generally overpredicted by the theory, however, due in some part to the assumption of infinitesimal displacements. The rigid-plastic requirement of large work input by the loads may conflict with the elementary theory's restriction to infinitesimal displacements. In plates and axially restrained beams, finite displacements are associated with stretching of the neutral axis or surface of the structure. Membrane forces must be induced, leading to geometric stiffening of the structure. The effect of finite displacements on the response of axially restrained beams has been examined for various beam section geometries and load pulse characteristics: Symonds and Mentel [68] considered rectangular sections under ideal impulses, Vaziri et al [72] extended Symonds and Mentel's work to rectangular pulses, and, most recently, the author [54,56] presented a solution for general doubly symmetric sections subjected to general pulses. The initial response of an axially restrained beam is governed by bending, similar to that predicted by the elementary theory. As the transverse displacements grow to the order of the beam depth, the bending capacity of the beam vanishes, the axial force in the beam grows to the section's plastic capacity and the response is that of a plastic string. At large displacements, this plastic string is much stiffer than the same beam in bending. Similar conclusions may be drawn from Jones' finite displacement analysis of rectangular plates [24]. The finite displacement theory may still overpredict the displacement response of a structure. The rigid, perfectly plastic material idealization neglects both strain hardening and sensitivity of the yield stress to the strain rate. For ductile metals such as mild steel,  Chapter 1. Introduction  9  the yield stress rises with the strain rate. An obvious and potentially large stiffening effect exists. Symonds and Jones [67] have shown that this effect can be accounted for in the analysis of an axially restrained beam by a simple, one-time correction to the static yield stress of the beam. In so doing, excellent agreement with experimental results may be obtained. The effects of strain hardening are, by comparison, small and so may be neglected [67]. Despite the simplifications afforded by the rigid-plastic material idealization, the analysis of structures subjected to blast loads can be very difficult, even for very simple structures such as beams and plates. As the level of complexity of the structure increases, the analysis quickly becomes intractable by closed-form methods. For example, the analysis of a simple beam grillage by Huang and Liu [22] required a numerical algorithm for its solution, even though the elementary theory was used. Such intractabilities are doubtlessly responsible for the absence of rigid-plastic analyses of blast loaded stiffened plates in the literature. In attempts to deal with more complex structures, approximate methods of analysis beyond those of rigid-plasticity have been developed. Among the most useful of concepts developed to this end is that of mode approximation.  1.2.4  Mode Approximations  The reader is familiar with modal analysis of linear elastic structures. Such structures have preferred patterns of deformation, or normal modes, within which they will respond in free vibration after the removal of external loads. Mathematically, these modes are characterized by velocity fields which are separable functions of the spatial and temporal variables. Using the elementary theory, Martin and Symonds [43] found similar modes for rigidplastic structures. When subjected to an impulsive load of arbitrary spatial distribution,  Chapter 1. Introduction  10  the response of a rigid-plastic structure approaches and finally corresponds to a modal response. Martin and Symonds concluded that the transient response of a rigid-plastic stucture would be well approximated by response in one well chosen mode. (Just as for an elastic structure, there are an infinite number of modes for a rigid-plastic structure.) The resulting mode form response was referred to by Martin and Symonds as a mode approximation. When finite displacements are considered, the structure is no longer linear and so the modes must change with the displaced configuration of the structure. Lee and Martin [38] introduced the idea of instantaneous mode approximations, whereby the response of a structure is approximated at successive instants by a sequence of modal responses which exist at those instants only. Lee [37] later developed an extremum principle by which these instantaneous modes may be determined.  1.3  P u r p o s e a n d Scope o f t h e P r e s e n t S t u d y  Stiffened plates are common structural elements which may, in many applications, be subjected to intense blast loads. Analytical methods which predict the dynamic response of stiffened plates to blast loads are therefore required. In particular, simplified methods of analysis suitable for preliminary blast-resistant design are needed. The purpose of this study is to develop such methods using the techniques of rigid-plastic analysis and, where necessary, mode approximation. Many of the observations of the preceding sections will be incorporated into the analyses as simplifying assumptions in the problem formulations. Further to the cause of simplicity, consideration will be limited to Mode I response (large ductile deformations) of stiffened plates.  Chapter 2 Dynamic Response of Beams and One-Way Stiffened Plates 2.1  Introduction  The displacement response of one-way stiffened plates comprised of several bays and subjected to high intensity blast loads has been investigated experimentally and numerically [19,20,21,34,33,59,60]. When loads of very high intensity (several times the static collapse pressure) were considered, relative displacements between the stiffeners and the plating were small. This result suggests that, away from the lateral edges, the stiffened plate behaves much like a singly symmetric beam with the plate acting as a large flange. A one-way stiffened plate with stiffener spacing a is considered (Figure 2.1). The edges of the plate and stiffeners are fully clamped with constraints against in-plane displacements. Initially at rest, the plate is subjected to a uniformly distributed blast load p(t), where t is the time measured from the original incidence of the load on the plate. The load is assumed to be a blast-type pulse (Figure 1.2) which satisfies the inequality  An interior portion of the panel, which includes one stiffener and plating one-half of 11  Chapter 2. Beams and One-Way Stiffened Plates  12  p— a •  2L  Pressure p(t) ~ae  ^  at  ac,  ^  Figure 2.1: One-way stiffened plate subjected to uniformly distributed pressure pulse. q(t) per unit length i X  q(t) = a-p(t) i  1  t  • mass m per unit length  i  L  Figure 2.2: Fully clamped beam subjected to uniformly distributed pulse loading.  Chapter 2. Beams and One-Way Stiffened Plates  13  the distance to each neighbouring stiffener, is assumed to behave as a beam of singly symmetric cross-section with mass per unit length m. The uniformly distributed pressure pulse is resolved to act through the beam's axis of symmetry with intensity q(t) — a-p(t), as shown in Figure 2.2. To obtain a relatively simple analytical solution, the following assumptions regarding the beam's material and geometric properties are made: 1. The beam material is rigid, perfectly plastic. 2. Strain rate effects are ignored. 3. Shear deformation and rotary inertia effects are ignored, and Bernoulli-Euler beam theory is used. 4. The deflections are finite but small compared to the beam span so that the square of the slope of the beam is small, i.e., (dw/dx)  2  <C 1.  As the beam undergoes transverse displacements w(x,t), x being the distance along the beam axis from the midspan, it will resist loading by some combination of axial force and bending moment stress resultants, N and M, located at the section centroid. The free body diagram of a small segment of the deformed beam is shown in Figure 2.3, where use of d'Alembert's principle is made and the inertia of the segment is included as an effective force. Assumption 4, above, allows the axial force in the beam to be approximated by its horizontal component: N y/l +  (dw/dx)  2  «  N.  (2.2)  A consequence of this approximation is that the axial force is taken to be constant along the span. Neglecting higher order effects, the equation of motion for the segment is found  14  Chapter 2. Beams and One-Way Stiffened Plates  x  9(0  t t f  z, w  ^dx M+^dx  N  m(w +  mw  dx  ^dx)  -— dx  Figure 2.3: Free body diagram of a segment of the deformed beam. to be  dM 2  ~d^  Ar +  N  0w  .  2  M  =  rnw(z,t)  .  -  (2.3)  q(t) ,  where a dot denotes differentiation with respect to time.  2.2  Interaction Relation—Stress Resultants  The equation of motion for the beam has been given in terms of the stress resultants M and N. These resultants are determined by the beam section interaction relation and its associated flow rule. Figure 2.4 shows a typical yield curve for a singly symmetric I-beam with a large top flange and a small bottom flange. Once again, iV and M are the axial force and bending moment acting at the section centroid. N and MQ are the 0  maximum values that may be developed within the section. Notice that, for the singly symmetric case, the section has its maximum moment capacity in the presence of axial  Chapter 2. Beams and One-Way Stiffened Plates  15  force. Specifically, this maximum moment occurs when the neutral axis coincides with the centroidal axis, resulting in an axial force of magnitude ATV . The parameter A may 0  then be considered a measure of the asymmetry of the section, where a doubly symmetric beam has A = 0.  Figure 2.4: Yield curve and associated flow rule for a singly symmetric I-beam. All deformations of the beam must occur within fully plastic sections (plastic hinges). A hinge will undergo rotation (positive in sagging) at the rate 6 and absolute centroidal extension at the rate e. Hogging (negative) hinges will likely form at the supports (x — ±L)  and sagging (positive) hinges at some presently unknown position (x = ±X). Field  quantities at these points will be denoted by the subscripts L and X, respectively. From the associated flow rule (Appendix A), the plastic deformation vector (./V e, 71^,0) of a 0  hinge must be outwardly normal to the yield curve at the considered stress state, as shown in Figure 2.4. Since the axial force TV is taken to be constant along the length of the beam, this specifies a unique set of plastic deformation ratios (e/8)  LX  for a given TV,  Chapter 2. Beams and One-Way Stiffened Plates  16  and, when the beam's kinematics are considered, specifies a relationship between the axial force N and the displacements w(x,t).  As deflection of the beam proceeds, extension  of the hinges will increase relative to rotation. The stress states will therefore move, with equal TV, along the yield curve to the right, represented generically by the points A and A'. Ultimately, when N — N , the moment reduces to zero, the beam behaves as 0  a plastic string, and a new mode of deformation is begun. In this way, the hinge stress states may be determined in terms of w(x,t) and substituted into the equation of motion. The results, however, are non-general and must be re-evaluated for every different section considered. Furthermore, the resulting differential equations are typically nonlinear and very difficult to solve.  Figure 2.5: Linear yield curve approximation. An approximation to the yield curve can be developed by inscribing the true yield curve with four linear segments, as shown in Figure 2.5. Previous studies [55,56] have shown that such an approximation may be used to uncouple the nonlinear problem into  17  Chapter 2. Beams and One-Way Stiffened Plates  two linear ones. Furthermore, the results will be general, applicable to any singly symmetric beam section and requiring only the determination of MQ, N , and the asymmetry 0  parameter A. Previous analysis of doubly symmetric beams showed that such an approximation has very little effect on the displacement response of the beam, and there is no obvious reason to expect the singly symmetric case to differ. Referring to Figure 2.5, stress states are determined uniquely by deformations (via the associated flow rule) only at points of slope discontinuity of the yield curve. And at these points the deformation ratios are constrained only to lie within a range of values, as indicated by the fan of outward normals existing at such a point. Immediately at the onset of finite displacements, the stress states will correspond to the points B and B' at the sagging and hogging hinges, respectively, where N M  X  -  M  L  (2.4)  =  \N ,  =.  2M /{\ + A) .  0  (2.5)  0  As deformation proceeds, the stress states at the hinges remain constant at these values as long as the plastic deformation vector at the support hinges remains in the fan of outward normals at point B'. Hence, for the above equations to be valid we must have  and *  AMI-A)  < (e/% <  iV (l 0  +  (2.7)  A)  For deformations outside of this range the stress states jump instantaneously to point C where N = N and M = 0, and the beam thereafter behaves as a plastic string. 0  By linearizing the yield curve of a singly symmetric beam section in this manner, the section has implicitly been simplified to the asymmetric sandwich section shown in Figure 2.6. The simplified section has a top flange of area A = \A(\ + A) located a t  Chapter 2. Beams and One-Way Stiffened Plates  y— Area A  18  t  TV(1 + A) 0  centroid TV(1 - A) 0  Area A = \A(l - A) b  Figure 2.6: Asymmetric sandwich beam section corresponding to the linear interaction relation. distance TV£,/TV(1 + A) above the section centroid and a bottom flange of area Ab — 0  \A{1  — A) located a distance MQ/NQ^I — A) below the centroid, where  A  is the total  cross-sectional area of the beam. This idealization of the section geometry will prove to be of further benefit in a forthcoming analysis of strain rate effects in the response of stiffened plates (Chapter 6). The above discussion has been for a beam with a large top flange and a small bottom flange. If, instead, the beam has a small top flange and a large bottom flange, i.e., the beam is inverted, the yield curves will be mirror images of those in Figures 2.4 and 2.5. However, the resultants TV and M y — ML and valid range of deformations will be unchanged. Bending resistance will dominate in the early stages of deformation, and axial force or string resistance in the later stages. Dynamic response of the beam, then, will be comprised of some initial plastic hinge mechanism phase, with the stress resultants in the hinges determined by the interaction relation (or yield curve) for the section, and a final plastic string phase during which the beam has no bending resistance.  19  Chapter 2. Beams and One-Way Stiffened Plates  (a)  (b)  Figure 2.7: Travelling hinge mechanism: (a) displacement field; (b) velocity field.  2.3 2.3.1  Hinge Mechanism Response Travelling Hinge Mechanism  It is assumed that the beam initially responds in the mechanism shown in Figure 2.7, with hinges forming at the supports and at some distance X{t) to either side of the midspan. Symonds [62] determined this mechanism to be valid for the linear bendingonly response of rigid-plastic beams subjected to blast-type pulse loads. In their analysis of axially constrained rectangular beams subjected to impulsive loads, Symonds and Mentel [68] assumed an initial response according to this mechanism and, although its admissibility was not explicitly checked, this mode of response showed good agreement with the experimental observations of Humphreys [23]. The velocity field for this mechanism is w(x, t)  W(t), W(t)  H0<x<X(t); L —x L-X(t)  , iiX(t)  < x < L,  (2.8)  20  Chapter 2. Beams and One-Way Stiffened Plates  where W(t) is the midspan displacement. Differentiating with respect to time gives the acceleration field: if 0 < x < X(t);  W(t), w(x, t) W(t) +  X(t)W(t)  L —x  L - X(t)  L - X(t)  if X(t)  (2.9)  <x<L.  The extension of the beam at finite deflections is e =  2  Jo \ dx J  \  (2.10)  dx  where use has been made of Taylor's expansion and higher order terms have been neglected. Differentiating with respect to time, the rate of extension is found to be ^ r (dw low Jo \dx  di \ dw dx)  L  (2-11)  and, upon substitution of the presently considered displacement and velocity fields, is reduced to  2WW  e = _  r n  .  n s  (2.12)  This extension occurs in a discrete manner at the plastic hinges such that e = 2e + 2e , x  L  where the subscript X denotes a field quantity at the travelling midspan hinges and the subscript L a quantity at the support hinges. The beam will also be rotated through the finite angles 0 at the travelling hinges X  and 9 at the support hinges. If these angles are defined as positive in sagging, their time L  rates of change are easily found to be 6*  -Or  W L-X  (2.13)  Combining equations (2.12) and (2.13) yields the beam's kinematic relation: (2.14)  Chapter 2.  Beams and One-Way Stiffened Plates  21  If the travelling hinges do not move outward (away from the midspan), the midspan segment 0 < x < X will remain flat with the transverse displacement w(x,t) = W(t). Integrating equation (2.3) with respect to x and imposing symmetry at the midspan (x = 0), the bending moment gradient is written as ^  ox  = [m^(<)- (f)l x. g  L  (2.15)  1  Since a plastic hinge is developed at the point x = X(t), the bending moment M  x  must be a maximum and the gradient dM/dx must vanish at that point. Clearly, these conditions can only be met if mW(t) = q(t)  (2.16)  and if the bending moment is constant along the length of the midspan segment. With the initial conditions W(0) = W(0) = 0, the midspan velocity and displacement are then given by W(t)  =  W{t) =  - fq(T)dT,  (2.17)  m Jo  —f  m Jo  q(r)(t - ) dr. T  .  (2.18)  Though rigid, the end segments X < x < L will be deformed. The beam curvature at a section will be permanent, induced by the earlier motion of the plastic hinge through the section at the time t [i.e., X(t ) = x]. This curvature will be of the magnitude x  x  ^u, • dx  2  =  M^) X(t ) x  =  w(t ) x  X{t ){L -x)'  K  x  '  ]  Recalling the acceleration field, equation (2.9), and making use of the midspan segment results, the equation of motion (2.3) is specialized to the end segments as dM 2  fl ™ 2  X(L-x)  [* ,  ,  . , / x - X N  Chapter 2. Beams and One-Way Stiffened Plates  22  Integrating twice with respect to x gives [M + Nw]  x  =  x  (2.21) Setting x = L, the equation of motion of the rigid end segments is determined: Mr.  N r^ Jy X ~ ~ q(T)(t-r)dr m Jo \X{L-X)  =  f (T)dr  -  q  (2.22)  lq(t)(L-Xf  Equation (2.22) can be further reduced to ! [ (  £  - * > ' j f > H  M  =  N  r  ft  + m Jo  q(r)(t-r)dr  (2.23)  which is then solved to give X(t) = L  6  ( M Y - M )t +  j\(r)(t  L  - rf dr] .  (2.24)  Recalling the bending response stress resultants, equations (2.4) and (2.5), and defining q to be the linearized estimate of the static collapse load, 0  % =  4Mo  (2.25)  (l + A)! ' 2  the hinge motion is given by x(t)  =  A(l + A)/V  0  rt  i  Jo  q{r)(t-TfdT  . L U -  (2.26)  Also recalling the valid ranges for the deformation vectors (e/0) , LX  equations (2.6)  and (2.7), and combining them with the kinematic relation, equation (2.14), the range of displacements for which the travelling hinge mechanism response is valid is found to be 2Mo 0 < W < N (i-vy 0  (2.27)  Chapter 2. Beams and One-Way Stiffened Plates  23  (a)  (b)  Figure 2.8: Midspan hinge mechanism: (a) displacement field; (b) velocity field. The upper bound of this range is, incidentally, just the depth of the simplified sandwich section of Figure 2.6.  2.3.2  Midspan Hinge Mechanism  At the time t the travelling hinges meet at the midspan and, following Symonds [62] x  and Symonds and Mentel [68], response is assumed to continue in the midspan hinge mechanism shown in Figure 2.8. The rigid half-beams of this mechanism are similar to the rigid end segments of the travelling hinge mechanism with X = X = 0. The previously derived kinematic relation, equation (2.14), remains valid and the equation of motion becomes dM dx 2  2  +  (Pw dx  N  =  w(t)(l- /L)  m  x  2  -  q(t)  (2.28)  Integrating twice and making use of the hinge stress resultants determined previously, the equation of motion is reduced to nt)  +  -^w(t)=  3  mL  2  3  2m  (2.29)  Chapter 2. Beams and One-Way Stiffened Plates  Initial conditions at the time t  24  are determined by the travelling hinge mechanism re-  x  sponse. The range of displacements for which this midspan hinge response may occur is again given by equation (2.27).  2.4  String Response  At the time t the midspan displacement is W = 2MQIN {\ — A ). The stress states have 2  s  Q  just jumped to the point C in Figure 2.5, so the beam has no bending stiffness and the axial force is equal to the section capacity  along the entire span. The beam therefore  responds as a plastic string with the equation of motion dw 2  , dw  ,.  2  -  r  = «<«)  . „ ,  ( -30) 2  and the initial conditions w(x,t ), w(x,t ) determined by the hinge mechanism response. s  3  In the strictest sense, this plastic string response can only occur so long as the deformations conform to the flow rule—the relative rates of extension and curvature must lie within the fan of normals to the yield curve at point C—at every section of the beam. At various times during the string response, regions of the beam may deform such that the limits of this fan are reached but not exceeded. In these regions, motion is no longer governed by equation (2.30), but by the flow rule condition just discussed. Such response is obviously very complicated and difficult to solve. However, in studying the response of an impulsively loaded beam, Symonds and Mentel [68] used bounding techniques to show that the effect of this flow constraint on the beam's response is relatively unimportant and so may be disregarded. Therefore, for all times t > t , motion is taken to be governed by equation (2.30) and s  the displacement and velocity conditions at t = t . If the displacement field of the beam s  Chapter  2. Beams and One-Way Stiffened  25  Plates  is represented by a Fourier series of the form  w{x,t)= £  W (t) c o s ^ ,  (2.31)  n  n=l,3,5,...  Z  L  the equation of motion becomes  £  \ " + -r-iT " w  c o s  w  n=lA5..A  i  m  J  L  2  ^ r = — £ - irrc o s  * ™ n=US,..P  L  -  2  32  2 j L  Considering the complicated displacement profile of the beam w(x, t ), a complete Fourier s  analysis would be tedious. However, Vaziri et al [72] showed that a good approximation to the solution can be found by considering only the first term of the Fourier series' so that  w(x,t)  A very simple equation of motion results: frW + r4mL ^ W M - 7rm ^  = W COS(TTX/2L).  2  (2-33)  Following Vaziri, the midspan displacement W and the kinetic energy of the beam at time  t are held constant through the sudden change in the deformation mode. The initial s  displacement condition is W(t ) = 2M /7V (1 - A ). If we denote the midspan velocity at 0  s  an instant before  t = t  s  2  0  as W(t~) and at an instant after as W(t+), equating the kinetic  energy through the transition gives W ( / ) ] / c o s ( g ) dx = £ [w(x,t:)] dx s  +  2  o  i  2  2  (2.34)  which is reduced to  W(tt)  =  \ljJ [w(x,t )} dx. o  L  7  2  (2.35)  Substituting in the velocity field for the initial response, equation (2.8), the initial velocity condition at time t = tt is found to be W(tJ)  =  W(t;)  1  W  ,  (2.36)  Chapter 2. Beams and One-Way Stiffened Plates  2.5  26  A d m i s s i b i l i t y o f the H i n g e M e c h a n i s m s  In the foregoing analysis, the beam was assumed to deform according to certain prescribed hinge mechanisms. The admissibility of these mechanisms is now considered. A velocity field is kinematically  admissible if it satisfies the velocity or displacement  constraints of the structure. In the present case, kinematic admissibility requires zero displacements at the beam supports and a continuous displacement field along the span. T h e velocity fields for b o t h of the hinge mechanisms satisfy these requirements. T h e present analysis has been further based on the assumption that, i n the initial travelling hinge mechanism, the hinges at x = X(t) move monotonically toward midspan. F r o m this assumption it was deduced that the midspan segment was undeformed and that the bending moments w i t h i n it were constant, neither of which is true if the hinge motion is at any time outward. Requiring that X < 0 and recalling equation(2.26),  the  hinge motion is monotonically inward if  (  2  '  3  7  )  A t the outset of the analysis, it was postulated that blast-type pulses would result in monotonically inward hinge motion and so consideration was limited to this class of pressure pulse. A n extreme example of a blast-type pulse is the step load q(t) =  q. m  T h i s pulse always satisfies the inequality of (2.37): a  t  q m t  ... A(l + A)iV ^  A(l + \)N gl  0  +  imM,  0  t  >  q  m  t  +  \2rnM,  * '  F r o m this result it might be suggested that the requirement of blast-type pulses is overly restrictive.  Since only the equality of (2.37) need be satisfied, the kinematics of the  beam seem to allow load pulses to increase i n intensity w i t h time. It must be noted,  Chapter 2. Beams and One-Way Stiffened Plates  27  however, that such a conclusion is derived from the use of the linear interaction curve (Figure 2.5) which gives the axial force N = XN throughout hinge response. If the true 0  yield curve were used instead, iV would initially be zero. The second terms on both sides of equation (2.37) would vanish, yielding (2.39) which is just the blast-type pulse condition, equation (2.1). While this condition is restrictive, particularly in the latter stages of hinge response, it will be retained because of its relative simplicity when compared to equation (2.37) and its conservative nature for any beam section. The stress field is said to be dynamically admissible if it is in internal equilibrium and in equilibrium with the external loads, where the effective d'Alembert forces due to inertia are included; and if the field nowhere exceeds the yield limit, it is said to be safe. For the assumed hinge mechanisms, the equations of motion have been derived from consideration of the dynamic equilibrium of the deformed beam, as stated in equation (2.3). The equilibrium conditions are therefore satisfied identically. To satisfy the yield condition, the stress state M(x,t),  N(x,t) must lie on or within the yield curve at every section x  throughout the response. For the linear approximation to the yield curve (Figure 2.5) and N constant in x, the bending moment is required to decrease from a value M(X, t) =  MQ{1 — A)/(l -f A) at the travelling or midspan hinge to M(L,t) = —MQ at the support. At no point are these hinge moments to be exceeded. To satisfy the yield condition, then, it is sufficient—though not necessary—that the bending moment decrease monotonically between the hinges: dM  dx  < 0,  X < x < L.  (2.40)  Chapter  2. Beams  and One-Way  Stiffened  28  Plates  The second derivative of the bending moment in the travelling hinge mechanism may be derived from equations (2.19) and (2.20):  (2.41) Recalling that the hinge motion must be monotonically inward so that X < 0, equation (2.41) may be rewritten as  (2.42) Since equation (2.26) is not readily inverted to give t for a general load history q(t), x  general results for the dynamic admissibility of the travelling hinge mechanism are not available. The dynamic admissibility can only be checked for a specified pulse shape, as has been done in Appendix B for impulsive loads. It is shown therein that, for highly unsymmetric beams, the yield condition may be violated in the latter stages of travelling hinge response, with the bending moment magnitudes in the neighbourhood of the supports exceeding il^,. Repeating such an admissibility check for dynamic (nonimpulsive) loads would be tedious, and has not been carried out. However, it may be shown that as the loading tends away from the impulsive case both X and \X\ decrease. Consideration of equation (2.42) reveals that  d M/dx , 2  2  and hence  dM/dx,  must then become more  positive, increasing the likelihood of violating the yield condition. The second derivative of the bending moment in the midspan hinge mechanism is given by equation (2.28). Integration yields the bending moment gradient: (2.43)  Chapter  2.  Beams  and  One-Way Stiffened  29  Plates  Near the midspan hinge, this gradient is l i m ^ o dM/dx  = — N\im ^ dw/dx. x  0  As soon as  the midspan hinge is rotated through a finite angle, giving the values l i m ^ o dw/dx and l i m ^ o dM/dx  <  0  > 0, the yield condition must be violated. Under no circumstances  other than A = 0 (for which there is no axial force during bending response) is the midspan hinge mechanism safe. It is found that, within even the limited framework of rigid-plastic beam analysis, the solution to the response derived herein is inexact. Still, the present method may be sufficiently accurate to be useful. The studies of impulsively loaded, axially constrained beams by Symonds and Mentel [68] and Symonds and Jones [67] made use of the presently considered hinge mechanisms, and similar limitations regarding admissibility must certainly apply to their results. Humphreys [23] observed deflection profiles which closely resembled the travelling hinge mechanism and, when corrections for rate-sensitive material behaviour were made [67], the predicted final displacements agreed closely with experimental results. Furthermore, the violation of the dynamic admissibility conditions might not incur errors of greater magnitude than those incurred by the present method's basic assumptions and approximations. These errors may in fact compensate. Heuristically extending the kinematic theorem of static limit analysis suggests that the effect of violating the yield condition is to artificially increase the load resisting capacity of the beam. But the effect of the linear yield curve approximation, which inscribes the true yield curve, is to weaken the beam. The cumulative effect of these and the other approximations can only be assessed by comparing the results of this method with those of more refined numerical formulations and, ultimately, experiments.  Chapter  2.  Beams  and One-Way  Stiffened  2.6  Example Applications  30  Plates  The equations of motion have been solved in Appendix C for a time segment during which the load pulse is decaying linearly. The solutions are of closed form, but for convenient use they have been coded into the FORTRAN 77 program RIPTAB [57]. Some examples of blast loaded, one-way stiffened plates are presently considered, for which the rigid-plastic results obtained from RIPTAB are compared with those from more exact numerical formulations. A five bay, T-beam stiffened steel plate, designed and constructed by DRES for experimental purposes, is considered in all the cases. The geometry of the stiffened plate, hereafter referred to as the DRES panel, is shown in Figure 2.9. The mass density of the panel material is taken to be p — 0.733 x 10~ lbs /in . 3  2  4  The static yield stress has been measured as 45,000 psi, but to account for the strain rate sensitivity of the panel it is herein taken to be cr = 54, 000 psi. The section properties 0  of the equivalent beam representation are calculated to be m = 8.84 x 10  -3  lb-s /in , 2  2  MQ = 1.07 x 10 in-lb, N = 6.51 x 10 lb, and A = 0.538. The static collapse load 6  0  5  is q = 1206 lb/in, corresponding to a collapse pressure of 33.5 psi, and string response 0  begins at the deflection W = 4.62 in. The fundamental period of elastic vibration of the equivalent clamped beam having an elastic modulus of 30 x 10 psi is 6.2 msec. This 6  period is of interest in assessing the validity of the rigid-plastic theory for a particular load case (Section 1.2.3).  2.6.1 Rectangular Pulses The DRES panel is subjected to two rectangular pressure pulses of constant intensity p  m  and of duration t — 2 msec (Figure 2.10). This pulse duration is significantly less than p  the fundamental period of elastic vibration (6.2 msec) of the DRES panel's equivalent beam representation. For the first pulse, p = 129 psi which is approximately four times m  er 2.  Beams  and One-Way  36 in  Stiffened  36 in  Plates  36 in  36 in  6.  2.95 in  Figure 2.9: DRES stiffened plate geometry.  P(*)t Pm  * Figure 2.10: Rectangular pressure pulse.  36 in  Chapter  2.  Beams  and  One-Way Stiffened  32  Plates  the static collapse pressure of the panel and for the second, p  m  = 258 psi, roughly eight  times the collapse pressure. These load intensities and durations are consistent with the requirements of rigid-plastic theory (Section 1.2.3). The midspan displacements due to these pulses, as predicted by RIPTAB, are plotted in Figure 2.11. During the first 2 msec of response, the midspan velocities gradually increase under the pulse loading. After the loads are removed, the velocities begin to decrease until the panel comes to rest at the time tj — 5.4 msec with the permanent midspan displacement Wj = 3.1 in for the first pulse, and at tf = 6.6 msec with Wj = 7.9 in for the second. Also plotted in Figure 2.11 are the midspan displacements predicted by the beam finite element program FENTAB [8,9] with the stiffeners located alternately on the loaded and the unloaded sides of the panel. As in the present study, the FENTAB analyses model the DRES panel as an asymmetric I-beam, using ten elements of equal length for the half-span. The material behaviour is elastic, perfectly plastic with an elastic modulus of E = 30 x 10 psi, and nonlinearities arising from finite displacements are considered. 6  Due to the presence of elasticity the panel does not come to rest; instead, the midspan displacement oscillates about some average value. With structural damping, this value would be the eventual permanent midspan displacement Wj: 2.8 inches for the first pulse, 7.8 inches for the second, irrespective of the stiffener orientation. Comparison of the results is in essence a check on the errors introduced in the present beam analysis by the rigid-plastic idealization, by the linear interaction relation, and by the response mode approximations. The cumulative effects of these errors are seen to be small for these cases, particularly as the load is doubled from four times the collapse pressure (p  m  = 129 psi) to eight times that value (p  m  = 258 psi). The orientation of the  panel (stiffeners above or below the plating) is observed to have no significant effect on the displacement response, which is in agreement with the present rigid-plastic analysis (page 18).  Chapter 2. Beams and One-Way Stiffened Plates  1 1  I  I  I  I  i  i  i  i  1  RIPTAR  -  FENTAB  -  stiffener  below  plating  FENTAB  -  stiffener  above  plating  258  psi  Pm =129  psi  8 7  /: // // /  6Qoo  //  //  /  -  /  '!/ /// ///  5-  .in  j i  -  •  //  i 1 1  /  0  1  I  ^  .sf ,/  •7  V  /  i  2  // i  3  i  4  Time t  1  5  I  6  1 7  1  8  I  9  10  (msec)  Figure 2.11: M i d s p a n displacement response of the D R E S panel to rectangular pulses of 2 msec duration.  Chapter  2.6.2  2.  Beams  and One-Way Stiffened  34  Plates  H O B Tests  The response of the D R E S panel under blast loads has been investigated by Houlston and Slater [19] and by Slater et al [60] in a series of experiments at the D R E S Heightof-Burst (HOB) facility. In these tests, the target D R E S panels were flush-mounted on a reinforced concrete foundation. The edges of the panel were bolted to the foundation with the objective of simulating the fully clamped boundary conditions assumed in the present study. Bare high-explosive charges were suspended and exploded above the panels in such a way as to approximate spatially uniform pressure loads. These loads were measured by pressure transducers located around the periphery of the panels, indicated in Figure 2.9 by the points PI through P4. In all cases, the load pulses were of a much higher intensity than the static collapse pressure of the D R E S panel and of durations much shorter than the panel's fundamental period of elastic vibration. HOB  #315  Trial #315 from this series of experiments is considered here. The measured pressures incident on the transducers PI through P 4 have been averaged into a spatially uniform representation which is shown in Figure 2.12 along with two approximations: a trilinear approximation which is used in the R I P T A B analysis, and an exponential decay approximation p(t) = (925psi)(l - t/3.3msec)  215  which is used in a F E N T A B analysis. The  response to either of these approximate pulses is not expected to vary appreciably from ;  the response to the measured pulse. The rigid-plastic midspan displacement response as calculated by R I P T A B is plotted in Figure 2.13. It is compared with the results from F E N T A B and the finite strip program F S T N A P S [34,33]. In these numerical analyses the material behaviour is taken to be elastic-plastic-strain hardening with the elastic modulus E = 30 x 10 psi and the strain 6  Chapter  2.  Beams  and One-Way  Stiffened  Time  Plates  t  (msec)  Figure 2.12: Measured pressure pulse and approximations for HOB #315.  Chapter  2.  Beams  and  One-Way  Stiffened  36  Plates  RIPTAB FENTAB FSTNAPS -  Stiffener midspan (B)  FSTNAPS -  Panel centre  Time  t  _  (A)  (msec)  Figure 2.13: Midspan displacement response of the DRES panel subjected to the HOB #315 pressure pulse.  Chapter 2. Beams  and  One-Way  Stiffened  Plates  37  hardening modulus ET = 18 x 10 psi. Apart from the strain hardening, the FENTAB 4  model is the same as for the rectangular pulse case, and comparison of its results with those of RIPTAB is again excellent. In the FSTNAPS analysis, the measured pressure pulse is used and one half of the panel is modelled using three strips for each stiffener and four strips for the plating between adjacent stiffeners, amounting to 16 strips [33]. A slightly higher static yield stress of cr = 54,400 psi is used, but this will have no 0  significant effect on the response. The displacement response of the panel centre, point A in Figure 2.9, and of the adjacent stiffener's midspan, point B, are considered. There is an initial disparity between the stiffener and plating response, but this disparity becomes relatively small as the panel comes to rest. FSTNAPS predicts the permanent midspan displacement of the stiffener midspan to be approximately 14.5 inches, and RIPTAB predicts 15.8 inches — values that agree to within ten percent. For this example, the panel response seems well approximated by beam response. HOB #327 Trial #327 is now considered. The pressures measured by the transducers P I , P3 and P4 are plotted in Figure 2.14. The pressures incident on the transducers P I and P3 are almost identical, with initial pressure spikes of approximately 1400 psi. While the pressure on the transducer P4 lacks this initial spike and lags by approximately 0.25 msec, it is otherwise similar. For use within the programs RIPTAB and FENTAB, the pulse has been approximated by a spatially invariant load function, which is also plotted in Figure 2.14. As compared with HOB #315, the HOB #327 pulse is approximately fifty percent more intense, but has only about one-fifth of the duration. The midspan displacement as calculated by RIPTAB is plotted in Figure 2.15, where it is compared with the results from FENTAB. Again, the results from the two beam solutions are very similar. RIPTAB predicts the permanent midspan displacement to be  Chapter 2. Beams and One-Way Stiffened Plates  1500  Approximation Transducer  1200  P1  Transducer P3 Transducer P4  oo  CL  900  CL CD 00 00 CD  600  300  0  -0.2  0.0  0.2  0.4  Time t  (msec)  0.6  0.8  Figure 2.14: Measured pressures and approximation for H O B #327.  Chapter  2.  Beams  and  One-Way  Stiffened  Time  Plates  t  39  (msec)  Figure 2.15: Midspan displacement response of the DRES panel subjected to the HOB #327 pressure pulse.  Chapter  2.  Beams  and  One-Way Stiffened  Plates  40  Wf = 3.0 inches after a response time of 4.7 msec. This response time is much greater than the 0.25 msec time lag between the onset of the pressures at P I and P4, so the spatially invariant load representation seems reasonable. FENTAB predicts a maximum midspan displacement of 3.4 inches after a similar response time, followed by an elastic oscillation of the midspan about a displacement of approximately 2.5 inches. HOB #338 The panel tested in trial #338 from this series of experiments is a modified version of the previously considered DRES panel. The plating thickness is reduced from 1/4 inch to 3/16 inch. Otherwise, it is identical to the DRES panel and is herein referred to as the DRESM panel. The pressures measured during trial #338 are plotted in Figure 2.16. The shapes of the pulses at the transducers are again very similar, although the pulse at P4 lags 0.25 msec behind those at P I and P3. A spatially invariant approximation of the pressures, also plotted in Figure 2.16, is used to represent the pulse within the analyses. The midspan displacements calculated by RIPTAB and FENTAB are plotted in Figure 2.17. The permanent midspan displacement calculated by RIPTAB is 9.0 inches, and the response time is 6 msec. From the FENTAB results, the permanent midspan displacement is estimated to be 8.1 inches. Once again, these predicted responses are within approximately ten percent of one another. Experimental Results  Although the analyses of the HOB tests are all in agreement with one another, they do not accurately reflect the actual measured responses of the panels [60]. The measured permanent midpanel displacement in trial #315 is approximately 4 inches, very much less than the average calculated value of 15 inches. On the other hand, the measured  Chapter  2. Beams  and One-Way  Stiffened  Plates  1 0 0 0  - 0 . 5  0 . 0  0 . 5  Time  1.0  t  1.5  2 . 0  2 . 5  (msec)  Figure 2.16: Measured pressures and approximation for HOB #338.  Figure 2.17: Midspan displacement response of the DRESM panel subjected to the HOB #338 pressure pulse.  Chapter  2.  Beams  and  One-Way Stiffened  Plates  43  permanent midpanel displacement from trial #327, 11 inches, is very much larger than the computed value of 3 inches. And although the comparison for trial #338 is reasonably good —- the predicted permanent midpanel displacement is 9 inches and the measured value is 11 inches — this might be little more than a chance occurrence. These large discrepancies between experiment and analysis may simply be due to errors in measuring the incident pressure pulse. The analyses, after all, were unanimous in their predictions. However, it is very likely that some response phenomena, unaccounted for by the analyses, occurred. In particular, the fully clamped boundary conditions assumed by the analyses were not realized in trial #327, wherein the ends of the panel were observed to slip inward by more than 1.5 inches. Furthermore, accounting for the rate sensitivity of the panel with a uniformly applied yield stress value of 54,000 psi cannot be expected to give reliable results. The effects of partial end fixity and material rate sensitivity within these and other experiments will be investigated in later chapters.  2.6.3  Isoresponse C u r v e s  The response of a structure to load pulses of a specific shape is conveniently represented by a set of isoresponse curves. An isoresponse curve is the locus of combinations of load parameters that produce the same value of some response parameter. For the present case of a blast loaded beam or stiffened plate, appropriate load parameters are the peak pressure p and the nominal impulse 7, and an appropriate response parameter is the m  permanent midspan displacement Wj. The program RIPTAB is easily modified to construct isoresponse curves for a given beam or stiffened plate subjected to pulses of a given shape. For purposes of illustration, this has been done for the DRES panel subjected to rectangular and triangular pressure pulses with peak pressure p and duration t (Figure 2.18). m  p  Chapter  2.  Beams  and One-Way  Stiffened  44  Plates  Pm  *- t  Figure 2.18: Rectangular and triangular pressure pulses. Isoresponse curves for rectangular pulses are plotted in Figure 2.19, where the nominal impulse is / =  p t. m  p  In each of these curves, three distinct realms of response are  observed; quasi-static response, impulsive response, and intermediate dynamic response. ' In the quasi-static realm, Wj is solely dependent on the peak pressure so that the curve is horizontal. For the relatively large values of / (and hence t ) in this realm, the panel p  comes to rest at a time tj < t , so increasing / (and hence t ) will have no effect on p  p  the response. Here the pulse acts as a quasi-static step load. As p  m  increases and t  p  decreases, the curve asymptotically approaches the value of / for which an ideal impulse results in the prescribed value of Wj (Appendix B). For large values of p , the permam  nent displacement is weakly dependent upon p , and the response is largely impulsive. m  Between these two extremes, in the "knee" of the curve, the response is fully dynamic and dependent upon the values of both p and I. m  Isoresponse curves for triangular pulses are plotted in Figure 2.20, where the nominal impulse is / =  \p t . m  p  These curves are similar to those for rectangular pulses, also  plotted in Figure 2.20 for comparison. At large values of p and small t , both sets of m  p  curves approach the impulsive solutions, since both the rectangular and triangular pulses approach an ideal impulse. However, the response to triangular pulses is never quite  Chapter  2.  Beams  and One-Way Stiffened  45  Plates  500  i  i  r  Isoresponse  400  Impulsive  curve  solution  oo  Q_ E  Q_  300  0)  v_  Z5 00 00  (D  200  o 0_  W =10  100  0  0  300  600  Impulse Figure 2.19: pulses.  in  f  J  I  I  I  900  I  I  I  1200  I  L  1500  (psi.msec)  Isoresponse curves for the DRES panel subjected to rectangular pressure  Chapter  2.  Beams  500  and One-Way  Stiffened  n—r  400  46  Plates  I  i  r  Triangular pulse  h  Rectangular pulse Q_  300 CD  =5 00 00  CD L_  Q_  200  a CD  •_  100  W =1 f  0  0  in J  300  L  600  Impulse  I  I  900  I  L  1200  J  L  1 500  I (psi.msec)  Figure 2.20: Isoresponse curves for the D R E S panel subjected to triangular and rectangular pressure pulses.  Chapter  2.  Beams  and  One-Way Stiffened  Plates  47  quasi-static, since changing the pulse duration must also change the pressure distribution, and the triangular pulse curves never quite flatten out. As the nominal impulse and, hence, the load duration become very large (t —• oo) the triangular pulse comes to p  resemble a step load and the triangular pulse curves approach the rectangular pulse curves asymptotically. In the realm of dynamic loading, the triangular pulse curves lie above the rectangular pulse curves. If two panels are subjected to pulse loads of equal nominal impulse, one to a rectangular pulse and the other to a triangular pulse, the triangular pulse must have a greater peak pressure than the rectangular pulse if the panels are to undergo the same permanent displacement Wj.  2.6.4  Equivalent Rectangular Pulses  In the preceding example, the response of a stiffened plate was observed to be dependent on the pulse shape. Characterizing a pulse by its peak load and impulse alone is inadequate. Youngdahl [73] has developed a means of reducing a pulse of general shape to an "equivalent" rectangular pulse; a pulse equivalent in the sense that it causes the same permanent displacement of the structure as does the original pulse. Youngdahl's method was developed for linear, single mode response of rigid-plastic structures, and so for the present stiffened plate/beam problem it is an approximation at best. Nevertheless, the method was shown to be of value in the analysis of doubly symmetric beams [54], and should be so herein. By Youngdahl, "equivalent" pulses have the same values of impulse and centroid (i.e., mean time). Nominally, these values are.given by / imean  = =  f p(t)dt, Jo  (2.44)  7 f" t P(t) dt .  (2.45)  tp  1 Jo In reality, the upper limits of the above integrals should be the time at the cessation  Chapter  2.  Beams  and  One-Way Stiffened  48  Plates  1000  Time t (msec) Figure 2.21: Trilinear and equivalent rectangular representations of the HOB #315 pulse. of response, tf. However, tf is not generally known a priori and so the nominal values will be retained. Therefore, a pulse p(t) has an equivalent rectangular pulse of duration t  pe  = 2£  mean  and intensity p = e  I/t . pe  HOB #315 Test The response of the DRES panel to the HOB #315 pulse is now re-examined by modelling the pulse in its equivalent rectangular form. The trilinear approximation of the pulse has an impulse of  I  = 936 psi • msec and has its centroid at the time  t  m e a  n  = 0.836 msec.  The equivalent rectangular pressure pulse therefore has an intensity of p = 560 psi and e  a duration of t  pe  = 1.67 msec, as shown in Figure 2.21. The midspan displacements  due to the two pulse representations are plotted in Figure 2.22. The response time of the panel, tf « 6.5 msec, is greater than the true pulse duration, t = 3.25 msec, p  so that the impulse and pulse centroid are given exactly by their nominal values from  Chapter 2. Beams and One-Way Stiffened Plates  49  Figure 2.22: Midspan displacement response of the DRES panel to the trilinear and equivalent rectangular representations of the HOB #315 pulse.  50  Chapter 2. Beams and One-Way Stiffened Plates  equations (2.44) and (2.45). Due to temporal differences in the pulse representations, there are slight differences in the two responses during the first 2 msec, but after this time the differences virtually disappear. Isoresponse Curves In comparing the isoresponse curves of the DRES panel under rectangular and triangular pulses, it was observed that characterizing the pulse by its peak load and impulse alone was inadequate. The response of the panel is also dependent on the shape of the pulse. In Youngdahl's approach, the pulse shape is represented by the pulse's time centroid, f  mean  , so that the pulse is now characterized by three parameters.  A triangular pulse of peak intensity p I  =  ^p t with its centroid at the time t m  p  therefore has a duration of t  pe  m  m e a  and duration t has the nominal impulse p  n  = ^ /3. Its equivalent rectangular pulse P  = |t and is of the intensity p — | p . In Figure 2.23, p  e  m  isoresponse curves are plotted for the DRES panel subjected to triangular pulses and their equivalent rectangular pulses. While these curves do not collapse identically, they do compare very well. There are no discernible differences between the responses in the impulsive realm, and only small differences in the dynamic realm. In the realm that approximates quasi-static response (large values of I and small values of p ) errors arise m  because the nominal impulse is used rather than the exact value. The pulse durations in this realm are longer than the response times. As the pulse duration is made longer, the nominal impulse becomes less accurate an approximation of the true impulse, and the "equivalency" of the rectangular pulse calculated as above is further compromised. Estimating the response times of the panel and reducing the impulse and centroid values appropriately would improve the accuracy of the rectangular pulse representation, but the accuracy of the results obtained with the simple procedure presently in use and the added difficulties involved in estimating the response times make doing so hardly  2.  Chapter  500  Beams  and  One-Way  n—r  Stiffened  51  Plates  n—i  1  1  1  r  Triangular pulse  400  Equivalent rectangular pulse Q_  CL  3  0  0  CD !^ 13  (/) if) CD  ol  _^ o  200  CD Q_  100  0 0  J  I  300  I  L  600  Impulse  i  I  Response ends before r e c t a n g u l a r p u l s e is r e m o v e d i  I  900  i  i  I  1200  i  i  1500  (psi.msec)  Figure 2.23: Isoresponse curves for the DRES panel subjected to triangular and "equivalent" rectangular pressure pulses.  Chapter  2.  Beams  and One-Way  Stiffened  Plates  52  justifiable. It seems that Youngdahl's method of determining equivalent rectangular pulses is applicable to the large displacement response of singly symmetric beams and one-way stiffened plates. As RIPTAB is restricted to blast-type pulses, this method will be very useful for representing pulses which do not outwardly resemble that type.  Chapter 3 Instantaneous Mode Response of Beams and One-Way Stiffened Plates 3.1  Introduction  In Chapter 2, the response of fully clamped beams and one-way stiffened plates subjected to blast-type pulses was determined using a  vector mechanics  approach; Newton's mo-  mentum axioms were applied to derive the equations of motion for the structure. For brevity, this solution will hereafter be referred to as RIPTAB, the name of the program within which the equations of motion are solved [57]. An  analytical mechanics  approach may also be used, wherein the work of the ap-  plied forces and the energy states of the structure are considered and the principle of virtual work is applied to derive the equations of motion [36]. The analytical mechanics approach is particularly useful in the derivation of approximate solutions of blast load response. Bounds on structural response [42,52,61] are commonly derived from energy considerations, as are  mode approximations.  In the mode approximation technique, the  structure is assumed to respond in a form which is analogous to a normal mode of a 53  Chapter  3.  Instantaneous  Mode  Response  54  linear elastic structure. The remainder of this chapter is concerned with seeking the approximate solution to the beam problem of Chapter 2 using the techniques of analytical mechanics and mode approximation.  3.1.1  M o d e A p p r o x i m a t i o n Technique  In their analysis of impulsively loaded structures using elementary (infinitesimal displacement) rigid-plastic theory, Martin and Symonds [43] observed two distinct stages of response. There firstly occurred a transient response phase wherein the velocity field changed shape in time. This form of response corresponds to the travelling hinge mechanism in rigid-plastic beam bending. Afterward, a mode response phase occurred. Corresponding to the midspan hinge mechanism in rigid-plastic beam bending, mode response is characterized mathematically by a velocity field which is a separable function of the spatial and temporal variables. For a structure loaded impulsively by an initial velocity distribution matching the mode response velocity field, Martin and Symonds recognized that the structure's response would be of mode form in its entirety. Furthermore, they were able to show that the separate responses of a structure to two independent impulsive loadings would approach one another as time progressed. (This proof is summarized in Appendix D.) It was then concluded and demonstrated that, by approximating the initial impulsive velocity distribution with a distribution corresponding to the mode response velocity field, the response of the structure could be adequately approximated. The resulting mode form response was referred to by Martin and Symonds as a mode approximation. Consider a rigid-plastic structure of density p and volume V. It has generalized coordinates x and displacements u. The structure is subjected to an impulsive load which imparts upon it an initial velocity u(x, 0) = v(x). The response to this impulse is  Chapter  3. Instantaneous  Mode  55  Response  characterized by the transverse velocity u(x, t). If the impulsive velocity is approximated by u*(x,0) = a $(x) where 3>(x) is a mode shape of the beam, the entire response of 0  the beam is approximated by the mode response u*(x,£) = a(i)$(x). The kinetic energy of the difference between the exact and approximate velocity fields at the time t is given by the function A(t) = J \p{ii - a$).(u - a * ) dV ,  (3.1)  and at the time t = 0 by °  A  =  Iv  2 / 9 ( v  ~ o * ) ' ( ~ o * ) dV . Q  v  (3.2)  a  Because the exact and approximate velocity fields approach one another as time progresses, the value A must be a measure of the greatest difference between the fields. 0  Martin and Symonds reasoned that the smaller the value of A , the quicker the conver0  gence of the exact and approximate solutions. For a given mode shape 3? then, the value of ct should be chosen so as to minimize A . Setting 0  0  /  «  0  0  = 0, it is determined that  pv.&dV  = f  0  dA /dct  •  (3.3)  Jv  Given a number of mode shapes for the structure, Martin and Symonds suggested that the response would be best approximated by the mode which provided the smallest A  0  after minimization with respect to a . 0  3.1.2  Instantaneous M o d e A p p r o x i m a t i o n s  Within the scope of linear elementary rigid-plastic theory, mode form solutions which are valid throughout the entire response of the structure are available. Symonds and Wierzbicki [69] refer to such responses as  permanent  mode form solutions. For nonlinear  problems, however, permanent mode forms do not exist. In such cases, it was proposed by  Chapter  3.  Instantaneous  Mode  56  Response  Lee and Martin [38] that a sequence of  instantaneous  mode form solutions, each existing  uniquely at a given instant during the response, may be found. For a structure with the generalized coordinates x and displacements u, the response at the time t is in the mode form u(x,i) = a(*)#(x;<),  (3.4)  u(x,<) = •/?(*) u(x,<).  (3.5)  Note that the vector of shape functions $ is time dependent and hence represents an instantaneous mode. As does a permanent mode, an instantaneous mode satisfies all the field equations (compatibility, kinetics, plasticity conditions) of the structure, though only for the instant and the configuration of the structure considered. As noted by Symonds and Wierzbicki [69], "The mode form motions of either type are 'natural' responses of the structure, in the same sense as the normal modes of linear elastic structures. Thus approximate solutions based on them may assist the analyst's insight into a dynamic plastic structural response, at least as well as the data furnished by wholly numerical methods." Response based upon a sequence of instantaneous modes is referred to as an instantaneous mode approximation. An extremum principle has been derived by Lee [37] for the determination of instantaneous modes of nonlinear structural response. This principle states that, for the class of structure encountered in the present study, the instantaneous mode $(x;t) renders the functional J(*)  =  -K{t)  = D(*,t)  - E(*,t)  (3.6)  Chapter  3.  Instantaneous  Mode  Response  57  a minimum, where the kinetic energy K(t) is independent of the variation in 3?. The term D(&, t) is the rate of energy dissipation within the structure and E(&,t) is the rate of external work being done by the loads on the structure. Minimizing the functional  J maximizes the rate of change of the structure's kinetic energy. By Lee, then, the instantaneous mode is the most flexible path in configuration space u which may be taken by the structure at the instant considered. Lee's principle is apparently contradicted by Symonds and Wierzbicki [69] who suggest that, during portions of the response of some structures, the extrema corresponding to the instantaneous modes may be maxima or saddle points. It will be shown in the following section, however, that when restricted to a certain class of shape functions the instantaneous modes of rigid-plastic beams always correspond to stationary minima. The derivation of Lee's extremum principle and further discussion of the extrema's properties are included herein as Appendix E.  3.2  Instantaneous M o d e S o l u t i o n for B e a m s  The fully clamped beam problem of Chapter 2 is now revisited. The approximate nonlinear response of the beam under blast loading is derived with use being made of the mode approximation techniques described above. The beam has mass per unit length m, spans a distance 2L, and is subjected to the uniformly distributed line load q{t). Details regarding the beam's geometric and material properties are listed in Section 2.1. 3.2.1  B e n d i n g Response  Selection of the Mode Shape As has been noted by Martin [41], variational principles are difficult to apply to the analysis of rigid-plastic structures. Owing to the presence of rigid regions and singularities  Chapter  3.  Instantaneous  Mode  58  Response  in the strain rate fields, the function which extremizes the considered functional can be quite complicated. Consequently, rigid-plastic modes are generally sought by a trial-anderror approach. In the dynamic analysis of rigid-plastic beams and one-way stiffened plates (Chapter 2), a very good approximation to the bending response was found in the use of the travelling hinge mechanism. (The midspan hinge mechanism may be treated as a special case of the travelling hinge mechanism.) This mechanism was discovered to be inexact, as the yield criterion may be violated in the presence of axial forces in the beam. It is amenable to analytical techniques, however, and the photographs of Humphreys [23] show that it closely resembles the true shape of the beam during bending. The travelling hinge mechanism will therefore be used to approximate the instantaneous modes of the beam during bending response. The instantaneous velocity and acceleration fields according to the travelling hinge mechanism are  where  V(t)  and  A(t)  w(x,t)  =  V(t)<p{x\t),  (3.7)  w(x\t)  =  A(t)<j>(x;t),  (3.8)  are the midspan velocity and acceleration, respectively, and  (f>(x;t)  is an instantaneous shape function given by 1,  </)(x;t)  if 0 < x < A"(*); L-x  K  L-X(t)  ,  \iX(t)<x<L.  (3-9)  The instantaneous mode acceleration field is continuous through the beam's span, while in RIPTAB there existed a discontinuity in the accelerations at the point x = X(t), as given by equation (2.9). It is further noted that, since the shape function <j> is time dependent, V(t) and A(t) are not merely time derivatives of the midspan displacement ti ^); rather, they are separate functions of time. 7  Chapter  3. Instantaneous  Mode  59  Response  Application of the Extremum Principle Making use of the above mode shape and the previously derived kinematic relations, equations (2.12,2.13), the energy rates and, in turn, the functional J may be written as functions of the hinge position X. The rate of external work being done by a uniformly distributed line load q(t) is E = 2 f q(t)w(x,t)dx Jo  = q(t)(L + X)V,  L  (3.10)  and the rate of energy dissipation in hinge rotations 0 and extensions e is D  =  2 (M 6 X  + M9)  X  L  M -M X  = 2V •  L  L  + Ne  + NW  Tr L/ — Jv  •  (3.11)  = lm(L + 2X)V\  (3.12)  = |m(X + 2X)VA.  (3.13)  T  The kinetic energy of the beam is K = 2 f \m[w(x,t)] dx 2  L  J  0  and its rate of change is rL  K = 2  mw(x,t)w(x,t)dx  Making use of these energy rates, the function J(X) — D — E is written as J(X)  =  M -M X  + NW  L  - q(t)(L + X) V  L-X  3K 2(M -M + NW) - g(t) (L - X ) V m ' (L- X)(L + 2X) / X  2  L  2  (3.14)  1 2  Differentiating with respect to the hinge position X yields dJ_ • dX dJ dX 2  2  =  J3K  [KM -M X  L  ~  V m  _ ~  f3l[ Q(M -M Vm '  + NW) - g(t)(L - X) } X 2  (L-X) (L  +  2  X  L  3 2  + NW)(L + 5X ) - g(t)(L - X) {L- X) (L + 2X) / 2  3  (3.15)  2X) / 2  5 2  4  (3.16)  60  Chapter 3. Instantaneous Mode Response  Setting the first derivative to zero, and imposing the constraint 0 < X < L, hinge positions X leading to extrema are found: J6(M  X  X(t)  =  -M  + NW)  L  if q{t) > 6(M - M + NW)/L ; X  0,  (- )  2  L  3  17  otherwise.  It is verified that these hinge positions do in fact correspond to minima by substituting them into equation (3^16) and observing that d J/dX  2  2  > 0.  These instantaneous mode results for the hinge position X are compared with those of RIPTAB, equations (2.24) and (2.26). The instantaneous mode hinge position depends only upon the applied load and beam configuration at the time considered, whereas, in RIPTAB, the hinge position depended more correctly upon the load and deformation histories of the beam. The correct dependence upon response history presumably tends to smooth out or dampen the hinge motions, making them less responsive to fluctuations or rapid changes in the load pulse than might be predicted by the instantaneous mode solution. With the hinge motions determined, the displacement response of the beam's midspan is now considered. Recalling that J(X) = —K, and making use of equations (3.13) and (3.14), the equation of motion of the beam is found: +  m(L  2  +  XL-2X ) ^ = 3 g(t)(L -X ) 2{M -M ) 2m' L + XL — 2X ' 2 W  2  2  X  2  L  2  In the foregoing analysis, the instantaneous mode bending response of the beam has been derived with reference to the hinge stress resultants M , M and N. These x  L  resultants are determined by the beam section interaction relation and its associated flow rule, and may be written in terms of the beam section capacities, MQ and N . This 0  Chapter  3. Instantaneous  Mode  61  Response  has been done in Section 2.2 for RIPTAB, and the resultants determined for bending response therein, i.e., M  x  —M  L  = 271^/(1 +A) and TV = AiV , remain valid for the 0  instantaneous mode solution. Equations (3.17) and (3.18) will be left in their present general form, however, so that they might be utilized for forthcoming problems involving different support conditions.  3.2.2  String Response  When the deflections in the fully clamped beam become large, the beam loses its bending resistance. The beam responds as a plastic string with the hinge stress resultants M = x  M  L  =  0 and  N = N. 0  Selection of the Mode Shape In RIPTAB, the plastic string response (Section 2.4) was assumed to take the form 7TX w(x,t)  = W(t)  cos — ,  and was then governed by a linear, second order differential equation, equation (2.33). This response is immediately recognized to be of a permanent mode form. As such, it may be included in the present solution without modification. Another possibility for approximating the string response mode shape is to retain the travelling hinge mechanism from the bending response. This might at first seem to be a poor approximation; the hinge mechanism in no way resembles the response of a plastic string. It should be realized, however, that the beam's transition to string response is not as distinct and instantaneous as has heretofore been assumed. A finite amount of time is required by the beam to change its configuration from that of a hinge mechanism with concentrated deformations at the hinges to that of a plastic string with smoothly distributed deformations. During the early stages of the so-called plastic string phase,  Chapter  3.  Instantaneous  Mode  62  Response  then, the hinge mechanism may provide as good an approximation as the cosine string mode. The equations of motion resulting from the use of the hinge mechanism are the same as those derived for the bending response, but with the hinge stress resultants set to M  x  = M = 0 and N = N , L  0  Mode Transitions The transition to string response, in particular to the sinusoidal mode response, involves a discrete change in the velocity field of the beam. It is not necessarily appropriate, therefore, to keep the midspan velocity of the beam continuous through this transition. In RIPTAB, the kinetic energy of the beam was conserved instead. Conservation of kinetic energy, or some other appropriate criterion, should also be applied through the bending-string transition of the instantaneous mode analysis.  3.2.3  Solution Algorithm  A closed form solution of the hinge mechanism response of the beam, governed by the coupled equations (3.17) and (3.18), is not forthcoming. A semi-analytical algorithm for the solution of the coupled equations is now developed. The problem is discretized by dividing the time domain following the onset of the blast load into relatively short time steps of duration At. A notation is adopted whereby the time at the start of the nth step is denoted by (n — 1) At = t _\ and the time at the n  end of the step by n At = t . n  At the beginning of any time step the hinge position X  = X(t+-i)  n  may be easily  determined from equation (3.17), provided that the midspan displacement known. This is certainly the case for the first step (ra = 1, t _ n  x  W(t _i) n  is  = 0) after the onset of  response. If the time step is short enough, no great error is incurred by holding this hinge  Chapter  3.  Instantaneous  Mode  63  Response  position fixed within the step. The hinge position then being a constant, the response is approximated by a permanent mode form <f> (x). The midspan velocity and acceleration n  become V(t) = W(t) and A(t) = W(t), respectively, and the equation of motion for the step is reduced to a linear, second order differential equation: W(t)  + u W(t)  = Q (t)  2 n  -  n  Q  (3.19)  0n  where -  w  Q n { t )  =  m  °  + x L-2xiy  ( 3  N n  2m(2* + L -2X1)  =  '  Xn  _ y n  {v  3(M  -  X  "  m(L  2  + XL n  "  ( 3  2 0 )  '  2 1 )  M) L  - 2X1)  '  '  [  The hinge mechanism response of the beam is then governed by an initial value problem comprised of the above differential equation and the initial conditions for the time step, W(t -i) n  and Wft^Li).  This initial value problem may be solved in closed form. In  particular, the response values W(t ) and W(t~) may be determined. These response n  values are in turn used as the initial conditions for the following time step, t < t < t x. n  n+  By this algorithm, the response of the beam is changed from an infinite sequence of instantaneous modes to a finite sequence of modes which may be considered permanent within small time steps. The start of each new time step is, in general, accompanied by a finite change in the beam's velocity field, as is the case at the onset of string response in the cosine mode. In RIPTAB, continuity of the beam's midspan velocity through the transition into string response was abandoned in favour of conservation of the kinetic energy, which provided a good estimate of the beam's response. In the present instantaneous mode solution, modes are chosen so as to maximize the rate of change of the kinetic energy at the instant considered. Therefore, with the kinetic energy  Chapter  3.  Instantaneous  Mode  64  Response  conserved through mode changes, one should obtain an upper bound to the exact rigidplastic solution. Actually, this bounding behaviour is lost because the assumed hinge mechanism is an approximation, albeit a good one, and because of the time discretization. Nevertheless, if small enough time steps are used, conserving the beam's kinetic energy through mode changes might be expected to overpredict the displacement response of the beam. The decision to conserve the kinetic energy through a mode change is a somewhat arbitrary one; conservation of momentum or any of a number of other equally valid conditions could as easily be applied. One such condition—one observed to yield good results with instantaneous modes [37]—was introduced earlier in connection with the linear mode approximation technique. Martin and Symonds [43] measured the difference between two velocityfieldswith the function A(i), which may be thought of as the kinetic energy of that difference. For a mode change occurring between the nth and (n + l)th time steps, this function is expressed as  (3.23) By this criterion, the two velocityfieldsare most similar when A ( i ) is minimized. Setting n  dA(t )/dW(t+) n  = 0, it is determined that / <j) (x) <f> (x) dx n  n+1  (3.24)  If the load q(t) is a blast-type pulse (Section 2.1) it is ascertained that X  n  >  X . n+i  Recalling the hinge mechanism shape function, equation (3.9), the midspan velocity after the mode change becomes W(tt)  = W(t~)  •  3(£ ~ Xj ) - (L 2(L + 2X )(L-X ) 2  +l  n+1  n+1  Xf n  •  (3.25)  Chapter  3. Instantaneous  Mode  65  Response  Unlike RIPTAB, the instantaneous mode solution has no kinematic admissibility constraint requiring monotonic hinge motion toward the midspan. The load pulse q(t) need not be blast-type and cases by transposing  X  > X  n+i  X  n  n  and  is admissible. Equation (3.25) is modified for such in the numerator.  X i n+  If the string response is modelled using the permanent cosine mode,  = cos(7rx/2L),  <j)(x)  the transition to string response at the time t is accompanied by the velocity change s  derived from equation (3.24): or  ^ = ^\ni-x(t -  w  -  (3  7)]  The algorithm is now summarized. First, the time domain of the problem is discretized into steps of duration At. This duration should be small enough so that the intensity of the load and the displacement of the beam, and hence the correct instantaneous hinge position, do not change significantly within the step. The response is approximated within each step in turn. During the hinge mechanism (bending or string) response, the following procedures are performed for any (the nth) time step: 1. The time step is entered with the values W(t _i), W ^ ^ l i ) , and X _i. n  2. The new hinge position, X  n  =.X(t^'_ ), 1  n  is calculated using equation (3.17).  3. The midspan velocity is changed from W(£ ~li) to n  Wft*^)  according to equa-  tion (3.25). 4. The initial value problem, comprised of equations (3.19—3.22) and the initial conditions  W(t -i) n  and W^nt-i), is solved.  5. A check is made for the onset of string response (if still in bending) or the cessation of response. 6. The response values at the end of the step,  W(t ) n  and  W(t~),  are determined.  26)  Chapter  3.  Instantaneous  Mode  66  Response  If in item 5 above the onset of string response is encountered at some time t , the hinge s  stress states are changed appropriately and the algorithm is restarted at that time. If the string response is taken to be in the permanent cosine mode form, the midspan velocity is changed from W(tj) to W(tf) according to equation (3.26). Thenceforth, the following procedures are performed for each time step: 1. The time step is entered with the values  and  W(t _i) n  W(t _i). n  2. The initial value problem, comprised of equation (2.33) and the initial conditions and W"(t„ti), is solved.  W(t _i) n  3. A check is made for the cessation of response. 4. The response values at the end of the step,  3.3  W(t ) n  and  W(t ), n  are determined.  Examples  The response of the DRES panel to several uniformly distributed blast loads will now be determined using the instantaneous mode solution (IMS) algorithm developed above. The stiffened plate, for which the properties are given in Section 2.6 and Figure 2.9, is once again modelled as a singly symmetric beam and the pressure pulses p(t) are resolved to act as line loads q(t) = p(t) x 36 in. The results determined herein will be compared to those from RIPTAB. In doing so, several key traits of instantaneous mode response—and of the solution algorithm used to determine such response—are illustrated.  3.3.1  Rectangular Pulses  The DRES panel is subjected to two rectangular pressure pulses of intensity p and m  of duration t = 2msec. For the first pulse, p v  m  = 129 psi, and for the second, p = m  Chapter  3.  Instantaneous  Mode  Response  Time  67  t  (msec)  Figure 3.1: Midspan displacement response of the DRES panel under rectangular pulses, as calculated by the IMS with hinge mechanism string response, the IMS with a cosine mode string response, and RIPTAB.  Chapter  3.  Instantaneous  20  "i  i  i  Mode  i  i  i  Response  i—r  i  r~i  i  i  i—i—i—i—i—i—r  = 258  IMS RIPTAB  14 X  c o  ft: o  12 10  Q_  <D  8  CD C  = 1 29  \ Onset of Jj^ string response  0 0  J  L J  I  I  I  I  I I  2  j  i  i  I  3  I  i  i  i  1  4  J  I  L  Time t (msec) Figure 3.2: Position of the travelling hinges under rectangular pulses.  Chapter  3.  Instantaneous  Mode  69  Response  258 psi. With the time step At chosen to be 0.2 msec, the response of the DRES panel to these pulses has been calculated by the IMS using both the hinge mechanism and cosine mode string response models (Section 3.2.2). The midspan displacement, W(t), and the travelling hinge position, X(t), are plotted in Figures 3.1 and 3.2, respectively. Also plotted in these figures are the results obtained by RIPTAB. The lower intensity pulse, having p = 129 psi ~ 4p , induces relatively small dis0  m  placements in the panel, as seen in Figure 3.1. A final midspan displacement of about 3.1 inches is calculated. This final displacement is less than that required for string response  (W = 4.62 inches), so the mode used for string response is a moot 'consideration. Consequently, only one curve is shown for the response according to the IMS. The midspan displacements calculated by the IMS and RIPTAB are in very close agreement, with the IMS prediction slightly higher than RIPTAB's. That the hinge position is held constant through each 0.2 msec time step in the IMS is readily observed in Figure 3.2. The IMS predicts faster convergence of the travelling hinges onto the midspan than does RIPTAB, but comparison of the hinge positions is reasonably good, nonetheless. For the higher intensity pulse having p = 258 psi m  8p , larger displacements are 0  induced and a significant amount of string response occurs. Where the permanent cosine mode is used to model the string response there is good comparison between the IMS and RIPTAB results, the IMS displacements again slightly in excess of RIPTAB's. The two solutions diverge at the time t — 2 msec. As the load is removed at this time, the p  IMS hinge position jumps approximately 12 inches to the midspan, whereas RIPTAB predicts a continuous hinge motion toward the midspan. The divergence of the midspan displacements may be attributed to these differences in the hinge position. Retaining the hinge mechanism throughout the string response in the IMS leads to still greater midspan displacements and poorer agreement with the results of RIPTAB— not too surprising a result, since RIPTAB uses the cosine mode to model string response.  Chapter  3.  Instantaneous  Mode  Response  70  The two IMS responses begin to diverge when string response begins at the midspan displacement W = 4.62 inches. Despite these differences, the IMS using the hinge mechanism to model string response still predicts a final midspan displacement that is within ten percent of the RIPTAB prediction.  3.3.2  H O B #315  Test  Test #315 undertaken at DRES' Height of Burst (HOB) facility is described in Section 2.6.2. The DRES panel is subjected to a blast pulse with a peak intensity of 925 psi which decays monotonically over an overpressure duration of 3.25 msec (Figure 2.12). The response of the panel has been calculated by the IMS with a time step of 0.2 msec and by RIPTAB. The calculated midspan displacement responses and the travelling hinge positions are plotted in Figures 3.3 and 3.4, respectively. Comparison of the responses predicted by the IMS and RIPTAB is much the same as for the rectangular pulse examples. One slight difference from the previous examples is a result of the loading. Because the load pulse decays sharply with time, the IMS predicts the travelling hinges to converge onto the midspan much more quickly than does RIPTAB. Hence, the IMS and RIPTAB predictions of the midspan displacement response begin to diverge very early. Despite this early divergence, and despite the very large displacements {Wj ~ 16 inches) for which modes based on the travelling hinge mechanism are inaccurate, the IMS and RIPTAB responses are still within approximately ten percent of one another.  3.3.3  S e n s i t i v i t y to t h e T i m e I n t e r v a l A t  The stability and convergence properties of the IMS algorithm with respect to the time step At will now be examined. In the previous examples, a time step of At = 0.2 msec  Chapter  3.  Instantaneous  Mode  71  Response  Time  t  (msec)  Figure 3.3: Midspan displacement response of the DRES panel under HOB #315 pulse, as calculated by the IMS with hinge mechanism string response, the IMS with a cosine mode string response, and RIPTAB.  Chapter  3. Instantaneous  35  ~i  Mode  r  i  Response  i  i  i  i  r  "i  i  i  r  IMS _ RIPTAB  30  25  Onset of string response  20  CD  CO C  10  0 0  J  I  I  I  1  J  Time  L  t  2  J  I  L  (msec)  Figure 3.4: Position of the travelling hinges under HOB #315 pulse.  Chapter  3.  Instantaneous  Mode  Response  73  was used within the IMS. Retaining the travelling hinge mechanism throughout the string response, the IMS response of the DRES panel to the HOB #315 pulse has also been calculated using time steps of 0.1 msec and 1.0 msec, in addition to 0.2 msec. The midspan displacement responses and travelling hinge positions as computed using these three values are shown in Figures 3.5 and 3.6, respectively. The hinge position being held constant within each time step, the coarseness of each time discretization is reflected graphically in Figure 3.6. By this hinge position plot, the 1 msec time step appears to be very coarse indeed, jumping from a value of X = 32 inches ' to X = 17 inches to X = 0 in only two steps. Such large jumps in the hinge position are doubtlessly due to the rapid decay of the load pulse [p(0) = 925 psi, p(l msec) = 333 psi, p(2msec) = 130 psi]. But inspection of Figure 3.5 reveals that even this very large time step yields midspan displacement results that are quite close to those yielded by the much smaller steps. Both the midspan displacement and the travelling hinge position converge rapidly with decreasing time steps onto the true instantaneous mode solution, that for the infinitesimally small time step. It therefore seems sufficient to leave the criterion for selecting the duration of the time step in a qualitative form: the duration of the time step should be such that the change in the position of the travelling hinges within the time step is small compared to the length L. Alternatively, this criterion could be expressed as: the duration of the time step should be such that the change in the load pulse intensity p(t) within the time step is small compared to the peak intensity p . m  However, even in such cases where this criterion is violated, e.g., At = 1msec for the HOB #315 pulse, relatively good approximations of the midspan displacement response may be obtained.  Chapter  3.  Instantaneous  Mode  Response  74  Figure 3.5: Midspan displacement response of the DRES panel under HOB #315 pulse, as calculated by the IMS with hinge mechanism string response, for different time steps At.  Chapter  3.  Instantaneous  35  n  r  Mode  "i  75  Response  i  i  i  i  30  i  i  i  i  i  At = 1.0  msec  At — 0.2  msec  At = 0.1  msec  r  25  X  c o  20  00  o  °-  15  ^  10  0  0  J  L  j  _  L  1  J  L  2  J  I  I  L  Time t (msec) Figure 3.6: Position of the travelling hinges under HOB #315 pulse for different time steps At.  Chapter 4 Instantaneous Mode Response of Stiffened Plates 4.1  Introduction  Attention is now directed towards the problem of a rectangular, orthogonally stiffened plate subjected to blast loading. Such a two-way stiffened plate is shown in Figure 4.1. Very few studies of the nonlinear response of two-way stiffened plates exist in the literature. Fewer still concern the dynamic nonlinear response. What has been studied to some extent is the small displacement plastic collapse of stiffened plates. One approach to the limit analysis of stiffened plates is to smear the stiffening beam sections with the plating, thereby modelling the panel as an orthotropic plate. Manolakos and Mamalis [39] used this approach. A lower bound on the collapse load of rectangular stiffened plates was obtained by considering the equilibrium of the smeared plate, and an upper bound was obtained using yield line theory and the well-known roof-shaped displacement field. A second, more commonly used approach is to model the panel as a grillage of beams, with the plating acting as a large upperflangefor the beams. This approach is analogous to that used in the one-way stiffened plate analyses of the preceding chapters. 76  Chapter  4. Two-Way  Stiffened  Plates  77  n stiffeners y  n  I m= 1  ii ii  II  :  H I .JI II II II  II II II II  II  r  i II ii  II  II  ||  IL_  II II II II  ir nr~" ii_ J I ir nr~" IIII _ J III ir ~ir  ||  ||  II  II  ||  _ll  II  J III  II  i IIn  I  §  2I„  m  Pressure p(t) W  I I  I  I  I I  I I  I  I  I  J  J  I  Figure 4.1: Two-way stiffened plate subjected to a uniformly distributed pressure pulse. Heyman [14,15,16] first investigated the collapse loads of beam grillages. Heyman examined square grids comprised of an equal number of evenly spaced beams in each direction, the ends of which were fully clamped. Torsion connections were assumed to exist at the supports and between the beams, and the interaction relation and flow rule for combined bending and twisting of the beams were determined. With equal point loads applied at each joint, Heyman derived upper and lower bounds on the loads required for collapse. For beams of open I-section, it was determined that the contribution of the torsional resistance to the load capacity of a grillage was negligible. In comparing his theoretical results to experiments on simply supported grillages with no torsion connections, Heyman found good agreement.  Chapter  4.  Two-Way Stiffened  Plates  78  By way of a finite difference formulation, Grigorian [10,11,12] determined a lower bound on the collapse loads of uniform, orthotropic, rectangular grillages subjected to uniformly distributed point loads at the joints. Several different boundary conditions were treated. The beams were assumed to be "twistless", i.e., the torsional resistance was neglected. Chowdhury [4] determined the linear static collapse loads of stiffened plates by modelling them as "twistless" beam grillages. Uniformly distributed transverse pressure loads were distributed to the beams as uniform line loads. The complete collapse modes of the grillages were generally characterized by lines of hinges located along the panel's edges and to either side of, and parallel to, the panel's centre lines. The effect of finite displacements on the static capacities of grillages was first investigated by Martin [40], who considered a fully clamped, square grillage with three beams running in each direction. Once again, equal point loads were assumed to act at each joint. Using Heyman's procedure [15], Martin determined that the small-displacement static collapse load of the fully clamped 3 x 3 grillage would be increased by less than ten percent if the torsional resistance of the square beams were to be included. A square beam may be regarded as an extreme case when assessing the effect of torsional resistance, as beams for which the breadth exceeds the depth do not constitute efficient bending sections and are unlikely to be used. Furthermore, the effect of torsion is expected to diminish with increasing displacements and membrane effects. Martin therefore neglected torsional resistance in deriving the relationship between the grillage's load capacity and its transverse displacements. Martin also tested several grillages composed of rectangular beams of varying depth and breadth. In these tests, full torsional connections were made at the supports and between the beams. For beams of fairly high elastic bending stiffness, the tests compared well with Martin's "twistless" rigid-plastic theory.  Chapter  4.  Two-Way  Stiffened  Plates  79  Only recently has the dynamic response of grillages subjected to blast loading been investigated. Huang and Liu [22], considering the bending-only response of rigid-plastic grillages, applied uniformly distributed blast-type line loads to what were termed the main beams. These main beams were supported by otherwise unloaded cross beams, the latter assumed to be much stronger than the former. Complicated travelling hinge mechanisms were considered within each span of the main beams, requiring the response to be solved numerically; this despite the assumption of bending-only response which limits the response to infinitesimal displacements. The techniques previously developed for the analysis of beams and one-way stiffened plates are now used to develop a finite displacement dynamic analysis of beam grillages and two-way stiffened plates subjected to blast loads. Beyond the complications of nonlinear geometry and material behaviour evident in the one-way case, the response of two-way stiffened plates is further complicated by coupling between the mutually orthogonal sets of beams. The degree of coupling depends on the relative strengths of these beam sets. A limiting case exists where one set is much stronger than the other; virtually no coupling is present and the problem reduces to that of- a one-way system. Another limiting case is where the strengths of the beam sets are nearly equal so that full coupling exists between them. It is suggested that this latter case corresponds to a well-designed panel, and hence is the case of greatest engineering interest and is the subject of investigation.  4.2  Problem Formulation  A rectangular plate spans a distance 2 L in one direction, the x-direction, and a distance X  2 L in the orthogonal y-direction. The plate is stiffened by n identical and evenly spaced y  x  beams running in the x-direction and by n identical and evenly spaced beams running y  Chapter  4.  Two-Way  Stiffened  Plates  80  Figure 4.2: Effective beam sections and distribution of the masses in the grillage model of a stiffened plate. in the y-direction. The edges of the plate and the ends of the beams are fully clamped. This two-way stiffened plate is subjected to a uniformly distributed pressure pulse, p(t), as shown in Figure 4.1. It is herein assumed that the stiffeners are densely spaced and are very strong in relation to the plating. Under these conditions, the response of the panel will resemble that of a grillage rather than that of one or a number of plates. Furthermore, the strengths of the x and y-direction beam sets are assumed to be similar to each other, thus ensuring full coupling between them and precluding response in any partial collapse mechanism. In a manner which is similar to that of the previous chapters, the stiffened plate is assumed to behave as a grillage of singly symmetric beams. For purposes of determining a beam's section capacities, M  0x  and N  0x  in an x-direction beam or M  0  and N  0  in a  Chapter  Two-Way  4.  Stiffened  Plates  81  y-direction beam, the beam is comprised of one stiffener and all of the plating within onehalf of the distance to each neighbouring stiffener (or adjacent boundary). The masses per unit length of the beams are taken to be just those of the stiffeners, m and m , and x  y  the mass per unit area of the plating, m , remains distributed as such. These effective p  beam sections and masses are illustrated in Figure 4.2 along with the stress resultants M , My, N and N acting in their positive senses. x  x  y  An implication of this grillage model is that each beam has access to the full uniaxial capacity of the plating. The normal stresses in the x-direction of the plating are independent of those in the y-direction. This situation corresponds to Johansen's well-known orthogonal yield criterion. In regions of contraflexure, the Johansen criterion overestimates the strength of a ductile metal plate, which is more accurately represented by the Tresca or von Mises criterion. Use of the Johansen criterion will simplify the analysis, however, and might not adversely affect the results of the forthcoming analysis. Other assumptions regarding the beams' material and geometric properties, listed in Section 2.1, are retained. In addition, the torsional capacities of the beams are neglected, in deference to the findings of Heyman [16] and Martin [40].  4.3  Instantaneous M o d e S o l u t i o n  Considering the complexity of the beam grillage in relation to a single beam, it seems unlikely that a simple closed form solution of its response to blast loads may be found. The concept of instantaneous mode response was introduced in Chapter 3, as was an extremum principle by which the instantaneous modes may be determined. This concept and extremum principle will now be applied to the dynamic analysis of beam grillages subjected to blast loads.  Chapter  4.3.1  4.  Two-Way  Stiffened  Plates  82  S e l e c t i o n of t h e M o d e S h a p e  The instantaneous mode shapes are generally determined by extremizing Lee's functional,  J(w), within a variation of the velocity field w(x, y, t). As was stated previously, however, this variational principle is difficult to apply to the analysis of rigid-plastic structures, and rigid-plastic modes are generally sought by trial-and-error. Experience in the analysis of rigid-plastic beam structures has provided some hints about what an instantaneous mode for a beam grillage might look like: • For the bending response of a single beam, the travelling hinge mechanism was found to closely resemble the true shape of the beam. • Chowdhury found the bending-only collapse mode of a beam grillage to be characterized by lines of plastic hinges located symmetrically about the panel's centre lines and, when clamped, along the panel's edges. Each beam in the collapse mode has the same trapezium-shaped velocity field as does a beam in the travelling hinge mechanism. Therefore, instantaneous modes of the grillage during bending response are likely to be well approximated by approximate mode shapes with these trapezium-shaped beam velocity fields. Furthermore, as the grillage is comprised of several beams which will not, in general, deform at the same rate or to the same extent, there is no well-defined transition to string or membrane response for the entire grillage. The beams will become plastic strings one at a time—or a pair at a time by symmetry—and those plastic strings will be restrained by the neighbouring beams which have retained some bending resistance. The instantaneous modes are therefore unlikely to change dramatically as individual beams lose their bending resistance. Hence the same approximate mode shapes will be used throughout the response. Compare this to the case of the single beam (Chapter 3) where  Chapter  4.  Two-Way  O  Stiffened  Plates  83  plastic hinge  j-T  C  x  Y(t)  L  i — n  Figure 4.3: Travelling hinge-line mechanism for a beam grillage. the travelling hinge mechanism was used to approximate a string response mode shape. Predicted midspan displacements using this mechanism were generally less than ten percent in excess of those predicted using a more correct sinusoidal mode shape. For reasons discussed above, errors incurred by a similar mode approximation in the grillage problem should be smaller. The instantaneous modes will be assumed to conform to a travelling hinge-line mechanism, shown in Figure 4.3. Lines of plastic hinges will be formed at a time-dependent distance X(t) to either side of the y-axis, at a time-dependent distance Y(i) to either side of the z-axis, and at the supports. The response of the grillage will be w(x,y,t)  =  V(t)<t>(x,t)il>(y,t),  (4.1)  w(x,y,t)  =  A(t)<f>(x,t)ib(y,t),  (4.2)  Chapter  where and  Two-Way  4.  Stiffened  Plates  84  and A(t) are the midpanel velocity and acceleration, respectively, and  V(t)  <f>(x,i)  are instantaneous shape functions given by  ij)(y,t)  if 0 < |x| <  1,  <j)(x,t)  L  - \x\  x  { ^{y,t)  ,  L -X(t)  ifX(t)  <  (4.3)  \x\<L , x  x  1,  = <  X(t);  ifO<|y|<F(*);  L y  I L  ~  Y(t)  -  y  , ify(«)<|y|<  l y l  (4.4)  V  The beams have been numbered sequentially from one edge of the grillage to its opposite. The zth beam running in the x-direction is located along the line y  {i)  2i  = L 11 f l -- n— +) \J V y  (4.5)  x  and the jth beam running in the y-direction is located along the line x  1  y  n  +  v  (4.6)  1  The values of the shape functions on these lines are L  x  L  x  - X(t)  n  x  x  L  y  2{n  +  y  — X(t)  n  +  y  - Y{t)  2i  n  1  if j >  2(n  x  where  j  X(t),  i.e., j  x  L -Y(t) y  n  x  + l-z) + l  i  Y  y  x  (4.7)  y  x  iyit); < i < n  — i ',  x  if i > n  x  is the number of x-direction beams that lie to the x  n -j \ n -j (t),  if i <  1  +  x  l-j)  if (  < j <  if  _  L  Xj<3x(t);  1'  +  y  1, . L  2j  —  Y  (4.8)  iy{t)i  outside  of the hinge line at  is the largest integer that satisfies n  y  Jx(t)  <  + 1  L x  X(t)  (4.9)  Chapter 4. Two-Way Stiffened Plates  85  and i is the largest integer that satisfies Y  M  i  )  <  .  * + i . h ^ m .  (4 10)  The kinematics of the travelling hinge mechanism for a single beam, given by equations (2.12-2.14), may be generalized and applied to the present grillage problem. In the zth x-direction beam, the rates of rotation of the travelling hinges, 0 \ and of the x  support hinges, 6^} , are • U  (t)  X  _  •(,) _ t&(0,?('•>,*) _  — ~°Lx  —  Y  T Ijrj.  V(t)^(t)  •—  T  v\  The hinge rotation rates on a y-direction beam are Ly  Y  ±j~,  Y  —  '  y\_  Ly  Y  The rate of extension of the zth x-direction beam is =  2 ^ 0 . ^ 0 , ^ , 0  =  2  ^ ,  r  o  (  0  ^ « ,  V  )  (  4  ,  3  )  and that of the jth y-direction beam is =  2»(xW.(M)«K«".0.0  ,  2  ^  w  {  x  U  K  o  ,t).  (4.14)  Ly — Y  The kinematics of the beams are summarized by (e/9)$ - {e/0)£  =  w(0j%t),  (4.15)  (e/8)P - (6/0)%  =  w(x^,0,t).  (4.16)  The midspan displacements of the individual beams are obtained through the following integrations: u>(0,yW,<) w{xl \0,t) j  =  =  /'V(r)^(T)dr, Jo  (4.17)  f Jo  (4.18)  V{T)^\T)dr.  Chapter  4.  Two-Way  Stiffened  Plates  86  T h e foregoing kinematic analysis describes the response of the grillage i n terms of three parameters:  Y(t).  the midpanel velocity, V(t), and the hinge line locations, X(t) and  These parameters are to be solved by applying Lee's extremum principle through  a suitable solution algorithm.  4.3.2  A p p l i c a t i o n of the E x t r e m u m P r i n c i p l e  B y Lee's extremum principle [37], an instantaneous mode extremizes the functional  J(w)  = D(w,t)  — E(w,t),  where the kinetic energy is independent of the variation i n  w(x, y, t). T h e term D is the rate of energy dissipation due to plastic deformations and E is the rate of external work being done by the loads. In assuming the mode shape to conform to the travelling hinge-line mechanism, the energy rate functionals are reduced to functions of X and Y. T h e rate of energy dissipation w i t h i n the ith. re-direction beam is  2MM +  + iVWe?  where N ^ is the axial force i n the beam and X  and M\ ' r  L x  are the bending moments  i n the travelling and support hinges, respectively. Time-dependency of the stress resultants and deformation rates is implied. U p o n recalling equations (4.11) and (4.13), this dissipation rate becomes  Similarly determining the energy dissipation rate i n a y-direction  beam, a n d summing  over a l l the beams, the rate of energy dissipation w i t h i n the entire grillage is D(X,Y,t)  =  V(X,Y,t)-V(t)  (4.19)  Chapter 4.  87  Two-Way Stiffened Plates  where 1  i—\ x + n  L  x  l=ly+l  — X  Ly  —  Y  x  j^i y + n  1  "™ + 2 £  -  M  Y  j)  - M  L  j)  r  Y  L  x  - X  + M'Wx^O) ^ •  (4.20)  The hinge stress resultants are determined by the beams' interaction relations and associated flow rules, and may be written in terms of the section capacities M , N , M 0x  Qx  0y  and N . This has been done in Section 2.2 for a fully clamped beam. The stress resulQy  tants will be left in their present form and the summations left unsolved, however, so that equation (4.20) may be utilized for forthcoming problems involving different support conditions. The rate of external work done by the uniformly distributed pressure is E(X,Y,t)  f  =  p(t)w(x,y,t)dxdy  Ly  J — Ly  =  J— Lx  p(t) V(t) \ J  <p{x, t) dx f  LX  — Lix  Ly  </>(y, t) dy .  (4.21)  J— Ly  After integration, the external work rate becomes E(X,Y,t)  = £(X,Y,t)-V(t)  (4.22)  where E(X, Y, t) = p(t) ( L + X)(L x  y  + Y).  (4.23)  The kinetic energy of the grillage, which is recalled to be independent of X and Y, is given by the general formulation K(t)  =  [  Ly  J — Ly  f * lm(x,y)[w(x,y,t)] dxdy, L  J — Lx  2  (4.24)  Chapter  4.  Two-Way  Stiffened  Plates  88  T h e mass per unit area of the grillage, m(x,y), is written through the use of the Dirac delta function as n  Tlx  y  = m + m ^8(y  m (x,y)  p  - y ) + m (,)  x  6(x - x )  (4.25)  w  v  Substituting this mass density into the kinetic energy integral and recalling the instantaneous m o d a l velocity field, equation (4.1), the kinetic energy becomes K{t)  =  V T  2  T  \ m .p 1  .  rL  x  tU  I * <f> dx f " xl> dy 2  J — Lx  +ro f><*W %  n  x  (4.26)  2  J—Ly rLx  m.g^)) / n  tfdx-r  x  2  rLv  tfdy  .  Finally, w i t h the integrals and summations calculated, the kinetic energy and its rate of change are K[t)  =  K(t)  =  -IC(X,Y,t).[V(t)} ,  (4.27)  fC(X,Y,t)-V(t)A(t),  (4.28)  l  2  w here  tC(X,Y,t)  = lm (L p  (4.29)  + 2X){Ly + 2Y)  x  - 2i  Y  +  2i  2J2  ,ta 3 X  \ x + 1  L -Y  n  (  y  2?  Lj.  —  j  X,  (L  y  +  2Y)  Once again, even though the hinge locations are present w i t h i n the above expression for the kinetic energy of the grillage, by Lee's extremum principle K(t) is held constant through the variation i n X and Y. It is now recalled that J(X,  Y, t) = D(X, Y, t) - E(X, Y, t). M a k i n g use of the above  energy rates and noting the relationship between V and K i n equation (4.27), the function  Chapter  4.  Two-Way  Stiffened  Plates  89  to be extremized with respect to X and Y is j(x,Y,t)  =  y^w-  V(X,Y,t)-S{X,Y,t)  (4.30)  y/>C(X,Y,t)  It is unlikely that the extrema of J(X, Y) may be found analytically. The nature of the extrema, however, may be inferred from the single beam analysis of Section 3.2.1, in which the extrema were shown to be stationary minima. The presently used travelling hinge-line mechanism is very similar to that used for the single beam. It is therefore likely that the present extrema are also stationary minima, provided they do not coincide with a beam line, i.e., provided that X ^ x^ and Y ^ y('\ Along the beam lines, the energy rate functions and, hence, J(X, Y, t) do not have continuous derivatives, so extrema along the beam lines are nonstationary. It was shown by Symonds and Wierzbicki [69] that nonstationary extrema must be minima. Therefore, a numerical procedure which searches for the minimum of the function J(X, Y, t) will be required.  4.3.3  Solution Algorithm: R I P T A B 2  The solution algorithm developed for the present grillage problem follows from that for the single beam case. The time domain of the response is divided into relatively short time steps of duration At. The notation of Section 3.2.3 is retained. The time at the start of the nth time step is denoted by t _ = ( — 1) At and the time at the end of the n  n  x  step by t = n At. n  It shall be assumed that, at the beginning of any time step, the displacement and velocity fields of the grillage,  w(x,y,  t _i) n  and  w(x,y,t _i), n  is certainly the case for the first step (n = 1, t _ n  x  are fully determined. This  = 0) after the onset of response, as  the grillage is initially undeformed and at rest. Through the interaction relations and flow rules, the hinge stress resultants are also known. The hinge positions and  Y  n  = Y(t^_ ) x  may then be determined by minimizing the function  X  J(X,  n  =  X(t*..i)  Y,t _ ) n  x  in  Chapter  4.  Two-Way  equation  (4.30)  Stiffened  Plates  90  using any of a number of numerical methods, e.g., gradient methods. The  hinge positions are then held constant through the duration of the time step. Within the time step, therefore, the response is of the permanent mode form with V(t) = W(t), A{t)  and  = W(t)  w(x,y,t)  =  W{t)<f> {x)i> {y),  (4.31)  w(x,y,t)  =  W(t)<j> (x)^ (y),  ( -32)  n  n  n  4  n  where <f> (x) and ^„(y) are obtained by inserting the values X(t) n  into equations  and  (4.3)  = X  n  and Y(t)  = Y  n  The displacements at any time t within the time step are  (4.4).  therefore w(x,y,t)  where AW(t)  =  w(x,y,t -i)  +  =  wix^y,^)  + <f, (x)tb (y)AW(t),  n  w(x,y,r)dT  f n  (4.33)  n  = ^ j / ' H^(r)(fT is the change in the midpanel displacement since the  beginning of the time step. The energy rates are now expressed in terms of this modal response. The rate of change of the kinetic energy of the grillage is K{t) = 2K, W(t) W(t), where K, is obtained n  by inserting X  n  £ (t) n  is D(t)  and Y  where S (t)  W(t)  n  V (t) n  n  n  (4.23).  E=  The energy dissipation rate  and  W(t),  =  Similarly, the external work rate is  (4.29).  from equation  = £(X ,Y ,t)  n  = V {t)  into equation  n  n  2> (*+_i) + n  t=i  [v {t)-V {tU)\ n  x ~r 1  L  n  , „  (4-34)  n  —X  x  L  n  y  ^ NQ,y V)-««(o,y ' ,U)] i)  i=iy+l  (,  (,  T  -  X  )  M  — Y  n  Chapter  Two-Way  4.  Stiffened  +  ^  1  n  9  \ i *  91  N^Hx^M-wjx^^t^)]  2J  j=l V  ,  Plates  L  x  -^n  ^3/  /V^h( :0),0,0- (x0),0,^)] 3  W  Recalling equations (4.7), (4.8) and (4.33), (4.35)  = 2>„(C_i) + fn*W(t)  V {t) n  where 4 —  L V i y / 2i \2 (') \L — Y J \n + 1/  /  X  y  n  n  i = 1  ^i/ — Yn \Lx — X J n  =1  + 1  x  - \n + 1J  L —Y j j y  y  (')  X _ t r  L —Xn  x  x  "  2  y  n  =  x + 1  Conservation of energy requires K(t) = E(t) — D(t) or )C W(t)  = S (t)  n  Noting that  W{t)  = AW(t),  n  - V (t+_r)  - F AW(t).  n  n  (4.37)  the equation of motion for the grillage can be written:  AW(t)  + u AW(t)  = Q (t)  2 n  n  - Q  0n  ,  (4.38)  where u  2 n  Qn(t)  =  (4.39)  7JK , n  = £n(t)/JC ,  (4.40)  n  Qon = V {t$_ )IK . n  X  N  (4.41)  The response of the grillage during the time step is then governed by an initial value problem comprised of a linear, second order differential equation and the initial conditions AW(t -i) n  —  0 and AW(t _i) n  = W(i^-i)-  in closed form. In particular, the values  This initial value problem may be solved  AW(t ) n  and W(t~) may be determined. The  Chapter  Two-Way Stiffened  4.  Plates  92  displacements and velocities at the end of the time step, w(x, y, t ) and w(x, y, t~), follow n  and provide the initial conditions for the next time step. The start of each new time step is generally accompanied by a finite change in the grillage's mode shape. The kinetic energy of the difference between the velocity fields before and after the mode change at the time t is n  fLy L/y  /  fLx fLix  I  -Ly  J — Lx  = if  m(x,y)  [w(x,y,t~)  - w(x,y,t+)}  ^ ^,y)[W{t-)<f> ^n-W{tt)K ii>n i]  LV  n  J — Ly  +  +  dx dy  2  dxdy  2  .  (4.42)  J— Lx  The two velocity fields are most similar when A ( f ) is minimized. Setting the derivative n  dA(t )/dW(t+)  to zero, it is determined that  n  fLy Liy  /  fLx fL,  x  \  ™,<f> <f> il; ip n  (•Ly  /  /  /  J—Lit  m[(j) ^ ] n+1  gration. Denoting the lesser and the greater of X and the lesser and the greater of Y and Y i  /  fLy •Ly Uy  /  n  m^> (j) ip ^ n  fLx J — Lx  «/ — L{L X  n  2  (L  x  2  X  — X) — —  L  Y1  E fri  2l \n Tl)  y  2  2  {Ly - Y ) - \{L  2  2  2  y  X  L  2  •-  —X) 2  - Y) 2  2  — Y\  2  X\ Ly  — Fx ' L -Y  (Ly -Y )-l(Ly-Y Y 2  y  y  Ly  2  L  x  + 2m  \  2  (4.44)  -X)  \(L  2  x  %  as X\ and X , respectively,  n+i  as Yi and Y , the numerator is  n+  x  2  dx dy  dx dy  n+1  - X ) - \{L _ Y?\ - , L — X\  2  m, (T.  + 2m.  n+1  2  and X  n  X  n+1  The numerator requires a lengthy inte-  n  rL  dx dy  n+1  J — Lx  The denominator is recognized as being JC +i.  Ly  n  (4.43)  tLx  J—Lx  -Ly  n+1  2  Ly - Yl  y  2  XX ,. ^ =1  +1  Ly n +l ' x  L -Yi y  +  2iYl  Chapter  4.  Two-Way  Stiffened  Plates  93  The algorithm is now summarized. First, the time domain of the problem is discretized into steps of duration At. This duration should be small enough that the intensity of the load and the displacement of the grillage, and hence the correct instantaneous hinge positions, do not change greatly within it. The following procedures are performed for the nth time step: 1. The time step is entered with w(x, y, t _i), n  2. The new hinge positions, minimizing the function  X  n  w(x, y, t _i), n  and  = X(t*-\)  X -\ n  and Y _i determined. n  = F(i "_ ), are calculated by  Y  n  n  l  1  of equation (4.30).  J(X,Y,t^_ ) l  3. The midpanel velocity is changed from  W(t~_ ) x  to  W(t^_ ) x  according to equa-  tions (4.43) and (4.44). 4. The initial value problem, comprised of equations (4.38-4.41) and the initial conditions AW(i _i) = 0 and AW"(*„_i) = n  W{t+_ ), x  is solved.  5. A check is made for a change in the stress states of the plastic hinges (according to the interaction relations) or the cessation of response. 6. The response at the end of the step, equation (4.33) and from  W(t~)  =  w(x, y, t ) n  and  w(x,y,t~),  is determined from  AW(t ). n  If in item 5 a stress state change is identified, the current time step is ended according to item 6 and a new step is begun. Thus, the time steps are not all of the uniform duration At initially chosen. This algorithm has been coded into the FORTRAN 77 program named RIPTAB2 [58]. The instantaneous mode solution for the response of beams and one-way stiffened plates, developed in Chapter 3, is incorporated into RIPTAB2 as the special case n = 1, n = 0. x  y  Chapter  4.  Two-Way  Stiffened  4.4  Examples  Plates  94  The only other investigation of the large displacement, dynamic plastic response of twoway stiffened plates known to the author is the numerical study conducted by Koko [35], in which the nonlinear super finite element code NAPSSE was developed. Two examples from Koko's thesis will be examined using RIPTAB2, and comparisons will be made between these results and those of NAPSSE.  4.4.1  DRES1B Panel  For investigative purposes, Koko analysed a hypothetical two-way stiffened version of the DRES panel. Shown in Figure 4.4 and designated the DRES1B panel, this two-way stiffened plate has the same overall dimensions as the DRES panel (see Figure 2.9) and is also fully clamped at the boundaries. In addition to the four transverse stiffeners, the DRES1B panel has one centrally located longitudinal stiffener of identical section. This section is somewhat lighter than that of the DRES panel stiffeners. The panel is made of an elastic-plastic-strain hardening material with an elastic modulus of 30 x 10 psi, a 6  small strain hardening modulus of 18 x 10 psi, a yield stress of 54,400 psi and a mass 4  density of 0.733 x 10~ lb • s /in . For analysis using NAPSSE, Koko [35] modelled each 3  bay of the panel,  e.g. BCFE,  2  4  by one 55-degree-of-freedom plate element and each beam  span, e.g. B C or BE, by one 10-d.o.f. beam element. Koko computed the response of the DRES1B panel when subjected to the pressure pulse from the HOB #315 test (Figure 2.12). Koko's NAPSSE analysis may not be of benchmark quality, due to his use of a relatively coarse finite element mesh, but it should at least provide a reasonable estimate of the panel's response. The response of the panel has also been computed using RIPTAB2, which assumes that the response is rigid, perfectly plastic. The results of these analyses are herein displayed and compared.  Chapter  4.  Two-Way  Stiffened  95  Plates  48 in  48 in  Section a-a:  0.28 in  0.25 in  5.395 in  1 2.95 in  0.5 in  Figure 4.4: DRES1B stiffened plate geometry.  Figure 4.5: Displacement response of the DRES1B panel under the HOB #315 pressure pulse.  Chapter  4.  Two-Way  Stiffened  Plates  Figure 4.6: Displacement profiles of the longitudinal stiffener  97  ABCD.  Chapter  4.  Two-Way  0  Stiffened  Plates  8  16  Distance  24  from centre  32 line y  40  48  (in)  Figure 4.7: Displacement profiles of the transverse stiffeners BE and CF.  Chapter  4.  Two-Way Stiffened  99  Plates  The RIPTAB2 predictions of the displacement responses of the beam intersection points B and C are plotted in Figure 4.5. The response is predicted to occur over a period of 6 msec, resulting in permanent displacements of 17.5 inches for point B and 10.6 inches for point C. During the initial 1.5 msec of the response, the displacements of  B  and C are identical; the travelling hinges in the beam  points, moving continuously from  X(0)  =  82 in to  ABCD  lie outside of both  = 65 in. This can be seen  X(1.5)  in the displaced profiles of the beam in Figure 4.6. The displacement profiles follow a smooth curve in the range  X(0)  < x < X(1.5).  At t = 1.5 msec, the travelling hinges  jump instantaneously to X — 35 in where they remain for the following 0.5 msec. This is evidenced by the small "kink" in the displacement profiles at x — 35 in. At t — 2 msec, the travelling hinges jump to X = 0 where they become stationary for the remainder of the response. The travelling hinge in the beams BE and CF move continuously from their initial locations at Y = 36 in to their final locations at Y = 3.5 in, as evidenced by the smooth displacement profiles shown in Figure 4.7. The reponse calculated by Koko using NAPSSE [35] is also plotted in Figures 4.5-4.7 for comparison. Positive aspects of this comparison are found in the beam displacement profiles of Figures 4.6 and 4.7. Beam  ABCD,  modelled with three elements, clearly  deforms according to the travelling hinge mechanism, the hinges moving continuously inward and bounding a flat midspan segment. The travelling hinge mechanism is less distinct in the one-element models of the beams BE and CF, but it is discernible. O f equal significance, beams BE and CF appear to deform according to (nearly) the same shape function so that  w(x ,y)/w(x ,y) BE  CF  fa w(x ,0)/w(x ,0). BE  CF  These response  characteristics are exactly those of the travelling hinge line mechanism (Section 4.3.1) assumed by RIPTAB2. Differences exist in that the hinge motions predicted by NAPSSE, in comparison to those predicted by RIPTAB2, seem damped. According to RIPTAB2,  Chapter  4.  Two-Way  Stiffened  Plates  100  the travelling hinges approach the midspan much more quickly, their locations being dependent only upon the instantaneous values of the loads and displacements whereas they are more correctly dependent upon the load and displacement  histories.  This observation  is consistent with that of Chapter 3 for the single beam case. Another aspect of the RIPTAB2 and NAPSSE computations for which there is good comparison is the response time of the panel. RIPTAB2 predicts the panel coming to rest after approximately 6 msec of response. Similarly, the peak displacements of the panel are calculated by NAPSSE to occur 5-6 msec after the onset of the pulse, after which a small amount of elastic oscillation occurs. Comparisons of the displacement magnitudes predicted by RIPTAB2 and NAPSSE are less favourable. The permanent displacements of the points B and C from NAPSSE are only 13.4 inches and 9.5 inches, respectively, compared with 17.5 inches and 10.6 inches from RIPTAB2. In part these rather large differences can be attributed to RIPTAB2's extremum principle seeking out the most flexible path for the structure. But it is likely that the discrepancies are in larger part due to the parameters of the problem. The formulation of RIPTAB2 assumes that the stiffeners are densely spaced and that the strengths of the x and y-direction beams are similar, whereas the DRES IB panel has only one longitudinal stiffener and each transverse beam is 3.5 times stronger than the longitudinal beam. Despite these differences, the basic characteristics of the response of DRES1B panel are the same for the NAPSSE and RIP TAB 2 analyses.  4.4.2  Square Stiffened P l a t e  A four metres square plate is stiffened by three identical and evenly spaced T-beams running in each direction, as shown in Figure 4.8. The edges of the plate and the ends of the beams are fully clamped. The panel material is elastic-plastic-strain hardening with  Chapter  Two-Way  4.  Stiffened  Plates  101  4m  II  II  II  c\r—E  m  n i  II  ii__  II  II HI  II i  I  r  ~  T  f I ? i—II—i  Pressure p{t)  I  I II  I  D  II  n  I  .007  4m  m\  III  i n  Section a-a:  I  I I  "  — I 0.128.0107 -J  mm l~l  Figure 4.8: Square stiffened plate. an elastic modulus of 210,000 MPa, a strain hardening modulus of 1250 MPa, a yield stress of 375 MPa and a mass density of 7900 kg/m . The latter two values are used in 3  the rigid, perfectly plastic RIPTAB2 model of the panel. The panel is subjected to a uniformly distributed rectangular pressure pulse having an intensity of 2 MPa and a duration of 2 msec. The transverse displacements of the points A, B and C (Figure 4.8) as calculated by RIPTAB2 are plotted in Figure 4.9, and the permanent displacement profiles of the beams A B D and B C E are plotted in Figure 4.10. According to RIPTAB2, the displacements of A, B and C are identical during the first 2 msec of the response. During this time, the travelling hinge lines are roughly stationary and are located to the outside of the beams BCE, as evidenced by the kinks in the beam displacement profiles (Figure 4.10) at 1350 mm from the panel's  Figure 4.9: Displacement response of the square stiffened plate.  Chapter  Two-Way  4.  Stiffened  Plates  103  300  E E  200  CD  E  ®  100  0 0  I  I  I  i  I  500 Distance  I  I  I  i  L_i  i  i  i  I  L  1000 1500 f r o m centre —line (mm)  2000  Figure 4.10: Maximum displacement profiles of the square panel's stiffeners. centre lines. Afterward, the hinge lines move inward of the beams B C E and are then located approximately 100 mm from the centre lines. The displacements of A, B and C diverge. The permanent transverse displacement of the midpanel point A, reached after a response time of 8 msec, is approximately 280 mm; the permanent displacement of B is 170 mm; and that of C is 110 mm. The NAPSSE analysis considers one quarter of the panel. Each bay of the panel, e.g. BCED,  is modelled by one 55-degree-of-freedom plate element and each beam span,  e.g. BC, by one 10-d.o.f. beam-element. The material parameters are as listed at the beginning of the section. Results from the NAPSSE analysis are also plotted in Figures 4.9 and 4.10. The agreement between the NAPSSE and RIPTAB2 predicted displacements of A, B and  C is very good. NAPSSE predicts the displacements of the three points to be similar  Chapter  4.  Two-Way  Stiffened  Plates  104  (though not identical) during the first 2 msec of the response, after which the displacements diverge as predicted by RIPTAB2. Both analyses predict response durations of approximately 8 msec. There is also reasonably good agreement between the predicted permanent (or, in the case of the NAPSSE analysis, maximum) beam displacement profiles. It is difficult to distinguish any evidence of hinging action in the NAPSSE profiles, which could easily be due to the rather coarse two-element discretization of the beams. Nevertheless, the predicted maximum displacements from NAPSSE and RIPTAB2 are close for all points on the beams.  0  Chapter 5 Partial End Fixity 5.1  Introduction  In the preceding chapters, stiffened plates subjected to blast loads were modelled by singly symmetric beams or by grillages comprised thereof. The solutions obtained for their response were based on the assumption of complete fixity, both axially and rotationally, of the beams at their boundaries. Substantial increases in the load-carrying capacities of beams are afforded by such boundary conditions, particularly by in-plane or axial constraints which induce load-resisting axial tensions in the beams. However, such fixities are idealized states which are very difficult to achieve in practice. The subject of partial end fixity of rigid-plastic beams, and the influence thereof on the static load-carrying capacity of such beams, has received some attention in the literature. Jones [24] conducted a theoretical investigation of the effects of axial displacements at the boundaries of rotationally fixed and free rectangular beams. On the basis that the extension of an axially restrained beam is proportional to the square of the maximum transverse displacement of the beam, he assumed that the axial displacement at the boundaries would likewise be so proportional. By varying the constant of proportionality, a beam with ends of some form of axial fixity ranging from completely free to completely fixed could be modelled. Jones concluded that "remarkably small in-plane displacements 105  Chapter  5.  Partial  End  Fixity  106  at the supports...can change the response from that of an axially restrained beam, with considerable membrane strengthening beyond the limit load for moderate lateral deflections, to that of a freely supported beam, with no increase in load-carrying capacity beyond the limit load." Whether these small in-plane end displacements will in fact occur was not discussed. Almost certainly they will not occur in the manner assumed by Jones. Nevertheless, it must be recognized that this investigation was largely exploratory in nature and that any shortcomings in the model do not invalidate a second conclusion: "It is therefore clearly imperative for designers to know precisely the characteristics of the axial boundary restraints." Hodge [17] recognized the limitations of Jones' model, and examined the problem with a more physical basis. Hodge assumed elastic axial restraints at the boundaries so that the in-plane end displacements were proportional to the axial force in the beam. By varying the stiffness of these restraints the boundaries could be modelled as axially free (zero stiffness) through to fully axially restrained (infinite stiffness). Hodge was able to establish a relationship between the load-carrying capacities and the transverse displacements of these elastically restrained beams. Tin-Loi [71] has recently extended Hodge's model by considering variable rotational restraints at the boundaries. If M is the moment capacity of the beam under a given axial force, yielding is assumed to occur at the supports under the moment pM, where 0 < p < 1. For certain structural configurations Hodge's elastic axial restraint model may well be the most appropriate. In particular, Hodge's example of a beam which is pin-supported atop two elastic columns [17] seems well suited to analysis by his approach. In many cases, however, the partial end fixity of the beam is not related to the characteristics of the supporting structures nor directly to those of the beam, but rather to the characteristics of the connections between the beam and its supports. These connections are typically  Chapter  5.  Partial  End  107  Fixity  bolts or welds, and it is often difficult to provide such connections with plastic capacities which meet or exceed those of the beam. Tin-Loi [71] accounted for reduced rotational capacities at the beam ends in his analysis, but not for reduced axial capacities. He, as well as the others, allowed the full membrane or string capacity of the beam to be developed at sufficiently large transverse displacements. Axial forces in the beam should be restricted to those permitted by the connections. It is proposed herein that these connections be modelled rigid-plastically in much the same manner as have the beams, with their own bending moment and axial force capacities and interaction relations. This chapter will outline the necessary modifications to RIPTAB2, the instantaneous mode solution for the response of stiffened plates derived previously. The discussion will be centred upon the single beam model which represents one-way stiffened plates. It is understood, however, that the results derived herein are applicable to any individual beam in the grillage model for two-way stiffened plates.  q(t) per unit length "  "  i ^  e.g. bolted connection 2L  Figure 5.1: Partially clamped beam subjected to a uniformly distributed load pulse.  Chapter  5.  Partial  End  Fixity  5.2  M o d i f i c a t i o n s to R I P T A B 2  108  A singly symmetric beam spans a distance 2L and is subjected to a uniformly distributed load pulse q(i). The ends of the beam are no longer fully clamped; rather, they are connected to rigid supports by way of bolts or welds, as shown in Figure 5.1. These connections between the beam and the supports are idealized as rigid-plastic links of zero length. The instantaneous mode response of a beam is assumed to be in the travelling hinge mechanism, wherein travelling plastic hinges are formed at a distance x = X(t) to either side of the midspan, and stationary plastic hinges are formed near the supports at x = L. If the bending moment and axial force capacities of the links exceed those of the beam, the stationary support hinges are located within the beam itself. In effect, the ends of the beam are fully clamped, and the response of the beam is as determined previously. If the capacities of the links are less than those of the beam, however, the support hinges are located within the links, and the capacities of these links govern the response of the beam near the boundaries.  5.2.1  Kinematics  Figure 5.2 shows the deformed configuration of a partially clamped beam. The travelling hinges in the beam undergo centroidal extensions at the rate e  x  and rotations  (positive in sagging) at the rate 0 . Plastic hinges are located within the support links, X  and these support hinges undergo centroidal extensions at the rate e and rotations at L  the rate 6 . The deformations of the support links, e and 0 , may alternatively be L  L  L  interpreted as boundary slippages, an axial displacement U of the section centroid and a rotation 0 as defined in Figure 5.2. Making use of the previously derived kinematic relations for the travelling hinge mechanism, equations (2.12-2.14), the rate of rotational  Chapter  5.  Partial  End  109  Fixity  Figure 5.2: Travelling hinge response of a beam with partial end fixity, slippage of the supports is  W  & = -e  = —  L  (5.i)  ,  and the rate of in-plane slippage is U  or  = e  L  = 8 (e/0) L  =  L  -  W L  (e/6) _ ' ^  L  v  (5.2)  u = = i {m -w] . iL  L  x  (5  .  3)  Equations (5.2) and (5.3) may be combined into the more general equation, U  where either T = 0 and T = x  2  =  (e/0)  L  K  L  _  x  2  )  (5.4)  .  or T i = 1 and T = 2  (e/0) . x  The rotational end  slippage rate is now fully determined in terms of the response parameters W and X. To determine the in-plane end slippages, however, either of the values be determined.  (e/0)  x  or  (e/0)  L  must  Chapter  5.2.2  5.  Partial  End  Interaction  110  Fixity  Relations  As plastic hinges are formed within the support links, the yield curve and flow rule for these links must now be considered in addition to those for the beam. Typical yield curves for the beam and links are shown in Figure 5.3. Only the M > 0 stress space is considered for the beam yield curve since the span hinges will be in sagging. Likewise, only the M < 0 stress space is considered for the support link yield curve since the hinges within the links will be in hogging. As introduced in Section 2.2, the yield curves are approximated by linear segments in order to linearize the equations of motion and to generalize the analysis. This approximation is shown in Figure 5.4. Three parameters are defined: the now familiar symmetry parameter A, a moment capacity reduction factor p, and an axial capacity reduction factor v. The latter two parameters represent the plastic capacities of the support links divided by the respective beam capacities. It is expected that the support connections in some way reflect the beam section geometry, so the beam symmetry parameter A is also applied to the support links. This expectation might also suggest that the reduction factors p. and v should be approximately equal, but for the sake of generality they are left as independent of one another. These reduction factors are limited to the ranges 0 <  p  <  0 <  v  <  1+  |AH  1  (5.5) (5.6)  since the plastic capacity of the beam cannot be exceeded. Four distinct cases of the linear yield curve approximation may be identified: (i) 0 < A < v, (ii) 0 < v < A, (iii) — v < A < 0, and (iv) A < —v. These four cases are illustrated in Figure 5.5. In contrast to a fully clamped beam (page 18), the response of a partially clamped beam is affected by the beam's orientation with respect to the direction of the pulse (i.e., A > 0 or A < 0). Within these cases are found seven possible stress states,  Chapter  5. Partial  End  Fixity  Figure 5.4: Linearized three-parameter yield curves.  111  Chapter  5. Partial  End  Fixity  M/Mo  (iv) A < -v Figure 5.5: Four cases of the linear yield curve approximation.  Chapter  5.  Partial  End  113  Fixity  AA' through GG', which are also illustrated in Figure 5.5. The existence and sequence of occurrence of the stress states varies between the four cases and with the relative values of  fi  and  v.  The stress state A A' exists for the case where 0 < X < u, and corresponds to the hinge stress resultants N  =  -AiV ,  M  0  = Mo,  x  M  L  =  -fiMo  y  v{l  "  A  +  From the associated flow rule for the section (Appendix A), the plastic deformation vectors must be outwardly normal to the yield curve at these points, requiring the relative rates of hinge deformation to be *  <(m*< v  JV„(1 - A) -  ' "  -  K  iV (l + A) ' 0  and { e / 6 ) L  ~  uN (l 0  + A) •  Recalling equation (2.14), the range of displacements for which this stress state is valid is  o <  W  < ^ \JLZJL  and recalling equation (5.2), the rate of change of the in-plane end slippage of the beam is •  HMQ  =  VN {1 0  W  + \)  L - X '  The parameters for equation (5.4) are then T! = 0 and T = 2  HMQ/vN (l Q  + A).  The analyses of the other six stress states are similar to that above. The sequence of occurrence and ranges of valid displacements for all seven possible stress states are summarized in Table 5.1, where a dot indicates the existence of a stress state for the particular case under consideration. For example, a beam with 0 < A < v responds with  Chapter  5.  Partial  End  114  Fixity  Table 5.1: Sequence of occurrence of the stress states for a beam with partial end fixity. Stress State  W  AA'  wmax  BB'  w  CC  w  v  0<  v  <A  -v  •  x  w  3  w  3  oo  •  •  •  •  < A< 0  A<  •  2  EE'  GG'  •  x  DD' FF'  0 < A<  •  • •  •  CO  Mo  v —p  Wo =  LKi + A) W  3  = Mo  N  0  i/(l - A)  -v  fj, — v  + 1+ A  the the hinge stress state AA' until the midspan displacement is W ; the stress state x  then jumps to DD', where it remains until the midspan displacement W is reached; and 3  finally the stress state FF' exists for all remaining response. The hinge stress resultants and the in-plane end slippage parameters, T i and T of equation (5.4), are summarized 2  in Table 5.2.  5.2.3  Integrating the E n d Slippage Rates  RIPTAB2 divides the response period of the beam into finite time steps of duration At. Within a time step, say the nth, the hinge position is held constant with the value X. n  A change in the stress state signals a change in the time step, so that the only  time-dependent terms within the expressions for the end slippage rates, equations (5.1) and (5.4), are W(t) and W(t).  Within the nth time step, then, integrating the end  Chapter  5. Partial  End  115  Fixity  Table 5.2: Stress resultants and end slippage coefficients for a beam with partial end fixity. Stress State  N/N M /Mo 0  AA'  -A  BB'  —v  x  v-A v{\  + A)  WV (1 + A) 0  Mo  1+A  7V (1 + A) 0  1 + \v 1- A  CC  Ma  7V (1 - A) 0  Mo  1 - At/ 1+A  DD'  7V (1 + A) 0  p+A  -A  EE'  WV (1 - A) 0  1-  FF'  Mo  v  N (l + A)  ITA  GG'  0  Mo  1-A  7V (1 - A) 0  rotation rate gives e(t)  =  r* •  / Q(r)dr  JO  =  0(^0  =  0(<„-l) +  +  II  , W(T) dr y iv — A n W(t)-W(t ^)  (5.7)  n  L  — vY  n  and integrating the in-plane end slippage rate gives U(t)  = = =  ftj^dr  Jo U{t _ ) n  x  + — 2 — +  W  /  ^ - ™ ^  W(T)W(r)dT {l  Tl  [W(0  - — 2 -  /  lV(r)dT  + W('n-l)] - T } . 2  (5.8)  Chapter  5.  Partial  End  5.3  Examples  116  Fixity  The modifications for partial end fixities, as outlined in the preceding section, have been coded into the FORTRAN 77 program RIPTAB2 [58]. Although RIPTAB2 is capable of modelling two-way stiffened plates, attention has been and will be focussed on the one-way case, as the effects of partial end fixities are more clearly illustrated when a beam is analysed in isolation. Some examples of blast-loaded DRES panels (Section 2.6) with partially clamped edges will now be considered.  5.3.1  Rectangular  Pulses  The DRES panel is subjected to two rectangular pulses of constant intensity p and of m  duration t = 2 msec. For the first pulse, p = 129 psi, and for the second, p = 258 psi. p  m  m  The edges of the panel are partially clamped, and so the rigid-plastic beam and support link model, RIPTAB2, is used. It was previously noted that the midspan displacement response of a partially clamped beam, in contrast to that of a fully clamped beam, is affected by the beam's orientation with respect to the direction of the load pulse, i.e., whether A > 0 or A < 0. The nature of this beam orientation effect is investigated in these examples. The pressure pulses are taken in turn to be incident on the plating side (A = 0.538) and on the stiffener side (A = —0.538) of the panel, so that four load cases are considered in all. Axially Constrained, Rotationally Free Edges  The edges of the DRES panel are initially assumed to be fully axially constrained but rotationally free. In the present beam model, these end fixities correspond to the values  p = 0 and v = 1. The midspan displacement responses and the travelling hinge positions as calculated by RIPTAB2 (with At = 0.2 msec) are plotted in Figures 5.6 and 5.7,  Chapter  5. Partial  15  i  End  i  i  117  Fixity  r~n  1—i  1  i — i — i  1—i  1—i  1  1  1  r~  RIPTAB2 - loaded on plating side  14  RIPTAB2 — loaded on stiffener side  13  FENTAB - loaded on plating side  12  FENTAB - loaded on stiffener side  1 1  p = 258 psi m  10 9 CD O _D D_ 00  Q c  D D_ 00  8 7  6 5 4 3 2 1 0 0  2 . 4  J  6  I  I  8  Time  I  I  10  I  I 12  I  I 14  t (msec)  L  16  J  I 18  L  20  Figure 5.6: Midspan displacement response of the rotationally free, axially restrained DRES panel under rectangular pulses of 2 msec duration.  Chapter 5. Partial End Fixity  35  "i  i  i  118  i  i  i  i  i  i  1  1  r  _ Loaded on plating side (A=.538) . Loaded on stiffener side  30  25  -  (A=-.538)  p =258 psi m  X  c o  20  00  o  a- • 15 <u oo c  E  p =129 psi m  10  0 0  J  L  1  I I -  L  J_  I  I  L  2  Time t (msec) Figure 5.7: Travelling hinge position in the rotationally free, axially restrained DRES panel under rectangular pulses of 2 msec duration.  Chapter  5.  Partial  End  119  Fixity  M/MQ i.  E  A  A'  F,F'  E'  (a) A = 0.538  (b) A =  Figure 5.8: Yield curves for the cases A = ±0.538, p  F, F'  ~ N / N  0  0.538 0 and  = 1.  v  respectively, for all four load cases. In the displacement plots of Figure 5.6, the panel is observed to be stiffer when loaded from the stiffener side (A < 0) than when loaded from the plating side (A > 0). The yield curves corresponding to the values A = ±0.538, p = 0 and v = 1 are plotted in Figure 5.8. Because p = 0, the support link yield curves have been reduced to the N/N  0  axes and the points of slope discontinuity C and D' (Figure 5.4) have vanished.  Therefore, the stress states C C and DD' do not exist. Referring to Figure 5.8 and Table 5.2, the sequence of stress states is AA' or EE',  M  L  = 0, followed by FF' where N = N and M 0  X  where N = — AiV , M 0  =M  L  X  = MQ and  = 0. When loaded from the  plating side (A > 0), the panel is therefore initially subjected to a compressive axial force which tends to drive the displacements rather than resist them. A second consequence of this compressive axial force is evident in Figure 5.7, in which the travelling hinges are seen to be driven outward to the supports rather than inward to the midspan. Such hinge motions preclude the modification of the original beam solution RIPTAB, for which they are kinematically inadmissible (Section 2.5).  Chapter  5.  Partial  End  120  Fixity  The beam finite element program FENTAB is capable of modelling pinned-axially constrained ends. The midspan displacements obtained from FENTAB are also plotted in Figure 5.6 for comparison with the RIPTAB2 results. Final midspan displacements once again agree to within ten percent, and the beam orientation effect predicted by RIPTAB2 is confirmed by FENTAB. These findings are in sharp contrast to those of the fully clamped beam example (Section 2.6.1, Figure 2.10) for which case RIPTAB and FENTAB confirmed that such an orientation effect did not exist. Sensitivity to End Fixities The sensitivity of the panel response to the end fixities // and v is now studied. The values of p and v are varied from 0 through 1, where p = v = 0 corresponds to simple supports with no axial restraint and p = v = 1 corresponds to fully clamped edges. Contours of the permanent midpanel displacement, Wj, are plotted for these variations in Figures 5.9-5.11. Figure 5.9 shows the results for p = 129 psi and A = 0.538. As expected, the final m  midpanel displacement increases with decreasing edge fixities p. and v, ranging in value from Wj = 3.11 in for the fully clamped case to Wj = 10.2 in for the simply supported, axially free case. The sensitivity of the response to p and v is seen to decrease with increasing values of these parameters, as evidenced by the increasing spacing between contours along the lines p = 0, v = 0 and p = v. Increasing the edge fixities from  p, = u = 0 to p = u = 0.2 decreases Wj from 10.2 inches to approximately 6.5 inches, whereas decreasing the fixities from p = u — 1 to p — u = 0.8 increases Wj from 3.11 inches to only about 3.5 inches. As the ratio ujp increases, so does the sensitivity of the response to v relative to that to  p,  the evidence of which is the increasing steepness of the slopes  (dp/dv)  of the  contours with increasing uj pi. This behaviour can be explained by consulting Tables 5.1  Chapter 5. Partial End Fixity  121  Figure 5.9: Variation of the permanent midpanel displacement of the DRES panel with end fixities p and v. loaded from the plating side (A = 0.538) by a rectangular pulse with p = 129 psi. m  Chapter  5.  Partial  End  122  Fixity  Axial  capacity  factor v  Figure 5.10: Variation of the permanent midpanel displacement of the DRES panel with end fixities p and v. loaded from the plating side (A = 0.538) by a rectangular pulse with p = 258 psi. m  Chapter  5.  Partial  End  Fixity  123  Figure 5.11: Variation of the permanent midpanel displacement of the DRES panel with end fixities fi and v. loaded from the stiffener side (A = —0.538) by a rectangular pulse with p = 129 psi. m  Chapter  5.  Partial  End  Fixity  124  and 5.2. In cases where A > 0 and p ^> v, all of the beam's response will occur in the stress state DD', in which bending resistance predominates and the value of v is unimportant. As v increases to where v > //, more and more of the response occurs in the stress states AA' (or BB') and F F ' , wherein axial effects are significant if not dominant; the sensitivity to v increases relative to that to p. These relative sensitivities are also tied to the level of displacement of the beam. For relatively small displacements (up to approximately 4 inches) the response is dominated by bending resistance and is insensitive to v. As Wj increases and axial effects become significant, the sensitivity of the response to v increases. Figure 5.10 shows the results for p  m  = 258 psi and A = 0.538. Even though the  displacements in the low-/x,^ corner of the plot are clearly beyond the limits of the present "moderately large" displacement formulation where (dw/dx) <C 1 (Section 2.1), 2  they are included because, equally clearly, they show the continuation and the logical conclusion of the trends observed in Figure 5.9. That is, for very large displacements, the response is insensitive to the rotational fixity p and almost entirely dependent on the axial fixity v. Figure 5.11 shows the results for p = 129 psi and A = —0.538, which corresponds to m  the panel being loaded from the stiffener side. Once again, Wj increases with decreasing  p. The effect of v on Wj is not as clear, however. The sensitivity of the response to the value of v will be examined by varying the value of v from 0 to 1. Consider first the range of values 0 < v < |A|, corresponding to the case A < —v. For  v = 0, linear bending-only response occurs and the p-Wj relationship is exactly that for the A > 0 case (Figure 5.9, v = 0). For values of v that are just slightly in excess of zero so that fi ^> v, Wj increases with increasing v. This surprising observation is explained with reference to Tables 5.1 and 5.2. For the case where A < — v and v <C /i, all of the response occurs in the stress state CC in which the beam is under a compressive force.  Chapter  5.  Partial  End  125  Fixity  This compressive force serves to drive the displacements. Increasing the axial capacity of the supports can only increase the compressive force and hence increase Wj. As v approaches \i in value, more and more of the response occurs in the stress state GG' in which the beam is under tension. The response of the beam returns to a more normal state in which W} decreases with increasing v. As v increases to \i < v < |A|, all of the response is in the state GG' which has no dependence on fx; hence the vertical Wj contours. Now consider the range of values |A| < v < 1 which corresponds to the case — v < A < 0. The behaviour is more as one would expect in this range, as Wj decreases with increasing p or v. Since the displacements are relatively small (Wj < 4 inches) for the present load case, response will occur predominantly in the stress states CC' and EE'. These states depend upon both \i and v, which is reflected in the Wj contours. However, once v becomes greater than p all of the response is in EE', wherein M  x  and N are  independent of /z and v but *(1 + |A|) so that dp. SM  L  du  ^v(\Jt\\\) =  _  ,|A|  K  ^  */ (l + |A|)' 2  It is recognized from the above derivatives as well as from Figure 5.11 that as v increases, the sensitivity of M (and hence of the response) to /i increases and that to v decreases. L  From these examples it is concluded that the rigid-plastic partial end fixity model developed in this chapter is reasonably well behaved. Generally, the midpanel displacement increases with decreases in \i or v, which is as to be expected. However, in cases where A < 0 and \i ^> v, there was the surprising observation that increasing the axial  Chapter  5.  Partial  End  126  Fixity  capacity of the support links, u, actually increases the displacement Wj. As this observation was shown to be consistent with the rigid-plastic model, the model itself must come into question. It is very unlikely that such behaviour exists in reality; it is more likely that this perceived phenomena arises within the model because of the omission of elastic effects or because of the linearization of the yield curves. Still, this strange behaviour occurs over a relatively small range of possible cases and is well removed from the fj, « u cases that are expected to exist in reality.  5.3.2  H O B #327  Trial #327 from the series of experiments conducted at the DRES Height-of-Burst facility [60] is revisited. This experiment was first examined in Section 2.6.2. The measured incident pressure loading is plotted in Figure 2.14. Fully clamped boundary conditions were not achieved in this trial, with the edges of the panel slipping inward by approximately 1.5 inches. A large permanent midpanel displacement of approximately 11 inches resulted. A preliminary analysis by RIPTAB2 using the static yield stress a = 45, 000 psi 0  indicates that such large displacements are only possible if fi and v are both quite small. So that comparison might be made with results obtained from FENTAB, the panel edges are assumed to be simply supported and axially free, i.e., fi = v = 0. The midspan displacements W(t) determined by the programs are plotted in Figure 5.12, and the in-plane centroidal end slippages U(t) are plotted in Figure 5.13. The RIPTAB2 and FENTAB predictions of the midspan displacements are in good agreement, aside from a large peak elastic displacement and rebound. RIPTAB2 predicts a permanent midspan displacement of 11.7 inches and FENTAB predicts 11.1 inches. Comparison between the predicted centroidal end slippages is similar, though better  Chapter 5. Partial End Fixity  Time  127  t  (msec)  Figure 5.12: Midspan displacement response of the simply supported DRES panel subjected to the HOB #327 pressure pulse.  Chapter  5. Partial  End Fixity  128  RIPTAB2 FENTAB -  centroid centroid  RIPTAB2 - top flange  0  15  30  45  60  75  Time t (msec) Figure 5.13: In-plane end slippage of the simply supported DRES panel subjected to the HOB #327 pressure pulse.  Chapter  5.  Partial  End  129  Fixity  still, with both programs predicting a permanent in-plane end slippage of 1.15 inches. An interesting feature of this end slippage is that the ends of the beam are initially (during the first three milliseconds) pushed outward along the centroidal axis, although they are eventually pulled back in as the transverse displacements become large. The RIPTAB2 and FENTAB results compare well with those from the experiment, with the permanent midspan displacements varying by approximately six percent. The measured in-plane end slippage was roughly thirty percent greater than the centroidal values predicted by the analyses, but it is unlikely that this measurement was taken at the centroid. More probably, the slippage of the plating was measured. A prediction of this plating slippage is obtained from the RIPTAB2 analysis. The centroidal axis of the panel is calculated to be 1.21 inches below the top surface of the plating or 1.085 inches below the plating centroid. The in-plane end slippage of the plating is then given by  U + 1.085 x 0 and is plotted in Figure 5.13. The permanent plating slippage predicted by RIPTAB2 is 1.43 inches, a value which is in good agreement with the approximate measured value of 1.5 inches. The analysis by RIPTAB2 appears to have accurately represented the HOB #327 test. Any conclusions regarding the validity of the RIPTAB2 model must be tempered, however, by the knowledge that the present results have come without due consideration of strain rate effects in the DRES panel. Strain rate effects, as will be seen in the following chapter, significantly strengthen a mild steel stiffened plate such as the DRES panel.  Chapter 6 Strain Rate Effects 6.1  Introduction  The material behaviour of a ductile metal is, in general, sensitive to the rate at which it is strained. At high rates of strain—of the order encountered in blast-loaded structures— the yield stress of mild steel can be significantly increased beyond its static value. Various yield stress-strain rate relationships have been proposed in the literature. Of these, perhaps the most commonly used is the uniaxial, perfectly plastic, empirical model suggested by Cowper and Symonds [5]:  1/r  (6.1)  % = 1 +  where a is the dynamic yield stress associated with a particular strain rate e, o~ is the d  0  static yield stress, and r and e are material parameters. Typical tested values for mild 0  steel are r = 5 and e = 40 sec , although recent tests of the mechanical properties of 0  -1  ship hull steels [45] yielded average values of r = 2 and e = 65 sec . Yield stress-strain 0  -1  rate curves using these values are shown in Figure 6.1. Stiffened plates are herein modelled as grillages of singly symmetric beams. These beams are in turn idealized as asymmetric sandwich beams as in Section 2.2. Referring to Figure 2.6, the sandwich beam has a top flange of area A = \A{\ + A) located a distance t  MQ/N (1 0  + A) above the section centroid and a bottom flange of area Ab = |A(1 — A) 130  Chapter  6. Strain  Rate  131  Effects  o  b b XI  0  200  400  Strain  r=5,  £ =40sec  r=2,  £ -65sec"  0  0  600  Rate £  _L  800  1 000  (sec ) - 1  Figure 6.1: Yield stress-strain rate curves for mild steel. located  MQ/N (1 0  — A) below the centroid, where A is the cross-sectional area of the beam  such that N = Ao . For purposes of illustration, attention will hereafter be focussed 0  0  upon the case of a one-way stiffened plate with fully clamped edges, for which the model is a fully clamped beam. The strain rate corrections derived for this single beam case are readily applicable to grillages representing two-way stiffened plates with partial end fixities.  6.2  Rate-Sensitive S e c t i o n B e h a v i o u r  Symonds and Jones [67] investigated the fully plastic section behaviour of rate-sensitive rectangular beams. Using equation (6.1) and assuming that plane sections remain plane, they determined the stress distributions within the section. Following Symonds' and  Chapter  6.  Strain  Rate  132  Effects  z  (a)  1  (b)  Figure 6.2: Strain rate and stress distributions for a fully plastic sandwich section in hogging near the supports during (a) bending response, (b) string response. Jones' procedure, the strain rate effects within fully plastic sections of the clamped sandwich beam will be investigated. Near the supports, a fully plastic section of the beam will undergo hogging curvature at the rate \k \, where k is negative, and centroidal strain at the rate ir.. The strain rate L  L  and stress distributions are shown in Figure 6.2, as is the location of the neutral axis:  z = \k \ ji L  e  Lb  L  L  below the centroid. The strain rates in the top and bottom flanges, e and Lt  respectively, may alternately be expressed as (  Mo  N (1 + Q  A)  Mo  ^o(l-A)  (6.2) (6.3)  Chapter  6. Strain  Rate  133  Effects  or  4 * • = y- +  (6.4)  N (l + X)z  4 *  =  f i -  0  L/  Mo ' 4 ^o(l-A)^  (6.5)  Recalling equation ( 6 . 1 ) and the idealized sandwich section geometry (Figure 2 . 6 ) , the section stress resultants are given by Mo\*L  1 =  Mo  - 1  1  \N (l-\*)i  2  0  0y  +  UL iV  |  1  =  A  0  +  (  l/r  ^'41  (1  2 1,JV (1-A )4, 2  + A 1 - A  + A)  0  1 - A  0  .(!_ A)  (6.6)  Mo J N (l - \>)z \ + Mo  1  1/r  L  A Q ( 1 ~ A )* + Mo ) a  + A-  \ I/"  —  Mo  L  -  (6.7)  J  \iz < Mo/N (l - A), and L  0  ML  Mo  1(4 2 \e  1  0  +  l/r  Mp(l-A)  (6.8)  ^(1-A')*J 1  Mo(l  + A) \ " ^Vo(l-A )z 1/r  2  l/r 2  V4,  (1  + A)  1  +  +(1-A)  if  2  L  >  MQ/N (1 0  —  L/  M.(l-A) J>&(I-A%;  1 -  l/r  M,(l + A) V  (6.9)  / r  N (l-\*)zJ 0  A). The above stress resultant formulas can be simplified by using the  binomial expansion and approximating for large r:  Chapter  6. Strain  Rate  134  Effects  l/r 1 ±  1± A  Mo  1/  _JV (1-A^ 0  r V  J  l/r I ±  A  2  M,(ITA  i  1 ±  ^(l-A )^,  (6.10)  Mo  r ' JV (1 - A*)z 0  (6.H) L  iV (l-A)' 0  The stress resultants are then approximated by l/r  Mo\k\  ML  Mo =  (6.12)  Mo\k \ L  A+  ^o(l-A )e 2  (6.13) 0  if z < Mo/N (l - A), and L  0  l/r  Mo  1  ML  (6.14)  ~Mo~ N_  -  1 +  ^r  (6.15)  if z > MQIN (1 — A). If the terms that are multiplied by l / r are neglected, the rateL  q  sensitive behaviour is clarified and the approximate stress resultants are M  L  =  L  -Mo  \N (1-W)i , (  = \N  n  Mp\k \ L  ^{N (l-X )i ) 2  0  (6.16)  0i  0  N  l/r'  Mo\k  0  \  l/r'  (6.17)  if z < Mo/N (l - A), and L  0  0,  Mr  N  =  K  l/r" 1 +  (6.18) (6.19)  Chapter  6.  Strain  Rate  135  Effects  Figure 6.3: Strain rate and stress distributions for a fully plastic sandwich section in sagging near midspan during (a) bending response, (b) string response. if z >  Mo/Noil-X).  L  Near the midspan, a fully plastic section of the beam will undergo sagging curvature at the positive rate k and centroidal strain at the rate e . The strain rate and stress x  x  distributions are shown in Figure 6.3. The strain rates in the top and bottom flanges, £  Xt  and i  Xb  respectively, may alternately be expressed as *xt  =  =  (Wl-  AT  f^°  AT  ,.  (6-20)  t  Mo  x J " * ,  (6-21)  Chapter  6. Strain  Rate  136  Effects  or 1  N (l  (6.22)  + \) \z \  0  x  Q  (6.23)  4f •  N (l-X)\z \ x  Retaining the assumption that the axial force N is constant along the length of the beam, the neutral axis must initially lie on the top flange line, as shown in Figure 6.3(a). An area of the top flange, A', is in tension with the stress <r, and an area A — A' is 0  t  in compression. The total area of the bottom flange, A&, is in tension. Later in the response, the neutral axis will move above the top flange, as shown in Figure 6.3(b). Approximating as before, the section stress resultants are found to be M  x  =  Mo  1+  l / r  2Mok  1  x  \N (1-X*)e  2  0  0/  l/r  N if \z \ = Mo/N (l x  0  =  N  + A), and  n  In  2 M  _ x) ( l  ;  if \z \>M /N (l x  0  0  M  °  =  *  k  (6.24)  - A+2 A  (6.25)  Uo(l-A )£ , 2  0  (6.26)  0,  x N  2  (rri) A  l/r'  N  n  1+  (6.27)  + X).  Even with the simplifications afforded by assuming r to be large, the beam section behaviour is complicated, varying temporally and spatially with the strain rates of the considered section. The rate-sensitive response of a beam is very nonlinear, so an exact analysis would therefore be very difficult. One or more approximate methods of analysis that may be used in conjunction with the rate-insensitive rigid-plastic analyses are desired.  Chapter 6. Strain Rate Effects  6.3  137  Rate-Sensitive H i n g e B e h a v i o u r  In rigid-plastic bending, the beam has been assumed to deform by way of point hinges of zero length. Strain rates cannot be calculated in such hinges, but were not previously required; only the relative rates of extension to rotation were needed. To estimate the strain rates in a plastically deforming region, a real-life hinge of finite length, some approximation to the geometry of the hinge is required.  |.  h + 2\z\  -j  Figure 6.4: Plastic hinge slip-line field for a rectangular section. Nonaka [47] assumed that hinges act like uniform slip-line fields according to quasistatic slip-line theory in plane strain. It is notable that these slip-line fields result in magnitudes of strain rate and stress that are constant through the beam depth, as shown in Figure 6.4 for a rectangular section. Thus, existing M , N relationships need only be modified by multipliers that are functions of the strain rate. Nonaka noted some theoretical defects introduced by this assumption. Most significantly, stresses are discontinuous across the rigid-plastic interfaces; the stress in the plastic region has magnitude a > o~ while that in the rigid segments cannot exceed o . d  0  Furthermore, the assumed slip-line field cannot transmit any transverse shear. Despite these defects, photographs taken by Nonaka of aluminum beams clearly show regions of  0  Chapter  6.  Strain  Rate  138  Effects  N (l + A) 0  Mo  N (l + A) 0  Mo  N (l - A) 0  / K ; \ y \ / \ y \ ; \ s \ ;  Mo  K(l-A) +  "  1*1  Figure 6.5: Plastic hinge slip-line field for an unsymmetric sandwich section. localized plastic deformation which bear close resemblance to the above slip-line field. Larger and less distinct deformation regions were observed in mild steel beams, presumably due to the greater degree of rate-sensitivity in steel. Nevertheless, this hinge representation seems to be a reasonable approximation. The slip-line field hinge representation is applied to the unsymmetric sandwich section. If the time rate of angle change through the hinge is 6, the magnitude of the strain rate in the flanges is  '^ „ ( 1 T A ) ±*||*| K  Mo  ±z  B  2  (6.28)  Putting this strain rate into equation (6.1), the rate-sensitivity correction factor for the stress magnitude in the flanges becomes = 1+  l/r  9 2i  (6.29)  0  The simple result is that any stress resultants or section capacities obtained from static  Chapter  6. Strain  Rate  139  Effects  plasticity need only be multiplied by cr /a in order to account for the effect of strain d  0  rate.  6.4  M o d i f i c a t i o n s to R I P T A B 2  Within the program RIPTAB2, the entire response of the beam is in the travelling hinge mechanism. The deformation of the beam is concentrated in plastic hinges at the supports and at a distance X(t) to either side of the midspan. The rate-sensitive hinge behaviour derived in the previous section is now applied to this mechanism, for which the kinematics have been determined in Section 2.3.1. Recalling equation (2.13), the rotation rates in the hinges are  W  L-X  which upon substitution into equation (6.29) give = 1+  l/r  W  2(L -  (6.30)  X)i  0  The RIPTAB2 algorithm divides the response period into short time steps. The travelling hinge position X is determined at the start of each step and held fixed throughout n  the step. At the time t -i, the beginning of the nth time step, the dynamic yield stress n  is  l/r'  W(*n-l)  2(L - X )e n  (6.31) J  0  It is consistent with the existing RIPTAB2 algorithm to hold the dynamic yield stress fixed at this value throughout the time step. Therefore, within the nth time step, the fully plastic section capacities M Q and NQ are given by U  ML Mo  N* 0n  ly  U  = 1+  l/r  W(f -l) w  2(L -  X )i n  Q  (6.32)  Chapter  6.  Strain  Rate  Effects  140  As already stated, the RIPTAB2 algorithm assumes that all deformations in the beam are concentrated at plastic hinges. During string response, however, deformations are actually distributed throughout the beam span, and RIPTAB2 might significantly overestimate the strain rates in the beam. The dynamic yield stress would necessarily be overestimated as well, although to a lesser extent since, for typically large values of  r and of the strain rate, the yield stress-strain rate curve flattens out (Figure 6.1) and the yield stress is only weakly dependent upon the strain rate. Overestimating the yield stress in the beam may in fact prove beneficial. As RIPTAB2 is based upon an extremum principle which seeks out the most flexible response path in configuration space, some artificial stiffening of the beam may improve the accuracy of the solution. This conjecture will be examined in forthcoming examples.  6.5  M o d i f i c a t i o n s to R I P T A B  Although coded in FORTRAN, RIPTAB is a closed form solution for the response of a beam under blast loading. It is most desirable to retain this closed form nature, which would be lost if the nonlinear strain rate effects were incorporated into the solution by continuously updating the yield stress. A simple approach to rate-sensitive analysis, applicable when most of the response occurs in a single given mode of deformation, is to use a uniform rate-sensitivity correction in conjunction with a rate-insensitive, rigidplastic analysis. Such a correction commonly takes the form of a constant dynamic yield stress associated with a governing strain rate. Perrone and El-Kasrawy [51] studied the dynamic response of simple rigid-viscoplastic structures under blast loads, of which a generic case is shown in Figure 6.6. Notably, the generic strain history closely resembles the midspan displacement histories encountered in the beam problem. Much of the response takes place at or near to the peak strain rate. As  Chapter  6.  Strain  Rate  141  Effects  e  *- t  £  -  t  Figure 6.6: Typical dynamic response of simple rate-sensitive structures subjected to blast loads. the area within the strain-yield stress curve is a measure of the energy dissipated in plastic deformation, Perrone and El-Kasrawy argued that the majority of the energy dissipation occurs at the peak strain rate, and, hence, one may assume to a good approximation that the yield stress is a constant, (JQ, corresponding to that rate. Perrone used this approach to determine the response of impulsively and dynamically loaded rings [49,51] and impulsively loaded annular sandwich plates [50]. Rate-sensitivity correction factors may also be applied in cases where two modes of deformation are prevalent. For impulsive loading of fully clamped beams, Symonds and Jones [67] used an effective dynamic yield stress corresponding to estimated strain rates at the onset of plastic string response. Strain rates at this point are representative of both the initial bending response and the final string response. The influence of rate-sensitive material behaviour may in some cases alter the mode of deformation from that predicted by rate-insensitive theory. Bodner and Symonds [3]  Chapter  6.  Strain  Rate  Effects  142  observed this phenomenon in cantilever beams subjected to tip impulses. For such cases, applying a uniform rate-sensitivity correction to a simple rigid-plastic analysis may be inappropriate and could lead to large errors. The introduction of rate-sensitivity to plastic beam bending will certainly result in some amount of change to the deformation mode, since the concept of a point hinge is inconsistent with rate-sensitive yielding [67]. Nonaka [47] conducted an experimental study of fully clamped beams which carry a concentrated mass at midspan and are subjected to blast loading at the midspan mass. In beams made of relatively rate-insensitive aluminum, plastic deformation was localized in small regions that correspond to plastic hinges. Somewhat larger and less distinct deformation regions were observed in rate-sensitive mild steel beams. Nevertheless, the gross characteristics of the steel beams' deformed shapes were very similar to those of the aluminum beams. Humphreys [23] studied the response of clamped steel beams subjected to uniform impulsive loading, and observed deformation shapes which correspond quite well to the rate-insensitive, rigid-plastic theory of Symonds and Mentel [68]. In light of these experimental results (the latter in paricular, since the structure and loading are similar to that of the present study) and the anticipated absence of mode changes in string response, it is felt that a uniform rate-sensitivity correction may in good conscience be applied to the dynamic rigid-plastic solution of Chapter 2. The methods developed by Perrone and El-Kasrawy [51] and by Symonds and Jones [67] will be modified for inclusion in RIPTAB.  6.5.1  Impulse-Momentum Method  Perrone and El-Kasrawy [51] developed an approximate analysis of the effect of strain rate on dynamically loaded structures. Their analysis is based on the assumption that most of the plastic response occurs in one mode of deformation at a substantially constant yield  Chapter  6. Strain  Rate  143  Effects  stress associated with the peak strain rate. Setting the yield stress to that constant value, OQ*, throughout the response should then introduce only a small error to the analysis. The peak strain rate is estimated to occur at the end of the load pulse duration, time t , and p  to be given by an impulse-momentum formulation. A blast pulse may be very intense and yet have so short a duration that it induces very little or no plastic string response in a fully clamped beam. The dominant travelling hinge mechanism response follows an impulse-momentum relation, equation (2.17). It therefore seems appropriate to use the impulse-momentum method for rate-sensitive analysis of such small displacement beam response. Bending response of the beam takes place in the travelling hinge mechanism, in which deformations are concentrated in plastic hinges. Use is again made of Nonaka's slipline field hinge model, so that the dynamic yield stress in the hinges is again given by equation (6.30). The midspan velocity at the end of the load pulse is found using equation (2.17): rt  P  Jo  dt.  m  (6.33)  Recalling equation (2.26) and substituting in the dynamic values of the section capacities, given by MQ/MQ  =  L-X(t )  N  0  */N  =  p  = <7o7<Zo = o*/ oi a  0  \/  0  6 tp  a  t n e nm  '2Mot + X  ftp  I  + 2m±i Jo p  q(t) dt[l  g e position is (6.34)  q(t)(t -tydt p  Combining equations (6.30) and (6.33-6.34) yields an equation that must be solved implicitly for  (r */cr : 0  0  °o*/°o  =  1 + 7«/^o)-  1 / 2 r  ,  (6.35)  where  l/r  \lo q(t)  dt  P  7  2mi  0  UMftp  \ 1+A  3XN  0  m  (6.36)  i q(t){t Jo P  p  -  tf-dt  )  Chapter  6.  Strain  Rate  Effects  144  The effective dynamic capacities MQ, N * and q^ are then substituted into RIPTAB in Q  place of their static counterparts.  6.5.2  Symonds-Jones M e t h o d  Symonds and Jones [67] developed an approximate analysis of the effect of strain rate on impulsively loaded, fully clamped rectangular beams which undergo significant string response. Estimating the strain rates associated with plastic deformation at the onset of string response, effective dynamic values of the moment and axial force capacities corresponding to these rates were found and used in the simple rigid-plastic theory of Symonds and Mentel [68]. This beam problem is very similar to that presently being studied, even though the section geometries differ. The bending deformation mechanisms of the present study (Section 2.3) reduce in the impulsive limit to those of Symonds and Mentel, and the string solution derived by those authors is herein approximated by a one-mode sinusoidal deflection shape. With modification to account for the differing section geometry, the strain rate analysis of Symonds and Jones should be applicable to the present problem of blast loaded beams and stiffened plates. To estimate the centroidal strain and curvature rates at the onset of plastic string response, the displacement and velocity fields shown in Figure 2.7 are considered at the time t . Deformations are taken to be spread uniformly over the outer segments of length s  L — X(t ), s  at  t  s  with no account taken of high local rates at the hinges. The curvature rates  are then estimated to be  rates to be become  i  x  = i  L  = e/2[L  k  = \k \  x  L  — X(t )]. a  =  20 (t )/[L x  s  — X(t )], s  and the centroidal strain  Recalling equations (2.12) and (2.13), these rates  Chapter  6. Strain  Rate  Effects  145  W(t )W(t ) s  =  £r  2Mo  s  [L-X{t )Y  W(t.)  N (l-\*)  a  [L-X(t )r  0  (6.38)  s  The midspan velocity W(t ) and hinge position X(t ) are determined from the ratea  s  insensitive rigid-plastic analysis of Chapter 2. Substituting the above rate estimates into equations (6.16-6.19), the stress resultants at the support hinges are found: M  -Mo  =  L  W(t )  271V  1 +  s  W (l-A )e 2  0  N  0  = AiV„ 1 +  s  N (l~V)£o  0  - A ), and 2  M  W  >  2MQ/N (1 0  (6.41)  0  =  L  N  if  (6.40)  [L-X(t.)Y  0  if W < 2MQ/N (1  l/r'  W(t )  2Mo  (6.39)  [L-X(t,)]t  =  N  n  W{t )  2Mo  1+  i  (6.42)  s  /V (l-A )e 2  0  l / r  \L-X{t )Y)  0  s  A ). Substituting the rate estimates into equations (6.24-6.27), and 2  —  determining A'/A by forcing the axial force N to be constant through the span, the resultants at the travelling hinges are determined for large r to be  N  =  1+  i-y  Mo  My  l + A.  \N  1+  n  l  N (l-\*)i 0  2M,  W (l-A )e 2  0  if W < 2Ad /N (l 0  0  >  2MQIN (1 Q  (6.44)  [L-X(t )}\  0  (6.43)  s  2  = 0  x  iV = W  [L-X(t,)]\  0  - A ), and  M  if  .l/r"  W ( M  2M)  (6.45)  N  0  1+  W(«.)  2^ ^(1-A*)e  0  [Z,-X(t,)]V  l/r'  (6.46)  — A ). These stress resultants are seen to correspond exactly to those 2  derived in the rate-insensitive analysis if the static section capacities, MQ and 7V , are 0  Chapter  6. Strain  Rate  Effects  146  replaced by the effective dynamic capacities, Af<* and N *, where Q  Ml  %L  =  w(t.)  =  V*  (647)  In summary, the procedure for a rate-sensitive analysis using the modified SymondsJones method is: 1. Perform a rigid-plastic analysis (Chapter 2) using the static section capacities, MQ and N , and determine at the onset of string response the midspan velocity W(t ) s  Q  and the travelling hinge position  X(t ). s  2. Put W(t ) and X(t ) into equation (6.47) to determine the dynamic section capacs  s  ities, MQ and N *. 0  3. Repeat the rigid-plastic analysis using the dynamic capacities. Since the initial rigid-plastic analysis is performed using the static section capacities, the midspan velocity W(t ) and the hinge position X(t ) are overestimated. This leads s  s  to overestimates of the strain rates and the dynamic section capacities. Conversely, equations (6.37) and (6.38) are based on deformations which are spread uniformly over the outer segments of the beam, leading to underestimates of the strain rates and dynamic capacities. Presumably, the errors from these approximations tend to cancel. Considering the weak dependence of yield stress on strain rate for large r, any residual error should be of relatively little consequence.  6.6  Example Applications  The approximate procedures derived in the preceding two sections have been coded into subroutines of the programs RIPTAB [57] and RIPTAB2 [58]. Examples of blast loaded  Chapter  6.  Strain  Rate  Effects  147  DRES panels (Section 2.6, Figure 2.9) are presented. The approximate rate-sensitive results from RIPTAB and RIPTAB2 are compared with those from numerical formulations.  6.6.1 HOB  H O B Tests #315  The response of the DRES panel to the HOB #315 pressure pulse, first investigated in Section 2.6.2, is now re-examined using the rate-sensitive analyses derived in this chapter. The panel material has the mass density p = 0.733 x 10~ lb • s / i n , the static 3  2  4  yield stress cr = 54, 000 psi, and the rate-sensitivity constants r = 5 and e = 40 sec . 0  0  -1  The HOB #315 pulse is approximated by a trilinear representation, shown in Figure 2.12, for use in the RIPTAB and RIP TAB 2 analyses. In the rate-insensitive RIPTAB analysis (Section 2.6.2), it is observed that significant string response occurs. The modified Symonds-Jones method (Section 6.5.2) is then employed to calculate an effective yield stress of a* = 101, 300 psi, and a second rigidplastic analysis is performed using this value. The midspan displacement response as calculated both with and without material rate-sensitivity is plotted in Figure 6.7. The predicted final midspan displacement is observed to drop from 15.8 inches to 10.8 inches when the strain rate effects are included. The results from RIPTAB2 analyses, with and without rate effects, are also plotted in Figure 6.7. The permanent midspan displacement predicted by RIPTAB2 is reduced from 17.6 inches to 11.8 inches upon the inclusion of rate effects. Also plotted in Figure 6.7 are the results obtained from two analyses using the finite element program FENTAB, one of which accounts for rate-sensitive material behaviour by updating at every time increment the yield stress for each element according to the  Chapter 6. Strain Rate Effects  148  Figure 6.7: M i d s p a n displacement response of the rate-sensitive D R E S panel subjected to the H O B #315 pressure pulse.  Chapter 6. Strain Rate Effects  149  Cowper-Symonds equation (6.1). In these FENTAB analyses, the panel is assumed to be loaded by the exponential decay pulse approximation (Figure 2.12), and the material behaviour is taken to be elastic-plastic-strain hardening with the elastic modulus E = 30 x 10 psi and the strain hardening modulus E = 18 x 10 psi. As before, the FENTAB 6  T  4  model uses ten elements for the" half-span. The RIPTAB and FENTAB results agree very closely. Differences between the analyses' results are of the same order for the rate-sensitive and rate-insensitive cases, suggesting that the differences are due more to the rigid-plastic beam approximations than the Symonds-Jones rate-sensitivity correction. The RIP TAB 2 results also agree with the RIPTAB and FENTAB results, particularly in the rate-sensitive case for which the comparison seems to be improved upon that in the rate-insensitive case; strain rate effects are slightly more pronounced in the RIPTAB2 analysis than in the RIPTAB and FENTAB analyses. It was suggested in Section 6.4 that concentration of deformations in plastic hinges during string response might lead to beneficial overestimates of the rate effects. This is apparently so. The rate effects within RIPTAB and FENTAB are further examined in Figure 6.8, in which the yield stress is plotted versus the midspan displacement. It is firstly noted that the yield stress computed by RIPTAB2 exceeds the constant value computed by RIPTAB through much of the response. That the yield stress be overestimated by RIPTAB2 was predicted in Section 6.4. Secondly, it is recognized that the majority of the beam deformations in RIPTAB2 occur with the yield stress being roughly constant and equal to its peak value. This is in agreement with the assertion of Perrone and El-Kasrawy (Section 6.5, Figure 6.6) which forms the basis for the uniform yield stress corrections used in RIPTAB. In Figure 6.9, the RIPTAB and RIPTAB2 results are compared with those from a ratesensitive finite strip analysis using FSTNAPS, which also updates the yield stress at every  Chapter 6. Strain Rate Effects  150  2.5  2.0 -  o  b  TS  b  0.5 h  0.0  2  4  Midspan  6  8  Displacement  10  W  12  14  (in)  Figure 6.8: Yield stress versus midspan displacement of the rate-sensitive DRES panel subjected to the H O B #315 pressure pulse.  Chapter 6. Strain Rate Effects  151  18  i  r  "i  r  RIPTAB  1 6  RIPTAB2 14  2  FSTNAPS  Stiffener  FSTNAPS  Panel  4  Time  midspan  centre  6  (B)  (A)  8  10  t (msec)  Figure 6.9: M i d s p a n displacement response of the rate-sensitive D R E S panel subjected to the H O B #315 pressure pulse.  Chapter 6. Strain Rate Effects  152  time increment. In the FSTNAPS analysis, the measured pressure pulse (Figure 2.12) is used and one half of the panel is modelled using 21 strips [33]. A slightly higher static yield stress of <J = 54,400 psi is used in FSTNAPS but this is of little consequence. As 0  in the previous rate-insensitive analysis, there is an initial disparity between the stiffener and plating displacements which diminishes and becomes small as the panel comes to rest. FSTNAPS predicts the final displacement of the panel centre to be 11.0 inches and that of the adjacent stiffener's midspan to be 11.2 inches. The RIPTAB result of 10.8 inches and the RIPTAB2 result of 11.8 inches are in good agreement with the FSTNAPS result. The above analytical and numerical predictions are still well in excess of the experimentally measured final midspan displacement of 4 inches. This discrepancy is made all the more striking by the knowledge that, in reality, the DRES panel material has the mechanical properties <r = 45,000 psi, r = 2 and e = 65 sec 0  0  -1  [45]. The value  <7 = 54, 000 psi was chosen to account for strain rate effects a priori. The panel is seem0  ingly modelled to be stiffer than in reality, yet the predicted displacements are more than two and one half times greater than those measured. No further insight into the cause of this discrepancy has been gained since Section 2.6.2. HOB #338 The rate-sensitive response of the DRESM panel (a DRES panel with a reduced plate thickness of 3/16 inches) in trial HOB #338 is now investigated. From the test data obtained at Defence Research Establishment Pacific [45], the material properties of the panel are revised as follows: <r = 45, 000 psi, r = 2, and e = 65 sec . RIPTAB and 0  0  -1  RIPTAB2 analyses are performed with the pressure pulse approximated by the trilinear representation shown in Figure 2.15. The midspan displacements calculated by RIPTAB and RIPTAB2 are plotted in Figure 6.10, in which they are compared with the results obtained from FENTAB. The  Chapter  6.  Strain  Rate  Effects  153  Figure 6.10: Midspan displacement response of the rate-sensitive DRESM panel subjected to the HOB #338 pressure pulse.  Chapter  6.  Strain  Rate  Effects  154  comparison of the results is not quite as good as in HOB #315, but still all predicted results are within approximately 25 percent of one another. Including rate effects, the RIPTAB analysis predicts a permanent midspan displacement of 8.4 inches, RIPTAB2 predicts 8.7 inches, and FENTAB predicts 7.2 inches. Of immediate interest, the rate effects appear to be of the same magnitude in all three analyses, although once again they are slightly more pronounced in the RIPTAB2 analysis. The measured midpanel displacement from this experiment is approximately 11 inches, which is significantly greater than the displacements predicted by the analyses. No boundary slippages were reported by Slater et al [60]. This discrepancy between experiment and analysis is inconsistent with that of HOB #315, in which case the measured displacements were very much smaller than those calculated.  6.6.2  Event M I S T Y P I C T U R E  A DRES panel was tested at the U.S. Defence Nuclear Agency's event MISTY PICTURE. The panel was flush-mounted on the ground. Heavy concrete walls were constructed above ground along two adjacent edges of the panel, forming a re-entrant corner so that the shock front would be reflected from the walls back onto the panel. Averaging the pressures measured at various points on the panel, the pressure pulse in Figure 6.11 was obtained by Houlston and Slater [21] and used as a spatially invariant load function for finite element analysis. The material properties obtained by DREP (see HOB #338) are used. The response to the pressure pulse of Figure 6.11 cannot be determined directly using RIPTAB, since the sharp pressure rises during the first 10 msec, consequences of the reflected shock wave, would result in outwardly moving hinges (Section 2.5). An alternate representation of the pressure pulse must be found.  Chapter  6.  155  Strain Rate Effects  200  1  i  i—i  i  i  i—i—i  i—i—i—i—i  Measured  to 150  Triangular  CL  CD  i  i  i  i—i  average  i  i  i  r  pressure  approximation  100  13  00 00  £  50  CL  0 0  J  I  I  L  20  Time  30  t  (msec)  Figure 6.11: Measured pressure pulse and triangular approximation for event M I S T Y PICTURE. The average measured pressure is initially approximated by a triangular pulse with a peak intensity of 130 psi and a duration of 50 msec (Figure 6.11), values chosen to provide an impulse equivalent to that of the measured pulse over the first 50 msec. Analysis by R I P T A B using this triangular pulse representation and excluding strain rate effects gives a final midspan displacement of 16.9 inches after 12 msec of response. Examining the first 12 msec of the pressure history, it is apparent that the triangular pulse is a poor representation of the average measured pressure; the impulse during this period is overestimated by almost 25 percent, and it follows that the response must also be significantly overestimated.  Chapter 6. Strain Rate Effects  200  "i  i  156  i  i  i  i  i  i  i  i  i  i  r  Measured average pressure Equivalent rectangular pulse  w 150 Q_  Q_ Q> i_  100  3 00 GO 0)  Q_  5  10  15  Time t (msec) Figure 6.12: Measured pressure pulse and equivalent rectangular representation for event MISTY PICTURE. There is no blast-type pulse which outwardly resembles the measured pulse over the first 12 msec. However, using the procedure developed by Youngdahl [73] and investigated in Section 2.6.4, a good estimate of the response can be found by loading the panel with a rectangular pulse having the same impulse and centroid as the measured pulse. If it is assumed that the previously estimated response time of 12 msec is reasonable, the measured pressure pulse has an impulse of 1110 psi • msec and has its centroid at 6.88 msec. The equivalent rectangular pulse is then determined to have an intensity of 81 psi acting over a duration of 13.75 msec. This rectangular pulse and the first 12 msec of the measured average pressure are shown for comparison in Figure 6.12.  Chapter  6. Strain  1 6  "i  Rate  i  i  Effects  i  i  i  157  i  i  i  i  i  i  i  i  i  i  i  i—i—i—i—i—i—r  RIPTAB RIPTAB2  14  .-  FENTAB ADINA - Stiffener midspan  12  ADINA - Panel centre  (B)  (A)  10  5  10  Time  15  t  20  J  L  25  (msec)  Figure 6.13: Midspan displacement response of the DRES panel to the MISTY PICTURE pressure pulse.  Chapter  6.  Strain  Rate  Effects  158  The displacement response is recalculated by RIPTAB using the rectangular pulse. An analysis neglecting strain rate effects predicts a final midspan displacement of 11.3 inches after 13.2 msec. Since significant string response occurs, the modified SymondsJones method is used to determine an effective yield stress of o~* = 54, 500 psi. A second rigid-plastic analysis, using this value of a^, predicts the panel coming to rest after 12.3 msec with a final midspan displacement of 8.9 inches. This response is plotted in Figure 6.13. The rate-sensitive displacement response calculated by RIPTAB2 is also plotted in Figure 6.13. Since RIPTAB2 is not restricted to monotonically inward hinge motions, the measured average pressure is used as input. A final midspan displacement of 10.8 inches after a response time of 13 msec is predicted.  boundary  Figure 6.14: ADINA finite element model of the DRES panel. Also plotted in Figure 6.13 are predictions by Houlston and Slater [21] using FENTAB and ADINA [2]. The FENTAB analysis is rate-sensitive, using the tested values of the material properties listed, previously, and models the panel with eight elements for the half-span. The ADINA analysis models an interior portion of the panel, shown in  Chapter  6.  Strain  Rate  Effects  159  Figure 6.14, by imposing conditions of symmetry at its edges. It accounts for strain rate effects by choosing a  priori  the yield stress  o  0  = 54,400 psi, a value which is very close  to the effective yield stress determined.above. Both of these numerical models use the measured average pressure as the input loading function. There is good agreement between the analyses. During the early transient response of the panel, the RIPTAB analysis appears to overpredict the displacement, but this is certainly due to the rectangular pulse representation, the intensity of which initially exceeds the measured average pressure. The similarity of the RIPTAB and FENTAB predictions is a further validation of the Symonds-Jones method. As is typical for a single beam/one-way stiffened plate analysis, RIPTAB2 overpredicts the permanent midspan displacement due to the use of a hinge mechanism to model string response. ADINA predicts little difference between the stiffener and plating displacements and compares well with the beam analyses, again validating the beam response assumption. The actual final displacements of the DRES panel were measured to be 14.25 inches at the panel centre and 14.0 inches and 13.2 inches at the two adjacent stiffener midspans [59]. It is firstly noted that the panel was not deformed symmetrically. Reflection of the shock front from the re-entrant corner resulted in asymmetric pressure distributions—the peak pressures being greatest near the re-entrant corner—rather than the uniform distribution assumed by the above analyses. Secondly, the measured displacements are up to 50 percent larger than those predicted. Tearing failure of the box sections which supported the panel led to large support rotations along the sides of the panel, relaxing the clamped boundary conditions. The panel displacements would certainly have been smaller had the integrity of the supports been maintained.  Chapter 7 Conclusion 7.1  Summary  In the present study, the large ductile deformation (Mode I) response of stiffened plates under blast loads has been investigated. In the course of this investigation, simplified methods of analysis which are suitable for the preliminary design of blast-resistant panels have been developed. Much of the simplification was afforded by modelling stiffened plates as singly symmetric beams or as grillages thereof. Analysis was further facilitated by the use of the rigid-plastic material idealization and mode approximation techniques. In Chapter 2, one-way stiffened plates having edges fully restrained against transverse and in-plane translations and rotations were modelled as fully clamped, axially constrained beams. The singly symmetric beam section included one stiffener and plating one-half of the distance to each neighbouring stiffener. Blast-type pulses of uniformly distributed line loads were assumed. The bending moment-axial force capacity interaction relation (or yield curve) was inscribed and approximated by four linear segments. As a consequence of the linearized yield curve, the response of the beam was divided into two distinct phases within which the critical beam stress states were constant and determined throughout. In the early, small displacement phase, the beam resisted the load primarily through bending, responding approximately as a mechanism in which plastic 160  Chapter 7. Conclusion  161  hinges formed at the supports and at a time-dependent distance to either side of the midspan. With time, the latter "travelling" hinges moved toward the midspan where they may have ultimately met and coalesced. Later, in the large displacement phase, the beam's bending resistance had vanished and its response was that of a plastic string. Following Vaziri et al [72], the displacement field of the plastic string was approximated by a simple cosine shape. Both of these phases of response were governed by linear differential equations which were solved in closed form. For convenience, the solutions were coded into the FORTRAN 77 program RIPTAB. Examples of a one-way stiffened plate subjected to various blast-type pressure pulses demonstrated good agreement between RIPTAB and beam finite element and finite strip solutions using elastic, plastic, strain hardening material behaviour. Of particular note was RIPTAB's prediction that the magnitude of displacement response is independent of the direction of loading, i.e., the downward displacement of a panel loaded on the plating side from above will equal the upward displacement of an identical panel loaded identically on the stiffener side from below; this despite the asymmetry of the beam model section. Results from the beam finite element program FENTAB [8,9] attested to this prediction. Poor comparisons between RIPTAB and experimental results were obtained in Chapter 2, due in part to the neglect of material rate-sensitivity and the assumption of fully clamped supports. Chapter 3 presented a second approach to the analysis of one-way stiffened plates subjected to blast loads. The beam model and linear interaction relation were again utilized, but the response was not solved in closed form. It was instead approximated as a sequence of "instantaneous" mode responses, for each of which the velocity field was a separable function of the spatial and temporal variables. Further approximating the instantaneous modes by the travelling hinge mechanism of Chapter 2, the locations of the hinges were at any time determined to maximize the rate of change of the kinetic energy of the beam. As the hinges were not assumed a priori to move inward to the  Chapter  7.  Conclusion  162  midspan, the load pulse was not restricted to be blast-type. This response was solved by an instantaneous mode solution (IMS) algorithm wherein the "instantaneous" modes were taken to be valid over time steps of small but finite duration At. It was observed that the displacement response as predicted by the IMS was insensitive to the choice of At. There was good agreement between the displacement responses computed by RIPTAB and the IMS, particularly where the IMS string response was modified to use Vaziri's cosine mode (as does RIPTAB) rather than the travelling hinge mechanism. Even without this modification, the IMS predictions of the final midspan displacement generally exceeded those of RIPTAB by no more than ten percent. In Chapter 4, the instantaneous mode approximation technique was applied to the analysis of a two-way stiffened plate under blast loading. The edges of the panel were assumed to be fully clamped with in-plane restraints, parallel stiffeners were identical and evenly spaced, and loading was in the form of a uniformly distributed pressure pulse. In a manner consistent with the one-way stiffened plate analyses, two-way stiffened plates were modelled as beam grillages and instantaneous modes were approximated by travelling hinge  line  mechanisms. The resulting numerical algorithm was coded into the  FORTRAN 77 program RIPTAB2 within which the one-way IMS algorithm exists as a special case. For a square stiffened plate with equal stiffening in its principal directions, the response predicted by RIPTAB2 was extremely close to that predicted by the super finite element program NAPSSE [35] which used elastic, plastic, strain hardening material behaviour. The comparison of results for the case of a rectangular plate with unequal stiffening in its principal directions was less satisfactory, but the super finite element results clearly showed a hinge line mechanism response corresponding to that assumed by RIPTAB2. Incomplete fixity of the stiffened plate's edges was accounted for in Chapter 5. The beam (and grillage) models were no longer rigidly connected to rigid supports. Rather,  Chapter  7.  Conclusion  163  the connections were made by way of rigid-plastic links having bending moment and axial force capacities which were independent of and less than the beams'. The linearized interaction relations of the links were incorporated into the analysis and the critical beam stress states were modified accordingly within the program RIPTAB2. For a oneway stiffened plate modelled by a beam having axially constrained but rotationally free ends, RIPTAB2 response predictions compared very well with FENTAB predictions. In contrast to the fully clamped case, the direction of loading affected the magnitude of the displacement response; the panel was stiffer when loaded from the stiffener side than when loaded from the plating side. A study of the sensitivity of the response to the end fixity gave the expected results that small displacement response is sensitive to the moment capacities of the links and that large displacement response is sensitive to the links' axial force capacities. An experiment in which a one-way stiffened plate exhibited large end slippages was analysed using RIPTAB2 and FENTAB with the panel modelled as a simply supported, axially free beam. The predicted midspan displacements and end slippages from both analyses compared very well with the experimentally measured values. It must be pointed out, however, that the analyses neglected rate effects in the panel. The rate sensitivity of ductile metals was accounted for in Chapter 6 using the Cowper-Symonds relation, which gives the dynamic yield stress of a material as a function of its static yield stress and the rate at which it is strained. Means of determining an effective yield stress to be used throughout an entire closed form RIPTAB analysis were presented. Within RIPTAB2, the dynamic yield stress was easily recalculated at each time step. RIPTAB and RIPTAB2 analyses of blast loaded, one-way stiffened plates which used these dynamic yield stresses compared very well with those of finite element and finite strip programs. Significantly, RIPTAB2 showed that the dynamic yield stresses in a panel are roughly constant through much of its displacement response, confirming the  Chapter 7.  164  Conclusion  validity of using a time-invariant effective yield stress in RIPTAB. Comparison between the rigid-plastic analyses and experimental results was inconsistent.  7.2  V a l i d i t y a n d L i m i t a t i o n s of the A n a l y s e s  The analyses derived herein account for many varied characteristics of the stiffened plate and its response to blast loading. But these analyses are approximate, incorporating many simplifying assumptions regarding the structure and the load and neglecting many of the response phenomena. Some of the approximations and their effects on the calculated response were discussed as they were introduced, e.g., the assumed hinge mechanisms in Section 2.5. Many others, however, were not. Qualitative assessments of these approximations and, where appropriate, potential means of circumventing them will now be presented.  7.2.1  Problem Geometry  Limitations on the analyses are imposed by assumptions regarding the geometry of the stiffened plate and of the load to which it is subjected.  With regards to the panel  geometry, the stiffeners running along one principal direction of the panel must be evenly spaced and identical to one another, and the panel must be supported symmetrically in both principal directions. A two-way stiffened plate must be approximately equally stiff in its principal directions. As these requirements correspond to the majority of stiffened plate designs, they are not considered to be severe restrictions. Of greater concern is the magnitude of the stiffener spacing. If this spacing is large in comparison to the panel dimensions, the assumption that all of the plating acts as large upper flanges for the stiffeners may be very inaccurate. The bending stiffness of the panel may in such cases be overestimated by the beam grillage model. On the other hand, the membrane capacity  Chapter  7.  Conclusion  165  of the plating would still be utilized and, at those points mid-way between stiffeners, may come into play earlier in the response and hence compensate for any loss in the bending stiffness. A parameter study requiring many numerical analyses by the finite element or finite strip method would be required to investigate these possibilities. Such a parameter study might also point the way to developing an effective flange breadth calculation for plating supported by widely spaced stiffeners. The loads imposed on the stiffened plate are assumed to be uniformly distributed. Variations in the spatial distribution of the load resulting from near-field explosions, from blast waves grazing across the face of the panel, or from waves reflected off nearby structures  (e.g.,  re-entrant corners) are not considered by the analyses derived herein.  To do so would require new analyses with different response mechanisms. It is not clear whether the effects of such load conditions are great enough or whether rigid-plastic analyses are accurate enough to warrant the consideration of spatially varying loads.  7.2.2  Material Properties  The methods derived herein are intended to be appropriate to the analysis of ductile metal stiffened plates. Ductile metals are typically elastic, viscoplastic, strain hardening materials. In this study they are idealized as being rigid, perfectly plastic with viscoplasticity (or rate sensitivity) accounted for by corrections to the yield stress. Elasticity and strain hardening are neglected. The rigid-plastic analyses herein have neglected elastic deformations on the assumption that they are small in comparison to plastic deformations. For this assumption to be correct, the work done by the blast load on the stiffened plate must be much greater than the maximum strain energy that can be absorbed by the panel in a wholly elastic manner. According to Bodner and Symonds [3], the ratio R of external work to maximum  Chapter  7.  Conclusion  166  elastic strain energy should have a value of at least three. This criterion is related in part to the intensity of the load. An extreme case is where the peak blast pressure is less than that required for static collapse. A rigid-plastic analysis predicts no deformation of the panel, whereas significant elastic, perhaps elastic-plastic, response does in fact occur. A second condition on the validity of neglecting elasticity is that the load pulse be of relatively short duration, usually meaning a fraction of the panel's fundamental period of elastic vibration. Symonds and Frye [66] demonstrated how a half sine wave pulse having a duration roughly equal to an elastic-plastic single-degree-of-freedom system's fundamental period could induce very large elastic deformations in that system. A rigidplastic analysis would in such a case be grossly in error. Symonds and Frye also showed, however, that the magnitude of this error decreases with increasing R. The rigid-plastic material idealization seems well-suited to the examples considered in this study. The HOB test blasts on the DRES panel (Section 2.6.2) all had peak intensities of several times the static collapse pressure, very short (essentially zero) rise times, and durations of less than one-half of the beam model's fundamental period. Both of the conditions for neglecting elasticity are satisfied and the rigid-plastic results are accurate. And despite the load pulse having a long duration and a finite (though irregular) rise to its peak intensity, event MISTY PICTURE (Section 6.6.2) could be analysed without regard for elastic response. It would seem for this case that the high peak intensity of the pressure pulse resulted in a high enough value of the ratio R to compensate for its duration effect. When response to pulses of low intensity and long duration must be determined, elasticity cannot be neglected, By Symonds' extended mode approximation technique [65], the elastic-plastic response of a beam is approximated by a sequence of pure elastic bending, plastic bending, plastic string, and elastic rebound modal responses. It should be relatively straightforward to incorporate the present rigid-plastic methods within an  Chapter  7.  Conclusion  167  extended mode analysis. Strain hardening is also neglected within the rigid-plastic analyses, but this omission is of little consequence. Ductile metals typically strain harden by less than ten percent, and much of this hardening occurs late in the stiffened plate's response. The numerical analyses of the HOB tests included strain hardening with little perceived effect. In presenting the program FENTAB, Folz et al [8] found very little difference between analyses using perfectly plastic and strain hardening materials. Considering the level of accuracy desired of (and realized by) the present analyses, it does not seem profitable to account for strain hardening within them.  7.2.3  B e a m Response  Having modelled stiffened plates as beams and having idealized their material behaviour as rigid-plastic, further assumptions were made regarding the response of these beams. It was firstly assumed that the transverse displacements of a beam, though finite, were small in comparison to the beam span so that the axial force in the beam, N, could be approximated by its horizontal component,  N yj\ + (dw/dx)  2  '  A RIPTAB analysis of the HOB #315 test resulted in a final midspan displacement of 16 inches with a cosine shaped displacement field over a 96 inch span. The maximum value of the beam slope at the supports was in this case equal to 0.52 and the horizontal component of the axial force was 0.89./V. Even for this very large displacement, the difference between the axial force and its horizontal component was little more than ten percent. The assumption of "moderately large" displacements does not seem overly restrictive.  Chapter  7.  Conclusion  168  The analyses derived herein do not account for shear effects in the beam. Shear forces can interact with the bending moments and axial forces to cause a section to yield and can induce plastic deformation by way of "shear slides" or "shear hinges" [63]. Although shear effects are usually unimportant for beams of compact section, e.g., rectangular beams, they can significantly alter the response of beams of non-compact section, e.g., I-beams and T-beams, which have relatively high ratios of bending capacity to shear capacity. As expected, the effects of shear diminish with increasing beam span-to-depth ratios [31,32,47,63]. Accounting for shear effects would once again require a completely new analysis: one based on response mechanisms which include shear hinges. Mode II (ductile tearing) response and Mode III (shear failure) response of the beam have not been considered. The beam material has been assumed to be infinitely ductile and, as noted above, shear effects have been disregarded. Jones [25] has derived critical velocities induced by impulsive loads which will lead to such failures in clamped rigidplastic beams. His analyses might be used to provide rough estimates of critical impulses for the presently considered beam models. Alternatively, they might be used as the basis for failure computations which would be contained within RIPTAB and RIPTAB2. Jones' Mode II analysis could easily be modified for implementation in these programs. His Mode III analysis, however, is based upon rigid-plastic shear deformation modes which would require their inclusion within RIPTAB and RIPTAB2. In light of the approximations discussed above and the many more introduced throughout this study, it is impossible to consider the analyses RIPTAB and RIPTAB2 as being valid for all cases of dynamic response of stiffened plates. But for the range of problems considered herein—for stiffened plates subjected to intense, uniformly distributed blast loads—these approximate rigid-plastic analyses yield displacement response predictions which compare very well with those of finite element analyses with the advantages of  Chapter  7.  Conclusion  169  greatly reduced modelling and computing times. Comparison of the rigid-plastic predictions with experimentally measured responses can in the best terms be described as inconsistent, but comparison among the experimental results alone can be described as such. There are many difficulties involved in blast response experimentation—not the least of which is measuring the incident pressures—which may make these experimental results no more reliable than the analyses. Refinements to the present analytical methods, if deemed appropriate by further study, might be made. In the meantime, however, RIPTAB and RIPTAB2 should be considered to provide estimates of the deflection response of blast loaded stiffened plates which are suitable for the preliminary stages of design.  Bibliography [1] J. Ari-Gur, D.L. Anderson and M.D. Olson, "Review of Air-Blast Response of Beams and Plates", 2nd Int. Conf on Recent Advances in Structural Dynamics, Southamp-  ton, England, April 1984. [2] K.J. Bathe, ADINA—A  Finite Element Program for Automatic Dynamic Incre-  mental Non-Linear Analysis, Report 82448-1, Acoustics and Vibration Laboratory, Dept. of Mechanical Engineering, MIT, Cambridge, Mass., 1975.  [3] S.R. Bodner and P.S. Symonds, "Experimental and Theoretical Investigation of the Plastic Deformation of Cantilever Beams Subjected to Impulsive Loading", J. Appl. Mech., 29(4), 1962, pp. 719-728. [4] M . Chowdhury, "Plastic Collapse Theory for Rectangular Grids", Int. J. Mech. Sci., 27(3), 1985, pp. 145-155. [5] G.R. Cowper and P.S. Symonds, Strain Hardening and Strain Rate Effects in the Im-  pact Loading of Cantilever Beams, Technical Report No. 28, Contract Nonr 562(10), Brown University, Providence, R.I., 1957.  [6] A.D. Cox and L.W. Morland, "Dynamic Plastic Deformations of Simply-Supported Square Plates", J. Mech. Phys. Solids, 7, 1959, pp. 229-241. [7] A . L . Florence, "Clamped Circular Rigid-Plastic Plates Under Blast Loading", J. Appl. Mech., 33(2), 1966, pp. 256-260. [8] B.R. Folz, M.D. Olson and D.L. Anderson, "Non-Linear Transient Beam Analysis  on a Microcomputer", Proceedings of the Third Int. Conf. on Computing in Civil  Engineering, Vancouver, B.C., 1988, pp. 341-348.  [9] B.R. Folz, Numerical Simulation of the Non-Linear Transient Response of Slender  Beams, M.A.Sc. Thesis, UBC, Vancouver, B.C., 1988.  [10] M . Grigorian, "A Lower-Bound Solution to the Collapse of Uniform Rectangular Grids on Simple Supports", Int. J. Mech. Sci., 13, 1971, pp. 755-761. [11] M . Grigorian, "The Plastic Design of Orthotropic Grids with Fixed Supports", Int. J. Mech. Sci., 14, 1972, pp. 197-203. 170  Bibliography  171  [12] M . Grigorian, "Plastic Design of Regular Orthotropic Grids with Two Adjacent Edges Fixed, Free, or Hinged", Journal of Strain Analysis, 7(4), 1972, pp. 279-284. [13] J. Henrych, The Dynamics Amsterdam, 1979.  of Explosion  and fts Use, Elsevier  [14] J. Heyman, "The Limit Design of Space Frames", pp. 157-162.  J. Appl.  Scientific Publishing, Mech.,  18(2), 1951,  [15] J. Heyman, "The Limit Design of a Transversely Loaded Square Grid", J. Mech., 19(2), 1952, pp. 153-158. [16] J. Heyman, "The Plastic Design of Grillages",  Engineering,  Appl.  176, 1953, pp. 804-807.  [17] P.G. Hodge, "Post-Yield Behavior of a Beam with Partial End Fixity", Int. Sci, 16, 1974, pp. 385-388.  J.  [18] H.G. Hopkins and W. Prager, "On the Dynamics of Plastic Circular Plates", (Journal of Applied Mathematics and Physics), 5(4), 1954, pp. 317-330.  Mech.  ZAMP  [19] R. Houlston and J.E. Slater, "Structural Response of Panels Subjected to Shock Loading", 55th Shock and Vibration Bulletin, Part 2, 1985, pp. 149-163. [20] R. Houlston and C G . Desrochers, "Nonlinear Structural Response of Ship Panels Subjected to Air Blast Loading", Computers Sz Structures, 26(1/2), 1987, pp. 1-15. [21] R. Houlston and J.E. Slater, "The Response and Damage of a Stiffened Panel Subjected to Free-Field and Confined Air-Blast Loading", Presented at SUSI 89 Conference, Cambridge, Mass., 1989. [22] Z. Huang and J. Liu, "Rigid-Plastic Analysis of Beam Grillages under Blast-Type Loading", Int. J. Impact Engng., 3(3), 1985, pp. 179-190. [23] J.S. Humphreys, "Plastic Deformation of Impulsively Loaded Straight Clamped Beams", J. Appl. Mech., 32(1), 1965, pp. 7-10. [24] N. Jones, "Influence of In-Plane Displacements at the Boundaries of Rigid-Plastic Beams and Plates", Int. J. Mech. Sci., 15, 1973, pp. 547-561. [25] N. Jones, "Plastic Failure of Ductile Beams Loaded Dynamically", J. Eng. for Ind., 98, 1976, pp. 131-136. [26] N. Jones, " A Literature Review of the Dynamic Plastic Response of Structures", Shock and Vibration Digest, 7(8), 1975, pp. 89-105.  Bibliography  172  [27] N. Jones, "Recent Progress in the Dynamic Plastic Behaviour of Structures, Parts I and II", Shock and Vibration Digest, 10(10), 1978, pp. 13-19. [28] N . Jones, "Recent Progress in the Dynamic Plastic Behaviour of Structures, Part III", Shock and Vibration Digest, 13(10), 1981, pp. 3-16. [29] N. Jones, "Recent Progress in the Dynamic Plastic Behaviour of Structures, Part IV", Shock and Vibration Digest, 17(2), 1985, pp. 35-47. [30] N. Jones, "Recent Studies on the Dynamic Plastic Behavior of Structures", Mechanics Review, 42(4), 1989, pp. 95-115.  Applied  [31] N . Jones and J. Gomes de Oliveira, "The Influence of Rotatory Inertia and Transverse Shear on the Dynamic Plastic Behavior of Beams", J. Appl. Mech., 46(2), 1979, pp. 303-310. [32] B. Karunes and E.T. Onat, "On the Effect of Shear on Plastic Deformation of Beams Under Transverse Impact Loading", J. Appl. Mech., 27(1), 1960, pp. 107-110. [33] M.R. Khalil, M.D. Olson and D.L. Anderson, namic  Response  of Air-Blast  Loaded  Plate  Large  Structures  Deflection,  Elastic-Plastic  by the Finite  Strip  Dy-  Method,  Structural Research Series Report No. 30, Dept. of Civil Engineering, UBC, Vancouver, B.C., 1987.  [34] M.R. Khalil, M.D. Olson and D.L. Anderson, "Nonlinear Dynamic Analysis of Stiffened Plates", Computers & Structures, 29(6), 1988, pp. 929-941. [35] T.S. Koko, Stiffened  Super  Plate  [36] H.L. Langhaar, York, 1962.  Finite  Structures, Energy  Elements  for Nonlinear  Static  and Dynamic  Ph.D. Thesis, UBC, Vancouver, B.C., 1990.  Methods  in Applied  Mechanics,  Analysis  of  John Wiley and Sons, New  [37] L.S.S. Lee, "Mode Responses of Dynamically Loaded Structures", J. 39(4), 1972, pp. 904-910.  Appl.  Mech.,  [38] L.S.S. Lee and J.B. Martin, "Approximate Solutions of Impulsively Loaded Structures of a Rate-Sensitive Material", Journal of Applied Mathematics and Physics (ZAMP), 21(6), 1970, pp. 1011-1032. [39] D.E. Manolakos and A.G. Mamalis, "Limit Analysis for Laterally Loaded Stiffened Plates", Int. J. Mech. Sci., 30(6), 1988, pp. 441-447. [40] J.B. Martin, "Large Deflection of an Encastre Rigid-Plastic Grid", chanical Engineering Science, 4(4), 1962, pp. 322-333.  Journal  of Me-  Bibliography  173  J.B. Martin, "Extremum Principle for a Class of Dynamic Rigid-Plastic Problems", Int. J. Solids Structures, 8, 1972, pp. 1185-1204. J.B. Martin and A.R.S. Ponter, "Bounds for Impulsively Loaded Plastic Structures", J. Engr. Mech. Div., Proc. ASCE, 98(2), 1972, pp. 107-119. J.B. Martin and P.S. Symonds, "Mode Approximations for Impulsively-Loaded Rigid-Plastic Structures", J. Engr. Mech. Div., Proc. ASCE, 92, 1966, pp. 43-66. S.B. Menkes and H.J. Opat, "Tearing and Shear Failures in Explosively Loaded Clamped Beams", Experimental Mechanics, 13(11), 1973, pp. 480-486. J. Morrison and J.P. Gough, Mechanical Response of Hull Plate Steels at Intermediate Strain Rates, Defence Research Establishment Pacific Technical Memorandum 87-5, Victoria, B.C., 1987. N.M. Newmark and R.J. Hansen, "Design of Blast-Resistant Structures", Shock and Vibration Handbook (Ed. C M . Harris and C E . Crede), McGraw-Hill, New York, 1961. T. Nonaka, "Some Interaction Effects in a Problem of Plastic Beam Dynamics, Parts 1-3", J. Appl. Mech., 34(3), 1967, pp. 623-643. C. Palomby and W.J. Stronge, "Evolutionary Modes for Large Deflections of Dynamically Loaded'Rigid-Plastic Structures", Mech. Struct. &: Mach., 16(1), 1988, pp. 53-80. N. Perrone, "On a Simplified Method for Solving Impulsively Loaded Structures of Rate-Sensitive Materials", J. Appl. Mech., 32(3), 1965, pp. 489-492. N. Perrone, "Impulsively Loaded Strain-Rate-Sensitive 34(2), 1967, pp. 380-384.  Plates",  J. Appl.  Mech.,  N. Perrone and T. El-Kasrawy, "Dynamic Response of Pulse-Loaded Rate-Sensitive Structures", Int. J. Solids Structures, 4, 1968, pp. 517-530. D. N. Robinson, "A Displacement Bound Principle for Elastic-Plastic Structures Subjected to Blast Loading", J. Mech. Phys. Solids, 18, 1970, pp. 65-80. C A . Ross, W.S. Strickland and R.L. Sierakowski, "Response and Failure of Simple Structural Elements Subjected to Blast Loadings", Shock and Vibration Digest, 9(12), 1977, pp. 15-26. R.B. Schubak, Finite Deflection Dynamic Analysis M.A.Sc. Thesis, UBC, Vancouver, B.C., 1986  of Axially  Restrained  Beams,  Bibliography  174  [55] R.B. Schubak, M.D. Olson and D.L. Anderson, "Simplified Rigid-Plastic Beam Analysis", J. Appl. Mech., 54(3), 1987, pp. 720-722. [56] R.B. Schubak, D.L. Anderson and M.D. Olson, "Simplified Dynamic Analysis of Rigid-Plastic Beams", Int. J. Impact Engng., 8(1), 1989, pp. 27-42. [57] R.B. Schubak, M.D. Olson and D.L. Anderson, User's Manual for RIPTAB:  Rigid-  Plastic Transient Analysis of Beams and One-Way Stiffened Plates, Dept. of Civil  Engineering, UBC, Vancouver, B.C., 1989.  [58] R.B. Schubak, M.D. Olson and D.L. Anderson, User's Manual for RIPTAB2: RigidPlastic Transient Analysis of Beams and Stiffened Plates, Dept. of Civil Engineering,  UBC, Vancouver, B.C., 1991.  [59] J.E. Slater and D.V. Ritzel, "Air Blast Loading and Structural Response of a Ship Stiffened Panel in a Re-entrant Corner at Event MISTY PICTURE", Presented at the U.S. Defence Nuclear Agency MISTY PICTURE Symposium, Adlephi, Maryland, 1987. [60] J.E. Slater, R. Houlston and D.V. Ritzel, Air Blast Studies on Naval Steel Panels: Final Report, Task DMEM-53, Defence Research Establishment Suffield Report No.  505, Ralston, Alberta, 1990.  [61] W.J. Stronge, "Accuracy of Bounds of Plastic Deformation for Dynamically Loaded Plates and Shells", fnt. J. Mech. Sci., 27(1/2), 1985, pp. 97-104. [62] P.S. Symonds, "Large Plastic Deformations of Beams Under Blast Type Loading", Proc. 2nd U.S. Nat. Cong. Appl. Mech., ASME, 1954, pp. 505-515.  [63] P.S. Symonds, "Plastic Shear Deformations in Dynamic Load Problems", Engineering Plasticity, (Ed. J. Heyman and F.A. Leckie), 1968, pp. 647-664. [64] P.S. Symonds, "The Optimal Mode in the Mode Approximation Technique", Mechanics Research Communications, 7(1), 1980, pp. 1-6.  [65] P.S. Symonds, "Finite Elastic and Plastic Deformations of Pulse Loaded Structures by an Extended Mode Technique", Int. J. Mech. Sci, 22, 1980, pp. 597-605. [66] P.S. Symonds and C.W.G. Frye, "On the Relation Between Rigid-Plastic and ElasticPlastic Predictions of Response to Pulse Loading", Int. J. Impact Engng., 7(2), 1988, pp.139-149. [67] P.S. Symonds and N . Jones, "Impulsive Loading of Fully Clamped Beams with Finite Plastic Deflections and Strain-Rate Sensitivity", Int. J. Mech. Sci., 14(1), 1972, pp. 49-69.  Bibliography  175  [68] P.S. Symonds and T.J. Mentel, "Impulsive Loading of Plastic Beams with Axial Constraints", J. Mech. Phys. Solids, 6, 1958, pp. 186-202. [69] P.S. Symonds and T. Wierzbicki, "On an Extremum Principle for Mode Form Solutions in Plastic Structural Dynamics", J. Appl. Mech., 42(3), 1975, pp. 630-640. [70] R.G. Teeling-Smith and G.N. Nurick, "The Deformation and Tearing of Thin Circular Plates Subjected to Impulsive Loads", submitted to Int. J. Impact Engng.. [71] F. Tin-Loi, "Post-Yield Behaviour of a Rigid-Plastic Beam with Partial Axial and Rotational End Fixities", Int. J. Mech. Sci, 32(7), 1990, pp. 623-630. [72] R. Vaziri, M.D. Olson and D.L. Anderson, "Dynamic Response of Axially Constrained Plastic Beams to Blast Loads", Int. J. Solids Structures, 23(1), 1987, pp. 153-174. [73] C.K. Youngdahl, "Correlation Parameters for Eliminating the Effect of Pulse Shape on Dynamic Plastic Deformation", J. Appl. Mech., 37(3), 1970, pp. 744-752.  Appendix A Associated Flow Rule for Bending and Stretching of Beams A singly symmetric beam section is considered. At a distance z below the centroidal axis of the section, the beam has a breadth of b(z) and a yield stress of cr , where normal 0  stresses u(z) are considered to be positive in tension. The beam is subjected to a tensile axial force N and is bent about an axis perpendicular to its plane of symmetry by a sagging moment M.  The section is deemed to be fully plastic when subjected to a  critical combination of the stress resultants M and N such that every fibre within the section is stressed to its yield value a . The locus of all such critical combinations of M 0  and N is referred to as the section's yield curve. When a section is fully plastic, it forms a hinge which may undergo a finite centroidal extension at the rate e and a rotation, positive in sagging, at the rate 9. These generalized strain rates are shown in Figure A . l , where it is assumed that the adjacent sections remain plane. The neutral axis n.a., on which fibres do not experience any axial extension, is at a distance ~z below the centroidal axis (shown as a negative quantity in Figure A . l ) . Thus, the following relationship between the extension and the rotation of the hinge is obtained: e =  -z9  176  (A.l)  Appendix  A.  Associated  Flow  Rule  177  |— %  (a)  »|.  (7 —«| 0  (b)  (c)  Figure A . l : (a) Singly symmetric beam section; (b) Fully plastic stress distibution corresponding to a plastic hinge; (c) Generalized strain rates. Integrating the stresses through the depth of the section, the axial force is found to be N  P  =  <T(Z) b(z) dz  J—C-x  r b(z)dz - r b{z)dz  Jz  J—c-i  (A.2)  which varies with the neutral axis position according to dN  dz  =  (A.3)  -2cr b{z). 0  The bending moment in the section is M  =  =  f  J-  zo(z)b{z) dz  2 C l  % jjf  z b(z) dz —  z b(z) dz  J  (A.4)  which varies with the neutral axis position according to dM  dz  —2a z 0  (A.5)  b(z)  The variation of bending moment with axial force is then given by the chain rule of differentiation to be dM  dM/dz  dN  dN/dz  =  z.  (A.6)  Appendix  A.  Associated  Flow  Rule  178  plastic deformation rate vector  yield curve N, e  Figure A.2: Relationship between the yield curve and plastic deformations of the beam section. When equation (A.l) is recalled, the generalized strain rates e and 9 may be related to the stress resultants M and iV by dM  9  dN  e  (A.7)  •1  In physical terms, this relationship states that the vector of plastic deformation rates (e, 9) must be outwardly normal to the yield curve of the section at that critical combination of stress resultants, plasticity  M  and Af. This relationship, known as the  for the beam, is shown graphically in Figure A.2.  associated  flow  rule of  Appendix B Response to Ideal Impulses Using the equations of motion derived in Chapter 2, the response of an impulsively loaded singly symmetric beam or one-way stiffened plate is determined. The pressure function for an ideal impulse is p(t) = I-S(t),  (B.l)  where 8(t) is the Dirac delta function. After the response according to the present theory is found, a check on the dynamic admissibility of the solution is performed.  B.l  Hinge Mechanism Response  The beam or stiffened plate will initially deform according to the travelling hinge mechanism. Recalling that q(t) = a • p(t), equations (2.17) and (2.18) give the midspan displacement and velocity as W(t)  =  W(t)  =  (B.2)  Iat/m,  (B.3)  Ia/m,  and equation (2.26) gives the position of the travelling hinges to be x(t)  =  3g  \  la  0  t + 179  A(l +  X)N Ia 0  (B.4)  Appendix  B.  Response  to Ideal  Impulses  180  The response proceeds as above until either the hinges meet at the midspan at a time t  x  or string response begins at a time t . Setting X(t ) = 0 yields s  t  2m  =  x  x  \)N Ia  A(l +  0  where  N (l  W(t ) s  -  0  P= Setting  A/? - 1 3(1-A)  MQ  (B.5)  \ )(Iaf 2  (B.6)  mqnMo  - A ) gives  = 2Mo/N (l  2  0  U  2m  =  N (l  MQ  - \ )Ia 2  0  (B.7)  '  Combining equations (B.5) and (B.7), it is found that string response begins before the travelling hinges meet (t < t ) if s  x  3(2 - A)  /3 > & =  (B.8)  1- A  If the above inequality is not satisfied, response proceeds in the midspan hinge mechanism according to the following initial value problem: 3%  (B.9)  2M>  N X(l la  + A)  0  W(t ) x  =  '1 +  A/3 3(1-A)  1  m  The solution is found to be W(t)  =  2Mo N \{1 0  +  A)  '1 +  2A/? cos [ui(t 3(1 - A)  - t) x  - </>i] - 1  (B.10)  where I3XN  (B.ll)  0  V mL  2  '  <j)\ = arctanW  \/3  3(1 - A) + A/3  (B.12)  Appendix  B.  Response  to Ideal  Impulses  The beam will come to rest at the time  181  tj — t  x  + 4>i/u>i. with  the midspan displacement  if the final displacement lies within the realm of hinge mechanism response, requiring that W < 2Mo/Ao(l - A ) or 2  f  0 < A =  (B.14)  If this is not the case, string response begins at the time t. =  t  +  x  1 T 1/(1 - A) - fa + arccos f  V  1  (B.15)  2Xp  +  3(1-A)  and the midspan velocity at the onset of string response is  B.2  String Response  If 0i < (3 < (3 , the travelling hinges will meet before the onset of string response, and 2  string response will be governed by the initial value problem  W  +  ^ W  = 0,  (B.17)  JV (1 - A ) 2  0  where the initial velocity condition is determined by substituting X(tj)  = 0 and equa-  tion (B.16) into equation (2.36). The midspan response is then solved: W(t)  = A ^ p )  V  1  +  ~ X W i - 1) • cos [u {t - t ) A  2  s  <t> ] 2  (B.18)  Appendix  B.  Response  to Ideal  Impulses  182  where No  7T U>  2\lmL  2  2  fa  and  (B.19)  '  2  i  arctan -\/2(2 7T "  t  s  \){P/Pi  (B.20)  - 1),  is given by equation (B.15). The beam will come to rest at the time  =  t -\-falu s  2  with the midspan displacement 2Mo  '  'l +  A)  No{l -  S  2  ^(2-A)(/3/ft-l).  (B.21)  If P > Pi, string response begins before the hinges meet at the midspan. Recalling equations (B.3), (B.4), (B.7) and (2.36), the initial velocity condition is now (B.22) The midspan response is 2Mo  ^)  =  7trf^V  1+  ^ (1  1-  A)  lyfpyp  • cos [u (t - t ) - fa] 2  s  ,  (B.23)  where w is given by equation (B.19), t is given by equation (B.7), and 2  s  fa = arctan ^ ( 1 - A) 1 - f^/V/? The beam will come to rest at the time W  f  f  =  2Mo  tj — t  '  s  +  fa/uio  i V ( l - A ) V'1 + 7 T^ ( 1 - A ) 0  2  2  (B.24)  with the midspan displacement -\4NP  (B.25)  The above results for the permanent midspan displacement of the beam are summarized in Table B . l . It is easily verified that these results, when applied to the limiting case A = 0, are exactly those obtained in the earlier analysis of doubly symmetric beams [54, page 50].  Appendix  B.  Response  to Ideal  Impulses  183  Table B . l : Permanent midspan displacement of a singly symmetric beam subjected to an ideal impulse. W,  P  Pl<P<  AoA(l + A) V 2MQ  P2  W-A ) 2  _ _2Mp _ ^ 1  P>P  2  3(2 - A) 2(1 - A)  Pi  B.3  f  2K  0<P<P\  3(1 - A)  'l + £ ( 2 - A ) ( / ? / f t - l )  +  _Sp( 1 _ ) A  3(2 - A) 1-A  ;  D y n a m i c Admissibility  The response of the impulsively loaded beam in the travelling hinge mechanism is reexamined, with attention given to the distribution of the bending moment. As stated in Section 2.5, the mechanism is dynamically admissible if the bending moment decreases monotonically from a value of MQ(1 — A)/(l + A) at the travelling hinge to — MQ at the support. (A sufficient condition, though not a necessary one.) This is so if  dM/dx  <  0  everywhere on the rigid end segment. For the impulsive load case, the second derivative of the bending moment in the segment X < x < L is given by equation (2.41) to be dM  _  dx  ~  2  2  AiV  X(t)(L-x) l a  '  [L-X(t)\  0  2  m  la X(t )(L-x) x  (B.26)  Appendix  B.  Response  where X(t )  to Ideal  Impulses  184  = x, with t determined by inverting equation (B.4). Using these results,  x  x  this formula can be rewritten as 2Ho  dM 2  dx  1  (B.27)  2  where 3(1 - \)L \p(L-xy'  V  2  (B.28)  L — x  (B.29)  L-X  Integrating from X to x, the bending moment gradient through the rigid end segment becomes dM  2Mo  dx  Since  ^(1  d M/dx 2  2  + X)(L  In  -X)[  increases with  x  1 + y/TTTj xA+^(i-€ ) VF+VJ  (B.30)  2  \£ +  (decreases with £), the maximum value of  dM/dx  must  occur at the support point x — L (£ = 0). The moment gradient at this point is dM  2Mo  dx  y/rjil + \)(L  x=L  -  X)  (B.31)  In  from which it is found that the moment distribution is dynamically admissible if rj > 0.169, or  A/? 3(1  -  A) V  L  <  < 5.92.  < » M2  This condition is least likely to be satisfied late in the response when X is minimized at the time t or t . If /? < /? , the hinges will meet at the midspan at the time t so that x  X(t ) x  s  2  x  = 0. A safe (dynamically admissible) moment distribution is then found to occur  if A < 1If  ,  (B.33)  1  ^1 + 5.92/V/? '  > fl , string response will begin at the time t with the hinge position X(t ) given 2  s  s  by substituting equation (B.7) into equation (B.4). A safe moment distribution is found  Appendix  B.  Response  to Ideal  Impulses  185  1 .0  UNSAFE  SAFE  0.6 CO  0.5  0  2  Normalized  Impulse  (3/(3  2  Figure B . l : Values of A and j3 for which the travelling hinge mechanism is statically admissible under impulsive loading. to occur if A < 0.62. The values of A and /? for which the travelling hinge mechanism is dynamically admissible are plotted in Figure B . l . Further integration gives the. distribution of the bending moment through the segment X  < x < L:  M  (B.34) The bending moment distribution has been calculated for various values of A for the case /? = 02, t = t = t . These distributions are plotted in Figure B.2. It is observed 3  x  that if A > 0.62 the bending moment must attain magnitudes greater than MQ near the supports, clearly violating the yield condition.  Appendix  B.  Response  to Ideal  Impulses  Figure B.2: Distribution of the bending moment in an impulsively loaded beam.  Appendix C Response to General Blast-Type Pulses C.l  Linear Representation of the Pulse  A general time-varying blast-type pressure pulse may be suitably represented by a number of segments within each of which the pressure decays linearly. The time at the beginning of one such segment (say the nth) is denoted as t  p  pressures corresponding to these times are p(t  p  and that at the end as t  )  n  ^)  p  and  =  n  p(t ^) p  n+l  The  K  n+1  The  = p( \ n+1  pressure pulse during such a time segment is shown in Figure C . l , and may be defined algebraically by p(t)  =  P  -  l  (C.l)  pt 2  where Pi  p  _ ?  2  ~  (n) (»+l)] +  p  +  l  p2  [(n+l) j(n)] t  +  >  (  Q  3)  (n) _ (n l) p  p  An+l)  +  An)  •  Using the equations of motion derived in Chapter 2, the response of a singly symmetric beam or one-way stiffened plate during the time segment is determined.  187  Appendix  C. Response  to General  Blast-Type  Pulses  188  Pit) Pi  \  1 P2  L  >+l)  >~  Figure C . l : Linearized representation of a general blast-type pressure pulse.  C.2 C.2.1  Hinge Mechanism Response Travelling Hinge Mechanism  Recalling that q(t) = a • p(t), we define q = p a and q = p a. The above pressure x  x  2  2  history and the equations (2.17) and (2.18) give the midspan displacement and velocity as  + W(t)  =  g l  W(tW)  " 2m  (t-ffl) -^(t-fW) , ~ 9i ~ fcfl 2m m  g 2 < ! , W )  +  v  a  p  0  and equation (2.26) gives the position of the travelling hinges.  3  (C.4) (C.5)  Appendix  C. Response  to General  Blast-Type  Pulses  C.2.2  Midspan Hinge Mechanism  189  Response in the midspan hinge mechanism during the nth time segment begins at the time t* which is the greater of t and t \ The response is governed by x  W{t)  with the initial conditions  +  p  n  -^W(t) mL  = ^  3  2  and  W(t*)  (C.6)  " * 2m  For the case of A ^ 0, solving the initial  W(t*).  value problem gives (t)  =  VA^TF  C O S M *  W(t)  =  u^A  sm[fa  W  + B  2  2  n -  -  -u.it-  + t*)}  (<?i  q  "2  \*  2  -  N  2t)L2  (C.7) (C.8)  where A  =  (Qi -%-  Wit*)  <l2t*)L  2  (C.9)  2AJV  0  B  u  x  U! -  Wit*)  +  qL 2\N 2  (CIO)  2  n  I3\N  0  mL  2  (Cll)  ' B  fa = arctan [ — ) + <  0;  Tr  if A <  0  otherwise.  (C.12)  If A = 0, the displacement term in equation (C.6) vanishes and integration gives the response to be Wit)  =  Wit*) + wit*)it - t*) 3(?i - ? o - 9 * * ) ^ ^2 (*-oa-^(*-n3, . 4m 2  Wit)  =  Wit*) +  3(gi  - q 2m 0  q t*) 2  it-n-^it-n  2  (C.13) (C.14)  Appendix  C.3  C. Response  to General  Blast-Type  Pulses  190  String Response  String response during the nth time segment begins at the time t** which is the greater of t, and t \ The response is governed by v  n  (C.15) with the initial conditions  and  W(t**)  w(t)  = VWTD*-  W(t)  =  u y/C  2  2  +  D  2  Solving the initial value problem gives  W(t**).  C O S M *  S  m[<f>  2  n  -  - <j> \ +  1  2  - w (< - t**)} a  6  (  1692jL2  /V 7r3  \  f  q  )L2  (C.16) (C.17)  0  where C  D  =  --  U)  2  W(t**)  u>  -  Wit**) 2  INQIT  16(  +  q t")L  -  9l  16q L 2  2  2  2  JV 7T  (C.18) (C.19)  3  0  2  4mL  (C.20)  2  $2 = arctan ( — I + < ,'D\  TT  ifC<0;  0 otherwise.  (C.21)  Appendix D Convergence of Impulsive Responses Using elementary rigid-plastic theory, Martin and Symonds [43] showed that the responses of identical structures subjected to independent impulsive loadings approach one another as time progresses. In doing so, they derived a functional, A , which has become the basis for the selection of modes for use in the mode approximation technique. This proof and derivation are repeated herein.  D.l  C o n v e x i t y C o n d i t i o n of P l a s t i c i t y  Consider a point in the structure subjected to the stresses cr and undergoing strains at the rate k. If cr lies on the yield surface, the associated flow rule dictates that the strain rate vector is outwardly normal to the yield surface, as shown geometrically in Figure D . l . If <r*is any other admissible stress state, the convexity condition of plasticity requires that the angle between the vectors cr — cr* and k be acute, their scalar product being {cr-cr*).k  > 0.  (D.l)  If a* itself lies on the yield surface and is associated with the strain rate k*, a second 191  Appendix  D.  Convergence  of Impulsive  Responses  192  'i,  e  Figure D . l : Geometric representation of the convexity condition of plasticity, application of the convexity condition yields (a - tr*).(e - e*) >  D.2  0.  (D.2)  C o n v e r g e n c e of S o l u t i o n s  A structure of mass density p is subjected to impulsive velocities v at time t = 0. At any time t the surface S of the structure is comprised entirely of a region ST upon which zero tractions are prescribed and a region S upon which zero displacements are prescribed. U  The response is given by the kinematically admissible velocity field u and corresponding strain rates e, and by the safe and dynamically admissible stresses cr. If at the time t a virtual displacement field 8u — u6t is imposed upon the structure, consideration of the principle of virtual work over the volume V of the structure yields (D.3)  Appendix  D.  Convergence  of Impulsive  193  Responses  If an identical structure is subjected to an impulse v* and the response is then given by ii*, e* and tr*, consideration of the principle of virtual work with virtual displacements  8u* — ii* St yields J pil*.u*dV  -  =  v  tr'.e'dV.  J  (DA)  The kinematically admissible virtual displacements in the above virtual work equations may be transposed: - J  pii.u*dV  -J PU*.udV V  =  J a-.k*dV,  (D.5)  =  J a*.kdV.  (D.6)  v  v  Combining equations (D.3-D.6) and recalling equation (D.2) yields -  / p(u-ii*).(u-u*)dV  =  (<r - cr*).(k  /  - k*)dV  >  0.  (D.7)  Defining the functional A(i) to be the kinetic energy of the difference between the responses i i and i i * , i.e.,  A(t) = j f j p ( u - u * ) . ( u - u * ) < f V ,  (D.8)  it is observed that dA IT  s  0  so that the responses i i and i i * must approach one another as time progresses.  <> M  Appendix E Lee's Extremum Principle for Instantaneous Mode Approximations E.l  Derivation of the E x t r e m u m Principle  Lee [37] derived an extremum principle for the determination of instantaneous modes for nonlinear structures. The derivation for rigid-plastic structures with "stress-type" boundary conditions is repeated herein. A structure having volume V and boundary surface S is subjected to the prescribed surface tractions To on the portion ST of S and to the prescribed velocities u on the 0  remaining portion S . The response is given by the dynamically admissible stress field U  cr corresponding to the tractions T and by the kinematically admissible strain rates k corresponding to the velocities u. Furthermore, the response is of the instantaneous mode form u(x,*) =  a(i)*(x;t),  (E.l)  u(x,<)  /?(*)u(x,i).  (E.2)  194  Appendix  E.  Lee's  Extremum  Principle  195  If at the time t a small kinematically admissible virtual velocity field 8u is imposed upon the moving structure, consideration of the principle of virtual work over the volume of the structure yields - J T.SudS  J^cr.SkdV  + J pu.8udV v  where p is the mass density of the structure.  = 0,  (E.3)  Because the field 8\i is kinematically  admissible, the boundary integral over the portion S vanishes (8u = 0 on S ) leaving U  U  only that over the portion ST on which T = To. Also recalling the mode relationship, equation (E.2), the virtual work equation becomes J <r.8edV  - j  v  s  T .8udS 0  + f3(t) J^pii.SudV  = 0.  (E.4)  The structure is comprised of a rigid, perfectly plastic material having a convex yield surface and an associated flow rule. Consider a point x subjected to the stresses cr and given a small increment of stress 8cr. If a lies within the yield surface, the strain rates k must be zero; so too if cr lies on the yield surface but 8a unloads the point to a new stress state within the surface. Non-zero stress states can only exist if <r lies on the yield surface and if 8cr constitutes neutral loading, i.e., 8<r is tangential to the yield surface. But, from the associated flow rule, the strain rates k are normal to the yield surface so that 8<r.k = 0. Therefore, the first term in equation (E.4) becomes J <r.8kdV  = j 8{cr.k)dV  v  v  = 8 f tr-edV  (E.5)  which is recognized as being the first variation of the time rate of energy dissipation, £)(u,t), in the structure. Over the portion ST of the boundary surface, the tractions T are specified and hence 0  must be independent of any variation. The second term in equation (E.4) is then f  T -8udS 0  = 8 [  T .udS 0  (E.6)  Appendix  E.  Lee's  Extremum  Principle  196  which, if the structure has "stress-type" boundaries such that u = 0, is recognized as 0  being the first variation of the rate of external work, E(u, t), being done by the tractions T. The third term in equation. (E.4) is manipulated by noting that \i.8u = |6(u.u). The scalar time function f3(t) is related to the true modal response of the structure and so is independent of the variation. A second scalar time function K*(t) which is also independent of the variation, i.e., 8K* = 0, is introduced in such a way that P(t) J pu.6udV  = 6{p(t)  v  §/>u.iW  [jf  - K*(t)  } .  (E.7)  The integral on the right hand side of equation (E.7) is recognized as being the kinetic energy,  K(u,t),  of the structure.  Combining equations (E.4-E.7), a variation may be written: 6{b(u,t)  - E{u,t)  + P{t)[K(ii,t)-K*{t)}}  -  0.  (E.8)  Lee's functional for a rigid-plastic structure with stress-type boundary conditions is defined as J(u,t)  = D(u,t)  -  E(ii,t),  (E.9)  which, according to equation (E.8), should be extremized in a variation of u subject to the constraint that  K(u,t)  —  K*(t),  i.e.,  that the kinetic energy of the structure at the  time t is independent of the variation. In this regard, P(t) may be looked upon as a Lagrange multiplier.  E.2  Properties of the E x t r e m a  Lee [37] considered a rigid-plastic material to be a limiting case of a viscous material having a positive-definite stress-strain rate tensor. In so doing, he ascertained that an  Appendix  E.  Lee's Extremum  Principle  197  extremum of the functional J(u, t) must in fact be a stationary minimum. By noting that  D — E = —K, Lee provided a physical interpretation of his extremum principle: that the instantaneous mode at the time t maximizes the rate of change of the structure's kinetic energy. In other words, the instantaneous mode is the most flexible path in configuration space u which may be taken by the structure at the time t. Symonds and Wierzbicki [69] investigated the properties of the extrema of a form of Lee's functional which included the kinetic energy constraint, J'(u,t)  = D(u,t)  - E{u,t)  +  They examined the difference between its value J'(u*,t)  fl[K(u,t)-  K*(t)}  t)  —  J'(u*,  t)  (E.10)  for the exact modal response and  that for an arbitrary neighbouring kinematically admissible response, this difference as A J ' = J'(u,  .  J'(u,t).  Denoting  the difference in the modal velocity fields as  u — u* = Su, and the difference in the strain rate fields as e — e* = Se, Symonds and Wierzbicki determined that AJ' =  fl  J^pSu.SudV  +  J  {a - er*).(e* + Se) dV .  (E.ll)  Both integrals in equation (E.ll) are known to be positive: the first by inspection, the second by the convexity condition of plasticity (Appendix D). The first integral is of second order in the small variations Su.. If the second integral is also of second order in small quantities Se, then the first-order variation SJ' = 0 and the extremum of J' is a stationary one. In this case, if fl > 0 as when the structure is accelerating under load, it is also verified that the second-order variation S J' > 0 so that the stationary extremum 2  is in fact a minimum. Otherwise, if  < 0 as when the structure is decelerating near  the end of its response, the extremum may be a minimum, a maximum, 'or a saddle point. If the second integral is not of second order, the extremum is a nonstationary minimum since A J ' « SJ' > 0. Symonds and Wierzbicki presented instances in which such nonstationary extrema arise.  Appendix  E.  Lee's Extremum  Principle  198  It is beyond the scope of the present study to reconcile the findings of Symonds and Wierzbicki with those of Lee, as the issue of these discrepancies has, to an extent, been rendered moot. Variations of the mode shapes of the beams and beam grillages are herein restricted to mere variations of the hinge locations, 8X and 8Y. In such cases, it has been shown by Palomby and Stronge [48] that the extremum will be stationary. Furthermore, it has been determined in the analysis of a single beam (Chapter 3) that the extremum is in all cases a minimum, and it is assumed that the same is true for a beam grillage.  

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