Applied Mechanics and Mathematics

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Selected Works in Applied Mechanics and Mathematics Reissner, Eric. Jones & Bartlett Publishers, Inc. 0867209682 9780867209686 9780585363530 English Mechanics, Applied--Mathematics. 1996 TA350.R45 1996eb 624.1/7 Mechanics, Applied--Mathematics.

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Selected Works in Applied Mechanics and Mathematics Eric Reissner

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CREDITS The following publishers graciously gave permission to reprint the articles listed. AIAA: J. Aeron. Sc. 4, © 1937, p. 539; 8 © 1941, p. 8; 16 © 1949, p. 516; 18, © 1951, p. 579 ASME: Proc. 5th Intern. Congr. Appl. Mech. © 1938, p. 134, p. 542; Proc. 1st Nat'l Congr. Appl. Mech. © 1952, p. 584; Journal of Applied Mech. 12, © 1945, p. 155; 40, © 1973, p. 75; 41, © 1974, p. 343; 47, © 1980, p. 189, p. 375, p. 189; 59, © 1992, p. 211 Birkhauser: J. Applied Mathematics and Physics 17, © 1966, p. 298; 23, © 1972, p. 58; 30, © 1979, p. 96; 32, © 1981, p. 105; 33, © 1982, p. 389; 34, © 1983, p. 114; 35, p. 194 Elsevier Science, Ltd.: Computer Methods in Appl. Mechs. & Eng. 85, © 1991, p. 200; Pergamon Press, Inc.: Journal of Mechs. and Phys. of Solids 6, © 1957, p. 246; Int. J. of Solids Struct., 1, © 1965, p. 450; 11, p. 82; 13, p. 366; Int. J. Non-Linear Mechs. 17, © 1982; Int. J. of Solid Struct. 21, © 1983; p. 121, p. 392; © 1995, p. 216. Interscience: Commun. Pure & Applied Math., 7, © 1959, p. 264. Macmillan Publishing Co.: Prog. Appl. Mech.; Prager Anniv. Vol. , © 1963, p. 275. MIT Press: Journal of Math and Physics 23, © 1944, p. 147; 25, © 1946, p. 32; 27, © 1948, p. 435; 29, © 1950, p. 437; 37, © 1958, p. 264; Studies Appl. Math. 49, © 1970, p. 176; 52, © 1973, p. 66. Oxford University Press: Qu. J. Mech. & Appl. Math., 21, © 1968, p. 300 Prentice Hall: Thin-Shell Structures: Theory, Experiment, and Design , Fung/Sechler eds., © 1974, p. 353. The Quarterly of Applied Mathematics: Qu. Appl. Math. 4, © 1946, p. 21; 10, © 1953, p. 173; 20, © 1962, p. 43. Society for Industrial and Applied Mathematics: J. Soc. Indust. Appl. Math. 4, © 1956, p. 237; 13, © 1965, p. 281. Springer Verlag: Math Anal. 111, © 1935, p. 131; Ingenieur Archiv. 7, © 1936, p. 491; 40, © 1971, p. 321; Mechanics of Generalized Continua, IUTAM Symp. © 1967, p. 453; Acta Mechanica 56, © 1985, p. 463; Proc. Intern. Conf. Comp. Mech. © 1986, p. 397; © 1987, p. 407; Computational Mathematics 1, © 1986; 5 © 1989, p. 478. John Wiley & Sons, Inc.: Communications on Pure and Applied Mathematics, Vol. 7, © 1959, p. 264.

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CONTENTS Preface A Biographical Sketch

xi xiii

Beams

1

Über Die Berechnung Von Plattenbalken

3

Least Work Solutions of Shear Lag Problems

8

Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential Energy

21

Note on the Shear Stresses in a Bent Cantilever Beam of Rectangular Cross Section

32

On Finite Pure Bending of Cylindrical Tubes

35

Finite Pure Bending of Circular Cylindrical Tubes

43

Considerations on the Centres of Shear and of Twist in the Theory of Beams

54

On One-Dimensional Finite-Strain Beam Theory: The Plane Problem

58

On One-Dimensional Large-Displacement Finite-Strain Beam Theory

66

Upper and Lower Bounds for Deflections of Laminated Cantilever Beams Including the Effect of Transverse Shear Deformation

75

Improved Upper and Lower Bounds for Deflections of Orthotropic Cantilever Beams

82

Note on a Problem of Beam Buckling

93

On Lateral Buckling of End-Loaded Cantilever Beams

96

On Finite Deformations of Space-Curved Beams

105

On Axial and Lateral Buckling of End-Loaded Anisotropic Cantilever Beams

114

A Variational Analysis of Small Finite Deformations of Pretwisted Elastic Beams

121

Plates

129

Über Die Biegung Der Kreisplatte Mit Exzentrischer Einzellast

131

On Tension Field Theory

134

On the Calculation of Three-Dimensional Corrections for the Two-Dimensional Theory of 143 Plane Stress On the Theory of Bending of Elastic Plates

147

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The Effect of Transverse-Shear Deformation on the Bending of Elastic Plates

155

Pure Bending and Twisting of Thin Skewed Plates

173

On Postbuckling Behavior and Imperfection Sensitivity of Thin Elastic Plates on a NonLinear Elastic Foundation

176

On the Analysis of First- and Second-Order Shear Deformation Effects for Isotropic Elastic 189 Plates A Tenth-Order Theory of Stretching of Transversely Isotropic Sheets

194

On Asymptotic Expansions for the Sixth-Order Linear Theory Problem of Transverse Bending of Orthotropic Elastic Plates

200

On Finite Twisting and Bending of Nonhomogeneous Anisotropic Elastic Plates

211

A Note on the Shear Center Problem for Shear-Deformable Plates

216

Shells

221

On the Theory of Thin Elastic Shells

225

A Note on Membrane and Bending Stresses in Spherical Shells

237

On Stresses and Deformations of Ellipsoidal Shells Subject to Internal Pressure

246

On the Foundations of the Theory of Thin Elastic Shells

253

The Edge Effect in Symmetric Bending of Shallow Shells of Revolution

264

On the Equations for Finite Symmetrical Deflections of Thin Shells of Revolution

275

Rotating Shallow Elastic Shells of Revolution

281

A Note on Stress Strain Relations of the Linear Theory of Shells

298

Small Strain Large Deformation Shell Theory

304

Finite Inextensional Pure Bending and Twisting of Thin Shells of Revolution

308

On Consistent First Approximations in the General Linear Theory of Thin Elastic Shells

321

On Pure Bending and Stretching of Orthotropic Laminated Cylindrical Shells

343

Linear and Nonlinear Theory of Shells

353

On Small Bending and Stretching of Sandwich-Type Shells

366

On the Transverse Twisting of Shallow Spherical Ring Caps

375

On the Effect of a Small Circular Hole on States of Uniform Membrane Shear in Spherical 385 Shells A Note on the Linear Theory of Shallow Shear-Deformable Shells

389

A Note on Two-Dimensional Finite-Deformation Theories of Shells

392

Some Problems of Shearing and Twisting of Shallow Spherical Shells

397

On a Certain Mixed Variational Theorem and on Laminated Elastic Shell Theory

407

On Finite Axi-Symmetrical Deformations of Thin Elastic Shells of Revolution

416

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Variational Principles

433

Note on the Method of Complementary Energy

435

On a Variational Theorem in Elasticity

437

On a Variational Theorem for Finite Elastic Deformations

443

A Note on Variational Principles in Elasticity

450

A Note on Günther's Analysis of Couple Stress

453

On a Certain Mixed Variational Theorem and a Proposed Application

457

On a Variational Principle for Elastic Displacements and Pressure

460

On Mixed Variational Formulations in Finite Elasticity

463

Some Aspects of the Variational Principles Problem in Elasticity

470

On the Formulation of Variational Theorems Involving Volume Constraints

478

Vibrations

489

Stationäre, Axialsymmetrische, Durch Eine Schüttelnde Masse Erregte Schwingungen Eines 491 Homogenen Elastischen Halbraumes Forced Torsional Oscillations of an Elastic Half-Space I

511

Complementary Energy Procedure for Flutter Calculations

516

Reihenentwicklung Eines Integrals Aus Der Theorie Der Elastischen Schwingungen

518

On Axi-Symmetrical Vibrations of Shallow Spherical Shells

523

Aerodynamics

537

A Contribution to the Theory of Turbulence

539

Note on the Statistical Theory of Turbulence

542

On Compressibility Corrections for Subsonic Flow over Bodies of Revolution

547

Note on the Theory of Lifting Surfaces

551

Boundary Value Problems in Aerodynamics of Lifting Surfaces in Non-Uniform Motion

558

Note on the Relation of Lifting-Line Theory to Lifting-Surface Theory

579

A Problem of the Theory of Oscillating Airfoils

584

Bibliography

589

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PREFACE It is a pleasure and an honor to write this brief preface to introduce our teacher, Professor Eric Reissner, some of whose works compose this volume. We hope to shed some light on not only Eric Reissner, as a contributor to the fields of applied mathematics and mechanics, but also on him as a generous and caring individual. As a biographical sketch written by him follows this preface we will limit ourselves to some thoughts of ours having to do with his influence on us, with the recognition which his work has received and with a brief personal assessment of what we believe to be the principal contributions of a man who is both Professor Emeritus of Applied Mathematics of the Massachusetts Institute of Technology and Professor Emeritus of Applied Mechanics of the University of California. All four of us were privileged to be taught the Theory of Elasticity and Theories of Plates and Shells by Professor Reissner at M.I.T. His lectures were always clear, incisive, and thorough, exposing both the subtlety of solid mechanics and the subtlety of his thinking. He demanded much of his students because he demanded so much from himself. Yet, for us, on the other side of this keen professional was a generous and caring friend, colleague and mentor. From the recognition which Eric Reissner has received in appreciation of his work we would like to mention the following: He was elected a Fellow of the American Academy of Arts and Sciences in 1950, and received the Clemens Herschel Award of the Boston Society of Civil Engineers in 1955. He was a Guggenheim Fellow during 1962. He received the von Karman Medal of the American Society of Civil Engineers in 1964, "for noteworthy contributions to the theory of elasticity and theory of plates and shells, and for outstanding papers on those subjects." Also in 1964, he received an Honorary Dr. Ing. degree from the University of Hannover, Germany, "in appreciation of his pathbreaking works in the field of elastomechanics." Later, in 1973, he received the Timoshenko Medal from the American Society of Mechanical Engineers, "for distinguished research and exceptional teaching in solid mechanics, especially in the theory of elastic plates." On the occasion of Reissner's receiving this award, Professor J. P. Den Hartog, a former student of Timoshenko, and a friend and colleague of Reissner at M.I.T. for more than thirty years, congratulated him for having "surpassed the master in the value of his life's contributions." Professor Reissner was elected a Member of the U.S. National Academy of Engineering in 1976. He became a Corresponding Member of the International Academy of Astronautics in 1979, and a full Member in 1984. A symposium in his honor was held on the occasion of his 65th birthday at the University of California at San Diego, and a volume of "Mechanics Today" (Pergamon Press) appeared in 1980, containing the papers presented at this symposium by his former students, colleagues and friends, from all over the world.

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After his retirement in 1978 he received the Structures, Structural Dynamics, and Materials Award from AIAA, in 1984, "in recognition of fundamental contributions to the aerospace community as a teacher and researcher in applied mathematics and mechanics of aircraft structures, and for the establishment of the Reissner variational principle," and a certificate of achievement from the Pressure Vessels and Piping Division of ASME, in 1987, "in testimony of his contributions to the Division membership by his pioneering work in the theory of plates and shells." This was followed by the receipt of the ASME Medal in 1988, "for eminently distinguished contributions to the practice of engineering through his research on plates and shells, structures, theory of elasticity, turbulence, aerodynamics, wing theory and mathematics and for his stewardship of numerous doctoral candidates," and by an Honorary Membership in ASME in 1991 "for his profound and lasting mark on international applied mechanics through over half a century of teaching and research and for wise counsel at the highest levels of ASME." In 1992, Eric Reissner became the seventh Honorary Member in the 70 year history of the German Gesellschaft für Angewandte Mathematik und Mechanik ''in recognition of his exceptional accomplishments in Applied Mathematics and Mechanics." It would be presumptuous of us to embark on a thorough assessment of Professor Reissner's work in this preface. His awards, and the citations thereof, indicate the specific seminal contributions that the community of mechanicians appreciates him for. We can only cite, from our perspectives, what we think are the contributions over a span of sixty years that will secure his place in the history of 20th century applied mathematics and mechanics: (i) the two-field variational theorem involving independent stress and displacement variations for linear as well as for finite elasticity; (ii) shear deformation plate theory, with resolution of the classical boundary condition paradox of Kirchhoff; (iii) his contributions to the subject of the center of shear; (iv) his 1949 seminal contribution to the nonlinear theory of shells; (v) his insights concerning the asymptotics of edge-zone and interior solution contributions for plate and shell boundary value problems; (vi) his 1965 contribution to variational theorems in elasticity, with rotations as additional independent variables. Apart from these contributions in the mechanics of solids his creative spirit touched on other topics as well. They included (i) statistical theory of turbulence; (ii) steady and unsteady aerodynamic lifting-surface theory; and (iii) analysis of finite span effects for wing divergence and flutter speed computations. This volume presents selected original research papers of Professor Eric Reissner in the various areas mentioned above. May it serve as a milestone and a beacon for future generations. SATYA N. ATLURI, GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTA THOMAS J. LARDNER, UNIVERSITY OF MASSACHUSETTS, AMHERST JAMES G. SIMMONDS, UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE FREDERIC Y-M. WAN, UNIVERSITY OF CALIFORNIA, IRVINE

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A BIOGRAPHICAL SKETCH I was born January 5, 1913 in Aachen, Germany, the son of Hans Reissner, then Professor of Applied Mechanics and Founder of the Aerodynamics Institute at the Aachen Technische Hochschule. That same year my father followed a call from his alma mater which meant that I grew up in Berlin during the period 19131936. My secondary school years were scholastically without distinction as I preferred to work on improving athletic skills. I was a member of field hockey teams, ran in the Potsdam-Berlin relay races, and for a while was the best shot putter among the 15 year olds in the Berlin area. Mathematics had always been my easiest academic subject. I became truly interested in it upon exposure to the elements of the calculus. The new concepts fascinated me and I remember supplementary studies from one of my father's old textbooks, authored by Serret and translated by Scheffers. After graduation, in 1931, with average grades, except in mathematics, physics and physical education, I matriculated at the Technische Hochschule Berlin. I first majored in Applied Physics, as this seemed the safest subject from the point of view of future employment prospects. However, I soon found out that I was not particularly disposed towards doing some of the things which went with becoming an applied physicist. On the other hand, I had no trouble at all with mathematics and mechanics courses, and so I moved from Applied Physics to Applied Mathematics at the end of the second year. In trying to combine ideas coming from different sources and courses I solved two problems which became published papers in 1934 and 1935 [1, 2]. My most influential professors were Georg Hamel and my father who taught me Theoretical and Applied Mechanics. Aside from this I have good memories of learning about complex variables from Ernst Jacobsthal, about differential equations from Richard Fuchs and about theoretical physics from Richard Becker. A one-semester leave in 1934, to attend the Zürich Institute of Technology, provided a valuable opportunity to take courses from Enst Meissner, Wolfgang Pauli and Georg Polya. Graduating with honors in the Fall of 1935 I spent the following six months expanding my Dipl. Ing. Thesis into a Dr. Ing. Dissertation, on the subject of forced vibrations of a mass supported over a finite contact area by an elastic halfspace, expanding on some classical work by Lamb on the corresponding mass-less point load problem [5, 78]. At that time the political developments in Germany became more and more unpromising. Several inquiries about opportunities abroad resulted in a one-year Mathematics scholarship at the Massachusetts Institute of Technology, and a one-year student visa allowing travel to the U.S.A. Before the year was over M.I.T. decided that it could use me as a research assistant in Aeronautics. This meant a permanent residence permit through the offices of the U.S. consulate in Niagara Falls. The aeronautics appointment lasted from 1937 to 1939. It included an opportunity for a Ph.D. in mathematics with an analysis of the aeronautical

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structures problem of tension field theory [15]. A subsequent instructorship in Mathematics was followed by promotions to Assistant Professor in 1942, Associate Professor in 1946, and Professor in 1949. As I think back to my more than thirty years at M.I.T. I must begin by recalling the importance of friendships with my colleagues H. B. Phillips (who brought me into the Department which he headed), Ted Martin, George Thomas and C. C. Lin. All of them contributed significantly to my development. In a temporal way, I have special memories of the period 1945 to 1960. It was during this period that those of my papers which are still often referred to were written. This period also included summer appointments with the Structures and the Dynamics Divisions of the Langley Field Laboratory of the National Advisory Committee for Aeronautics (1948, 1951), with Ramo-Wooldridge in Los Angeles (1954, 1955) to help solve problems in the design of the Atlas missile, and with Lockheed in Palo Alto (1956, 1957) who were then concerned with the development of the Polaris. A highlight was a two months Symposium on Structures and Elasticity at the University of Michigan in 1949. My assignment was to present the Theory of Elasticity, together with S. Timoshenko giving lectures on the Theory of Thin Plates, and R. V. Southwell on Advanced Airplane Structures. Also in 1949 I was asked to be Consulting Mathematics Editor for the Addison-Wesley Publishing Company, then a very small new organization. The fact that the ensuing series of books included a Calculus text by George Thomas and a text on Advanced Calculus by Wilfred Kaplan meant that this assignment, which lasted until 1960, resulted in significant economic benefits. As the years went by I became more and more conscious of the fact that my research and teaching interests belonged to the Engineering Sciences rather than to Mathematics. This being the case, I accepted in 1970 an appointment as Professor of Applied Mechanics to participate in the growth of this field at the new San Diego campus of the University of California. There I joined in the efforts of Y. C. Fung, J. W. Miles, W. Nachbar, S. S. Penner and other younger, capable and friendly colleagues. It turned out to be a truly uplifting and refreshing experience. In conclusion, I would like to express thanks to my friends and one-time students Satya Atluri, Fred Wan, Tom Lardner and Jim Simmonds. Their help in bringing the project of this Volume to fruition has been important. Besides, our personal contacts over the years and our joint studies from time to time have helped me to stay involved in the adventure of seeking new insights in the field of applied mechanics to this day. And, as far as keeping me involved personally and professionally is concerned, I feel an obligation to acknowledge my long-ago students Bob Clark at Case Western, Millard Johnson at the University of Wisconsin, Jim Knowles at Cal Tech, Sud Nair at Illinois Tech, W. T. Tsai at Long Beach State and the late Hubertus Weinitschke of the University of Erlangen-Nürnberg. A word about the contents of this Volume. In selecting papers for inclusion I was guided by the wish to consider the significance of the work at the time it was done, relative inaccessibility except in this place, in some instances a preference for co-authored efforts, and finally a preference for brief articles, in order to be able to include as many of them as possible. Finally, and foremost, I must express a sense of deep gratitude and affection to my wife Johanna. Her support and influence during our many years of harmonious togethernesswhich included the raising of our son John, and our daughter Evahave been of inestimable value. E.R.

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BEAMS While still in high school and having just learned the geometrical significance of first and second derivatives of a function, I asked my father what significance there might be to derivatives of higher order than two. Naturally, he mentioned that the fourth derivative of the deflection curve of a beam would be proportional to the intensity of the load distribution responsible for this deflection. I learned a good deal more about this subject during my first year at the university as a student in my father's mechanics course. As luck would have it, a student summer job involved the inadequacy of elementary beam theory for T-beams with very wide flanges. I read an analysis of this problem by von Karman who considered the behavior of the flanges as a problem of the theory of plane stress. However, instead of determining the constants of integration in a Fourier series solution by means of transition conditions between flange and web, von Karman pursued a more elaborate route by way of minimizing the strain energy of the flange-web combination. Finding out that it was simpler to solve the problem without use of the minimum energy condition resulted in my first published paper [1]. As a graduate student assistant three years later, I was asked to work on the related problem of shear lag in box beams. I recognized that it was a good enough approximation for the shear lag problem to replace the elementary uniform crosswise distribution of axial normal stress by a parabolic distribution and to look for an approximate Rayleigh-Ritz type solution with the help of a minimum energy condition. I first used the principle of least work [24] and later the principle of minimum potential energy [43]. After this I never lost my interest in beam problems of a non-standard nature. An unexpected result for the Saint Venant distribution of shear stress in plate-like rectangular cross-section beams [44,48] was followed by a shell-theoretical analysis of von Karman's problem of bending of curved tubes [60, 74, 223] and by an analogous analysis of Brazier's non-linear effect of cross section flattening for the bending of straight tubes [83, 132, 138]. This in turn was followed by a consideration of the center of twist problem for the torsion of cantilevers [93], by a consideration of torsional vibrations of pre-twisted beams [108], and by a solution of the torsion problem of circumferentially non-homogeneous tubes, which included as special cases the classical results for both open and closed cross section tubes [122, 173]. An interest by my doctoral student W. T. Tsai led me to the problem of how non-symmetrical beams should be loaded in order to have bending without twisting [187, 188]. While I had known of the text book solution for thin-walled open cross section beams, and also of attempts to deal with the problem in the context of Saint Venant's flexure analysis by such people as G. I. Taylor and E. Trefftz, I had felt

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uncomfortable with the premise of these approaches, which involved either displacement specifications for an (in my mind) arbitrarily chosen point in the supported cross section, or an energy specification without appeal to the form of the support condition. I came to the conclusion that for a rational definition of a center of shear it was necessary to begin with a suitable system of mixed loading boundary conditions for a three-dimensional formulation of the problem. Subsequent to this it was then possible to utilize the Saint Venant flexure assumptions in a Rayleigh-Ritz sense for an approximate determination of the coordinates of this center. I returned to the problem once more fifteen years later by way of deducing beam-theoretical results as consequences of approximately analyzing elastic plates of variable thickness [263, 272, 280]. This included, in particular, the derivation of beam equations accounting for anti-clastic curvature effects in addition to warping stiffness effects in the sense of Vlasov, with this permitting in particular a quantitative appraisal of the influence of Poisson's ratio on the location of the center of shear. At about the same time I became interested, as a consequence of studying finite rotation shell theory, in a consideration of the one-dimensional space-curved beam problem [190, 191, 225]. My approach here, the same as for the shell problem, was the reverse of what was usually done. Instead of beginning with the geometry of finite deformations I began with what was for me the easier problem of stating equilibrium equations for the deformed structure. After that I used the principle of virtual work in an inverse fashion to establish a system of virtual strain displacement relations. While the step from virtual to actual relations is elementary in the linear theory, in finite-deformation theory this step required a judicious non-linear differential equations integration scheme. I arrived at a system of beam equations which represented a generalization of Kirchhoff's rod theory, the generalization consisting in allowance for force-deformational effects, in addition to the classical momentdeformational effects. An occasion to apply these equations was the appearance of a paper on lateral buckling in which the first Kirchhoff-rod-theory based solution of this problem, in a 1904 paper by H. Reissner, was used once more, with a critical comment on the neglect of one of two pre-buckling deformation terms in the 1904 paper. I reexamined the problem [214], hoping to find fault with this comment but came to the conclusion that the criticism was in fact justified, although the effect of the neglected term was quite small compared to the effect which had not been neglected. In the course of this study, I then also used the force-deformational terms in my equations for a determination of the transverse shear effect on the value of the classical Michell-Prandtl buckling load. A continuation of my concern with the problem of lateral buckling, led on to a note, jointly with my son John Reissner, on the consequences of constitutive coupling of torsion and bending [237], and later on to results through use of the equations of three-dimensional finite-deformation elasticity concerning refined one-dimensional lateral buckling equations incorporating warping stiffness in addition to bending and twisting stiffness [240, 259, 265].

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Über Die Berechnung Von Plattenbalken [Der Stahlbau 7, 286288, 1934] Einleitung Die übliche Biegungstheorie der Träger mit gerader Mittellinie geht von der Voraussetzung aus, daß ein in einer Hauptträgheitsebene der Querschnitte wirkendes Biegungsmoment quer zu dieser Ebene konstante Spannungsverteilung erzeugt. Im allgemeinen führt diese Annahme auch zu keinen unzulässigen Widersprüchen mit den Ergebnissen der Elastizitätstheorie. Es ist jedoch seit langem bekannt, daß die erwähnte Annahme bei Plattenbalken und Kastenträgern einigermaßen breitem Gurt auch näherungsweise nicht mehr zutrifft. Man hat in diesen Fällen den Begriff der "mittragenden Breite" eingeführt, worunter man diejenige Gurtbreite versteht, mit der bei der Annahme konstanter Spannung nach der Breite hin sich dieselbe maximale Biegungsspannung ergeben würde, wie diejenige des Plattenbalkens mit nach der Seite abklingenden Spannungen. Eine rationelle Methode zur Berechnung der mittragenden Breite bei durchlaufenden T- Trägern hat zuerst Prof. v. Kármán angegeben 1). Vorausgesetzt wird dabeiwas auch hier geschehen solldaß die Plattenstärke klein ist im Vergleich zur Trägerhöhe, und daß die Biegungssteifigkeit der Gurtplatte senkrecht zu ihrer Ebene zu vernachlässigen ist gegen die des Steges2). Es wird also angenommen, daß in der Platte ein ebener Spannungszustand herrscht. Dieser Spannungszustand ist offenbar abhängig von der Belastung und von den Abmessungen des Systems. Den Zusammenhang zwischen Steg und Platte berücksichtigt v. Kármán mit Hilfe des Prinzips vom Minimum der Formänderungsarbeit. Zahlenbeispiele nach dieser Methode für verschiedene Lastverteilungen rechnete Dr. Metzner 3). Es ergab sich aus diesen Rechnungen, daß die tragende Breite längs der Trägerachse durchaus nicht immer konstant, sondern von der Momentenverteilung abhängig ist. Erweiterungen der Theorie auf Kastenträger, auch auf Fälle nicht durchlaufender Träger finden sich in zwei Arbeiten von Prof. G. Schnadel 4). Im folgenden soll zunächst eine Methode angegeben werden, mit der ebenfalls der elastische Zusammenhang zwischen Steg und Gurt berücksichtigt, die Aufgabe aber auf ein reines Randwertproblem der Spannungsfunktion der Gurtplatte zurückgeführt wird. Auf diesem Wege können die formelmäßigen Ergebnisse der bisherigen 1Th. v. Kármán, Die mittragende Breite. A. FöpplFestschrift 1924. S. a. S. Timoshenko, Theory of Elasticity, S. 156. M cGraw-Hill Book Comp. Inc. New York und London 1934. 2In einer späteren M itteilung wird gezeigt werden, daß es möglich ist, die Aufgabe in gewissen Fällen auch ohne diese einschränkende Voraussetzung streng zu lösen. 3W. M etzner, Die mittragende Breite. Lufo IV, 1929. 4G. Schnadel, Die Spannungsverteilung in Flanschen dünnwandiger Kastenträger. Jahrb. d. Schiffbautechn. Ges. 1926.Die mittragende Breite in Kastenträgern. Werft, Reederei & Hafen 1928.

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Arbeiten mit sehr wenig Rechenaufwand erhalten werden. Weiter ergibt sich die prinzipielle Möglichkeit, diejenigen Näherungsverfahren zur Lösung von Randwertaufgaben anzuwenden, welche die Angabe sämtlicher Randbedingungen durch die Randwerte der gesuchten Funktion und ihrer Ableitungen erfordern (Ritzsches Verfahren, Methode der Differenzenrechnung usw.). In einem zweiten Abschnitt wird eine genauere Theorie aufgestellt, die insbesondere für Träger mit einer gegenüber der Spannweite nicht mehr kleinen Steghöhe von Bedeutung sein kann. Ferner wird gezeigt, wie man auch aus ihr durch Grenzübergang zu kleinen Steghöhen die alten Ergebnisse erhalten kann. I Einfache Theorie. Steg Als Balken Hier ist die folgende Aufgabe zu lösen: Gegeben nach Bild 1 ein Steg, in dem der Charakter der Spannungsverteilung nach der üblichen Näherungstheorie, und eine Gurtplatte, in der ein ebener Spannungszustand vorausgesetzt werden soll. Die Berücksichtigung des elastischen Zusammenhangs erfolgt in der Weise, daß man an der Anschlußstelle StegGurt die Dehnung in der Platte derjenigen Dehnung gleichsetzt, die dort herrschen würde, wenn man einen Plattenbalken vor sich hätte von der Gurtbreite 2O und der Breite nach konstanter Spannung.

Bild 1.

Nun lassen sich bekanntlich die Spannungen Vx, Vy, W eines ebenen Spannungszustandes folgendermaßen als Ableitungen einer Spannungsfunktion F schreiben:

wobei F der folgenden Differentialgleichung genügen muß:

Den Zusammenhang zwischen den Dehnungen und den Spannungen gibt das verallgemeinerte Hookesche Gesetz, wenn man die Verschiebungen in der x- bzw. yRichtung mit u bzw. v, die Winkeländerung mit J bezeichnet, folgendermaßen:

wobei E den Dehnungsmodul, G den Schubmodul und m das Querkontraktionsver-

Page 5

hältnis bedeutet. Zwischen E und G besteht überdies die Gleichung

Das Koordinationssystem möge nach der in Bild 1 angegebenen Weise gewählt werden. Die Randbedingungen für die Anschlußstelle StegGurt können ein für allemal angegeben werden. Aus Symmetriegründen folgt, daß die Verschiebung quer zur Stegachse verschwinden muß.

Zur zweiten Bedingung werde die Aussage über die Dehnung längs der Trägerachse gemacht. Es ist unter den gemachten Voraussetzungen:

wobei M(x) das Biegungsmoment und W(x) das Widerstandsmoment des Trägers mit der vollmittragenden Breite 2 O ist. Andererseits ist O durch die folgende Gleichung definiert.

welche ausdrückt, daß der Inhalt der nach der Seite abklingenden Gurt-spannungsfläche einer ideellen rechteckigen Spannungsfläche gleichgesetzt wird. Das Widerstandsmoment wird, wie man leicht ausrechnet,

oder, wenn man nach O auflöst,

Aus (8a) und aus der Definitionsgleichung (7) der tragenden Breite bekommt man das zugeordnete Widerstandsmoment

Wenn man (9) in die Randbedingung (6) einsetzt, erhält man schließlich als Randbedingung aus (6)

und unter Berücksichtigung der Beziehungen (3) und (1)

Page 6

Für die tragende Breite O ergibt sich aus (7) unter Berücksichtigung von (8) die folgende Gleichung:

Durchführung Für Einen Besonderen Fall Nimmt man als Spannungsfunktion F die M. Lévysche Lösung der biharmonischen Differentialgleichung

mit Q = nS/l, so läßt sich durch sie einmal, wie v. Kármán gezeigt hat, der Spannungszustand in der Gurtplatte eines durchlaufenden Trägers von der Stützweite 2 l, der ein ebenfalls periodisches Moment von der Form

aufzunehmen hat, darstellen Man muß dann für den durchlaufenden Träger mit überall positiver Belastung an den Stützpunkten, d.h. für x = 0 und x = 2l aus Symmetriegründen fordern

Aus der Form der Spannungsfunktion ergibt sich damit, daß ebenda

Man kann aber auch, wie G. Schnadel zuerst bemerkt hat, den Spannungszustand in der Platte eines gelenkig gestützten Trägers von der Spannweite l darstellen, denn (10) erfüllt die Bedingung

für x = 1/2l und x = 3/2l in jedem Gliede der Spannungsfunktion für sich. Man erhältals zweite Randbedingung an denselben Stellen

d.h. die Lösung ist streng, wenn durch Versteifungen an den freien Rändern für die Erfüllung der Gl. (18) gesorgt wird, was in der Praxis oft der Fall ist (Man kann sich diesen gelenkig gestützten Träger auch als Teil eines durchlaufenden Trägers vorstellen mit periodischer, abwechselnd positiver und negativer Belastung. Beschränken wir uns hier für die weitere Durchführung auf den Fall des unendlich breiten Gurtes, so werden wegen des Verschwindens der Spannungen für y = f

Drückt man die Bedingung v(x, 0) = 0 mit Hilfe von (3) durch die Ableitungen der Spannungsfunktion aus, so erhält man folgenden Zusammenhang zwischen An und Bn

Page 7

Damit nimmt die Spannungsfunktion die folgende Gestalt an

Die vierte Randbedingung, Gl. (11), des stetigen Überganges vom Gurt auf den Steg ist die folgende:

also:

damit erhalten wir aus Gl. (12) die folgende Bestimmungsgleichung für O

welche also erlaubt, den Plattenbalken nach der elementaren Theorie mit Vx unabhängig von y zu berechnen, wenn die sich daraus ergebende ideelle Gurtbreite O eingeführt wird. Für M(x) = M cos Qx, eine Momentenverteilung, wie sie sich sehr angenähert für den gelenkig gestützten Träger unter gleichmäßiger Volllast ergibt, wird z.B., mit m = 10/3

Formel (24) findet sich bereits in der Arbeit von Herrn v. Kármán, der mit ihrer Hilfe feststellt, daß für eine einfach harmonische Momentenverteilung O = const. wird (was man übrigens bei der gewählten Spannungsfunktion unmittelbar aus (6) und (7) ersehen kann, so daß dieses Resultat unabhängig von der Randbedingung (11) ist), und daß die tragende Breite durch die späteren harmonischen Glieder nicht unerheblich vermindert werden kann. Es ist möglich, aus Gl. (22) die folgende schärfere und wie es scheint bis jetzt unbekannt gewesene Folgerung zu ziehen, daß es Momentenverteilungen gibt, für die im gefährlichen Querschnitt die tragende Breite beliebig klein wird. Hinreichend dafür ist die genügende Kleinheit von

d.h. bei Spitzen in der Momentenfläche ist die Materialausnutzung besonders schlecht.

Page 8

Least Work Solutions of Shear Lag Problems [J. Aeron. Sciences 8, 284291, 1941] Introduction It is a well known fact that the distribution of bending stresses in thin-walled box-beams cannot be obtained from the customary theory of bending of beams when the lateral extension of such structures is of the order of magnitude of their spanwise extension. The elementary beam theory assumes a uniform distribution of spanwise normal stress at any transverse section and consequently fully efficient chord members, but the shear deformation of these members leads to a distribution of normal stress which, at a given beam section, has its maximum in general at the side webs, decreasing toward the middle of the chord. Neglecting this effect amounts to an overestimation of the strength of the beam. It has been found that the dimensions of box-beam-like airplane-wing structures are often such that this shear-lag effect is appreciable. It may happen that the stress in the middle of the sheet amounts to only 60 per cent of the edge stress. To determine the magnitude of the effect, theoretical and experimental investigations of the problem have been carried out, giving the desired information for some important cases and showing furthermore which mathematical problem is to be solved in any given case (see the references at the end of this paper). The main difficulty consists in the fact that the stress problem is two-dimensional and attempts to solve its fundamental equations, with a variety of assumptions concerning the elastic properties of the cover sheets, have been successful for certain arrangements only. It seems, therefore, desirable to find a way to reduce the shear-lag problem to a one-dimensional problem, in the sense that an equation be established for a quantity indicating the amount of shear lag at every cross-section of the beam. Such an equation will have to contain parameters depending on the dimensions of the structure and the distribution of the load. It need not be an exact result of the theory of elasticity so long as it is certain that the analysis retains the essential characteristics of the problem and gives numerical results in close agreement with the exact results. The purpose of this paper is to derive such an equation for the class of box-beams symmetrical about span-wise vertical and horizontal planes through the neutral axis of the beam. There is, however, no inherent difficulty in generalizing the results to include unsymmetrical beams as well, although the corresponding derivations will be less simple than the ones presented here. Also given are applications of the fundamental equation to the actual solution of a series of shear-lag problems. The starting point for the method developed here was the fact that in all symmetrical cases investigated so far the shape of the curve representing the

Page 9

distribution of normal stress across the beam seemed to be very nearly parabolic. If one makes the assumption that these curves should be true parabolas, distinguished from each other only through the values of their vertex curvature, all that remains to be done is to establish an equation for the spanwise variation of this vertex curvature. The most convenient way to do this appears to the author to be the application of a minimum energy principle. A distribution of stress in the horizontal (or nearly horizontal) cover sheets is assumed, at every cross-section parabolic in the spanwise normal stress and, moreover, satisfying the equilibrium conditions for every element of the sheet. The linear side-web normal stresses are determined in such a way that cover sheet and side-web normal stresses coincide along the flanges. Furthermore, the condition is imposed that the resultant moment of the spanwise normal stresses at every section about a transverse horizontal axis equals the external bending moment at that section. When these conditions are satisfied there remains only one unknown quantity, the vertex curvature of the normal stress parabola, and this quantity may be determined by minimizing the internal work of the structure. This minimum condition is shown to reduce to an ordinary second order differential equation, with constant coefficients for beams of constant cross-section, and with variable coefficients for tapered beams. Formulation of the Problem A cantilever box-beam is considered, with rectangular doubly symmetrical cross-section, acted upon by a given distribution of bending moments (Figure 1).

Fig. 1.

The assumption of parabolic spanwise normal stress in the coversheets is expressed by writing,

For the normal stress in the side webs one has

Continuity of the stresses demands

Page 10

The condition of moment equilibrium is expressed by the equation

Introducing Vs, from Eqs. (2) and (3) into Eq. (4), there follows a relation involving sheet stresses only,

Eq. (5) serves to express s0 in terms of the vertex curvature 2s/w2 of the normal stress parabola, giving

It shall be assumed that the parameter m has the same value all along the span. Thus one may write Eq. (1)

The least work condition will serve to determine the quantity s. To apply this condition, it is first necessary to find the remaining sheet stress components, W and Vy, which must be in equilibrium with Vx as given by Eq. (7). Then it is necessary to establish the expression for the internal work W of the entire beam and finally the stresses have to be introduced into W and W be made a minimum. The Distribution of Stress in the Sheets and the Internal Energy of the Bent Beam The sheet stresses have to fulfill the following equilibrium conditions of generalized plane stress,

It is well known that Eqs. (8) and (9) may be satisfied in terms of stress functions in the following manner,

or

Eq. (11) is preferable when the transverse normal stress, Vy, need not be considered, and it can be shown that this is generally permissible in the shear-lag problem. From Eqs. (7) and (8) one obtains by integration for the shear stress

Page 11

In this formula an arbitrary function of integration has been eliminated by the condition that W must be antisymmetric about the axis y = 0. With Eq. (11) for W there follows from Eq. (9) for Vy,

*

In Eq. (13) an arbitrary function of x has been determined by the condition that Vy(x, ± w) = 0. Eqs. (7), (11) and (12) are satisfied by taking as stress function

where

are the sheet stress and the sheet stress resultant of the elementary beam theory. To obtain an expression for the internal work in terms of the stresses, it is necessary to agree on the elastic properties of the sheet material. It shall be assumed that the cover sheets are of non-isotropic material. In this way it is possible to account in a convenient way for the influence of closely spaced transverse and longitudinal stiffeners and it also shows that neglecting the work of the transverse normal stresses Vy corresponds exactly to a limiting case of the orthotropic stress-strain relations. Writing the stress-strain relations in the form

where v x and v y are Poisson ratios, the virtual work per unit of sheet area, t(VxGH x + VyGH y + WGJ), gives for the elastic energy of the two sheets

For the existence of Ws it is necessary that the following relation is satisfied between the elastic constants,

Adding to the energy stored in the sheets, as given by Eq. (18), the energy of the two side webs,

which with Eqs. (2) and (3) becomes

*This equation for Vy may be used to estimate quantitatively the magnitude of the transverse normal stresses, associated with the parabolic distribution of the spanwise stresses determined subsequently.

Page 12

there is given in

the total energy of the box-beam, provided the conditions of support at the root section and the stiffening of the tip section are such that the edge stresses at these sections are prevented from doing work during a virtual displacement. Such conditions at the root section require vanishing spanwise and transverse displacements, or, instead of the second condition, vanishing shear (which is the condition the available exact solutions fulfill). * At the tip section these conditions require vanishing spanwise normal stress and vanishing transverse displacement (as in the case of the available exact solution) or instead of the displacement condition, the condition of vanishing shear. It is, however plausible that with the exception of the condition mentioned above * the possible contribution to the total work due to flexibility of tip and root ribs is relatively small and may therefore be disregarded. The expression for the internal work will be further simplified by neglecting the work of the transverse normal stresses Vy compared with the work of Vx and W. It seems plausible that this omission is in general of no great influence in the final result** (see also references 6, 7, 21). From Eqs. (18) and (19) it follows that neglecting Vy amounts to putting

in the stress-strain relations. This is exact for the case of a sheet rigid in transverse direction. In this connection it is noted that the presence of rather narrowly spaced transverse stiffeners would in any event tend to give the sheet an effective Ey which is greater than Ex. The Least Work Condition Neglecting Vy in Eq. (18) and writing Ex = E the total work W is

and with Vx and W from Eq. (11)

According to the rules of the calculus of variations the condition for a minimum of W is that the variation GW, vanishes. Now

*For those cases where the spanwise displacement of the sheet at the root section is not completely restrained, due for instance to the presence of a retractable landing gear, it will be necessary to add a term to the total work expression, to account for the bending energy of the transverse stiffener. **It would be possible to find the solution without this simplifying assumption, by means of a fourth order equation instead of the second order equation derived in what follows, and thus to determine quantitatively its effect.

Page 13

and since G (dH) = d(GH) there follows, by integration by parts,

The second and the last integral in Eq. (25) may be combined to

Observing that from the equilibrium condition Eq. (5), which on account of Eq. (11) can be written in the form,

there follows

it is seen that the integral Eq. (26) vanishes. The condition of least work is thus reduced to

From Eq. (14) follows

and

Since at the tip Vx(0, y) = 0 it follows further that

and with that

Page 14

Introducing Eqs. (30) and (31) into Eq. (29) one has

The integration with respect to y may be carried out, and since Gs(x) is arbitrary the integrand of the first term and the second term has to vanish separately, resulting, together with Eq. (32), in the following differential equation and boundary conditions for s(x):

With the solution of Eq. (36), subject to the boundary conditions Eq. (37), there is given the solution of the shear-lag problem for symmetrical box-beams under symmetrical (bending) loads. It is evident that the solution for antisymmetrical (torsional) loads for the same beam may be obtained in a completely analogous way. Some added considerations seem necessary to generalize the solution to the case of unsymmetrical beams. The nature of the analysis is, however, such that the possibility of this generalization is apparent. Of interest is the value of an effective sheet width weff. which, with the help of Eq. (7), is expressed in the form

In what follows Vx and weff. will be calculated for some typical beams of constant cross-section, for different loading conditions. These calculations permit one to draw some useful general conclusions, as will also be shown. It should be noted that the integration of Eqs. (36) and (37) in closed form is also possible for tapered beams of the sort that w = w0(x + x 0), t = t 0(x + x 0)n, that is for beams with a law of sheet-thickness and beam-width variation such that if the beam were continued to the left of x = 0, w and t would simultaneously become zero with exception of the case n = 0 where the sheet thickness is constant. The integration is possible by assuming the solutions of the homogeneous equation (36) in the form s(x) = (x + x 0)r. A further noteworthy result consists in the fact, that when the sheet stress resultant S(x) = tVb is constant, then s(x) = 0 satisfies Eqs. (36) and (37) and there is no shear lag. The explanation of this follows from the equilibrium condition Eq. (8) which shows that for (wtVx/wx) = 0 the sheet shear stress W vanishes and consequently no shear deformation occurs. Therefore, one can say that, in principle, it is possible to design box-beams of the type considered, with fully efficient chord members.

Page 15

Solution of the Shear-Lag Equation for Beams of Constant Width and Sheet Thickness Assuming w and t constant,* Eqs. (36) and (37) become

where O2 and N are defined by

The solution of the system of Eqs. (39) and (40) is found, by the method of variation of parameters, in the form

where [ = x/l. This formula is valid for distributed as well as for concentrated loads if it is understood that a concentrated load has to be considered as limiting case of a distributed load, the limiting process to be carried out after the integrals have been evaluated. The function s(x) and with that according to Eqs. (7), (12) and (13) the stress pattern in the coversheets will here be determined explicitly for the following typical loading conditions: 1. A concentrated load P at the tip section 2. A uniformly distributed load p0 3. A concentrated load P at a distance l1 from the tip. One obtains

*The solutions thus derived will be applicable also to the case of piecewise constant thickness t, if along the sections where t is discontinuous there is continuity of tVx and tW, which according to Eqs. (7) and (12) means continuity of ts(x) and tds/dx.

Page 16

To represent the results of these formulas graphically it is convenient to write Eq. (7) for the spanwise stress in the form

Numerical Examples Eqs. (43) to (49) shall be evaluated, assuming the following dimensions

From Eqs. (6), (38) and (41) results

With these values of the parameters the stresses in the middle of the sheet and at the edges of the sheet have been calculated and are represented in Figures 2 to 4, together with the corresponding curves for the stress without shear lag. The graphs show that shear lag is most pronounced near the built-in end (the most highly stressed section) of the beam. It is further noteworthy that in the case of concentrated load application at midspan there occurs an appreciable sheet stress at the point of load application, which along the flanges is of opposite direction to

Fig. 2.

Page 17

Fig. 3.

the corresponding stress at the root section (see also reference 14). The elementary theory does not account for this stress. The most important characteristics of these graphs are, however, that the shear lag at the built-in end is almost the same for the uniformly distributed load and for the concentrated load at midspan, while for the beam with tip load there is considerably less shear lag. This suggests that the effective sheet width depends, for beams of constant cross-section, on the distance of the center of gravity of the load curve from the built-in end rather than on the span length of the beam. A formula expressing this fact will now be derived. According to Eq. (38) weff. is, for beams with constant coversheet thickness t, given by

with, if it is furthermore assumed that the width w is constant, s(x) from Eq. (42). At the built-in end of the beam, weff. depends on s(l), which may be transformed into

For values of N in the practical range one may with good approximation neglect the term sinh NK under the intergral and put tanh N = 1. Thus

Page 18

Fig. 4.

Writing

it is seen that in the dimensionless ratio

the quantity L represents the distance of the center of gravity of the d2Vb/dx 2-curve from the built-in end of the beam. For beams of constant cross-section this is identical with the distance of the center of gravity of the load curve from the built-in end. Introducing Eqs. (54) and (41) into Eq. (50) there follows

*Instead of as indicated in Figure 5, equation should read: m = (Is + 3Iw)/I.

Page 19

Fig. 5.*

and in an analogous way

Eqs. (55) and (56) are general formulas for the amount of shear lag in beams of constant width and cover sheet thickness, with or without taper in height. They indicate in which way shear lag depends on the ratio of tension and shear modulus, on the relative magnitude of coversheet and side web stiffness and on the ratio of sheet width 2w and the distance L of the center of gravity of the Vbcc-curve from the root section of the beam. Figure 5 gives (weff./w) root as a function of w/L when E/G = 8/3 and for various values of the stiffness parameter m. It may be added that solutions of the numerically discussed problems could also have been obtained by an exact method (references 7, 22), although only in the form of not very rapidly converging infinite trigonometric series. Corresponding exact solutions can also be obtained for beams with isotropic coversheets (see references 1, 11, 13, 19, 21). These same exact methods are, however, not suitable for the treatment of tapered beams, while the present method, as shown, remains usable. Also, the general expressions, Eqs. (55) and (56), are mainly due to the fact that the present approximate solution has a considerably simpler form than obtainable exact solutions. References 1Chwalla, 2Cox,

E., Die Formeln zur Berechnung der ''vollmitragenden Breite" dünner Gurt und Rippenplatten , Der Stahlbau, 9, 7378, 1936.

H. L., Smith, H. E., and Conway, C. G., Diffusion of Concentrated Loads into Monocoque Structures, R. and M. No. 1780, 1937.

Page 20 3Cox, H. L., Diffusion of Concentrated Loads into Monocoque Structures, III; General Considerations with Particular Reference to Bending Load Distributions, R. and M. No. 1860, 1938. 4Cox,

H. L., Stress Analysis of Thin Metal Construction, Journal of the Royal Aeronautical Society, 44, 231282, 1940.

5Duncan, 6Ebner,

W. J., Diffusion of Load in Certain Sheet-Stringer Combinations, R. and M. No. 1825, 1938.

H., and Koeller, H., Über den Kraftverlauf in längs und querversteiften Scheiben, Luftfahrtforschung, 15, 527542, 1938.

7Younger,

John E., Metal Wing Construction, Part II, A.C.T.R. Series No. 3288, Material Division, U.S. Army Air Corps, 1930.

8Kuhn,

P., Stress Analysis of Beams with Shear Deformation of the Flanges, N.A.C.A. Technical Report No. 608, 1937.

9Kuhn,

P., Approximate Stress Analysis of Multi-Stringer Beams with Shear Deformation of the Flanges , N.A.C.A. Technical Report No. 636, 1938.

10Lovett,

B. B., and Rodee, W. F., Transfer of Stress from Main Beams to Intermediate Stiffeners in Metal Sheet Covered Box Beams , Journal of the Aeronautical Sciences, 3, 426, 1936.

11Metzner,

W., Die mittragende Breite, Luftfahrtforschung, 4, 120, 1929.

12Reissner,

H., Über die Berechnung der mittragenden Breite , Z. Ang. Math. Mech., 14, 312313, 1934.

13Reissner,

E., Über die Berechnung von Plattenbalken, Der Stahlbau, 7, 282284, 1934.

14Reissner,

E., Beitrag zum Problem der Spannungsverteilung in Gurtplatten, Z. Ang. Math. Mech., 15, 359364, 1935.

15Reissner,

E., On the Problem of Stress Distribution in Wide-Flanged Box-Beams , Journal of the Aeronautical Sciences, 5, 295299, 1938.

16Reissner,

E., The Influence of Taper on the Efficiency of Wide-Flanged Box-Beams , Journal of the Aeronautical Sciences, 7, 353357, 1940.

17Schade,

H., Application of Orthotropic Plate Theory to Ship Bottom Structure , Proceedings, 5th International Congress, Applied Mechanics, pp. 144149,

1938. 18Schapitz,

E., Feller, H., and Koeller, H., Experimentelle und rechnerische Untersuchung eines auf Biegung belasteten Schalenflügelmodells, Luftfahrtforschung, 15, 563576, 1938.

19Schnadel, 20Sibert,

G., Die Spannungsverteilung in den Flanschen dünnwandiger Kastenträger, Jahrb. d. Schiffbaut, Ges. 27, 207291, 1926.

H., Effect of Shear Lag upon Wing Strength , Journal of the Aeronautical Sciences, 6, 418, 1939.

21von Kármán, 22Winny,

Th., Die mittragende Breite, August-Foeppl Festschrift, Berlin, pp. 114127, 1924.

H. F., The Distribution of Stress in Monocoque Wings , R. and M. No. 1756, 1937.

Page 21

Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential Energy [Qu. Appl. Math. 4, 268278, 1946] 1 Introduction Let us consider a thin-walled box beam of web height 2h and cover sheet width 2w which is bent in such a way that one of the cover sheets is in tension while the opposite cover sheet is in compression (Figure 1). In elementary beam theory the assumption is made that the normal stress in the cover sheets does not vary in the direction across the sheet. Because of the shear deformability of the cover sheets this assumption of elementary beam theory is often seriously in error for wide beams. In aeronautical engineering this effect is known under the name of shear lag.

Fig. 1. Sketch of spanwise element of box beam with doubly symmetric cross section.

Page 22

In recent papers, 1,2 shear lag in box beams has been analyzed by an application of the theorem of least work which is the basic minimum principle for the stresses. The present paper contains an application to the problem of shear lag of the theorem of minimum potential energy, which is the basic minimum principle for the strains.3 It is shown that application of the theorem of minimum potential energy to the present problem leads to simpler and more general results than the application of the theorem of least work. While the least-work method furnishes the stresses in box beams with no cut-outs, application of the minimum-potential-energy method furnishes, in a simpler manner, the stresses in beams without or with cut-outs. It also furnishes beam deflections, and is equally convenient for beams supported in statically determinate or in statically indeterminate manner. Application, in the manner described below, of the minimum-potential-energy principle to the problem of bending of thin-walled box beams leads to a differential equation for the beam deflection which is a generalization of the relation z" = M/EI; this differential equation contains an additional term proportional to the fourth derivative of z which takes into account the shear deformability of the cover sheets. As the order of the differential equation in this theory is higher than the order of the differential equation of elementary beam theory, boundary conditions appear in addition to those of elementary beam theory. These additional boundary conditions are different for beams with cut-outs and for beams without cut-outs. The manner of application of the results obtained in the present paper is shown by solving explicitly the following four examples. 1. Simply supported beam. Load distributed according to a cosine law. 2. Cantilever beam with uniform load distribution. Cover sheets fixed at the support. 3. Cantilever beam with uniform load distribution. Cover sheets not fixed at the support. 4. Beam with both ends built in. Uniform load distribution. For the sake of simplicity, it is assumed in what follows that the cross sections of the beams are rectangular and doubly symmetrical. It also is assumed that there is no spanwise variation of cross-sectional properties. The author believes that the way in which the principle of minimum potential energy is here applied to the problem of shear lag will prove useful in other problems of structural mechanics. As an example of such future application, the theory for combined torsion and bending of beams with open or closed cross sections is mentioned. 2 Formulation and Solution of Problem In the following, we analyze a box beam of doubly symmetrical rectangular cross section, composed of cover sheets, sidewebs and flanges. A given distribution of loads is applied to the sidewebs, acting normal to the plane of the cover sheets (Figure 1). To this load distribution there corresponds a distribution of bending moments M(x). The spanwise coordinate being x, 1E. Reissner, Least work solutions of shear lag problems, Journal of the Aeronautical Sciences, 8, 284291 (1941). 2F. B. Hildebrand and E. Reissner, Least work analysis of the problem of shear lag in box beams, N.A.C.A. Technical Note No. 893 (1943). 3For a formulation of these theorems see for instance I. S. Sokolnikoff and R. D. Specht, Mathematical theory of elasticity, M cGraw-Hill Book Co., Inc., New York, 1946, pp. 275287.

Page 23

let y be the coordinate in the plane of the cover sheets perpendicular to the x direction, and z(x) the deflection of the neutral axis of the beam. The potential energy of the bent beam may be considered as composed of three parts. The first part is the potential energy of the load system. This may be written in the form

the integral being extended over the entire length of the beam.4 The second part is the strain energy of sidewebs and flanges. This may be written in the form

the quantity Iw denoting the principal moment of inertia of the two sidewebs and flanges. The third part is the strain energy of the two cover sheets. If it is assumed that the normal strains in the chordwise direction in the sheets may be neglected, as discussed in the reference given in Footnote 1, then the strain energy of the two sheets is given by the integral

where the quantity t denotes the cover sheet thickness, and where E and G are the effective moduli of elasticity and rigidity. Spanwise normal strain H x and shear strain J are then expressed in terms of the spanwise sheet displacement u as follows

The theorem of minimum potential energy states that the total potential energy

becomes a minimum for the correct displacement functions u and z, if only such displacement functions are compared which satisfy all conditions of support and continuity imposed on the displacements. Direct application of this condition by means of the calculus of variations leads to a partial differential equation for u and to a complete system of boundary conditions. In what follows, an ordinary differential equation for the beam deflection z and boundary conditions for it are obtained instead. This is done by making a suitable approximation for the sheet displacements u and by applying the rules of the calculus of variations to the resultant approximate expression for the potential energy function. A reasonable assumption for the spanwise sheet displacements is

4Equation (1) implies that the beam is supported in such a manner that the end forces and moments can do no work. This restriction shortens the developments slightly.

Page 24

The second term on the right of Eq. (6) represents the correction due to shear lag. Instead of the vanishing chordwise variation of the sheet displacements of elementary beam theory, we now assume a parabolic variation. The relative magnitude of the function U is a measure for the magnitude of the shear lag effect. The form of the correction is such that continuity of the displacements along the flanges, that is along y = ±w, is preserved. Denoting differentiation with respect to x by primes, we obtain the following expressions for the strains in the sheets from Eqs. (6) and (4):

On the basis of Eqs. (7) and (8) the following expression for the strain energy of the sheets is obtained:

In Eq. (9) the integration with respect to y is carried out. Setting

we have

Substituting Eqs. (11), (2) and (1) into Eq. (5), we obtain the following expression for the potential energy of the system

Differential equations and boundary conditions for z and U are obtained by making

Thus, with x 1 and x 2 denoting the ends of the interval of integration,

As Gz" and GU are arbitrary in the interior of the interval (x 1, x 2) the terms multiplying them must vanish. This gives the following two differential equations

Page 25

The integrated portion of Eq. (14) defines the boundary and transition conditions for the function U. At a section where the sheet is fixed, GU = 0 and

At a section where the sheet is not fixed and consequently GU is arbitrary,

Transitions conditions for adjacent bays with different stiffness are:

The above boundary and transition conditions are in addition to those imposed on z and M in elementary beam theory, as may be verified by repeated integration by parts of the term containing Gz" in the integral of Eq. (14). 3 The Modified Beam Equation and Its Boundary Conditions By eliminating the quantity U from Eqs. (15) to (19), we obtain a system of relations containing the beam deflection z only. The differential equation for z is derived by differentiating Eq. (16) and substituting Uc from Eq. (15). There follows

When the shear deformability of the sheets is neglected, that is when it is assumed that G = f, Eq. (20) reduces to the well known result of elementary beam theory. Equation (20) may be written in the alternate form

With the help of Eqs. (15) and (16), the boundary condition (17), which holds when the sheet is attached to the support, is transformed into

Similarly, the boundary condition (18), which holds when the sheet is not attached to the support, becomes

The continuity conditions (19) may be transformed in an analogous manner. The values of the sheet stresses may be obtained from Eqs. (9) and (10). From Eq. (9) it follows that the flange stress is given by

For the application of the results it may be noted that the differential equation (21) can first be solved for the value of z" which, according to (24), gives directly the

Page 26

approximate value of the flange stress Vg. The magnitude of the deflection z can then be found from the value of z" as in elementary beam theory. For the evaluation of the solution we define the following two parameters

With (25) and (26) the differential equation (21) becomes

The boundary condition at an end section where the sheet is attached to the support becomes

and the boundary condition at an end section where the sheet is not attached to the support becomes

4 Examples of Applications (Figure 2) 1 Simply Supported Beam. Load Distributed According to a Cosine Law Designating the span length of the beam by l and assuming the origin of the coordinate system at the center of the beam, we consider the moment distribution

A particular solution of Eq. (27) is

As Eq. (31) satisfies the boundary condition (29) and the conditon of vanishing deflection at the ends of the beam, it is the complete expression for the deflection function. When 1/k = 0, Eq. (31) reduces to the expression for z in the case where shear lag is not taken into account. The factor

expresses the effect of shear lag on deflection and flange stresses. 2 Cantilever Beam with Uniform Load Distribution. Cover Sheets Fixed at Support Assuming that, contrary to what is indicated in Figure 2, the free end of the beam has the coordinate x = 0 and the fixed end of the beam the coordinate x = l, we may

Page 27

Fig. 2. Diagrammatic sketches of beams analyzed as examples of application of the theory.

write the moment distribution in the form

The differential equation (27) then becomes

Page 28

Solving for z", we find

Satisfying the boundary condition (29) when x = 0 and (28) when x = l, we obtain

According to Eq. (24), the flange stress at the fixed end of the beam becomes

We take for a numerical example

so that according to Eqs. (25) and (26)

and we find

By application of the least work method1,2 a factor 1.186 is obtained instead of the factor 1.190 in Eq. (40). The deflection of the beam is obtained from Eq. (36) by integrating twice and making z(l) = zc(l) = 0. In the present case, the correction due to shear lag for the maximum deflection is about ten percent. 3 Cantilever Beam with Uniform Load Distribution. Cover Sheets Not Fixed at Support Moment distribution and differential equation are given by Eqs. (33) and (34). The constants of integration in (36) are determined by satisfying Eq. (29) for x = 0 and for x = l. There follows

Taking again Is/I = .5, we should have, for the flange stress at the supported end, a value twice as large as the stress according to elementary beam theory for a beam with sheet attached to the support. In the present solution the factor 2 is replaced by n = 1.714. This indicates that with the assumed parabolic chordwise variation of sheet displacement the condition that at the support of the beam the sheet is free of stress is only approximately satisfied. The same difficulty arises in methods which incorporate the ability of the sheet to carry normal stresses as effective width

Page 29

contributions to the strength of stiffners.5 This difficulty is not serious when the main purpose of such ''cut-out" calculations is the determination of the distance over which the cut-out is effective and its effect on the over all beam stiffness.6 The localization of the effect of the cut-out may be seen by writing (41) in the form

This equation indicates that the influence of the cut-out is small as soon as the distance l x satisfies the inequality

Thus, the wider the sheet and the smaller the value of the shear modulus G, the farther away does the effect of the cut-out extend in the spanwise direction. The magnitude of the beam deflection is obtained from (41) in the form

which determines the constants of integration such that z(l) = zc(l) = 0. For the deflection at the free end of the beam, we have

For a beam with dimensions as in (38) and (39), Eq. (45) becomes

This indicates that for a beam with dimensions as given shear lag due to lack of sheet restraint at the supported end of the beam is responsible for a thirty percent increase of the maximum beam deflection as compared with the result of elementary beam theory for a beam fully restrained at the supported end. This increase of deflection of thirty percent compares with one of hundred percent which is obtained if the contribution of the cover sheets is neglected. 4 Beam with Both Ends Built-In. Uniform Load Distribution The distribution of bending moments may be written as

5P. Kuhn and P. Chiarito, Shear lag in box beamsmethods of analysis and experimental investigations, N.A.C.A. Technical Report No. 739 (1943). 6Exact solutions of problems of this kind have been obtained by F. B. Hildebrand, The exact solution of shear-lag problems in flat panels and box beams assumed rigid in the transverse direction, N.A.C.A., Technical Note No. 894 (1943).

Page 30

The value of M0 is determined by the load intensity, the value of M1 in this statically indeterminate problem has to be determined from the displacement boundary conditions. The boundary conditions are

For these boundary conditions the moment distribution is not affected by shear lag, provided the moment distribution is symmetrical about the mid-span section of the beam. Indeed, the differential equation (27) may be integrated to give

the limits of integration being so chosen that Eq. (51) satisfies the conditions of zero slope and zero vertical shear at the mid-span section. In view of (49) and (50), Eq. (51) implies

regardless of whether or not shear lag is taken into account. A considerably less simple proof of the same fact by means of the least work method has been given in the reference quoted in Footnote 2. For the moment distribution of Eq. (47) there follows, from (52),

and hence

With this value of M and the requirement that z" be an even function of x, Eq. (27) is solved in the form

The constant C2 is determined from Eq. (50). There follows,

Taking a beam five times as long as wide, that is l/2w = 5, and assuming the remaining parameters as in (38) and (39), we obtain the following expressions for the flange stresses at the built-in section and at the center section of the beam

Page 31

These results agree to within a fraction of a percent with the corresponding results obtained by the least work method.2 It is worthy of note that, for this beam with both ends built-in, shear lag is considerably larger than for a cantilever beam with the same load, same width and half the span of the beam with both ends built-in. If both beams had the same span, the discrepancy would be even larger. The deflection z of the beam is obtained from (56) and (48) in the form

Corresponding to the stresses of Eqs. (57) and (58) we find for the deflection at mid-span

Shear lag in this beam is thus responsible for an almost fifteen percent increase in deflection. This percentage increase of deflection, while appreciable, is considerably smaller than the percentage increase of maximum flange stress.

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Note on the Shear Stresses in a Bent Cantilever Beam of Rectangular Cross Section* [J. Math. & Phys. 25, 241243, 1946] In this note we wish to report on some calculations which we have made concerning the distribution of shear stresses, according to the St. Venant theory, in bent cantilevers of rectangular cross section.1 Referring to Figure 1, our main result is the fact that for sufficiently wide beams (b/a >> 1) the component of stress Wyz(a, K) not only is of the same order of magnitude as the stress Wxz(0, b) for which values have previously been calculated,1 but exceeds Wxz(0, b) in magnitude.2 We list first the exact expressions for Wxz(x, y) and Wyz(x, y) in a form convenient for our purposes and deduce from them simplified expressions for Wxz(0, b) and Wyz(a, y) which ensure accuracy for all decimals computed, when b/a t 4. On the basis of these formulas we calculate for a number of values of b/a in the range (2, f) (i) values of Wxz(0, b), (ii) values of the coordinate y = K for which Wyz(a, y) is greatest, (iii) values of Wyz(a, K). The numerical results obtained are collected in Table I. Exact expressions for the shear stresses may be written in the following form, with 4ab = A,

*With G. B. Thomas. 1For an exposition of this theory see S Timoshenko's book on Theory of Elasticity (pp. 285288, 292298, M cGraw Hill, 1934). 2We are indebted to Prof. S. Timoshenko and to Prof. H. Reissner for the information that this fact appears not to have been noted previously.

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Fig. 1. Diagram showing dimensions of cantilever beam and nature of shear stress distribution.

From Eqs. (1) and (2) are derived the following approximate expressions which are exact within the accuracy of our calculations when b/a t 4:

Substituting in Eq. (3) the series sums

this equation becomes

To obtain the maximum of Wyz(a, y) we set (wWyz(a, y)/wy)y=K = 0. According to Eq. (4) the maximum condition becomes

Page 34

Carrying out the summation in (6) we have

or

Equation (8a) may also be written in a form which gives directly the distance of the maximum location from the edge in units of the thickness 2a,

Introducing Eq. (a) in Eq. (4) we obtain

Equation (9) is less simple in appearance than Eq. (5) but is readily evaluated as the remaining series converges rapidly. From (9) and (5) we obtain the following limit relation

which shows that for wide beams (plates) the component of shear parallel to the face of the plate reaches a value which is almost 35 percent higher than the value reached by the component of transverse shear. The way in which this limit state is approached is apparent from the values given in Table I which has been computed on the basis of Eqs. (5), (8b), and (9). In the calculations the value of Q has been taken as .25. Table I

0 2 4 6 8 10 15 20 25 50 f

1.000 1.39(4) 1.988 2.582 3.176 3.770 5.255 6.740 8.225 15.650 f

0.000 0.31(6) 0.968 1.695 2.452 3.226 5.202 7.209 9.233 19.466 f

0.000 0.22(7) 0.487 0.656 0.772 0.856 0.990 1.070 1.123 1.244 1.347

0.000 0.31(4) 0.522 0.649 0.739 0.810 0.939 1.030 1.102 1.322 f

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On Finite Pure Bending of Cylindrical Tubes [Österreichisches Ing. Arch. 15, 165172, 1961] Introduction We are concerned in what follows with a rather general and simple formulation of the problem of finite pure bending of thin-walled cylindrical tubes of arbitrary cross section. Our analysis contains as special cases Brazier's analysis of the flattening instability of circular cross section tubes1 and our own refinement of Brazier's theory, which in turn is a special case of results for toroidal tubes with circular cross section.2 The physical basis of the present work is similar to that of our earlier work. Certain simplifications arise through the consideration of initially straight tubes instead of toroidal tubes. Further simplifications are due to an appropriate use of the variational theorem for displacements in elasticity. Formulation of the Problem We consider a cylindrical tube with cross section before deformation specified by middle surface equations x = x(s), y = y(s), where s represents circumferential arclength, and by a wall thickness function h = h(s). The originally cylindrical tube is deformed through the application of end moments Mx and My. These moments Mx and My result in a curving of the axial fibers of the tube, with curvature radii Rx and Ry. As long as elementary beam theory applies, the radii Rx, Ry and the moments Mx and My are related by equations of the form

In these formulas E = E(s) is the modulus of elasticity for axial stress, and the origin of the x, y-system of coordinates is chosen such that

Associated with the uniform curving of axial fibers is a deformation of the cross section of the tube which changes x into x + u and y into y + v and which is neglected in elementary beam theory. It is of the essence in what follows that this cross sectional deformation may be assumed to take place without meridional extension of the middle surface of the tube (Figure 1). 1L. G. Brazier, Proc. Roy. Soc. A, 116, 104114, 1927. 2E. Reissner, Proc. 3rd U.S. Nat. Congr. Appl. M ech. 5169, 1958, and J. Appl. M ech. 26, 386392, 1959.

Page 36

Fig. 1. Element of tube cross section before and after deformation.

The linear displacements u and v define an angular displacement E which in turn defines a circumferential bending strain N, given by

The components u, v and E are connected through two relations which may be read from Figure 1, and which are

In addition to the circumferential bending strain N we have an axial direct strain H , which through a stress strain relation of the form V = EH , enters into the formulas for moment components Mx and My. These moment components are defined with reference to the deformed cross section, as follows:

In order to express H in terms of Rx, Ry, u, v we make use of the fact that pure bending of the tube takes place in such a way that plane sections perpendicular to the axis of the undeformed tube are deformed into plane sections perpendicular to the curved axis of the deformed tube. This means that H is given by the formula

Introduction of (7) into Eqs. (6) leads to expressions for Mx and My which are the extensions of (1) and (2) and which are

Page 37

In order that the cross sections of the tube are free of resultant forces, we must further have

which generalizes Eqs. (3) consistent with the step from (1) and (2) to (8) and (9). In order to evaluate the basic formulas (8) and (9) it appears necessary to determine the functions u and v in their dependence on the geometry of the cross section and on the values of Rx and Ry. It will be seen in what follows that a somewhat simpler procedure is possible in which H and E are determined rather than u and v. Derivation of Differential Equations We base our derivation on the requirement that, for given values of Rx and Ry, the strain energy of the bent tube be a minimum. We take as expression for strain energy, per unit of axial tube length

where H and N are given by (7) and (4), where C = Eh and where D is a circumferential bending stiffness function which for isotropic homogeneous tube materials is given by D = Eh3/12(1 v 2). The quantity 3 s is to be made a minimum subject to the constraint Eqs. (5). Considering the form of H and N in (7) and (4) we may evaluate the minimum condition without explicit use of the displacement components u and v by writing as constraint condition

Introducing N from (4) and introducing the constraint condition (12) by means of a Lagrange multiplier F the condition of minimum strain energy assumes the following form

The variational Eq. (13) is equivalent to two differential equations of the form

which must hold, together with the constraint Eq. (12). Boundary conditions for the system (12), (14) and (15) are the conditions of periodicity in s for H , F, E and dE /ds. In view of one of these periodicity conditions we have satisfied.

so that the condition of no resultant force over the tube cross section is automatically

Equations (8) and (9) for Mx and My may now be written, through the use of (6) and appropriate integration by parts, in the alternate form

Page 38

We may finally reduce the two first-order Eqs. (12) and (14) to the second order differential equation

and thereby reduce the problem to the two simultaneous second order Eqs. (15) and (17) for E and F. We note that these two equations may be written in the form

and

Equations (15c) and (17c) may be used as the starting point of an expansion procedure which will now be discussed. Expansion in Powers of 1/Rx and 1/Ry A formal expansion procedure for (15c) and (17c) which is suggested by the appearance of these equations upon writing cos E = 1 1/2E 2 ± . . ., sin E = E 1/6 E 3 ± . . . consists in setting

where Fn and E n are homogeneous of degree n in the quantities 1/Rx and 1/Ry. In particular

Introduction of (18) to (20) into (15c) and (17c), written as

Page 39

leads to the following system of successive differential equations

Solution of (21) to (23), and of the corresponding subsequent systems, is carried out by direct integration. We find from (21) and from the periodicity condition in view of the fact that cos M = dx/ds and sin M = dy/ds,

and

From (22) follows

with corresponding expressions for dE 11/ds and dE 02/ds. It is apparent that, in general, further integrations must be carried out numerically. Corresponding expansions for applied moments, as defined by (16) are of the form

and

With (19), (20) and (25) it is found that Mx may be written as

Page 40

where the Xjk and Yjk are suitable constants. An analogous expression may be deduced for My. The following observations may be made: 1. Equation (29), when specialized to the case of the homogeneous constant-wall thickness, circular-cross-section tube, and without any of the terms represented by dots or the terms with 1/Ry, reduces to the formula of Brazier for the moment Mx in terms of the curvature 1/Rx. 2. As long as the calculations of Mx and My do not go beyond third-degree terms in R-1 they may equally well be considered to be based on a linear system of differential equations, of the form

Associated with this linear system are non-linear expressions for Mx and My of the form

3. Higher degree terms than those displayed in (29) may have an effect of the order of ten percent or more in the range of practical interest of the theory. The fact that this may be so becomes apparent even without explicit calculations upon introduction of appropriate non-dimensional variables and parameters. Equations for the Circular-Cross Section Tube We designate the radius of the circular cross section by b and write the coordinates of the middle surface of the tube in terms of a polar angle [ , as follows:

Therewith,

and Eqs. (15) to (17) become

Page 41

If we limit ourselves in what follows to the case that D and C are independent of [ then, because of symmetry, we may further limit ourselves to the consideration of the case 1/Ry = 0 and My = 0. Writing Rx = R we have then

and

If we further set

and indicate differentiation with respect to [ by primes, then the system (39), (40) assumes the form

and Eq. (41) becomes

where EI = SCb3 is the bending stiffness factor of the tube according to elementary theory. Equation (45) may be written in the alternate form

in which a dimensionless applied moment m appears as a function of the dimensionless curvature parameter D. Equation (45c) may be simplified by introducing (44) and by integrating by parts. In this way there follows the relation

Equations (43) and (44) may be solved by expansion in powers of D 2. We set

and expand both cos E and sin E in powers of D 2. In this way we obtain a system of

Page 42

successive differential equations of which we list the first seven equations, as follows:

The solutions of (48) to (52) must be periodic of period 2S in [ . We list below the form of these solutions for Eqs. (48) to (50).

In terms of these expansions we have for the dimensionless moment m,

Introduction of (53) to (55) results in the following explicit formula for m,

Equation (57) is in agreement with our previous result for this problem and reduces, upon omission of all terms except the first two, to Brazier's result. In order to delineate the range of values of D which is of practical interest, we determine the value of D for which flattening instability occurs. This value of D is obtained by solving the equation dm/dD = 0. From (57) follows for this value according to Brazier mB

mC

while if D 5 is retained in (57) we obtain

. Corresponding values of m are = 1.086 and = 0.998. These numerical data indicate that the exact value of m for which dm/dD = 0 may differ from Brazier's value by an appreciable amount. They also indicate that it will be of interest to obtain numerically more accurate solutions of (43) and (44) than given here, for values of D 2 in the range from two to three.

Page 43

Finite Pure Bending of Circular Cylindrical Tubes * [Qu. Appl. Math. 20, 305319, 1962] Introduction The present paper is concerned with the determination of stresses, deformations and stiffness of originally straight circular tubes in pure bending. The non-linear problem of determining the stiffness of such a tube as a function of the applied moment and the determination of a critical moment for which flattening instability occurs has originally been discussed by Brazier [1]. An alternate more precise formulation of the problem of flattening instability of circular cross-section tubes is contained in a recent paper by one of the present authors [2], as a special case of results for pure bending of general cylindrical tubes. In this same paper approximate solutions of the non-linear differential equations of the problem were obtained as expansions in powers of a dimensionless parameter D. It was found that the first terms of these expansions give the results of linear theory and that consideration of two terms gave the results of Brazier [1]. It was further found that consideration of three terms lead to results which differed from Brazier's to the order of ten per cent. Since the calculation of additional terms in the D-series becomes progressively more complicated, an alternate determination of the results is of interest. The present paper presents such an alternate determination, involving the iterative solution of a system of two simultaneous non-linear integral equations. In addition to this, the previous three-term D-series are extended by the calculation of fourth terms. Our calculations lead to the note-worthy conclusion that Brazier's results for flattening instability are quite close to the results of precise calculations based on the equations given in [2], in the sense that consideration of three and even four terms in the D-series lead to results which are further from the correct results (in the critical D-range) than the results based on only two terms in the D-series. In addition to these conclusions for the problem of the flattening instability, we obtain in what follows quantitative results for the non-linear behavior of stresses and deformations in the tube. We find, in particular, that when the applied bending moment is of the order of the critical moment, the order of magnitude of the secondary circumferential wall bending stressesassociated with the flattening of the cross-sectionis the same as the order of magnitude of the primary longitudinal direct fiber stresses in the tube. Basic Equations It has been shown previously [2] that the problem of pure bending of a tube with cross-section before deformation given by x = b sin [ , y = b cos [ for *With H. J. Weinitschke.

Page 44

0 d[ , d 2S is associated with two simultaneous non-linear differential equations for a stress function variable F and an angular displacement variable E of the following form

In these equations R is the radius of curvature of the originally straight axis of the tube, D = Ebh3/12 is the circumferential wall bending stiffness factor and 1/A = Esh is the axial stretching stiffness factor of the tube.1

Fig. 1. Notation.

Equations (1) are to be solved in the interval 0 d[ d 1/2S subject to the following boundary conditions,

To be determined are in particular the applied moment M given by

axial fiber stress Vg and the circumferential bending stress Vb given by

1In our earlier paper [2] it had been assumed that D = Ebh3/12(1 Q2) which is the appropriate circumferential stiffness factor for small cylindrical bending. Considering that in the range of practical interest we will have cylindrical bending with relatively large deflections a stiffness factor D without the term (1 Q2) seems more appropriate.

Page 45

and cross sectional flattening and bulging displacements given by

Equations (1) to (3) are made non-dimensional by setting

Indicating differentiation with respect to [ by primes, we now have the differential equations

with boundary conditions

The dimensionless moment m becomes, after an integration by parts

and dimensionless stresses may be obtained in the form

Two alternative dimensionless stress quantities may be defined as follows. One definition makes use of the maximum fiber stress tube bent to a radius R if there were no flattening effect. Introduction of

into Eqs. (4) and (6) leads to the formulas

As flattening makes the tube more flexible than it would otherwise be, we expect that the ratio of A second non-dimensionalization makes use of the maximum fiber stress if there were no flattening effect. Introduction of

which would exist in the

will decrease as D increases.

, where I = Sb3h, which would exist in the tube subject to an applied moment M

into Eqs. (4) and (6) leads to the formulas

In these equations we consider D a function of m which is defined by means of Eq. (9). Whether or not

will increase or decrease with increasing m will

depend on the shape of the curve for as function of [ and cannot be predicted without numerical calculations. Numerical calculations are also needed for a comparison of the magnitude of the secondary bending stresses Vb with the magnitude of the primary direct fiber stress Vg, in their dependence on D or m.

Page 46

Expansion in Powers of D The boundary value problem (7) and (8) may be solved, as in [2], by expansions

Expanding sin E and cos E in terms of D 2 we obtain a system of successive linear differential equations, of which we list the first seven as follows:

We require that the boundary conditions (8) be satisfied identically in D. The functions g0, g2, g4, E 2, E 4 have been calculated in [2] and are listed here for completeness sake. The functions g6 and E 6 as well as formulas for stresses and displacements have not been obtained before. We find

Insertion of Eq. (12a) into Eq. (9) gives an expression for the moment function m of the form,

Substitution of (17a), (17c), (18b) and (20) and subsequent integration gives the relation

The partial sum obtained by omitting the last listed term in Eq. (22) agrees with the result given in [2]. Retaining only the first two terms on the right side of Eq. (22) gives the result of Brazier. A quantitative discussion of the dimensionless moment curvature relation (22) is given further on in conjunction with a discussion of the corresponding relation obtained from the numerical solution of the integral equation.

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For the calculation of stresses in accordance with Eqs. (9a, b), (10) and (11) we need the following expressions for the derivatives gc and E c:

which follow from Eqs. (12) and (17) to (20). Expressions for the flattening and bulging displacements follow from Eqs. (5), (12) and (17) to (20) in the form

We note that while for sufficiently small D we have displacement v.

, it is found that for increasing D the flattening displacement u increases more rapidly than the bulging

4 Integral Equation Formulation and Numerical Solution The differential equations (7) with boundary conditions (8) may be reduced to the following system of integral equations

where x = [ /1/2 S,

,

and Gg, GE are Green's functions given by

(in what follows the bars on , will not be written, for simplicity's sake.) Numerical solutions of adequate accuracy of these integral equations may be obtained by a combination of iteration and numerical integration. Values of the dimensionless moment m and of the displacements u and v are calculated by introducing the solutions g(x), E (x) into the integrals in Eqs. (3) and (5). Dimensionless stresses in accordance with Eqs. (9a, b), (10) and (11) are obtained by calculating dE /dx and df/dx in terms of the integrals which follow, rather than by numerical differentiation of the discrete values of g and E obtained from the solution

Page 48

of Eqs. (27) and (28). We find, in dimensionless form

where

In order to describe the iteration scheme to be used, we write Eqs. (27) and (28) in the form

where the integral operator K1 depends on E only and where K2 is linear in g. The most straightforward iteration of (39) is expressed by gn+1 = K1[E n], E n+1 = D 2K2[gn, E n]. Using the iteration gn+1 for the calculation of E n+1, a more rapidly converging iteration scheme for (39) is

The two equations (40) can be written as one equation as follows

Numerical calculation shows convergence of (41) for values of D 2 up to about 5. For larger D 2, examples of both oscillations and steady increase in magnitude of successive iterates were obtained, that is, the iteration scheme diverges. In order to obtain solutions for larger values of D 2, the iteration scheme is modified by introduction of two ''relaxation parameters" O and P as follows2

Clearly, if the sequences gn, E n converge, they converge to a solution of (39). The relaxation parameters O and P are allowed to depend on D 2. With this scheme and with appropriate choice of O, P, the speed of convergence was considerably increased as compared with the iteration (40), and convergence was induced for values of D 2 for which (40) diverges. In this way solutions up to were obtained; the range 2A similar iteration scheme using one relaxation parameter was employed by Keller and Reiss in solving a system of non-linear difference equations [3].

Page 49

could probably be extended to still larger D 2 by proper choice of O, P, although solutions beyond a critical value (see Section 5) are physically less interesting. Table 1 shows some numerical results. The special case O = P = 1 is identical with the iteration (40). A proof for the convergence of the modified scheme (42) for values of D 2 > 1.8 has yet to be obtained. Table 1. Number of iterations for 0.1% accuracy of solutions of (39) D2 P= O= 1 O = 1, P = 3/4 O = 3/4, P = 1 O = 3/4, P = 3/4 2.6 6 8 9 6.0 34 10 18 9.0 oscill. 32 20 16 13.0 div. div. 32 20.0 div. div. oscill.

O = .65, P = .60

27 33

Some numerical solutions of Eqs. (39) have previously been calculated by G. L. Brown [4], who used Simpson's rule for numerical integration combined with an iteration equivalent to (40). He found, using interval lengths 'x = 1/5, 1/9 and 1/16 that the latter was not small enough to draw conclusions on the accuracy of the results obtained. 5 Discussion of Results The integral equations (39) have been solved for values of D 2 up to 25. These solutions are in the following referred to as "numerical solutions." A comparison of the D-expansion with the numerical solutions shows that the D-expansion solutions are accurate almost up to the critical value D c, defined by dm/dD = 0 (see Figures 24). For larger values of D, they become quite inaccurate. As a check on the numerical solutions, Eqs. (7) were approximated by a finite difference system

which was solved for E i, gi i = 2, . . ., N, the values g0, g1, E 0, E 1 being given from the solution of the integral equations. If the solutions of Eqs. (43) differed by more than one unit in the third figure from the solutions of Eqs. (39), the latter solutions were recalculated with a smaller spacing h, and the check via Eqs. (43) was repeated for that spacing. The moment curvature relation is shown in Figure 2. What is of particular interest is the value D c, for which flattening instability occurs. For this value of the dimensionless curvature, the moment m attains its maximum value mc. The numerical values for D c, and mc when retaining 2, 3, or 4 terms in Eqs. (22), and the corresponding values D c, mc from the numerical solution are given below. The numbers of the last column of Table 2 were obtained by interpolation from a large scale plot of the dimensionless moment-curvature relation near the critical point (D c, mc). Figure 3 shows the maximum values of the dimensionless direct stress Vgb/Esh and bending stress Vbb/Ebh as defined by Eqs. (9a, b). The maximum bending stress

Page 50

Fig. 2. Dimensionless moment curvature relation.

Fig. 3. Dimensionless maximum direct and bending stresses.

occurs at the neutral plane, that is Vb,m = Vb(0). The bending stress Vb(1/2S) at the farthest distance from the neutral plane is slightly less in absolute value than Vb(0) and is also displayed in Figure 3. For small D bending stresses are negligible with respect to direct fiber stresses. For values of D approaching D c, the bending stresses become of the same order of

Page 51 Table 2. Critical curvatures and moments 2 terms 3 terms 1.513 1.422 Dc 1.086 0.998 mc

4 terms 1.468 1.034

numerical solution 1.526 1.063

Fig. 4. Dimensionless maximum flattening and bulging displacements.

magnitude as the direct stresses. It is interesting to note that the maximum direct stress is not attained at

but for the somewhat larger value

.

Values of the dimensionless maximum flattening and bulging displacements of the cross section are displayed in Figure 4. For small values of D, the two displacements are nearly identical. As in the D-expansions (Eqs. (25) and (26)), flattening increases faster with increasing D than bulging, the rate of increase being strongest near D c, for both flattening and bulging. Next we compare our results with those of elementary linear beam theory. In Figure 5 the ratios plotted against the dimensionless moment m/mc. In Figure 6, the stress ratios a given moment, the direct stress produced according to the nonlinear theory is

and

and

with J = Es/Eb, according to Eqs. (11) 3 are

are plotted against D. We conclude from these graphs that for

3The calculations were carried out on the IBM -709s at Western Data Processing Center, Univ. of Calif., Los Angeles, and at Computation Center, M .I.T., Cambridge.

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Fig. 5. Stress ratios as functions of dimensionless moment m/mc.

Fig. 6. Stress ratios as functions of dimensionless curvature D.

slightly larger than the value given by the linear theory which neglects the flattening of the cross section. On the other hand, for a given curvature of the central axis of the tube, the nonlinear theory stress is less than that given by the elementary beam theory. Finally, we consider briefly the stress distribution over the cross-section of the tube and its deviation from the elementary (linear) stress distribution. As is seen from

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Fig. 7. Dimensionless direct stress vs. dimensionless distance d/b from neutral axis for several values of D.

Fig. 7, this deviation is quite small for values of DdD c. For larger values the distribution becomes more markedly nonlinear, in fact, for D > 2.5 the maximum fiber stress is no longer attained in the outermost fiber. References 1. L. G. Brazier, "On the flexure of thin cylindrical shells and other thin sections," Proc. Roy. Soc. A, 116, 104114 (1927). 2. E. Reissner, "On finite pure bending of cylindrical tubes," Österr. Ing. Arch., 15, 165172 (1961). 3. H. B. Keller and E. L. Reiss, "Iterative solutions for the nonlinear bending of circular plates," Comm. Pure Appl. Math., 11, 273292 (1958). 4. G. L. Brown, "On the numerical solution of two simultaneous non-linear differential equations arising in elasticity," M. S. Thesis, Mass. Inst. of Technology, June 1960. 5. L. Collatz, "Numerische Behandlung von Differentialgleichungen" 2. Auflage, Springer 1955.

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Considerations on the Centres of Shear and of Twist in the Theory of Beams* [Muskelisvili 80th Anniversary Volume, pp. 403408, Moscow 1972] 1 Introduction The considerations which follow resulted from endeavours to understand, appreciate and reconcile different statements in the literature on the subject of the centre of shear and of the centre of twist, and on the conditions under which these centres would or would not coincide. The outcome of our considerations was an approach which to us seemed quite different from what had been done previously, and rather more appropriate to the question, as we hope to make evident in what follows. Specifically, our first object is to show that coincidence of the centres of twist and of shear may be taken to be no more than a natural consequence of a reasonable formulation of the problems of torsion and of flexure in the theory of beams. Our second object is to show that an explicit, approximate determination of the location of these centres may be based on the Saint-Venant solutions of the problems of torsion and flexure (which by themselves are known to leave these centres arbitrary), in conjunction with a direct-methods-of-the-calculus-of-variations-type use of the principle of minimum complementary energy. 2 A Formulation of the Problems of Torsion and Flexure We consider a linear elastic prismatical body, with boundaries defined, in an (x, y, z) co-ordinate system, by means of a cylindrical surface f(x, y) = 0 and two parallel planes z = 0 and z = L. We designate displacements by ux, uy, uz and stresses by Vx, Vy, Vz, Wxy, Wxz, Wyz and we assume that the usual three-dimensional homogeneous equations of linear elasticity hold. We further assume that the cylindrical boundary portion of the body is free of tractions and that the plane boundary portion z = 0 is fixed. In regard to the plane boundary portion z = L we assume the absence of normal tractions while at the same time its tangential displacement components correspond to a plane rigid body translation and rotation, i.e., we stipulate

where U, V, and : are given constants. It is evident that we may, in conjunction with the above set of prescribed boundary conditions, consider a set of overall tractions consisting of transverse *With W. T. Tsai.

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forces P and Q and of a torque T defined by

In view of the linearity of the problem, P, Q and T will come out to be linear combinations of U, V and : and it may be concluded that, inversely, U, V and : are related to P, Q and T in the form

It may further be concluded that Eqs. (3) retain their form for such generalizations of the stated problem as are obtained upon replacing the given conditions of end fixity at z = 0 by homogeneous linear relations of the form gi(ux, uy, uz, Vz, Wxz, Wyz) = 0 for i = 1, 2, 3, and by replacing the condition Vz = 0 for z = L by any homogeneous linear relation g(uz, Vz) = 0 for z = L. 3 The Centre of Twist and the Centre of Shear The possibility of defining points in the cross-sections of a prismatical beam which may be designated as centre of twist and as centre of shear, respectively, depends on the rationality of the concept of the cross-sections of the beam rigidly translating and rotating in their own planes, at least approximately. We propose here to sharpen these definitions by confining them to cross-sections which are prescribed to translate and rotate rigidly in their own planes, i.e., to the end crosssection of the prismatical beam for which the boundary conditions (1) and its mentioned generalizations are stipulated*. Definition of Centre of Twist With end cross-section displacements prescribed in the form ux = U y:, uy = V + x: we define the co-ordinates x T, yT of the centre of twist at those values of x and y for which ux = uy = 0, while at the same time P = Q = 0, i.e.,

Introduction of Eq. (3) into Eq. (4) expresses yT and x T in terms of influence coefficients, as follows:

*Given that the end cross-section translates and rotates rigidly, interior cross-sections may or may not also translate and rotate rigidly, exactly or approximately. To the extent that they do, by virtue of geometrical and material properties of the beam, one may ask for the dependence of the location of the centres of twist and of shear on distance from the end section of the beam, as has been done sometime earlier, in 1955, by the first-named author, in regard to the centre of twist.

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Definition of Centre of Shear We define co-ordinates x s, ys of the centre of shear as the co-ordinates of the point of intersection of the lines of action of the end forces P and Q for the case that (1) there is no rotation of the end section, and (2) the total torque about the point x = y = 0 (and any other point in the cross-section) is due to the forces P and Q. Setting in accordance with the above definition, in Eqs. (3)

we obtain from the third equation in (3) the relation

Since (7) must hold for arbitrary ratios P/Q we have as expressions for ys and x s

Conditions for Coincidence of Centre of Twist and Centre of Shear Inspection of Eqs. (8), (5) and (3) reveals that a sufficient condition for the coincidence of the centre of twist and the centre of shear is that the influence coefficient matrix C in Eqs. (3) be symmetric. Since this matrix will be symmetric provided a strain energy function exists for the beam problem, coincidence of the two centres is established in the foregoing for most cases of practical interest. Conversely, we may expect that the centre of twist and the centre of shear, as defined in the present account, will in general not coincide with each other for beams with material and/or support condition properties of such nature that a strain energy function does not exist for them. 4 The Principle of Minimum Complementary Energy for the Problems of Torsion and Flexure We know that among all states of stress and displacement which satisfy the differential equations of equilibrium and the given surface traction conditions for f(x, y) = 0 and z = L, the state which also satisfies the stress-displacement differential equations and the given displacement boundary conditions for z = 0 and z = L is determined by a variational equation G3 = 0 where

Here W is the complementary energy density of the material of the beam, and Eq. (9) may be appropriately generalized for more general boundary conditions for z = 0 and z = L, of the kind noted in the paragraph following Eqs. (3). In accordance with our earlier discussion it is our object to use the equation G3 = 0, on the basis of suitable approximative assumptions for the state of stress, in order to obtain approximate values of the integrals

, in terms of the parameters U, V and : in (9).

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To utilize (9) in a practical sense it suggests itself that we limit ourselves to cases for which part of the approximate assumptions for the state of stress consists in stipulating, as in Saint-Venant's theory of torsion and flexure, that

We further assume that the material of the beam is homogeneous and transversely isotropic so that, with (10),

where E and G are independent of x, y, z.

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On One-Dimensional Finite-Strain Beam Theory: The Plane Problem [J. Appl. Math & Phys. (ZAMP) 23, 795804, 1972] Introduction The following is concerned with a consistent one-dimensional treatment of the class of beam problems dealing with the plane deformation of originally plane beams. Our principal result is a system of non-linear strain displacement relations which is consistent with exact one-dimensional equilibrium equations for forces and moments via what is considered to be an appropriate version of the principle of virtual work. Having a consistent system of equilibrium and strain displacement equations it is further necessary to stipulate, or rather to establish by means of an appropriate set of physical experiments, an associated system of constitutive equations. We discuss the nature of this aspect of the problem, including a solution of its linearized version, but without arriving at the solution of the general problem. The principal novelty of the present results is thought to be a rational incorporation of transverse shear deformation into one-dimensional finite-strain beam theory. A case may be made that the theory, with this effect incorporated, is of a more harmonious form than the corresponding classical theory, where account is taken of finite bending and stretching, while at the same time it is postulatedfollowing Euler and Bernoullithat the transverse shearing strain is absent, with the corresponding force being a reactive force. As an application of the general work a solution is given of the problem of circular ring buckling, including consideration of the effects of axial normal strain and of transverse shearing strain on the value of the classical Bresse-Maurice Lévy buckling load. Kinematics of Beam Element We consider an element ds of a one-dimensional beam with equations x = x(s) and y = y(s) before deformation. We designate the tangent angle to the beam curve by M0 and write cos M0 = x c(s) and sin M0 = yc(s), where primes indicate differentiation with respect to s. We note that M0 is also the angle between the normal to the beam curve and the y-axis. Due to deformation the points x = x(s) and y = y(s) of the undeformed beam curve are changed to x(s) + u(s) and y(s) + v(s). We now assume that transverse elements which were originally normal to the beam curve do not necessarily remain so but end up enclosing an angle 1/2S F with this curve. At the same time we designate the angle enclosed by such an element and the y-axis by M. We then have a geometrical situation as shown in Figure 1. We note in particular, in addition to the angle F, the relative change of length e of the beam curve element ds, and the

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Fig. 1.

change of the angle M0 into an angle M, and we read from the deformed beam element, as relations between F, M, e, u and v,

Dynamics of Beam Element We now consider the deformed beam element, with normal and shear forces N and Q and with a bending moment M, in accordance with Figure 2. Together with this we assume force load intensities px and py and a moment load intensity m, per unit of undeformed beam curve length, also in accordance with Figure 2.

Fig. 2.

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We then read from Figure 2 as component equations of force equilibrium in the directions of x and y,

At the same time we obtain as equation of moment equilibrium

We note, for future use, the possibility of deducing from (2a, b) the relations

where n = pxcos M + py sin M and q = pycos M pxsin M are components of load intensity in the directions of N and Q, respectively. Constitutive Equations We postulate that the material of the beam is elastic and that we have the existence of axial and transverse force strains H and J and of a bending strain N, in such a way that constitutive equations for beam elements may be written in the form

We are ignorant, at this point, not only in regard to the form of the functions g in (4), but also in regard to definitions for the components of strain H , J and N which enter into the constitutive equations (4).* Defining Equations for Strain In order to obtain equations for strain we consider a virtual work equation of the form

and we stipulate, as Principle of Virtual Work, that equation (5) be equivalent to the dynamic equations (2) and (3) in the interior of the interval ( s1, s2), given that G H , GJ and G N are appropriate expressions for virtual strains. Since we know the form of the dynamic equations but do not at this point know expressions for virtual strains we use equation (5), in conjunction with (2) and (3), to deduce expressions for virtual strains. Introduction of (2) and (3) into equation (5) gives a relation of the form

*However, we expect that H|e, J|F and N| McMc0, for sufficiently small strain.

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and in this we may now consider N, Q and M as arbitrary differentiable functions of s. In order to utilize (6) we integrate by parts, thereby eliminating all derivatives of N, Q and M as well as the boundary terms on the right. In this way we obtain

The arbitrariness of N, Q and M means that (7) implies the virtual strain displacement relations

It remains to take the step from virtual strain displacement relations to actual strain displacement relations. One of these actual strain displacement relations follows directly from equation (9) in the form

A correspondingly simple derivation of expressions for H and J is clearly not possible through direct use of (8a, b). Remarkably, we may obtain H and J by using (8a, b) in conjunction with the geometrical relations (1). To do this we observe that equations (1) imply the following relations between virtual quantities

We now use (11a, b) in order to eliminate Guc and Gvc in (8a, b). In this way we obtain

The form of (12a, b) is such that we can now go from virtual strains to actual strains. The results are

Having (13a, b) we can further express H and J in terms of u, v and M. Introduction of (13a, b) into (1a, b) gives first

and then, by inversion

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We finally note the possibility of rewriting the moment equilibrium equation (3) somewhat more simply with the help of the strain components H and J as in (13), in the form

Observations on the Problem of Experimentally Derived Constitutive Equations In order to see the nature of the problem of experimentally establishing the nature of the functions g in equations (4) we consider the problem of an originally straight beam, with x = s, y = 0 and M0 = 0, fixed at the end x = 0 and subject to given displacements u(a) = ua, v(a) = v a and M(a) = Ma at the other end. We assume absent distributed loads and have then from equations (2a, b)

where Xa and Ya are two constants of integration the mechanical significance of which is evident. To proceed further we consider the moment equation (3 *) as a differential equation for M, by writing

and by considering the constitutive equations involving N and Q partially inverted in the form

so that M = f M (H,J,N) = f k (N,Q,M') = g(M,M'). The resultant second-order equation for M must be solved subject to the boundary conditions M(0) = 0 and M(a) = Ma, with which M = M(x; Xa, Ya, Ma). Having M we find u and v from (14a, b). The boundary conditions for u and v are satisfied upon setting

We now measure X a, Ya and Ma as functions of ua, v a, Ma, and of a, giving a set of three relations from the form of these three experimentally determined functions

,

and

the form of the desired three functions gN, gQ, gM in equations (4).

The Linear Case We consider a range of stresses and strains within which

with a view towards determining the elements

, . . .,

From equations (17) follow the linearized relations

, etc. The remaining task then is to deduce

of the three by three matrix [C].

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and the moment equation (3*), again with boundary conditions M(0) = 0 and M(a) = Ma, is reduced to

Equations (19) for the translational edge displacements become

In order to solve the problem as stated in (20) to (23) we partially invert (20) in the form

and write (22) in the form

, with solution

We then have further, from (24),

and, upon making use of (23a, b),

We now stipulate knowledge of a matrix [B], as a result of experiment, such that

Having (26) to (28) we may then successively determine the elements of the matrix [C*] in terms of the elements of [B]. To see this we write

and have then from the relation Ma = BMNNa + BMQQa + BMMMa that

from which

,

and

follow in succession in terms of elements of [B].

We next introduce (26c) into (27a, b) and compare the resultant relations with corresponding relations in (28). In this way we obtain the remaining six elements etc. of the matrix [C*] in terms of the elements of [B]. Finally, having [C*] we find the elements of [C] by returning from (24) to (20). Buckling of Circular Rings As an application of the foregoing we consider the classical problem of in-plane buckling of a circular ring of radius R, subject to a uniform normal pressure p. We wish to obtain a buckling-load formula which

,

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incorporates the effects of (1) the symmetrical deformation of the ring prior to the onset of buckling, (2) axial strain associated with the buckling mode, (3) transverse shearing strain associated with the buckling mode. We will be concerned, in particular, with the question of appropriate constitutive equations. Inspection of Figure 2 indicates that for uniform normal pressure p, per unit of deformed beam curve, we have as expressions for the load intensity components q and n in the force equilibrium equations (2*a, b)

together with an absent moment load intensity m in equation (3*). We further have, with N as in equation (10) and with RdM0 = ds, that Mc = R1 + N. Therewith the equilibrium equations (2*a, b) and (3*) may be written in the form

In complementing (31) by constitutive equations we have no difficulty in deciding that suitable relations involving N and H are of the form

In stipulating a relation involving J we find it necessary to concern ourselves with the question whether J would be determined by the force Q tangential to the deformed cross section or by a force Q* normal to the deformed centerline. Evidently, we have Q* given in terms of Q and N by the relation Q* = Q cos F N sin F or, approximately, by Q* = Q NJ. If we stipulate that J = BQ* we arrive at a relation for J in terms of Q and N, of the form J = BQ/(1 + BN).* If we use Q instead of Q* at the outset we have instead that J = BQ. We may subsume both relations to one of the form

and consider in the end the two limiting cases O = 0 and O = 1. Having equations (31) and (32) we now consider the stability of the state

for which, evidently, in view of (31b) and (32b)

We now write

and linearize (31) and (32) in terms of Q, M, J, N, N1 and H 1 so as to have.

*This, together with (32b), is effectively equivalent to constitutive equations of the form Q = (J/B) + (HJ/C) and N = (H/C) + (J2/2C).

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Equation (32a) remains as is and equations (32b) and (32c) become*

We now use (32a), (34) and (37) to write (36a, b, c) as a system of equations for N1, Q and N, as follows

It is evident that (38b), differentiated once, my be written with the help of (38a) and (38c) as one second-order differential equation for Q. Appropriate solutions, for a complete ring, will be of the form Q = cos ns/R where n = 2, 3, . . . From this follows as the equation for possible values of P,

Equation (39) may be written as a cubic equation for PR2/D, involving axial-strain and transverse shear-strain parameters k H = CD/R2 and k J = BD/R2. We will here limit ourselves to a discussion of the case k H = 0, with k J{ k, for which the cubic equation reduces to a quadratic of the form

The smallest positive value of P follows from this for n = 2. We consider in particular the cases O = 1 and O = 0. When O = 1 we have from (40), in agreement with a recent result by Smith and Simitses**

When O = 0 the solution is

*We note the possibility that C and B, as well as D in equation (32a), may be considered to depend on HP. **J. Eng. M ech. Div., ASCE 95, EM 3, 559569 (1969).

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On One-Dimensional Large-Displacement Finite-Strain Beam Theory [Studies Appl. Math. 52, 8795, 1973] Introduction In what follows we consider once more the classical problem of large displacements of thin curved beams. As regards the literature on this problem prior to about 1930 we may refer to the Historical Introduction and to Chapters 18 and 19 of the 4th Edition of Love's Mathematical Theory of Elasticity. Subsequently, further attempts at improving the theory have been made by various investigators, but as far as could be ascertained none have dealt with the problem in the manner described in what follows. Briefly stated, we consider a large-deformation theory of space-curved lines, with the cross sections of the lines acted upon by forces and moments. We take as basic the differential equations of force and moment equilibrium for elements of the deformed curve. We then stipulate a form of the principle of virtual work, and use this principle so as to obtain a system of strain displacement relations, involving force strains and moment strains in association with the assumed cross-sectional forces and moments. Having a one-dimensional description of stress states and strain states we complete the formulation of the problem by postulating a system of one-dimensional constitutive equations which, in the end, must incorporate the consequences of suitably designed experiments or of the one-dimensional consequences of a theory of beams treated as a three-dimensional problem. The advantages of the present development, in comparison with work by others, seem to this writer to consist in the explicit consideration of axial and transverse force strains in place of the usual early introduction of inextensibility and of the Euler-Bernoulli hypothesis. We have previously considered linear beam theory [2], as well as the nonlinear theory of plane deformations of plane beams [5] in this manner. The present step to the general nonlinear theory suggested itself in the course of related work on thin shells [4], which in turn was suggested by work of Simmonds and Danielson [6] and by earlier work on the nonlinear symmetrical problem of the shell of revolution [3]. It is to be emphasized that the present theory, as far as it goes, is strictly along classical lines without consideration, for example, of Vlasov's concept of bi-moments. It remains to be seen in what way the present approach may be generalized so as to apply to more general non-classical one-dimensional nonlinear beam theories. Geometry and Statics of Beam Elements We consider a space curve with equation r = r(s) before deformation. We take the parameter s to be the length of arc, measured from a given point on the curve, and have then a tangent unit vector

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t = rc(s), where the prime here and in what follows indicates differentiation with respect to s. Having t we introduce two mutually perpendicular unit normal vectors n1 and n2, such that t × n1 = n2, n1 × n2 = t and n2 × t = n1, with generalized Frenet formulas

where U1, U2 and Ut are given functions of s. Due to deformation the curve r = r(s) changes into a curve R = r(s) + u(s), with tangent vector Rc(s) where |Rc(s)| = 1 + H , with H z 0, in general. Having the curve R = R(s) we now stipulate the existence of external forces and moments p ds and m ds acting on the element |dR|. We further stipulate the existence of internal forces and moments P(s) and M(s) acting over ''cross sections" of the curve. Consideration of the changes of P and M in going from s to s + ds, as well as of the principle of action and reaction then gives as equations of force and moment equilibrium of the elements of the deformed beam the two vector relations

Equations (2) are of course equivalent to six scalar component equations. In order to obtain such component equations, we introducefollowing Simmonds and Danielson's idea in treating thin-shell theory [6]a triad of as yet unspecified mutually perpendicular unit vectors T, N 1 and N2 in terms of which

and

with corresponding decompositions for p and m. In reducing (2) to scalar form through introduction of (3a, b) we make use of differentiation formulas for T, N 1 and N2 of the form

with r1, r2 and rt depending on how the triad (T, N i) is defined. The Equation of Virtual Work and Virtual Strain Displacement Relations We introduce in association with the external forces and moments p ds and m ds, virtual translational and rotational displacements G R = G u and G M. At the same time we introduce in association with the internal forces and moments P and M virtual force and moment strains G JDS GG N. With this we write as equation of virtual work for any segment (s1, s2) of the beam

and we designate as principle of virtual work the statement that with G J and G N suitably given in terms of arbitrary G u, G M and their derivatives, Eq. (5) is equivalent to the two vector equilibrium Eqs. in (2).

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For present purposes Eq. (5) is used, not to deduce (2) on the basis of given G J and G N, but rather to deduce G J and G N, on the basis of knowing the form of (2). To do this we introduce p and m from (2) in terms of P and M and write (5) in the form

where now P and M may be considered as arbitrary quantities. Integration by parts to eliminate Pc and Mc, and use of the relation Rc × P · G M = Rc × G M · P, then results in two relations, which may be designated as virtual strain displacement relations. These are

where Rc = t + uc. In order to translate (7) into a system of scalar relations we now define scalar components of virtual strain by means of the following two expressions

These, in conjunction with (3a, b) imply as expressions for virtual strains,

and it remains to use Eqs. (7) for the determination of G Jt, G Ji, G Nt and G Ni, in such a way that from these the determination of Jt, Ji, Nt and Nt becomes possible. Derivation of Implicit Strain Displacement Relations The crucial step in what follows is the tentative assumption of an implicit representation of the force strains Jt, and Ji, in terms of the triad (T, N i), as follows

From this follows as relation for (G u)c = (G R)c = G (Rc),

Introduction of (10), (11) and (9a) into the first Eq. in (7) gives after some cancellations,

Equation (12) is satisfied, identically in Jt and Ji, by further setting

Equations (13) in turn are solved for GM by considering the cross products T × G T and Ni × G Ni in conjunction with the canonical expansion formulas for vectorial

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triple products. In this way we obtain

Having Eq. (14), we now determine (G M)c in order to determine G N from the second Eq. in (7). The nature of the calculations which are required is as follows

The introduction of (17) and (18) into (15), with corresponding expressions for the terms (N1 × G N1)c and (N2 × G N2)c, gives after a remarkable series of cancellations the simple relation

A comparison of (19) and (9b) gives further

Inasmuch as Nt and Ni should vanish for the case of no deformation, that is for the case that rt = Ut and ri = Ui, we deduce from (20) as implicit expressions for the components of moment strain,

Intrinsic Equations of Beam Theory Having Eqs. (21) together with (2), (3a, b), (4) and (10) it is now easy to state an intrinsic form of the beam problem, that is a complete system of differential equations without reference to displacement variables, as soon as it is agreed to stipulate a system of six constitutive equations of the form

Besides these six scalar equations, we have another set of six scalar equations upon introducing the component representations (3a, b) and (10), in conjunction with the differentiation formulas (4), into the vectorial equilibrium Eqs. (2). These six additional scalar equations are

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with ri, and rt in terms of Ni and Nt as in (21), and

The form of Eqs. (23) and (24) is as expected, except for the appearance of the nonlinear JP-terms in (24). Derivation of Explicit Strain Displacement Relations Considering that all components of strain depend on the choice of the orthogonal triad (T, N i) we must, in order to obtain an explicit set of strain displacement relations, connect this triad with the triad (t, ni), through a suitable set of angular displacement parameters. Geometrically speaking, the step from (t, ni) to (T, N i) involves an angle of rotation T, about an axis defined by a unit vector \. The associated transformation formula, for any vector A in terms of the corresponding vector a is, in accordance with Hamel [1], the Rodriguez formula *

Having (25) for T and Ni in terms of t and ni, respectively, we find components of force strain Jt and Ji on the basis of Eq. (10) in the form

in their dependence on T and on the components of u and \. In order to obtain corresponding expressions for Nt and Ni, we need the coefficients rt and ri in the differentiation formulas (4). In order to obtain these we now consider the problem of differentiating T and Ni as given by Eq. (25). It will be sufficient to describe the details for the case of the vector T. Straightforward differentiation of the formula for T which follows from (25) gives

In the first line on the right we may introduce tc from Eqs. (1) and make use of the fact that (25) also gives the Ni in terms of the ni. Furthermore we take account of the fact that Tc must be perpendicular to T. In this way we see that (27) is equivalent to a formula

*We note that this formula plays an important role in Simmonds' and Danielson's treatment of shell theory [6], and also in Tameroglu's treatment of small-strain EulerBernoulli type curved beam theory [7].

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where

It is now necessary to determine :. We do this, essentially in the manner in which Hamel [1] determines the "angular velocity" of A, by considering two special choices of t, namely t = \ and t perpendicular to \. When t = \ we also have T = \ and furthermore t·\ = 1, t·\c = 0, and \ × t = 0. Equation (29) becomes then

To proceed from (30) we use the unit-vector identity \ × (\c × \) = \c. In this way (30) may be written as

Equation (31) shows that : must be of the form

with E a scalar function of T and Tc which remains to be determined. To find E we now consider the case that t and T are perpendicular to \. When this is the case Eq. (25) implies as relation between t and T

We introduce (33) as well as (32) into Eq. (29) and obtain as equation for the determination of E

We transform this equation with the help of appropriate expansion formulas for triple and quadruple vector products on the left, and obtain after a number of cancellations the remarkably simple relation

Introduction of (35) into (32) gives

It would now be possible to introduce Eq. (36) as it stands into Eq. (28) and then proceed with the determination of rt and ri. A considerably more convenient result follows upon first transforming Eq. (36) through the introduction of a modified (non-unit) axis-of-rotation vector O, given by

Insertion of (37), together with \c = gOc + gcTcO, into Eq. (36) gives

We now determine g(T) so as to make the term with Tc in (38) disappear. This gives

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and therewith, after some further transformations

where O = O1n2O2n1 + Ot t, with O1, O2, Ot being three independent angular displacement parameters, and c an arbitrary constant. It is convenient, in order to establish direct contact with known results of linear theory to set

We now introduce Eqs. (40) and (41) into Eq. (25) for T and Eq. (28) for Tc and obtain after some further simple transformations

and

with analogous formulas for Ni and

.

Having Eq. (42) for T and the corresponding equations for N1 and N2 we could now, upon writing u = uini + utt obtain expressions for the components of force strain Jt and Ji in terms of ui,ut,Oi and Ot in accordance with (26). Inasmuch as all that we need, for the satisfaction displacement boundary conditions, are expressions for T and the Ni as well as an expression for u we limit ourselves here to writing as expression for u, on the basis of (10)

with T and the Ni taken from Eq. (42).* In order to evaluate (42) and (44) we need of course to determine the angular displacement vector O and for this purpose we now deduce expressions for the moment strains Nt, N1, N2 in terms of the components of the angular displacement vector O. Combining the first of Eqs. (4) with Eq. (43) and the second relation in (21) we have

Combination of the second equation in (4) with the expression for

which corresponds to (43) and with the first relation in (10) gives further

A simple direct calculation shows that (45a, b) may be written, alternately, in the *For cases for which the effect of force strains on the deformation of the beam is negligible Eq. (44) reduces to the simple form

.

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form

Equation (46) suggests expressing Ni and Nt, in terms of the corresponding expressions of linear theory. Setting

we will have,

which, except for notation and sign conventions, agrees with the expressions for moment strains of linear theory in [2]. Introduction of (47) into (46) gives as expressions for the components of moment strain of the present nonlinear theory

Moment Strain for In-Plane Deformation of Plane Beams For this case we may set 1/U2 = 1/Ut = 0 and O2 = Ot = 0. Therewith

and

and therewith, according to (49a)

In order to express N1 in terms of the angle of rotation T, as in the independent treatment of the plane problem [5], we consider that for this case the unit rotation vector \ equals the unit vector e2. Furthermore, O = O1n2 = O1e2 and therefore, in accordance with Eqs. (40) and (41),

and then

which is in agreement with the result in [5]. References 1. G. Hamel, Theoretische Mechanik , 103107, Springer-Verlag, 1949. 2. E. Reissner, Variational considerations for elastic beams and shells, J. Eng. Mechanics Division, Proc. Amer. Soc. Civil Eng., EMI, 2357, 1962. 3. E. Reissner, On finite symmetrical deformations of thin shells of revolution, J. Appl. Mech. 36, 267270, 1969.

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4. E. Reissner, Linear and non-linear theory of shells, Thin Shell Structures, pp. 2944, edited by Y. C. Fung and E. E. Sechler, Prentice-Hall, 1974. 5. E. Reissner, On one-dimensional finite strain beam theory: The plane problem, J. Appl. Math. and Phys. (ZAMP) 23, 795804, 1972. 6. J. G. Simmonds and D. A. Danielson, Non-linear shell theory with finite rotation and stress function vectors, J. Appl. Mech. 39, 10851090, 1972. 7. S. Tameroglu, Finite theory of thin elastic rods, Techn. and Scientific Res. Council of Turkey, Appl. Math. Division, Rpt. No. 4, July 1969.

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Upper and Lower Bounds for Deflections of Laminated Cantilever Beams Including the Effect of Transverse Shear Deformation [J. Appl. Mech. 40, 988991, 1973] 1 Introduction The considerations which follow were originally motivated by the semi-elementary treatment of plane-stress solutions of the problem of the isotropic, homogeneous elastic cantilever beam of narrow rectangular cross section, fixed at one end and acted upon by a transverse force at the other end, as in [1]. Of specific interest here is the tip deflection formula

where the term having the numerical factor k accounts for the influence of transverse shearing strains on the deflection of the beam. In deriving a formula of the foregoing type by semi-elementary means, that is, by use of simple polynomial solutions of Airy's differential equation as in [1], it is necessary to make certain assumptions concerning the approximate formulation of the conditions of end fixity of the cantilever, with the value of k depending on the assumptions which are made. In what follows we propose to remove this ambiguity by the use of upper and lower-bound results for the ratio V/P, obtained through suitable application of the principles of minimum potential and complementary energy of the theory of elasticity. Having once formulated a procedure for deriving such bounds for homogeneous beams it is readily evident that the same procedure is applicable for nonhomogeneous, laminated beams. The results which follow are therefore stated for this more general class of problems. 2 Formulation of Problem We consider states of plane stress in a rectangular region with boundaries x = 0, x = a, and y = ± c, this region representing a beam of unit width. We assume that the boundary portion x = 0 is loaded by a force P in the negative y-direction, that the boundary portion x = a is fixed, and that the boundary portions y = ± c are traction-free. Writing as differential equations for stress and strain the usual relations

and

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the associated system of boundary conditions is here taken in the following form:

At the same time the force P is given by the integral

In the semi-elementary formulation of this problem, referred to previously, the displacement boundary condition for x = 0 is replaced by a stress boundary condition W(0,y) = (3P/4c)(1 y2/c2), in order to conform with the type of solution which is used. With this type of solution it is also not possible to satisfy the displacement conditions (4) as they stand. Instead, all remaining arbitrariness in the semi-elementary solution is removed by postulating and satisfying a system of conditions of the form *

The formulation of the boundary-value problem is completed by the statement of stress-strain relations which for a linear isotropic homogeneous medium are of the form

where E = 2(1 + Q)G, with E, Q, and G being constants [1]. In what follows we consider the problem subject to a system of relations of the form:

where B is a given function of Vx, Vy, W (and of x and y), and we obtain explicit results for the case that (9) reduces to

where E, Q, Ey, and G are given even functions of the coordinate y. For the formulation of the solution procedure which is to be used, we require a statement of the relations inverse to (9) as well, in the form

*Alternately, consideration is given to the possibility of using, instead of (wu/wy)a,0 = 0, a condition (wv/wx)a,0 = 0, although, strictly speaking, a condition of this kind would not appear to be prescribable "from the outside." We note, as a possible alternate condition which would be prescribable, the relation

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with a corresponding inverse statement of (10). 3 Upper and Lower Bounds for the Work Done by the Force P Certain transformations of a functional I which enters into the statement of both the principle of minimum potential energy and the principle of minimum complementary energy lead to the conclusion that, with conditions as stated in Eqs. (1)(6), (9), and (9c), we have as inequalities for the work quantity PV,

In this where

where ,

, and

and

are differentiable functions which satisfy the displacement boundary conditions (4) and (5b), and

are differentiable functions which satisfy the equilibrium differential Eqs. (1) and the stress boundary conditions (3) and (5a), with

The equal signs in these inequalities apply whenever , , and (5), together with (9) and (9c).*

,

,

.

, respectively, are the actual solution functions of the boundary-value problem stated in (1)

4 Application of Formula for Bounds In order to apply the system of inequalities we first express the force P as a function of the given displacement (which it must be considering Eq. (6), and the fact that the term V in (5b) is the only nonhomogeneity in the equations of the boundary-value problem). We will not consider the general case, and assume instead that the constitutive equations are the linear Eqs. (10). This will make P a linear function of V, and we write

with a view toward obtaining upper and lower bounds for the flexibility coefficient C, with the help of the system (11), written now in the form

where, consistent with (10),

*Equation (1) may be obtained in the same way as a related, more general Eq. in [2]. A direct specialization of some interest of the result in [2] is possible for the case that the stress boundary condition (5a) is replaced by a displacement condition u(0, y) = y: where : is a given constant. [We note that the signs in front of the last integrals in Eqs. (3.5) and (7.3), and in front of the second integral in (6.3) in [2] should be reversed]. Inequalities of a similar nature have been stated earlier by others, in particular by C. Weber, but as far as we know not for the problem which is here under consideration.

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and

with

,

, and

given in terms of

and

as in (2).

Lower-Bound Calculation We now choose, guided by the results of the semi-elementary theory, as expressions for

and

where O is an arbitrary constant. We have then

so that all three displacement boundary conditions are satisfied by the differentiable functions

and .

Introduction of (16) and (17) into (2) gives as approximations for strains to be used in Ã,

and therewith

where

We now determine O from the condition

, in the form

Introduction of (22) into (20) gives as the smallest value of

which is compatible with the assumed state of displacement

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Introduction of (23) into (13) gives

as the desired lower-bound formula for C. Upper-Bound Calculation We now choose, again guided by the results of the semi-elementary theory, and in accordance with Eqs. (1), (3) and (5a),

with P a constant parameter which remains to be determined. Introduction of (25) into Eq. (14) gives

We now determine P from the condition

and, with this, obtain as the largest value of

in the form

which is compatible with the assumed state of stress,

Introduction of (28) into Eq. (13) gives as the desired upper bound for C

In comparing the inequalities (24) and (29) with the approximate formula stated in the Introduction it is found that both inequalities provide appropriate generaliz-

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ations of the factor Pa3/3EI in the semi-elementary result, provided we agree to omit the effect of Poisson's ratio in the definition of E*. We are interested, particularly, in the terms with G in Eqs. (24) and (29), which are a measure of the relative importance of the effect of transverse shear deformation, and list explicit results for two special cases: Homogeneous Beams We assume that E and G are independent of y and have then

These values may be compared with the result of the semielementary calculation based on the substitute boundary conditions (7), which gives a factor 3/2 in place of the numerical factors on the right of (30a) and (30b). We note in particular, in view of certain important consequences of this observation, which however will not be discussed in this paper, that the shear correction terms (30) become of relative order of magnitude unity for the case that G is so small relative to E as to make G/E of the same order of magnitude as the ratio c2/a2. Sandwich-Type Beam The simplest way to appraise the effect of transverse shear deformation is to consider a beam consisting of a uniform shear-resisting core layer with negligible resistance to longitudinal normal stress, and two face layers sufficiently thin to assume negligibility of their bending stiffness [3]. Mathematically, this means that we consider a beam with the moduli E and G given as functions of the thickness coordinate y by the relations

where t 0 eine andere Bedeutung zulegt, kann man (3) auch als Randbedingung für eine elastisch eingespannte Platte ansehen.Es sei noch die Bedeutung der Konstanten in (4) angegeben: U 0 ist der Plattenradius, N die Biegungssteifigkeit der Platte, aU 0 der Belastungsradius [das Koordinatensystem ist so gewählt, daß M(P) = 0] und wU 0 die Biegungsfläche in wahrer Größe. 3A. Föppl, Ber. d. Kgl. Bayr. Ak. d. Wiss. 1912; S. 155. 4A. Clebsch, Theorie der Elastizität fester Körper. Leipzig 1862.

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Um diese zu finden, wird ausgegangen von der bekannten Tatsache, daß sich jede Lösung von (1) in der Form

mit

darstellen läst. Also ist auch

eine Lösung von (1). Man erkennt, daß sie die Bedingungen (2) und (4) bereits im Ansatz erfüllt, wenn g(z) eine für r d 1 regulär analytische Funktion ist. Die Bedingung (3) dient dazu, diese Funktion zu bestimmen. Es ist zweckmäßig, die Bedingung (3) etwas umzuformn. Wegen (2) ist

Damit wird

Beachtet man, daß

so erhält man mit (5) aus (3) die folgende Gleichung zur Bestimmung von g(z)

(3a) ist in bekannter Weise als eine Differentialgleichung für g(z) aufzufassen5). Ihr allgemeines Integral ergibt sich mit einer willkürlichen Konstanten c und unter Weglassung einer bedeutungslosen rein imaginären additiven Konstanten zu

Wegen der vorausgesetzten Regularität von g(z) ist c = 0. Wenn man (8) in (5) einsetzt, wird die gesuchte Biegungsfläche

5Auf diesen direkten Schluß hat mich freundlicherweise Herr Prof. G. Hamel hingewiesen. Ursprünglich 'war die Lösung gefunden worden durch Ansetzen einer Potenzreihe für g(z), deren Koeffizienten sich aus (3a) bestimmen, und Identifizierung der Reihe mit der Funktion (8).

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Nun läßt sich das Integral in (9) in allen den Fällen geschlossen ausintegrieren, in denen Q eine rationale Zahl ist, also praktisch immer, und damit hat man in (9) die gesuchte Lösung der Aufgabe in geschlossener Form. Für den Grenzfall Q = 0 z. B. erhält man in reeller Schreibweise

Für Q = 1/3 ist

Für die Durchbiegung unter dem Lastangriffspunkt (Biegungspfeil) ergibt sich daraus

Man kann als einfache Verallgemeinerung des vorstehenden Lösungsgedankens den folgenden Satz aussprechen: Die Randwertaufgabe der Bipotentialtheorie für den Kreis lä E t sich im Falle gegebener Funktionsrandwerte auf eine Randwertaufgabe der Potentialtheorie für den Kreis zurückführen, deren Art von der zweiten Randbedingung für die Bipotentialfunktion abhängt. Dieser Satz kann mit Vorteil noch für andere technisch wichtige Aufgaben aus der Theorie der durch eine oder mehrere Einzelkräfte und stetige Belastung beanspruchten Kreisplatte verwendet werden. Es ist beabsichtigt, dies an anderer Stelle auszuführen.

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On Tension Field Theory [Proc. 5th Intern. Congr. Appl. Mech. pp. 8892, 1938] 1 Introduction Tension field theory has been developed by H. Wagner 1 to describe the state of stress in a certain type of thin-walled structures after buckling has set in. It substitutes for the non-linear problem of a large-deflection theory of elasticity a simplified linear theory which in a number of practically important cases yields results in good agreement with experimental data. The reasoning leading to tension field theory is best explained by means of a definite example. Consider a strip of thin sheet supported perpendicular to the plane of the sheet and acted upon by uniform shear along the edges in the plane of the sheet. Up to a certain intensity of the shear load, a uniform plane state of stress is produced in the sheet. If the load is increased beyond that intensity buckling occurs. However if the distance of opposite edges of the sheet is kept constant the shear load can have an intensity many times that of the buckling load without failure of the structure as such. What happens is that wrinkles are formed in the sheet, the wave length of which decreases with increasing load, and that the sheet is mainly stressed in tension in the direction along the wrinkles while the compressive stress perpendicular to the wrinkles which causes the wrinkling becomes small compared with the tensile stress. Neglecting this compressive stress and also the bending stresses induced by the deformation out of the plane of the sheet against the tensile stress, one has to study types of plane states of stress for which at each point the only principal stress component which is different from zero is a tension, while the strain component perpendicular to the direction of the non-vanishing principal stress component is not uniquely determined by the stress at that point. In the present paper Wagner's formulation of the problem is modified in such a way, that it appears as a special case of a more general problem in plane stress, in itself of interest and hitherto not considered. It deals with the theory of elasticity of anisotropic media, the curvilinear anisotropy of which depends on the boundary conditions of the problem. In this new form the tension field is analyzed by straightforward calculus avoiding the lengthy geometrical considerations of Wagner. As an application of the theory a solution of the following problem is here given: A flat sheet of circular ring form, stiffened along inner and outer edges, is stressed by twisting one edge with respect to the other, the axis of the applied torque being perpendicular to the plane of the sheet. This is the first solved problem in tension field theory where non-parallel tension lines occur. 1H. Wagner, Zeitschr. Flugtechnik u. M otorluftschiffahrt 1929.

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2 Formulation of the General Problem In this paragraph is given the mathematical formulation of the following problem: To determine the state of stress in a material of such curvilinear anisotropy that the axes of elastic symmetry are always tangent to the lines of principal stress. The anisotropy of such a material thus depends on the conditions which are prescribed along its boundaries. Tension field theory is obtained as a particular case of this problem in which one of the two different moduli of elasticity has the value zero. Introducing as system of coordinates [ , K the orthogonal system of the lines of principal stress in which the line element has the form

the following system of equations has to be solved. 1. The equations of equilibrium for the principal stress components V[ and VK

2. The stress-strain relations. Calling u and v the displacements in [ and K-direction, E[ and EK the moduli of elasticity and G the modulus of rigidity and assuming no lateral contractions, these relations are

3. The relation between h1 and h2 which expresses the fact that the system ([ , K) is orthogonal

It seems difficult to solve this system if not either E[ = EK (isotropy) or one of the E's say

The latter case shall here be discussed. 3 Solution of the Equations of Tension Field Theory Introducing the assumption (5) into Eqs. (3) there follows

and with that from Eqs. (2) wh1/wK = 0, h1 = h1([ ). It amounts only to a change of scale along the curves K = const. if one puts

Geometrically the relation (7) means that the lines K = const. are straight (Figure 1).

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Fig. 1.

Furthermore it follows from (2) that

where g is an arbitrary function. From (4) follows

where g1 and g2 are arbitrary. For the calculation of the displacements u and v two cases are conveniently distinguished. (a) The case that the straight lines K = const. are parallel, and hence also the lines [ = const. are straight. Then g1(K) = 0 and by scaling appropriately one may put g2(K) = 1. The displacements result from Eqs. (3) in the form

Here h and k are two more arbitrary functions. (b) The case that the lines K = const. are not parallel. In this case a change of scale makes possible to put g1(K) = 1 which means that the variable K is identical with the varying angle D between the lines K = const. and a fixed straight line. From (3) results then

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In (8), (9) and (10) the four unknown quantities V[, h2, u and Q are expressed in terms of four arbitrary functions g, g2, h and k. Since these arbitrary functions depend on the variable K whereas the boundary conditions are given in terms of a fixed coordinate system, say a Cartesian system (x, y), relations must be established between the system ([ , K) and the system (x, y). For this purpose the following equations must be used

The integration of this system of equations is effected by a geometrical consideration. Since the lines K = const. are straight, one may write

This introduced into (11) gives for m and n

and from (13)

Hence

where dD/dK = g1(K) In (8), (9), (10) and (15) there is given the general solution of the tension field problem in terms of four arbitrary functions which have to be determined by boundary conditions. 4 The Stresses in a Sheet of Circular Ring Form Wrinkled under the Influence of Shear Stresses Acting along the Edges (Figure 2) As an application of the preceding general results this problem is solved rigorously. The tension lines start from the inner edge, and because of symmetry, each makes along the inner edge the same angle E with the radius from the origin. This angle E cannot be assumed but must be determined by means of the boundary conditions. Introducing as independent variables [ and K the distance [ along the tension lines from an origin whose position depends on the angle K of this line with the x-axis, one finds the following relations between the coordinates x, y, the polar

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Fig. 2.

coordinates r, M, and the coordinates [ , K

For the stresses one obtains from (8) the principal stress

and the radial and tangential stresses

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For the displacements in radial and tangential direction ur and uM results

Equations (18) and (19) are the most general expressions for stresses and displacements compatible with the assumed tension lines. If one is only interested in a stress distribution independent of the angle variable M = KU + S/2, that is, independent of K, these expressions are simplified to

and

For the determination of the four constants R0 = r0 sin E , g0, h0 and k 0 serve two boundary conditions at the inner edge r = r0 and two conditions at the outer edge r = r1. It shall be assumed that the shear load is applied by means of two stiffening rings of cross sectional area A0 and A1 which have moduli of elasticity E0 and E1. The two rings are necessary to prevent the sheet from collapsing after wrinkling has started. Assuming that the tangential displacement of the outer edge is zero and of the inner edge equal to and expressing the fact that the radial deformation of the stiffening rings under the influence of the uniformly distributed radial load has to equal the radial deformation of the sheet along its edges, one has the conditions

Further discussion of the problem is here restricted to the case of rigid stiffening rings, so that in (23) and (25) one puts

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That is, ur(r1) = 0 and ur(r0) = 0. Conditions (22) and (23) lead with (21) to

From (24) and (25) follows

Equations (27), (28) and (29) are three linear homogeneous equations for g0, h0, k 0 which cannot all be zero. Therefore the determinant of this system has to vanish, that is

Fig. 3.

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Developing and introducing the expressions for h2(r) and R0 from (16) this becomes the following equation to determine the angle E

which angle thus is seen to depend on the ratio r0/r1. Since (32) cannot be solved explicitly, E has been determined numerically as function of r0/r1 and the result plotted in Figure 3. It is seen that E varies between 45° and 90°; if the diameter of the hole approaches the diameter of the outer edge, the sheet behaves approximately like a straight strip and E equals 45°; if the diameter of the hole is small compared with the diameter of the sheet the tension lines depart under right angles from the radius vector. For very small r0/r1 where the numerical

Fig. 4.

Page 142

evaluation of (32) is inconvenient the following approximate relation holds

Having obtained E , one also knows g0, and with that the stresses in terms of the tangential edge displacement

.

Of especial interest is the ratio between radial stress and tangential stress along the edges since the radial stress for prescribed shear determines the dimensions of the stiffening rings. This ratio is, according to (20),

The values of the ratios (34) are plotted as function of r0/r1 in Figure 4.

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On the Calculation of Three-Dimensional Corrections for the Two Dimensional Theory of Plane Stress (Excerpt) [Proc. 15th Semi-Annual Eastern Photoelasticity Conf. pp. 2331, 1942] Formulation of the Problem In this note a method is developed which permits the approximate solution of the following class of boundary value problems of the theory of elasticity. A body, bounded by one or more cylindrical surfaces with parallel generators and by two parallel planes perpendicular to the axes of the cylindrical surfaces, is acted upon by tractions applied to the cylindrical portion of the boundary, the boundary tractions being parallel to the bounding planes and moreover non-varying in the direction of the axes of the cylinders. Mathematically the problem consists in finding solutions of the following system of nine equations:

for the nine unknowns Vx, Vy, Vz, Wxy, Wxz, Wyz, u, v, w, subject to the following boundary conditions

In these conditions 2h is the distance between the bounding planes, gi(x, y) = 0 the equation of the ith cylindrical bounding surface, s the arc length along the circumference of the cylinders and px and py the x and y-components of the boundary tractions. The two-dimensional theory of plane stress can be described as a method of approximately solving the system of Eqs. (1)(4), based on the following

Page 144

assumptions:

It is known that these assumptions are in general consistent only with the system of differential equations (1)(3) when the value of Poisson's ratio, Q, is zero. Otherwise it is necessary to disregard two of the shear stress strain relations, Eqs. (2f) and (2g), since it is impossible to satisfy them for a state of stress obtained from Eqs. (1), (2a)(2d) and (4)(6). The usefulness of the approximate theory lies in the fact that experimental evidence and mechanical intuition indicate that in many cases the results of this approximate theory are very close to the corresponding exact results. However, the same evidence and intuition indicate also that the results of the approximate theory may be seriously in error in such regions of the body where appreciable changes in stress occur over distances which are of the same order of magnitude as the thickness 2h of the body. Such changes occur for instance in the edge zone of holes (cut-outs) when the diameter of the holes is of the order of magnitude of the thickness 2h. It is for a quantitative analysis of such effects that it is desirable to calculate three-dimensional corrections for the two-dimensional theory of plane stress. Since it is believed that an exact solution of the three-dimensional problem presents very great analytical difficulties, an approximate method for its solution is here developed which reduces the three-dimensional problem mathematically to one in two-dimensions while retaining the three-dimensional mechanical characteristics of the solution. The method employed here is an appropriate application of the Principle of Least Work. Method of Solution The approximate analysis of the three-dimensional effect may be based on replacing the assumptions, Eqs. (6), of two-dimensional stress by the following assumptions:

while the assumptions, Eqs. (5), of plane stress are replaced by

In Eqs. (7) and (8) the functions S, T, s and t depend on x and y only while the functions g, which are to be determined suitably, depend on z only. The first set of conditions here imposed on the stresses of Eqs. (7) and (8) is that they have to satisfy the equilibrium conditions (1) and the boundary conditions (3) and (4). From the equilibrium conditions (1) follows that g1 and g2 depend on g and that t xz, t yz and sz depend on sx, sy and sz in the following way,

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(where dashes denote differentiation with respect to z) and

The boundary conditions (3) are satisfied by prescribing

and the boundary conditions (4) imply the following set of five conditions

In view of the special form of the expressions (7) and (8) it is still not in general possible to satisfy all the stress strain relations exactly. Since, however, Eqs. (7) and (8) are more general expressions than Eqs. (5) and (6) imply, it will be possible to satisfy the stress strain relations more nearly than is done in the two-dimensional theory of plane stress. The way in which this closer approximation is obtained is the following. Use is made of the basic minimum principle for the stresses to determine the functions of Eqs. (7)(10) by an application of the direct methods of the calculus of variations. The minimum principle can be stated in the following form: ''Among all possible states of stress which satisfy the equilibrium conditions in the interior of the body and the stress boundary conditions on the surface of the body, the correct state of stress makes the difference of the strain energy and of the work of the unprescribed boundary stresses a minimum." In the case that all boundary conditions are stress boundary conditions the strain energy itself is a minimum for the correct state of stress. The more general minimum principle permits, however, extension of the present results to problems in which displacement as well as stress boundary conditions are prescribed. On the basis of the equilibrium state of stress (7) and (8) a "best" approximation will here be obtained by determining the arbitrary functions sx, sy and t xy such that the strain energy of the body becomes as small as is possible with expressions for the stresses of the form Eqs. (7) and (8). For the function g(z) which determines the shape of the stress curves over the thickness 2h of the body the following assumption is made

This expression for g satisfies the surface conditions (11). It will be shown that application of the minimum energy principle leads to a system of simultaneous partial differential equations for a stress function I from which the average stresses Sx, Sy and T are derived and for the stress corrections sx, sy and t xy. The advantage which this system of equations possesses compared with the basic system of Eqs. (1) and (2) is that the number of independent variables is

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reduced from three to two and that it is of a form which permits an explicit solution of the boundary value problem for the cases that the solid is bounded by two parallel planes (problem of the infinite strip) or by two concentric circular cylinders (problem of the annulus). The latter case includes the problem of a circular hole in an infinite sheet which seems to be of the greatest practical interest among those which may be analyzed by the method of this paper.

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On the Theory of Bending of Elastic Plates [J. Math. & Phys. 23, 184191, 1944] Introduction In this paper there is established a system of differential equations of the sixth order for the linear problem of bending of thin plates. The form of these equations is such that results obtained by their application coincide with the results of the classical theory of thin plates except for narrow edge zones. On the basis of the present equations it is possible and necessary to satisfy three boundary conditions along the edges of a plate while, as is well known, the classical theory leads to two boundary conditions only. The history and significance of the problem has recently been discussed by J. J. Stoker. 1 Formulation of the Problem We consider an elastic body bounded by two parallel planes and by a cylindrical surface perpendicular to the two planes. The distance h between the two parallel planes is assumed to be small compared with the remaining linear dimensions of the body, which because of this order of magnitude relation may be called a "plate." A coordinate system (x, y, z) is chosen such that the faces of the plate are the planes z = ± h/2. The cylindrical boundary of the plate may be given by equations of the form x = x(s), y = y(s) where s stands for the circumferential arc length. It is assumed that the two faces of the plate are free of shear stress while the normal traction Vz is a given function of x and y. The resultant of the surface stresses Vz is balanced by stresses distributed over the cylindrical boundary of the plate. We begin by assuming that the bending stresses are distributed linearly over the thickness of the plate,

By means of the differential equations of equilibrium we obtain as expressions for the transverse shear stresses

The shear stress resultants V are given in terms of the stress couples M and H,

1Mathematical problems connected with the bending and buckling of elastic plates. Bulletin of the American M athematical Society, 48, pp. 247261 (1942).

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For the remaining normal stress component there results

if it is assumed that the vertical loads p are distributed over both faces of the plate as follows

Comparison of equations (5) and (4) gives

Substituting equations (3) in equation (6) we have

To obtain further relations for the three stress couples it is necessary to make use of the stress strain relations. This can be done in various ways. For the present purpose it is convenient to do this by means of Castigliano's Theorem of Least Work. For simplicity's sake it may be assumed that there are prescribed along the cylindrical portion of the boundary either the values of Vn, and or Wns, and or Wzs (in such a way that equations (1) and (2) remain satisfied) or vanishing of the work of the stresses Vn and or Wns and or Wzs. The Theorem of Least Work then states that among all statically correct states of stress the state of stress which also satisfies the stress strain relations and the displacement boundary conditions is characterized by the condition that the variation of the strain energy vanishes. Taking isotropic materials the strain energy is given by

Substituting the values of the stresses from equations (1), (2) and (4) the integration with respect to z may be carried out. With

2The difference between this condition and the condition that the surface loads are applied to one face of the plate only is not important for the present purposes.

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there results

The variation of 3 according to equation (10) is to be made equal to zero in such a way that the equilibrium equation (7) remains satisfied. Introducing according to the rules of the calculus of variations a Lagrangian multiplier w(x, y) we than have

Equation (11) is the basic relationship from which every other result will be deduced. It may be remarked that the results of the classical theory of plate bending are obtained if in equation (11) one neglects the strain energy of the transverse shear and normal stresses. The main point of the present work is recognition of the fact that application of the variational principle without this neglect leads to a system of equations for which three conditions can and have to be satisfied along the edges of the plate. This is due to the fact that retention of the transverse shear stress terms increases the order of the resultant system of differential equations. The Differential Equations and Boundary Conditions of the Theory Carrying out the variations in equation (11),

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Integrating by parts and rearranging,

In equation (13) the line integral is taken along the cylindrical boundary of the plate, s and n referring to the tangential and normal direction, respectively. It is seen that the variational equation (13) implies three differential equations and three boundary conditions. The first two differential equations may be solved for Mx and My. If this is done there follow, with the notation,

as differential equations,

and as boundary conditions

In addition to the differential equations (15) a fourth differential equation is the equilibrium condition (7). Before showing that the system of equations (15) and (7) is indeed of the sixth order, the significance of the three displacement boundary conditions may be explained. The second parts of equations (16) express the fact that the plate is supported in such a manner that the edge moments and forces can do no work. For this to be so it is necessary that the linear elements perpendicular to the middle

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surface of the plate do not change direction when the plate is stressed. Because of the effect of shear deformation this is not equivalent to the condition of no change of slope of the middle surface of the plate. Equation (16a) states what the change of slope of middle surface has to be in order that Mn may do no work. Equation (16b) states that when w = 0 along the boundary the distribution of edge twisting moments has to be such that

.

Transformation of the Differential Equations (7) and (15) Differentiating equation (15a) twice with respect to x, equation (15b) twice with respect to y and equation (15c) twice with respect to x and y there follows by addition, because of equation (7),

where

Observing equation (6) we obtain as differential equation for the deflection w,

Clearly, the second term on the right of equation (17) is insignificant unless p changes appreciably within distances of the order of the plate thickness. As the deflection w satisfies a fourth order equation as in the classical theory the additional two orders must be contained in equations (15). Adding equations (15a) and (15b) there follows, in view of equation (6),

With the help of equations (18) and (7) one may transform equations (15) into three equations each one of which contains only one of the three quantities Mx, My and H. Taking first equation (15a) in the form

and substituting w2H/wxwy from equation (7)

Taking now My from equation (18) and rearranging there results

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A corresponding equation (19b) is obtained for My. The equation involving H only follows from equation (15c) which may be written

Substituting Mx + My from equation (18) there follows

The three equations (19) together with equations (18), (17) and (7) may for the present be considered as the final form of the system of differential equations governing the problem. The "Edge Effect" Equations of Plate Theory For the analysis of the edge effect under consideration various terms in the differential equations are unessential. Omitting these terms there remains as the relevant system of equations

The edge effect terms in these equations are those having h2 as a factor. If they are omitted the customary equations of plate theory remain. The integration of the above system of equations is effected by first finding w from (20), by substituting the result in (22) to (24) whence Mx, My and H are obtained as combinations of particular solutions and complementary functions, Mx = Mx,hom + Mx,part , and so on. Equations (20) and (7) are two equations connecting the three complementary functions,

As an illustration consider a plate of the shape of a semi-infinite rectangular strip (0 d y d a, 0 d x). Assume that the edges y = 0, a are simply supported and that the edge x = 0 is acted upon by a distribution of bending and twisting moments and of

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vertical shear force in the following manner.

From equation (20) there is obtained as a suitable expression for w,

As we assume

we deduce from equations (22) to (24), except for negligibly small terms,

From equation (25) follows

From equation (26) follows

whence in view of equation (29)

Thus, there are three constants of integration by means of which the three boundary conditions (27) may be satisfied. Connection with the Theory of Moderately Thick Plates3 Considering in equations (19) the terms with h2 as small correction terms one might approximate these small terms by substituting in them the relations between Mx, My, H and w which hold if 3See A. E. H. Love, The Mathematical Theory of Elasticity, Cambridge, 1934, pp. 465487.

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the correction terms are omitted. If this is done there follows

Corresponding formulas may be found in Love's Treatise on page 473 when there is no surface load p. The only difference between Love's formulas and the present formulas is a factor (1 + Q/8) in the small fourth derivative terms.4 The important difference between equations (31) and (19) is that by the simplification of equations (19) to equations (31) the sixth order problem is reduced to a fourth order problem with resultant loss of the possibility to satisfy the three boundary conditions of the problem. It may be remarked that the approach to the theory of moderately thick plates and to more general problems which has been initiated by G. D. Birkhoff5 and which consists in series developments of the solutions in terms of a thickness parameter does not appear to be suitable for the analysis of the edge effect6 which is the main concern of the present paper. It may be of considerable interest to determine the inner reason for this difference between the variational approach as employed here and the approach by way of the thickness-parameter series solutions. In conclusion it may be stated that it is entirely possible by means of the variational method to obtain more accurate solutions than the one here obtained. One may for instance use instead of equations (1) more general equations of the form

where now Mx and Px and the corresponding quantities occurring in Vy and Wxy are to be determined by application of Castigliano's theorem. Instead of the two-term expressions of equation (32) one can, in principle, also use suitable n-term expressions. It is probable, however, that the calculations necessary for such extensions of the present results soon become quite involved. 4Note that in Love the plate thickness is 2h while here it is h. 5Circular Plates of Variable Thickness , Phil. M ag. (Ser. 6), 43, pp. 953962 (1922). 6J. N. Goodier, On the problems of the beam and the plate in the theory of elasticity. Trans. Roy. Soc., Canada, 33, Sect. 3 (1938).

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The Effect of Transverse-Shear Deformation on the Bending of Elastic Plates [J. Appl. Mech. 12, A69A77, 1945] Introduction It is well known that the classical theory of bending of thin elastic plates normal to their original plane permits the satisfaction of fewer boundary conditions along the edges of a plate than can in reality be prescribed. For instance, along a free edge, one has the three conditions of vanishing vertical force and of vanishing bending and twisting couple. Kirchhoff (1) has shown that the assumptions underlying the classical theory are responsible for a contraction of the three conditions mentioned into two conditions, which are vanishing bending couple, and vanishing of the sum of vertical force and edgewise rate of change of twisting couple. The meaning of this reduction in the number of boundary conditions has been explained by Thomson and Tait (2). The history of the problem has been discussed by Love (3) and recently again by Stoker (4). Because of the simplifying assumptions made in the development of the classical theory, it may lead to consequences such as the following. 1. There occur concentrated reactions at the corners of simply supported plates of polygonal shape. 2. Treatment of the St. Venant torsion problem of a rod with narrow rectangular cross section by means of plate theory, while leading to a fairly accurate torque-twist relation when the width of the plate is larger than, say, 10 times the thickness of the plate, does furnish insufficient information regarding the distribution of stress over the cross section. 3. Results for the magnitude of the stress concentration at the edge of holes in transversely bent plates become uncertain when the diameter of the hole is so small as to be of the order of magnitude of the plate thickness (5,6). In the present paper, a theory of bending of plates is developed which, to a considerable extent, is free of the limitations just described. In this theory, three boundary conditions can and must be satisfied along the edges of a plate. The theory is applied (a) to the torsion problem of the rod with rectangular cross-section where very good agreement is reached with the results of the exact theory; (b) to the stress-concentration problem of the plate with circular hole. Here considerable deviations from the results of classical plate theory are obtained as soon as the diameter of the hole is less than 3 times the thickness of the plate. The manner in which the equations of the theory are obtained consists in an application of Castigliano's theorem of least work, combined with the Lagrangian multiplier method of the calculus of variations. The physical basis of the present

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results forms the device of not discarding the energy of the transverse shear stresses, in contrast to what is done in the different derivations of classical plate theory. The results here obtained for flat plates may be extended so as to apply to curved shells. Derivation of Fundamental Equations As in the standard theory of thin plates, it is assumed that the bending stresses are distributed linearly over the thickness of the plate

In Equations [1] Mx and My are the bending couples, Hxy the twisting couple, h the thickness of the plate (which in what follows is assumed to be uniform), x, y are co-ordinates in the middle plane of the plate, and z the thickness co-ordinate (Figure 1).

Fig. 1. Orientation of stress resultants and stress couples, and stress variation over plate thickness.

From Equations [1], there are obtained, by means of the differential equations of equilibrium, expressions for the transverse shear stresses which satisfy the conditions that the faces of the plate are free from shear stress

The shear stress resultants Vz and Vy depend upon the stress couples:

Substituting Equations [2] in the remaining differential equation of equilibrium, there results for the transverse normal stress the expression

which satisfies the loading conditions

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The shear-stress resultants and the intensity of the vertical loading p are related by the equation

Combination of Equation [3] and Equation [6] results in one equation for the three quantities Mx, My, Hxy. To obtain further equations, use has to be made of the stress-strain relations. In view of the simplifying assumptions made in the writing of expressions for the stresses, this cannot be done in an exact manner. It is done in a rational manner in what follows by means of Castigliano's theorem of least work. This theorem states that, among all statically correct states of stress, the state of stress which also satisfies the stress-strain relations and the displacement boundary conditions is characterized by the condition that the variation of the following expression vanishes:

In Equation [7] the double integral extends over the cylindrical portion of the surface of the elastic body under consideration, un and us are displacement components parallel to the plane of the plate in normal and tangential direction, and w is the displacement component normal to the plane of the plate. Substituting for the stresses from Equations [1], [2], and [4], and carrying out the integration with respect to z where possible there follows, with

It is consistent with the assumption of linear bending stress distribution to assume that the displacements un and us vary linearly over the thickness of the plate and that w does not vary over the thickness of the plate

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The line integral in Equation [8] then becomes

where for a plate with built-in edges

For a plate with free edges

,

and

are unprescribed.

The variation of 3 according to Equation [8] is to be made equal to zero in such a way that the equilibrium Equation [6] remains satisfied. According to the rules of the calculus of variations, this is accomplished by multiplying Equation [6] by a Lagrange multiplier O and by combining Equations [8] and [6] in the following manner

Carrying out the variations

Integrating by parts in the last integral we have further

Substituting this in Equation [11], it is seen that, when GVn is arbitrary, then

As the same result can be obtained for any interior curve, it may be concluded that the Lagrange multiplier O is to be identified with the plate-deflection function w. The variations of Vx and Vy, according to Equation [3], depend upon the variations of Mx, My, and Hxy. Integrating in Equation [11] further by parts there

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follows

Equation [13] is the fundamental relationship of the present theory. From it follow the differential equations of the theory which hold in addition to the equilibrium equations and also the ''natural" boundary conditions of the problem, which under the assumptions made are either stress- or displacement-boundary conditions. The stress-boundary conditions, which make the variations in the line integral vanish, are

The displacement-boundary conditions are

There are to be prescribed either Equation [14a] or Equation [15a], either Equation [14b] or Equation [15b], and either Equation [14c] or Equation [15c]. The significance of Equations [14] is evident. The same is true for Equation [15c]. Equations [15a] and [15b] indicate that, due to the effect of shear deformation, normal and tangential line elements in the middle surface do not remain perpendicular to the linear element which was before deformation perpendicular to the middle surface. The double integral in Equation [12] is equivalent to three differential equations. They are, if the first two are solved for Mx and My

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In addition to the three equations just given for the six unknowns Mx, My, Hxy, Vx, Vy, w, there hold the three equilibrium Equations [3] and [6]. In its present form, this system of equations is not readily solved. It will next be shown that it can be transformed in such a manner that the way to its solution is clear. The first of the equations in its final form is the equilibrium Equation [6]

By means of Equation [I], Equations [16] are changed into

Equations [II] to [IV] will be used to determine Mx, My, Hxy when Vx, Vy, and w are known. From Equations [II] to [IV], there is next derived a system of two equations for Vx, Vy, and w. According to Equations [3], Vx and Vy are combinations of derivatives of Mx, My, and Hxy. In view of this, there follows from Equations [I] to [IV], by differentiation and combination:

In the foregoing equations

Equations [I], [V], and [VI] may be solved simultaneously for Vx, Vy, and w. Once this is done the stress couples Mx, My, and Hxy are obtained from Equations [II] to [IV] without further integrations. The equations of the standard theory of thin plates are obtained by disregarding on the left of Equations [II] to [VI] all but the first terms. The possibility of satisfying three instead of two boundary conditions in the present theory derives from the presence of the 'V terms in Equations [V] and [VI].

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Introduction of Stress Function Considering now Equations [I] to [VI] when p = 0, it is seen that Equation [I] can be satisfied by means of a stress function F

Substituting Equations [17] in Equations [V] and [VI] these may be written in the form

Equations [18] are Cauchy-Riemann equations for the functions D 'w and F (h2/10) 'F. Consequently, with two conjugate harmonic functions I and \, there is obtained

From

there follows

where \ 1 is the general solution of

Thus, the stress function F is a combination of a harmonic function and of a function defined by Equation [22]. And if the harmonic contribution to F is taken as the imaginary part of a complex function g(x+ iy) then D 'w is the corresponding real part of the same function g. From

it follows further that w itself is a biharmonic function, the same as in the classical theory of plates without surface loading. Once the solution of the homogeneous system of equations is found, it is only necessary to obtain a particular integral to take care of the load function p. The fact that in this formulation of the problem the only differential operators which occur are the invariant operators G and G 10/h2 indicates that explicit solutions of the theory may also be found in terms of plane polar and elliptical co-ordinates. Before doing this, there will first be discussed, as an example of the scope of the theory, a relatively simple example of a plate problem with rectangular boundary.

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Torsion of a Rectangular Plate A rectangular plate of length 2l and width a is considered. The two sides y = ± a/2 are free of stress while the two sections x = ±l are assumed to rotate without distortion and to be free of normal stress. The condition of distortionless rotation means that line elements perpendicular to each other before deformation remain so after deformation so that along the rotated end sections

. With this stipulation, the boundary conditions, Equations [14] and [15], become

To satisfy Equation [24a] let

As it is expected that the stresses will come to be independent of x and odd in y, the solution of Equation [V] is taken in the form

and as Vy vanishes all along the edges, the solution of Equation [VI] is taken as

From Equations [II] to [IV] follows then

As yet unsatisfied is the boundary condition, Equation [25b]. Substituting Equation [30] in Equation [25b]

The only nonvanishing stress couples and resultants are then

From Equations [32] and [33], there are obtained the values of the shear stresses by means of Equations [1] and [2], substituting the value of D and the

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relation E = 2(1 + v)G

According to customary thin plate theory, corresponding expressions for the stresses would be

except at the edges y = ± a/2 where the stresses Wxz may be assumed to become infinite in such a way as to be equivalent to concentrated forces. In the present theory, according to Equations [34] and [35], the stress Wxy is substantially constant over the width of the plate, except near the edges y = ±a/2 where it decreases to zero within a distance of the order of magnitude of the plate thickness h. The stress Wxz has its largest values when y = ±a/2 and drops down nearly to zero values a distance away from the edges y = ±a/2 which is again of the order of magnitude of the thickness h. The results of Equations [34] and [35] may be compared with the results of the St. Venant torsion theory, and the agreement is remarkably close even for plates so thick that the designation of "plate" is no longer appropriate. Taking first the case of a square cross section, there follows

In the exact theory, the numerical factors would be equal to each other and have the value 1.35 (ref. 7). It is of some interest to note that the average of the two values, 1.33, is remarkably close to the exact value. If one did not know the exact value, it would have suggested itself to consider this average as the true approximation rather than either one of the values in Equation [36]. For a plate twice as wide as thick, there results for the maximum shear stress

which differs by less than 2 per cent from the exact value 1.86. The limiting values of the stresses for very large values of a/h are Wxy(0, h/2) = 2GT(h/2) and Wxz(a/2, 0) = 1.58GT(h/2). An expression for the resultant torque is obtained from

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As

it follows that, as in the exact theory, the stresses Wxy and Wxz contribute in equal measure to the value of T. With Hxy from Equation [32]

The values of k 1, according to Equation [40] compare with the exact values of k 1 and the values of k 1 according to customary plate theory as follows: a/h k1 k 1,ex k 1,pl

1 0.139 0.1406 0.333

2 0.228 0.229 0.333

f 0.333 0.333 0.333

For values of a/h which are larger than three, k 1 of Equation [40] becomes (1/3)(1 .63h/a) which is a well known approximation formula. Polar-Co-Ordinate Solutions of Equations of the Theory Introduction of a stress function, according to Equations [17], and the subsequent integration of the system of equations in terms of the functions I, \, and \ 1, as defined by Equations [19] to [23], inclusive, indicates the way to obtain explicit solutions in polar co-ordinates r, T. The shear-stress resultants are now expressed in terms of the stress function F as follows

As before

and

where I + i\ = g(x + iy) and now

For D 'w may be written

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The conjugate of this is

Suitable solutions of Equation [22] are

In Equation [44], In and Kn are the modified Bessel functions (8). The functions In become rapidly large for large values of their arguments, while the functions Kn become rapidly small for large values of their argument. For small values of the argument, In stays finite and Kn becomes infinite. Equation [42] is integrated to

The starred constants depend upon the unstarred constants as follows

For each term in the trigonometric series, Equations [43], [44], and [45], there are six constants of integration, so that three boundary conditions can be satisfied along both edges of a circular-ring plate. In order to evaluate the foregoing solutions, it is necessary to obtain expressions for the stress couples Mr, MT, and HrT, which correspond to Equations [II] to [IV] for Mx, My, and Hxy. This may be done as follows: Observe that, in Equations [II] to [IV], the shear-stress resultants Vx and Vy occur in the same manner in which the displacement components u and v occur in the components of strain H x, H y, Jxy. This suggests that, in the equations for Mr, MT, and HrT, the quantities Vr and VT may occur in the same manner in which radial and circumferential displacement components occur in the expressions for H r, H T, and JrT. The correctness of this statement may be proved by deriving the equations of the theory directly, introducing curvilinear co-ordinates before applying Castigliano's theorem. This calculation is omitted in the present paper.

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Equations [II] to [IV] then have the following analogues in polar co-ordinates

Substituting the stress function F from Equations [41], there may be written, for the case of absent surface loads

The foregoing results permit a number of applications to problems involving circular boundaries. Two of these are made in what follows. Bending and Twisting of an Infinite Plate with Circular Hole The solution of these problems by means of classical thin plate theory has been given by Goodier (5). They have been investigated by Bickley (9) as problems of the theory of moderately thick plates. Experimental results (10) have confirmed the results of thin plate theory for a hole diameter plate thickness ratio of about seven. Taking first the case of plain bending, the boundary conditions in the present theory are

Instead of Equations [47] and [48], there may be written

The conditions at infinity suggest that the following expressions for w and F be taken

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From Equation [52] there follows for the shear-stress resultants with, according to (8)

For the stress couples, there results after some calculations

Substituting Equations [54], [56], and [58] in the boundary conditions, Equations [49], and [50], and with the notation

the following expressions for the constants in Equations [51] to [58] are obtained

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The stress-concentration factor of the problem is obtained from the value of the tangential edge stress couple

The value of this function is greatest for T = S/2

For large values of

the following asymptotic expressions for K2 and K0 may be used (8)

Hence

which is in agreement with the result obtained by means of standard thin plate theory (5,6). For small values of P the function K2 becomes infinite of a higher order than the function K0 and consequently

It is noteworthy that in the limit of vanishing hole diameter the value of the stress-concentration factor in bending becomes almost twice as large as in the limit of vanishing plate thickness, and moreover equal to the value of the stress-concentration factor in plane stress. For intermediate values of a/h the value of k B has been calculated by means of tables for the functions K2 and K0 (8). Figure 2 contains a graph of k B versus 2a/h. It is seen that even for holes 3 times as wide as the plate is thick the value of k B, according to the present calculations, is still more than 10 per cent greater than the value obtained by the application of standard plate theory. That taking into account the shear deformability of the plate leads to higher values of the stress-concentration factor than not taking into account this effect becomes physically evident when it is recognized that the assumed loading condition of the plate would lead to independent states of plane stress in every layer of the plate for an ideal orthotropic material offering no resistance to the transverse shear stresses Wrz, WTz, that is, for a material for which Grz = GTz = 0. In contrast to this the results of the customary theory may be thought of as exact results for a material for which Grz = GTz = f.

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Relatively simple expressions are obtained for the shear-stress resultants Vr and VT, by substituting the constants B2 and D2 in Equations [54] and [55]

For large values of a/h, for which the first terms of the asymptotic expressions of K2 and K0 may be used, this reduces to

Letting r = a + nh and consequently , it follows from the asymptotic formulas for Vr and VT that, for very thin plates and for a distance of the order of magnitude of the plate thickness away from the edge of the hole, the shear-stress resultants have the values

These expressions coincide with the expressions which according to the standard theory of thin plates hold throughout the plate (5, 6).

Comparison of Equations [71] and [73a] shows that the resultant Vr increases from its true edge value zero very rapidly to the edge value theory.

of thin plate

Also noteworthy is the behavior of the function representing VT. From Equation [72] it follows

This shows that, in the present theory, the edge value of VT is of opposite sign from the value according to Equation [73b]. Moreover, for thin plates, VT(a, T), according to the present theory, is of an entirely different order of magnitude than according to the usual plate theory. For given plate thickness, its value no longer decreases

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with increasing hole diameter. Substituting

there results for the maximum transverse edge shear stress

Thus, with transverse-shear deformation taken into account, there are portions of the plate where the transverse shear stress is of the same order of magnitude as the primary bending stress V0, no matter how thin the plate may be. From Equation [70] there follows for the variation of edge shear with diameter-thickness ratio

which compares with the constant value 4M0/(3 + Q)a, according to the standard theory. Figure 2 contains a graph of this function. Plate Subject to Pure Twist The results for this case may be obtained from the preceding results by superposition of a state of plain positive bending about the y-axis, as given, and a state of plain negative bending about the x-axis. Hence, for

Fig. 2. Stress-concentration factors and edge shear-stress resultant versus ratio of hole diameter to plate thickness.

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pure twist, every stress and displacement quantity gT is expressed as follows in terms of the corresponding quantity gB for plain bending about the y-axis:

From Equation [66] there follows

Values of k T as function of the ratio 2a/h are plotted in Fig. 2. Limiting values of k T are

which agrees with the result of standard thin plate theory, and

which is the same value which occurs when adjacent layers of the plate can slide freely with respect to each other. It is evident that, for the twisted plate, the effect of transverse shear deformability is still greater than the same effect for the plate subject to plain bending. Inspection of Equations [69] and [70] reveals that the transverse-shear-stress resultants for the plate subject to pure twist have exactly twice the magnitude of the corresponding stress resultants for the plate subject to plain bending. Consequently, the value of the maximum transverse shear stress is now

where V0 now represents the undisturbed maximum shear stress parallel to the plane of the plate. Remarks on Further Stress-Concentration Problems It is apparent that, by an application of the general results of the present paper, still further stress-concentration problems may be solved, for which there will be significant deviations from the results obtained by means of the classical plate theory. Of these may be mentioned (a) the plate subject to uniform transverse shear at infinity and the couples necessary for equilibrium (5, 6); (b) the same problems for a plate with rigid or elastic circular inclusion (6); (c) the same problems for a plate with circular hole reinforced by an elastic ring; (d) the plate with elliptical hole (5, 6). For this the additional problem arises to calculate the solutions of the equation

in elliptical co-ordinates which take the place of the modified Bessel functions in the solution for the polar co-ordinate system; (e) a circular plate with a small hole at the center and a linear (hydrostatic) load distribution.

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Remarks on Accuracy of Solutions of Stress-Concentration Problems It is not possible to make with certainty statements regarding the accuracy of the numerical results obtained. The results are as accurate as it is possible under the assumed variation of stress over the thickness. For the problem of torsion of a rectangular plate, comparison with the known exact solution indicates a surprisingly high degree of accuracy of the result here obtained. For the problem of the plate with the circular hole, such a comparison is not possible as the exact solution of the three-dimensional problem is not known. A way to determine the accuracy of the present solutions would be the following: Instead of the linear bending-stress distribution take a more general expression for the bending stresses containing third powers of the thickness co-ordinate z. Determine the corresponding transverse shear and normal stresses and again apply Castigliano's theorem. If the more accurate results thus obtained are in good agreement with the results based upon the linear bending-stress distribution, those may be assumed to be final from a practical point of view. However, the author would like to state as his belief that the present results regarding stress-concentration factors and transverse-shear forces are such that a more accurate analysis would indicate changes in values which are no more than 20 per cent of the difference of the values obtained here and the values obtained from standard thin plate theory. Thus, if the standard plate theory gives a value of 1.5 and the present theory a value 2.0 then it is believed that the actual value lies in between 1.90 and 2.10. A version of the contents of the first third of the present paper, which differs formally from the present improved version, together with the discussion of some points not considered here, has been published elsewhere (11). Bibliography 1. ''Über das Gleichgewicht und die Bewegung einer elastischen Scheibe," by G. Kirchhoff, Journal für reine und angewandte Mathematik , vol. 40, 1850, pp. 5188. 2. "Treatise on Natural Philosophy," by W. Thomson and P. G. Tait, Oxford University Press, 1867. 3. "A Treatise on the Mathematical Theory of Elasticity," by A. E. H. Love, fourth edition, The Macmillan Company, New York, N. Y., 1927, pp. 2729. 4. "Mathematical Problems Connected With the Bending and Buckling of Plates," by J. J. Stoker, Bulletin, American Mathematical Society, vol. 48, 1942, pp. 247261. 5. "The Influence of Circular and Elliptical Holes on the Transverse Flexure of Elastic Plates," by J. N. Goodier, Philosophical Magazine, series 7, vol. 22, 1936, pp. 6980. 6. "The Influence of the Shape and Rigidity of an Elastic Inclusion on the Transverse Flexure of Thin Plates," by M. Goland, Trans. A.S.M.E., vol. 65, 1943, pp. A69 to A-75. 7. "Theory of Elasticity," by S. Timoshenko, McGraw-Hill Book Company, Inc., New York, N. Y., 1934, p. 248. 8. "A Treatise on Bessel Functions and Their Application to Physics," by A. Gray and G. B. Mathews, second edition, prepared by A. Gray and T. M. MacRobert, London, Eng., 1931. 9. "The Effect of a Hole in a Bent Plate," by W. G. Bickley, Philosophical Magazine, series 6, vol. 48, 1924, pp. 10141024. 10. "Stress Concentration Around an Open Circular Hole in a Plate Subjected to Bending Normal to the Plane of the Plate," by C. Dumont, National Advisory Committee for Aeronautics, Technical Note No. 740, December, 1939. 11. "On the Theory of Bending of Elastic Plates," by E. Reissner, Journal of Mathematics and Physics, vol. 23, 1944, pp. 184191.

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Pure Bending and Twisting of Thin Skewed Plates [Qu. Appl. Math. 10, 395397, 1953] 1 Introduction The following is concerned with the problems of pure bending and twisting in the theory of transverse bending of thin plates. Known results for rectangular plates of uniform thickness will be extended to skewed plates. It was shown by Kelvin and Tait that the problem of St. Venant torsion of a thin rectangular plate with edges parallel to axes x and y has the solution w = Txy where w is the deflection function of the plate and T the (constant) angle of twist per unit length. Within this theory, which neglects transverse shear deformation, the torque is applied to the plate by means of concentrated forces of suitable direction which act at the corners of the plate. We will show here that Kelvin's and Tait's solution is readily extended to skewed plates of uniform thickness. In so doing we obtain, in particular, the influence of the angle of skew on the torque-twist relation for the plate. We also obtain a solution for the problem of pure bending of a skewed uniform plate. We find that pure bending of the skewed plate is associated with a twisting deformation the relative magnitude of which depends on the angle of skew. 2 Formulation of the Problem Let x, y be mutually perpendicular directions in the undeflected middle surface of the plate. The differential equation for a uniform plate bent by edge forces and moments only is of the form

where w is the deflection of the plate. Stress couples Mx, My and Mxy, defined in the usual way, are

For what follows we have no need for the corresponding expressions for the transverse stress resultants. We consider plates bounded by two straight lines x = ±l and by two straight lines y = ±1/2c x tan /. The angle / is the angle of skew of the plate, 2l is the span of the plate and c is the chord of the plate. Expressions for bending moment Mn and twisting moment Mnt acting along the edges y = ±1/2c x tan / follow by means of the usual transformation formulas for

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plate bending couples,

In addition to this we need Kelvin's and Tait's result that there occur concentrated forces P at the corners of the plate given by

and that the effective transverse edge stress resultant Rn is of the form Vn + wMnt/ws. 3 Choice of Deflection Function We shall find that for the problems of twisting and bending which are here considered it is sufficiently general to assume a deflection function

where A, B and C are constants. We then have a uniform distribution of stress couples

vanishing transverse stress resultants Vx and Vy, and vanishing effective edge stress resultants Rn. 4 Twisting of Skewed Plates The following boundary conditions must be satisfied

In addition to this we have that the applied torque T is given by

Equations (7), (8) and (9) are three simultaneous equations for the determination of the three constants A, B and C. We obtain

and therewith

The meaning of this result becomes somewhat more transparent if we introduce a new chordwise coordinate K counted from the centerline y = x tan / of the plate, by setting

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We then have

We may define an effective angle of twist T per unit length by means of the expression

Combination of (13) and (14) gives the following result for this effective angle of twist

Equation (15) indicates that the skewed plate has a smaller torsional rigidity than the unskewed plate and in which way the torsional rigidity varies with the angle of skew /. 5 Pure Bending of Skewed Plate We denote the applied moment by M. The following boundary conditions must be satisfied

In addition to this we have the condition of vanishing torque, or of vanishing corner forces P, which becomes

From equations (16) to (18) we obtain the following expressions for the coefficients A, B and C in w

The deflection w is now

In terms of the chordwise variable K defined by (12) this becomes

We see from (21) that the skewed plate has a smaller bending stiffness than the unskewed plate and that moreover the applied bending moment produces also a torsional deformation.

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On Postbuckling Behavior and Imperfection Sensitivity of Thin Elastic Plates on a Non-Linear Elastic Foundation [Stud. Appl. Math. 49, 4557, 1970] 1 Introduction The considerations which follow are presented as a contribution to the understanding of the asymptotic theory of postbuckling behavior of imperfection sensitive nonlinear elastic structures, in particular for cases for which a multiple mode linear buckling analysis applies. Specifically, a study of Hutchinson's paper on imperfection sensitivity of externally pressurized spherical shells [1], treated within the scope of Koiter's general theory [2,3], suggested to the author that he should look for a qualitatively similar problem of a simpler nature than the spherical shell problem, and that he should endeavour to derive the relevant results by asymptotic expansion considerations applied to a differential equation formulation without making use of potential energy concepts. Our simplified version of the spherical shell problem concerns a uniform infinite elastic plate, on a non-linear elastic foundation, and in a state of uniform twodimensional hydrostatic compression prior to the onset of buckling in the event that the plate is without imperfection. It is found that in order that this plate problem be qualitatively similar to the spherical shell problem the non-linear foundation should be a quadratic foundation. It is further found that with such a quadratic foundation it is allowable to base the work on the simple Kirchhoff form of plate theory, with no account being taken of the non-linear interaction between stretching and bending of the middle plane of the plate. In addition to the plate with quadratic foundation support, we also consider the case of a plate on a cubic foundation. We find that the shift from quadratic to cubic foundation has two important consequences. The first of these is that now postbuckling behavior is effectively of the single-mode type. The second is that now it is generally necessary to base the analysis on the non-linear von Karman type plate equations in which interaction between midplane stretching and bending is taken into consideration. 2 The Linear Problem We are concerned with the differential equation

where D, N and k 1 are positive constants where

is a given function of x and y and w is to be determined, subject to suitable boundary or periodicity conditions.

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We consider for the problem of buckling, that is for

, solutions of the form

where A is an arbitrary constant and have then that N must have a value NB given by

We define the buckling load of linear theory NBL as the smallest value of NB as a function of a and b and we designate the corresponding values of a and b by aL and bL. We then have, from wNB/(a2 + b2) = 0

In regard to the problem of imperfection sensitivity, we consider the case that

is given by

and that w is again given by (2.2). We then have as relation between A and

which is meaningful provided N is sufficiently small compared to NB. Of importance for what follows in relation to the above are the usual superposition possibilities associated with linearity and also that for a plate of given finite dimensions the effect of boundary conditions becomes insignificant provided the foundation constant k 1 is large enough to cause buckling in ripples the wave lengths of which are small compared to the linear dimensions of the plate. 3 Non-Linear Buckling. Quadratic Foundation We consider the equation

where D, N and k 1 are as before and where k 2 and h are additional constants, k 2 having the same dimension as k 1 and h having the same dimension as w. We assume that k 2 is of the order of magnitude of k 1 and positive. The constant h may be taken to be the thickness of the plate. Having the results (2.2) to (2.4) of linear buckling theory, we wish to obtain information on the effect of the non-linear term in (3.1). The simplest way that this effect could manifest itself would be by making the arbitrary amplitude constant A of linear theory a function of NB and k 2. The actual situation comes out to be considerably more complicated.

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Equation (3.1) may be modified by introducing non-dimensional independent and dependent variables in the form

where L remains to be determined. Inspection of the resulting equation for g suggests setting

whereupon equation (3.1), with (2.4), becomes

Since we are concerned with values of N near to NBL, we rewrite (3.4) with a parameter O defined by

to read as

Inspection of equation (3.6) indicates that results of interest for small values of O should be obtainable from an expansion of the form

where g1, g2, etc. and their derivatives are O(1) (at most). Introduction of (3.7) into (3.6) leads to the sequence of equations

etc. We propose to show that the first two equations of this sequence directly lead to the desired results concerning initial postbuckling behavior. Evidently, equation (3.8) has solutions of the form

where the Ai are arbitrary and the D i and E i are required to satisfy the relation

A comparison of (3.10) and (3.11) with (2.2) to (2.4) indicates that breaking off the expansion of the solution of the non-linear problem at this stage has led to results equivalent to those obtained by direct consideration of the linear buckling problem. Introduction of equation (3.10) into (3.9) and observation of (3.11) gives next

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where

The decisive step in our procedure is now to stipulate that the right hand side of (3.12a) must in the end not contain any of the terms cos D i[ cos E iK with , or in other words that the solution g2 must be free of secular terms. To see the meaning of this statement we consider successively the effect of assuming a one-mode, a two-mode or a three-mode combination for g1. One-Mode Solution We have on the right of (3.12a)

The only possible way in which all additive terms cos D 1[ cos E 1K can be made to disappear is by setting A1 = 0. This means there is no one-mode solution of the linear problem which can be extended into the non-linear domain by our procedure. Two-Mode Solution We now have on the right of (3.12a)

where both A1 and A2 are different from zero and where

.

In trying to match terms coming from the non-linear portion in the above to the terms of the linear portion we find one possibility, upon setting

whereupon

so that

Furthermore, equation (3.14) may be written in the form

The conditions of no secular terms in g2 are then two in number, namely

giving

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Introduction of (3.20), (3.17), (3.7) and (3.5) into (3.2) leads to the conclusion that possible two-mode solutions of the nonlinear problem are such that

representing a severely limited subset of the class of solutions A1 cos D 1[ cos E 1K + A2 cos D 2[ cos E 2K of the corresponding linear problem. We note finally that associated with (3.21) is a formula for the dependence of N/NBL on the value wc for which w is numerically largest. We find from (3.21) that

Three-Mode Solution We now have

with A1A2A3z 0, to insure the three-mode property. In order to make possible the disappearance of secular terms in g2 we must now find terms among the non-linear portion of g which annul all three linear portions. The evident symmetry properties of the expression on the right of (3.23) suggest that we should try to balance the linear A3-term by all or part of the non-linear A1A2-term. In order that this be possible, we must equate D 3 to either D 1 + D 2 or D 1D 2 and at the same time equate E 3 to either E 1 + E 2 or E 1E 2. Considering that at the same time

we find that we must use once the plus sign and once the minus sign, i.e.

Introduction of D 1 and E 1 from (3.25) into (3.24) leads to the result that the possible solutions of (3.24) and (3.25) are a one-parameter set which may be written in the form

Introduction of (3.26) into the terms with A1A2 in (3.23) shows that one of the conditions for the elimination of secular terms in g2 comes out to be

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Having taken care of the linear A3-term in (3.23) it is next necessary to take care of the linear A2 and A1 terms. Inspection of the A1A3 and A2A3-terms in (3.23) together with the relations (3.26) shows that elimination of all cos D 2[ cos E 2K-terms and cos D 1[ cos E 1K-terms in g is accomplished upon setting

The system (3.27) and (3.28) has the solutions

which are to be substituted in

Equation (3.30) allows the derivation of a formula for N/NBL, analogous to (3.22), differing from (3.22) only insofar as the numerical factor is concerned. Four-or-More-Mode Solutions Evidently the most general solution g1 of the form (3.10), with (3.11), may be written as

It would seem to be of interest to establish the possible forms of the function A(T) such that combination of (3.31) with (3.9) does not result in the production of secular terms in g2. 4 Imperfection Sensitivity. Quadratic Foundation The differential equation for w is now

In addition to non-dimensional variables as in (3.2) we introduce a non-dimensional initial deflection

in the form

Equation (4.1) can now be written as

In attempting an asymptotic expansion of the form (3.7) for the solution of (4.3) an important element of the procedure comes out to be the necessity of associating the effect of

(rather than

with the determination of g2 rather than g1. This means that in undertaking the start of an asymptotic expansion we must stipulate that

) where

and

are O(1) (at most).

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Introduction of (4.4) and (3.7) into (4.3) leaves equation (3.8) for g1 as before while equation (3.9) for g2 is changed to

In solving (4.5) we distinguish between two types of contributions to the imperfection function . One contribution does not contain any of the terms which can occur in g1 and therefore the question of secular terms in g2 does not arise. The other contribution does contain terms also occurring in g1 and consequently secular terms must be eliminated. Presumably, it is this second contribution which is critical insofar as imperfection sensitivity is concerned and in what follows consideration is limited to this second contribution, in conformity with the previous work in this field. Writing then

and taking g1 as in (3.10) we now have as equation for g2

Having (4.7) the analysis from here on proceeds as for the non-linear buckling problem. Specifically we will have a two-mode solution where

and we will have a three-mode solution where

with the D i and E i as in (3.15,16) and (3.26), respectively. The nature of the further discussion of the problem of the imperfect plate is illustrated for the two-mode case with (4.8)

. We now have from the second relation in

and from the first relation in (4.8)

To see the physical meaning of this result we revert back from the dimensionless functions

and in accordance with (3.7) and (3.2)

We further set

and g to

and w by writing in accordance with (4.4) and (4.2)

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and we may then consider the variation of [ 1 and [ 2 as a function of N/NBL, for given We take first the case A2 = 0 and

.

. For this we find

We then take the case A1 = 2 and

. For this we find

From equations (4.15) we obtain the existence of a critical bifurcation-stress N* and of an associated dimensionless deflection two N versus [ 1-curves, that is by

, given by the intersection of the

It is apparent that the above results for the quadratic-foundation plate problem are analogous to Hutchinson's results for the externally pressurized spherical shell [1]. The analogy becomes even more evident if we write equations (4.8) and (4.9) in terms of the quantities [ i and

as defined in (4.14). Equations (4.8) then become

and equations (4.9) become

For a complete analogy with Hutchinson's equations we should on the right of (3.8c) and (4.9c) have

rather than

. This can be accomplished formally by

changing the definition of in equation (4.4) by writing instead of (4.4) the relation . Since, however, the entire analysis is based on using the first step in an expansion in powers of the quantity O, the results obtained in this manner would not seem to be more accurate than the results based on (4.8c) and (4.9c), in that region of N/NBL-values for which the analysis based on the first step of the expansion procedure is relevant. 5 Postbuckling Behavior and Imperfection Sensitivity of Plate on Cubic Foundation. Elementary Theory We now consider the equation

and in it set in extension of equations (3.2) and (4.2)

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This transforms (5.1) into the relation

We take L again as in (3.3) and in addition dispose of w0 in a manner consistent with what was done for the quadratic-case by setting

With O as in (3.5) we can now write equation (5.3) in the form

which differs from (4.3) in the exponent of the non-linear term only. Postbuckling Behavior. Setting now set

we attempt an expansion in powers of O analogous to equation (3.7). In order to obtain a system of successive equations as in (3.8) and (3.9) we must

This gives

We may now proceed in the same manner as in equations (3.10) to (3.12) for the quadratic case by considering successively one-mode, two-mode and three-mode combinations for g1. We limit ourselves here to the one-mode case because in contrast to what happens for the quadratic foundation, a one-mode solution is possible for the cubic foundation. One-Mode Solution The right-hand side of the equation

, as g is of the form

In this all additive terms cos D 1[ cos E 1K can now be made to disappear by setting

The relation between deflection w and load N is obtained upon introducing g|O1/2g1 and w0 from (5.4) into equation (5.2). We find

and from this

Imperfection Sensitivity In analogy to the step from (4.3) to (4.4) we now set

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giving as equation for g2

Restricting consideration again to the one-mode case, we set in (5.13)

and have then as condition for the absence of secular terms in (5.13) the relation

In view of (5.12), (5.14) and (5.2) this is associated with the representation

If we designate the coefficients on the right of (5.16) by [ 1 and imperfection

, respectively, we obtain from (5.15) as load deflection relation, as affected by the initial

From this follows for the critical load N* and the associated dimensionless deflection

,

We add the remark that a one-mode solution also applies for the case of a ''quadratic" foundation with k 2w2 replaced by k 2|w|w. 6 A Remark on Mixed Quadratic-Cubic Foundation Effects We set in the non-linear buckling problem, as governed by an equation of the form

w = (k 1/k 2)h g([ , K), as in (3.2), and O = 1 N/NBL, as in (3.5). This transforms equation (6.1) into

The conclusion which follows concerns itself with the effect of foundations for which

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Assuming, as in (3.7), that g = Og1 + O2g2 + . . ., we obtain from (6.2) a sequence of equations of which we list the first three,

It is apparent from (6.4) that for the mixed quadratic-cubic foundation, with the restriction (6.3), initial postbuckling behavior is effectively governed by the quadratic contribution to the foundation effect, the cubic contribution having an influence which is small of higher order. 7 Postbuckling Behavior of Plate on Cubic Foundation. Rational Theory In place of equation (5.1) we consider the von Karman finite deflection equations

We introduce dimensionless variables as in (5.2) and write further

Choosing

and setting

equations (7.1) and (7.2) may be written in the form

Expanding again in powers of O a sequence of systems of equations is obtained, of which the first two are

Further consideration will be limited to the case of a one-mode solution

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of the first equation in (7.8). From the second equation in (7.8) it follows then that

Introduction of (7.10) and (7.11) into (7.9a) gives as equation for g2

Avoidance of secular terms in g2 now requires that A1 be subject to the condition

or

which reduces to (5.9) when U = 0. Introduction of (7.14) into (7.10) and (7.3) gives further, in modification of (5.10),

and then

It is evident from (7.16) that consideration of the non-linear finite plate deflection terms results in an effect which counters all or part of the effect of the cubic foundation term in (7.1) and that this countering effect depends on the aspect ratio of the wave pattern through the factor , which varies between one half and one, and on the value of the parameter U. For a homogeneous plate with differing stretching and bending moduli Es and Eb we have from (7.5) with D = 1/12(1 Q 2)Ebh3 and C = 1/Esh,

Therewith equation (7.16) becomes

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showing imperfection sensitivity whenever k 3/k 1 > 4/3(1 Q 2)Es/Eb, and absence of such sensitivity when k 3/k 1 < 4/3(1 Q 2)Es/Eb. References 1. J. W. Hutchinson, Imperfection Sensitivity of Externally Pressurized Spherical Shells. Journal of Applied Mechanics 34, 4955, 1967. 2. W. T. Koiter, On the Stability of Elastic Equilibrium, Thesis Delft, 1233, 1945. 3. W. T. Koiter, General Equations of Elastic Stability for Thin Shells. Donnell Anniversary Volume, 187228, Houston, Texas, 1967.

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On the Analysis of First- and Second-Order Shear Deformation Effects for Isotropic Elastic Plates [J. Appl. Mech. 47, 959961, 1980] Introduction In what follows we consider once more briefly the problem of transverse shear deformations for isotropic plates, within the framework of the two-dimensional sixthorder theory as derived from three-dimensional theory by a variational method [2] or, alternately, by means of self-contained two-dimensional considerations [3]. Specifically, we are here concerned with the fact that it is possible to distinguish between first and second-order shear deformation effects, with the determination of the first-order effects depending on a rational analysis of edge-zone behavior and with the second-order effects requiring no such analysis [5]. As regards the nature of the two kinds of effects we note, in particular, that the second-order effect is a natural generalization of Timoshenko's analysis of shear deformation in beams while the first-order effect disappears in a specialization of the plate problem to the corresponding problem of the beam. As regards the objects of this Note, these are as follows. Recent considerations by Simmonds [4], including a description of results by Goldenveiser [1] concerning the asymptotic derivation of a fourth-order plate theory in which first-order shear correction terms are accounted for by a modification of the classical Kirchhoff boundary conditions, make it seem worthwhile to indicate that results of the same nature are in fact implied by the writer's sixth-order two-dimensional plate theory. * Our results on modified Kirchhoff boundary conditions in [3] were stated for the case of straight edges only. It seems desirable to present a derivation of the corresponding results for the case of curved edges inasmuch as edge curvature brings with it a significant supplementary term in the relevant formulas. Two-Dimensional Plate Equations in Polar Coordinate Form We depart from our earlier Cartesian-coordinate statement of sixth-order two-dimensional theory for plates which are two-dimensionally isotropic and homogeneous [3] and rewrite these (for the case of absent distributed surface loads) with reference to polar coordinates r, T in the form

*We note that we did not think of this possibility in the presentation of our earlier work [2], and that our later analysis by a direct two-dimensional approach [3] led us to this possibility without consciousness of its relation to Goldenveiser's results.

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In this we have

with D and B being transverse bending and shear deformation factors, and w and F being solutions of the differential equations

The factors D and B are, for the case of a plate which is also homogeneous in thickness direction

Therewith, for this case

and then

for a three-dimensionally isotropic material as considered in [2].

Asymptotic Analysis Given a circular ring plate with inner edge r = a we consider the system of stress boundary conditions,

and, alternately, the system of displacement boundary conditions,

The possibility of an asymptotic analysis is given upon making the fundamental assumption

*The restatement of this expression involves a useful transformation with the help of the differential equation (7b).

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and upon making use of the order-of-magnitude relations

with these depending on restrictive assumptions of the form

, etc.

It follows from (12) and (6) that now

and, if we designate X-contributions to Mrr, etc., by a superscript i, the boundary conditions (9) may be written in the form

for r = a, while the boundary conditions (10) may be written in the form

Having the systems (14) and (15) we now proceed to deduce from them a system of abbreviated relations, in such a way that terms of relative order 1/Oa are retained, while terms of relative order 1/(Oa)2 are being disregarded . To accomplish this reduction, it is necessary to stipulate at the outset a suitable order-ofmagnitude relation between the dependent variables X and F. Inspection of the system (14) indicates that a reduction of this system is accomplished upon stipulating that

Therewith equations (14a, b) become, except for terms of relative order 1/(aO)2,

with the relevant expressions for unchanged.

,

,

now involving w rather than X directly, as a consequence of equation (13), and with equation (14c) remaining

The corresponding order-of-magnitude stipulation regarding X and F for the system (15) is

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Therewith the F-term in (15c) is of the same order of magnitude as the X-term, with the F-contribution in (15b) now being of relative order 1/Oa. At the same time, because of (13), equations (15b, c) may be written in terms of w rather than in terms of X, as follows:

Having the systems (17a, b) and (14c), and (19a, b) and (15a), we now use these for the derivation of equivalent systems which are of such nature as to allow a sequential determination of w and F, with the Z-problem being the desired generalization of Kirchhoff's problem in which first-order transverse shear deformation terms are taken into account entirely by modification of Kirchhoff's boundary conditions. In order to derive from the given systems of three boundary conditions for w and F separate systems of two conditions for Z and one condition for F we make use of the differential equation ’2FO2F = 0 in the asymptotic form F,rrO2F = 0, from which it follows that F = g(T) e-O(ra) and therewith, except for terms of relative order 1/Oa,

The introduction of (20) into (17a, b) changes these relations into

for r = a. Equation (21b) may be rewritten in the form

A substitution of this in (21a) and (14c) then gives as modified Kirchhoff boundary conditions, involving w alone

for r = a.* It is evident from (23a, b) and (22), in conjunction with the differential equations (7a, b), that the asymptotic determination of w and F up to terms of relative order 1/Oa is now in fact of a sequential nature. The previous result [3] for the case of a straight boundary follow from (23a, b) by first setting (),T/a = (),2 and by then going to the limit a of, with the term 2/Oa in (23b) disappearing in this process. *We note that for the case of a three-dimensionally isotropic plate, which is the case considered by Goldenveiser [1, 4], we have in the foregoing corresponds to a factor 0.630 . . . in Goldenveiser's three-dimensional asymptotic analysis.

. The numerical factor

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The analogous reduction of the displacement boundary conditions (19a, b) comes out as follows. We first combine (20) and (19b) in the form

and then use this relation in equation (19a) so as to obtain as second displacement boundary condition for w alone, in complementation of (15a),

for r = a. Equations (25), (24), and (15a) reduce directly to the corresponding conditions in [3] for the case of a straight boundary. References 1 Goldenveiser, A. L., "The Principles of Reducing Three-Dimensional Problems of Elasticity to Two-Dimensional Problems of the Theory of Plates and Shells," Proceedings of the 11th International Congress on Applied Mechanics , 1966, pp. 306311. 2 Reissner, E., "The Effect of Transverse Shear Deformation on the Bending of Elastic Plates," J OURNAL OF APPLIED MECHANICS, Vol. 12, 1945, pp. A69A77. 3 Reissner, E., "On the Theory of Transverse Bending of Elastic Plates," International Journal of Solids and Structures, Vol. 12, 1976, pp. 545554. 4 Simmonds, J. G., "Recent Advances in Shell Theory," Proceedings of the 13th Annual Meeting Soc. Eng. Science, 1976, pp. 617626. 5 Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, 1959, pp. 98104.

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A Tenth-Order Theory of Stretching of Transversely Isotropic Sheets * [J. Appl. Math. & Phys. (ZAMP) 35, 883889, 1984] Introduction In what follows we return once more to a problem which concerned us a long time ago [1,2], having to do with the approximate determination of three-dimensional Poisson's ratio corrections to the two-dimensional generalized plane stress theory of stretching of sheets. Our reasons for returning to this problem are in part due to a belated recognition that a 1949 joint manuscript on this subject, which had been accepted for publication, subject to stylistic revisions, had in fact never been resubmitted with these revisions. Going beyond this, subsequent variational developments [4] and more effective insights concerning the nature of interior and edge-zone solution contributions in the two-dimensional analysis of transverse bending of plates [5] made it likely that a reconsideration of the problem at this time would result in an analysis both simpler and more insightful than our earlier work. We add to these reasons the observation that the first paper on the subject, which may be considered effectively as a preparatory effort to the second named author's early analysis of transverse bending of shear-deformable plates [3], appears in a now nearly unavailable publication [2]. The present paper limits itself to the formulation of the general problem, and to its reduction to a system of Laplace operator differential equations. A subsequent paper (by R.A.C.) will be a reconsideration of a specific half-plane boundary-value problem previously considered in [1], with results going somewhat beyond what had been obtained earlier. Derivation of System of Two-Dimensional Differential Equations We consider a uniform homogeneous transversely isotropic layer bounded by two parallel planes, z = ±c, and by cylindrical surfaces, gi(x, y) = 0, with there being two such surfaces for the problem of an infinite layer with a single hole. We assume that the boundary portions z = ±c are traction free and that the cylindrical boundary portions are acted upon by an equilibrium system of prescribed tractions, *With R. A. Clark.

,

,

, with

and

being even in z and

being odd in z.

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Differential equations and boundary conditions of the problem as stated are known to be the Euler equations of a variational equation GI = 0, where

with E, G, Ez, positive, with Q 2 < 1 and displacement variations [4].

< 1 Q to ensure strain energy positive-definiteness, and with independent interior and boundary stress and

We use this variational problem to derive a tenth-order two-dimensional system of sheet equations by stipulating approximate stress distributions

where

We note that the function Z has been chosen in such a way as to ensure satisfaction of the stipulated traction conditions for z = ±c and that, moreover, equations (2) and (3) are consistent with the homogeneous equilibrium equations for stress, provided the nine coefficient functions in (2) and (3) are subject to the two-dimensional differential equations

An introduction of (2) to (4) into (1), with the supplementary stipulation that the z-dependence of the traction functions (2) to (4), and with defining relations

and

is consistent with the stipulations in

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for weighted displacement averages Ux, Uy, Vx, Vy and W, gives, upon integration with respect to z

In this the constitutive coefficients C, B, A are given by

The variational equation GI = 0 now implies as Euler differential equations the five equilibrium equations (5) and (6), together with nine constitutive equations of the form,

and, as Euler boundary conditions,

We note that the above results, except for the introduction of specific displacement averages, are in essence equivalent to the results in [2], upon setting Ez = E, Q z = Q and 2(1 + Q)G = E, and equivalent to the results in [1], upon replacing Q 1/E = Q z/Ez in [1] by the constitutive parameter

in equation (1).

Reduction to a System of Laplace Operator Equations Then tenth-order system (5), (6), (10) to (13) may be uncoupled in terms of three functions :, \ and M so as to have one second-order equation for : and two fourth-order equations for \ and M, respectively, with all three equations having derivatives in Laplace operator form

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only. It is found that of these three functions M represents the interior solution contribution and : and \ edge-zone solution contributions to the complete solution of the given system of differential equations, in the range of parameter values for which it is appropriate to make this kind of distinction. The first step of the reduction is satisfaction of the three equilibrium conditions in (5) in terms of stress functions K, : and \, as follows:

The second step involves writing the constitutive equations (11) in the inverted form,

with a corresponding expression (17c) for Ryy, and then to use the equilibrium equations in (6) to deduce as expressions for Sx and Sy,

A comparison of (18a, b) with (16) then leads to the conclusion that \ and : depend on Vx, Vy and T in the form

The third step depends on the use of the constitutive equations in (12), in conjunction with equations (19b) and (16), to deduce a differential equation for :, of the form,

where, on the basis of (9), CRBs = (2c2/21)(E/G). The fourth step consists in deducing from the constitutive equations in (10), in conjunction with the stress-function relations in (15) and (16) through use of the conventional compatibility equation involving Ux,x, Uy,y and Ux,y + Uy,x , the further differential equation,

Equation (21) is evidently equivalent to a representation for K in the form

where

and where M is a solution of the biharmonic equation

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The fifth and final step is based on the relation,

which is a ready consequence of (12), (19a) and (16). To make use of (24), we note that equation (17a) and its permutation (17c), together with (19a), result in the relation

Introducing (25) as well as T from (16), Nxx + Nyy from (15) and K from (22) into equation (13), we have, after some rearrangement,

Eliminating W between equations (26) and (24) and taking (23) into account, we obtain as a second fourth-order differential equation, which, remarkably, involves \ only,

The coefficients Ai, in this are given by

Given the way in which the half thickness c enters into the coefficient functions in (20) and (27), it is evident that, as long as E, G and Ez are of the same order of magnitude, the functions : and \ represent edge-zone solution contributions provided that all characteristic in-plane linear dimensions which enter into the formulation of boundary-value problems are large compared to the length c. At the same time it is evident from (23) that under the same circumstances the function M represents the interior solution contribution. We complement our reduction of the differential equations of the problem by the observation that, while equation (16) directly expresses the transverse stress measures Sx, Sy and T in equation (3) in terms of : and \, equation (15) in conjunction with (22) directly expresses the in-plane stress measures Nxx, Nyy, Nxy in terms of M and \. Expressions for Rxx, Ryy and Rxy in terms of the stress functions are more complicated. They may be obtained by first using (19a) to write (17a,c) as

with Vx and Vy following from (12), (16), (26) and (22). The results are

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where

We note that the terms with ’2M in (31) include as special cases known results for the exact theory of plane stress for isotropic materials [6]. This means that the approximate results as obtained here include exact results of three-dimensional elasticity subject to the limitation that the form of the prescribed boundary conditions is such as to be compatible with the stipulation that : = 0 and \ = 0 throughout. References [1] R. A. Clark, On the theory of generalized plane stress. M. S. Thesis. Massachusetts Institute of Technology, 1946. [2] E. Reissner, On the calculation of three-dimensional corrections for the two-dimensional theory of plane stress . Proc. 15th Semi-Annual Eastern Photoelasticity Conference, 2331 (1942). [3] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69A77 (1945); 13, A252 (1946). [4] E. Reissner, On a variational theorem in elasticity. J. Math & Phys. 29, pp. 9095 (1950). [5] E. Reissner, On the analysis of first and second-order shear-deformation effects for isotropic elastic plates . J. Appl. Mech. 47, 959961 (1980). [6] S. Timoshenko and J. N. Goodier, Theory of elasticity, 2nd Ed., 241244 (1951).

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On Asymptotic Expansions for the Sixth-Order Linear Theory Problem of Transverse Bending of Orthotropic Elastic Plates [Computer Methods in Applied Mechanics and Engineering 85, 7588, 1991] 1 Introduction Given the existing interest, from a computational point of view, in the boundary layer aspects of the sixth-order theory of transverse bending of shear deformable elastic plates [13] we are concerned in what follows with asymptotic expansions, in extension of earlier results for the leading terms of such expansions, which were deduced in a less systematic manner for isotropic [4, 5] and orthotropic [6] plates. The starting point of our analysis is a system of three simultaneous differential equations for the deflection W of the plate in conjunction with two transverse shear stress resultants Qx and Qy, with this system having been established, for isotropic plates, in [7, 8], and for orthotropic plates in [9]. With our earlier work having been concerned with appropriate versions of the canonical stress boundary condition problem and the canonical displacement boundary condition problem we here consider a mixed problem which has the two canonical problems as limiting cases. As regards our method of analysis we note in particular our approach to the task of appropriate scaling of the equidimensionalized version of the boundary layer portions of our three dependent variables, for the purpose of obtaining a better understanding of the importance of distinguishing 'soft' and 'hard' support relative to the matter of a smooth transition from the sixthorder theory to the fourth-order Kirchhoff theory, in the limit of vanishing plate thickness. In this connection we also note our introduction of the concept of 'almost soft' support as a consequence of the present approach to this transition problem. For convenience sake our considerations are limited to the problem of a semi-infinite plate with a straight edge, with this limitation enabling us to dispense with road blocks in the path of clarity without affecting the significance of our results. 2 The Sixth-Order Boundary Value Problem With x and y coordinates in the plane of the undeflected plate, the differential equations for transverse shear stress resultants Q, stress couples M, rotational displacements ) and deflection W are, for a plate with linear orthotropic behavior,

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For present purposes it is assumed that the values of the coefficients D, A and B are independent of x and y and that the load intensity P is a given function of x, y. For a solution of (1)(4), in the domain fd x d 0, fd y df, we here stipulate a system of boundary conditions of the form

As regards the statement of conditions for y = ± f, it is sufficient in the present context to note the possibilities of decaying or periodic behavior. The barred quantities in (5) are given functions of y, subject to the requirement that the semi-infinite plate be in overall equilibrium. The quantity b is a characteristic length such that P ,y = O(P/b),

, etc., and the quantities C are weighting factors with constraints

When Cw = Cx = Cy = 0 the mixed conditions (5a, b, c) reduce to a system of stress conditions. When CQ = CM = CT = 0, the mixed conditions reduce to a system of displacement conditions. Of particular interest in what follows are the limiting cases CT = 0 and Cy = 0, with the former representing 'hard' support and the latter 'soft' support in the sense of the considerations in [13]. To facilitate our approach to the solutions of the boundary value problem (1)(6) we rewrite (3) and (4) in the form

and we introduce (9a, b, c) into (2) so as to have as two equations for the three dependent variables W, Qx, Qy:

In place of using (10) and (11) as they stand we use (1) to eliminate Qy,xy in (10) and Qx,xy in (11) so as to finally have

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Equations (12) and (13), in conjunction with (1), are our basic system of three simultaneous differential equations for W, Qx and Qy.1 The associated boundary conditions in terms of W, Qx and Qy follow upon introducing (8) and (9a, c) into (5a, b, c), and (9a, c) into (6). Remark For the case of isotropy with Dx = Dy = D, Dv = vD, Dt = 1/2(1 v)D, Bx = By = B and Ax = Ay = A, eqs. (12) and (13) reduce to the compact form

and these, with V written in place of Q, become eqs. (V) and (VI) in [7], upon setting

3 Equi-Dimensionalized Form of the Boundary Value Problem We retain the variables Qx and Qy and we introduce equi-dimensionalized variables w, I x, I y, mx, my, mt and p as follows:

We further write

with D, E and P being dimensionless quantities of order unity and with h as the thickness of the plate. Equations (12), (13) and (1) therewith take on the form

The introduction of (17)(19) into (8) and (9a, b, c) gives

1Equations (12) and (13) have previously been derived in [9]. However, in place of using them in conjunction with (1), they are there used in conjunction with a third equation of higher order which follows from (1) upon the introduction of the differentiated Q-terms from (12) and (13) into (1).

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while the use of (17) and (18) in (5a, b, c) changes these conditions into

with a corresponding change in (6). 4 Interior and Edge Zone Solution Contributions In order to effect a solution decomposition into interior and edge zone contributions we introduce three dimensionless independent variables [ , K, ], and a dimensionless small parameter H by writing

With this we assume that the solution of the system (20)(22) will be of the form

At the same time we consider the possibility that the load intensity function p may be of the form

in such a way that there are no changes of orders of magnitude in connection with differentiations with respect to [ , K and ]. The introduction of (27) and (28) into the system (20)(22) leaves as differential equations for the determination of interior and edge zone solution contributions two separate systems of the following form:

and

The introduction of (27) and (28) into (23a, b) and (24a, b, c) gives, in analogy to (27),

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where

and

Given (29)(39) the remaining task is to establish a rational procedure for the formulation of boundary conditions for interior and edge zone solution contributions, on the basis of the conditions in (25a, b, c) for [ = ] = 0 and the corresponding conditions for [ = ] = f. 5 Parametric Expansions for Interior and Edge Zone Solution Contributions An inspection of the system (29)(31) shows the evident possibility of parametric expansions of the form

with the leading terms in these expansions being equivalent to the consequences of the classical fourth-order theory. However, it turns out that it is necessary, in connection with the nature of the boundary conditions (25a, b, c), to use in place of (40) the more general expansions

with distinct recursion relations following from (29)(31) for terms involving even and odd powers of H . Corresponding expansions for the solutions of (32)(34) are of a somewhat less elementary nature and the rational resolution of the difficulty in connection with the negative powers of H represents an essential element of our analysis. Assuming, for simplicity's sake, that pe = 0 in what follows it is necessary to take account of two salient facts, as follows. 1. We do not at this stage know the orders of magnitude of the edge zone contributions relative to the interior contributions. 2. The orders of magnitudes of we,

and

to result in a sequence of second-order problems, which complement rationally the sequence of fourth-order problems for wi, The form of (32)(34) makes it appear that expansions in accordance with the stated two requirements should be

with the exponent n remaining at our disposal.

relative to each other must be such as and

.

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The introduction of (42a, b, c) into (32)(34), with pe = 0, indicates that in order to accomplish our objective it is essential that we first consider (33), written in the form

With

following from this sequentially we next obtain

After this it remains to obtain

sequentially, on the basis of writing the homogeneous equation (34) in the form

on the basis of (32), written in the form

We complement (43)(45) by a restatement of (38) and (39a, b, c) in the form

In conjunction with the above we have, upon introducing (41) into (36) and (37a, b, c),

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With this and with (27) we now rewrite the boundary conditions (25a, b, c) for x = 0 as conditions for [ = ] = 0, in the form

In order to utilize the above it is necessary to appropriately dispose of the edge zone solution contribution exponent n. An appropriate disposition will be one which is associated with the possibility of having self-consistent systems of conditions for the successive terms of the fourth-order interior expansions as well as for the successive terms of the second-order edge zone expansions. We expect that such a self-consistent system will be unique, but we will not, here, furnish a mathematical proof of the validity of this expectation. Restricting, for the present, attention to the question of the leading terms in the two types of expansion we see that, in general, the appropriate choice for n will be

With n = 1 the boundary conditions for the leading expansion terms become

To see that the system (52a, b, c) is not always self-consistent it suffices to consider the special case CT = CQ = 0. For this case we have that (52a, b, c) becomes a system of three conditions for the solution of the fourth-order interior problem, with no condition for the second-order edge zone problem. An inspection of (50c) suggests that when CT = 0 we should replace (51) by the stipulation

With this value of n and with CT = CQ = 0 the leading term version of (50a, b, c) reduces to the form

In connection with the special case n = 2 and CT = CQ = 0 we note the following generalization of this case. If we assume

with cT = O(1) and cQ = O(1) then it follows from (7) that

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With this we obtain, in place of the two conditions in (54a),

with (54c) in conjunction (54b) again being of a self-consistent nature. Remark When n = 1 we will have the order of magnitude relations

with these being consistent with a previously observed stress concentration result [6]. At the same time it is seen, on the basis of (42c), (46a, b) and (47a, b) that the edge zone contributions to displacements and bending stress couples come out to be of a smaller order of magnitude than the associated interior contributions. For the exceptional cases with n = 2, order of magnitude as

and

and

dominate

and

, respectively, in addition to

dominating

, while

.

6 Explicit Determination of the Leading Terms of the Edge Zone Solution Expansions We deduce from (43) and the prescribed solution behavior for ] = f that

with K0 and K1 being arbitrary functions. The introduction of (56) into (44) gives

The introduction of (57) into (45) gives

with this being consistent with the known result that we{ 0 for the case of isotropy, for which E vy = vE xx and 2E ty = (1 v)E xx. With (56)(58) we then have further that

and

and the introduction of (59) and (60) into (46a, b) and (47a, b, c) gives

with (59)(62) being of the essence in what follows.

and

come out to be of the same

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7 Kirchhoff and Non-Kirchhoff Interior Solution Results We depart from a restatement of the here relevant portions of (50a, b, c) with

, etc., as follows:

The results which are the principal purpose of our analysis are obtained by solving (63c) for

, in the form

and by using this relation to obtain from (63a, b) as a system of two boundary conditions for the interior solution contribution

Given that the nature of the stipulated parametric expansions implies that the following possibilities.

is of the same order of magnitude as

and

, we have to consider separately

It follows from (64) that for this case

with (65a, b) reducing to the well-known conditions

for the fourth-order theory of Kirchhoff. Given the designations in [13] of the boundary conditions with Cy = 0 as conditions of 'soft' support, it suggests itself to designate the boundary conditions with Cy = O(H ) as conditions of 'almost soft' support.

We now have from (64) that, necessarily,

with (65a, b) reducing to

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It follows from (71a) that in this range of values of the weighting factors Cy and CT the interior solution contribution of the sixth order theory is not in agreement with the Kirchhoff solution of the problem, unless CQ = O(H ).

We now have from (64)

and from (65a, b)

For this 'almost hard' case of support to be self-consistent we must have CQ = O(H ). If we stipulate that CQ = cQH then (74a) becomes, with Cw = 1,

with (74b) remaining unchanged. For (74ac, b) to determine an interior solution of the Kirchhoff type it is evidently necessary to have cQ = 0 and CM = O(H ), whereupon, in place of (74ac, b)

The contents of (75) in conjunction with (73) represent a generalization of the corresponding result in [4, 5], by way of the presence of the parameter cT in (73). 8 Explicit Inclusion of First-Order Transverse Shear Deformation Effects Given the fact that the differential equations for the determination of w0, Qx0, Qy0 and w1, Qx1, Qy1 are of the same form, it is possible to deduce, on the basis of (50a, b, c), a reduced system of boundary conditions for the direct determination of

with

We here limit ourselves to showing this for the case n = 1 for which we can write (50a, b, c) in the form

where, in accordance with (61) and (62), again

, etc. We limit ourselves further by requiring that our results be consistent with the conditions of Kirchhoff, by

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stipulating almost soft support conditions, now in the form

We then have, on the basis (77c), that

for [ = ] = 0, and therewith from (77a, b) as a system of generalized Kirchhoff conditions for the determination of the interior solution

Equations (80a, b) are a generalization of our results in [4, 5] which were obtained for the special case Cy = Cw = Cx = 0. References [1] D. N. Arnold and R. S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21 (1990) 281312. [2] I. Babuska * and S. T. Scapolla, Benchmark computation and performance evaluation for a rhombic plate bending problem, Internat. J. Numer. Methods Engrg. 28 (1989) 155179. [3] B. Häggblad and K.-J. Bathe, Specifications of boundary conditions for Reissner/Mindlin plate bending finite elements, Internat. J. Numer. Methods Engrg. 30 (1990) 9811011. [4] E. Reissner, On the theory of transverse bending of elastic plates, Internat. J. Solids and Structures 12 (1976) 545554. [5] E. Reissner, On the analysis of first and second-order shear deformation effects for isotropic elastic plates, J. Appl. Mech. 47 (1980) 19591961. [6] E. Reissner, Asymptotic considerations for transverse bending of orthotropic shear deformable plates, J. Appl. Math. & Phys. (ZAMP) 40 (1989) 543557. [7] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech. 12 (1945) A69A78, 13 (1946) A252. [8] E. Reissner, On the theory of bending of elastic plates, J. Math. & Phys. 23 (1944) 184191. [9] K. Girkmann, Flächentragwerke, 5th Edition (Springer, Wien, 1959) 583610.

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On Finite Twisting and Bending of Nonhomogeneous Anisotropic Elastic Plates [J. Appl. Mech. 59, 10361038, 1992] Introduction In what follows, we utilize an intrinsic form of the equations for small finite deflections of plates, based on the original Kirchhoff displacement version rather than on the von Karman deflection-stress function version, in order to obtain generalizations of known exact solutions for the problem of twisting and bending of rectangular plates. This problem was first stated and solved, on the basis of von Karman's equations, for isotropic transversely homogeneous plates of constant thickness by Reissner (1957). Subsequently, solutions were obtained for isotropic plates of variable thickness of the symmetrical double-wedge type by Bisplinghoff (1957) and Simmonds (1958), and of the symmetrical lenticular type, independently by Mansfield (1959) and Simmonds (1958). A solution for orthotropic plates of constant thickness has been given by Chen (1974). We now consider anisotropic transversely nonhomogeneous plates with constitutive coupling of stretching and transverse bending and show that for this class of problems, it is again possible to effect a reduction of the problem to a system of two simultaneous linear second-order ordinary differential equations, with the significant nonlinear effects coming from the coefficients as well as from the right-hand sides of these equations. In the present analysis, the boundary conditions at the loaded ends of the plate are, as before, prescribed in a global rather than in a local sense, so as to make possible a one-dimensional solution procedure. We complement our one-dimensional analysis by the statement of a two-dimensional problem with a suitable system of local boundary conditions, where it remains to establish that the one-dimensional result does in fact represent the interior portion of an asymptotic solution of the two-dimensional problem. Differential Equations and Boundary Conditions We begin with a statement of the equilibrium and strain displacement equations for small finite deflections in the form

In this, x and y are coordinates in the undeflected midplane of the plate; N and M are midsurface parallel stress resultants and couples; u, v, and w are base plane

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parallel and perpendicular displacement components; and N and H are bending and stretching strains, respectively. We here associate these equations with constitutive relations of the form

For what follows it is of importance to transform the above system to an intrinsic form by deducing from the strain displacement relations (3) and (4) the three compatibility equations

The differential equation system (1), (2), and (5) to (8) is to be solved for a rectangular region |x| d a, |y| d b, with the edges y = ±b being traction-free and with the edges x = ±a acted upon by twisting moments T and transverse bending moments M. In the statement of these boundary conditions, account must be taken of the nature of the effective transverse edge stress resultants

and of the transverse corner forces 2Mt,(±a, ±b), in accordance with Kirchhoff. The transverse corner forces are equivalent to base-plane perpendicular forces. The transverse resultants Qe are distinct from base-plane perpendicular resultants P which are

The conditions for the traction-free edges are

and the conditions for the loaded edges are here stipulated to consist of three integrated moment conditions

and of three integrated force conditions

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Remark The analysis which follows turns out to be not applicable to the more general problem for which

in place of the conditions with F = 0 and Ms = 0 in (12) and (13). The One-Dimensional Semi-Inverse Problem We restrict attention to plates for which the elements of the constitutive matrices are independent of x and we attempt a solution of the given boundary value problem with stresses and strains also independent of x. We then have, on the basis of (1) in conjunction with (11), and on the basis of (7),

with k and T as arbitrary constants. With (15), and with the omission of all x-derivative terms, Eqs. (2) and (8) take on the form

where the primes indicate differentiation with respect to y. With Py as in (10), and with (15) and (9b), the solution of (16) has to satisfy the boundary conditions

Of the six global traction conditions in (12) and (13), the four homogeneous conditions are readily shown to be automatically satisfied on the basis of the contents of (16) and (17). The two remaining conditions may be transformed in a similar way, so as to come out in the intrinsic form

In order to solve (16) and (17), for the purpose of determining M and T as functions of k and T, use has to be made of a semi-inverted form of the constitutive Eqs. (5) and (6), in conjunction with (15), as follows:

The last of these relations is needed only for the purpose of an eventual determination of the displacement components u and v in terms of the solution of the intrinsic problem.

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Explicit Form of the Semi-Inverse Problem without Constitutive Coupling With the stipulation of a vanishing C-matrix in (5) and (6), and with (15), we now have as constitutive relations

where Byx = Bxy, etc., and

where Dyx = Dxy, etc. The differential equations in (16) become two simultaneous equations for My and H x upon deducing from (23) that

and upon writing (24b) in the form

The elimination of H x then leaves as a fourth-order differential equation for My:

The introduction of (26) into (24a, c) gives an expression for Mx and Mt to be used in the determination of M(k, T) and T(k, T) on the basis of (18) and (19)

Remark The same as for the orthotropic case, with Bxt = Byt = 0 and Dxt = Dyt = 0, we have that (27) is an equation with constant coefficients for plates with constant values of the constitutive coefficients. The same for the isotropic case in Mansfield (1959) and Simmonds (1958), it is found that for lenticular cross-section plates with

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Eq. (27) has the particular solution

where now

For this case the solution of the homogeneous Eq. (27) is not needed, inasmuch as Myp by itself satisfies (17). A Local Boundary Value Problem The physical significance of the one-dimensional solution of the problem with the global boundary conditions (12) and (13) depends on its being an asymptotic interior solution portion, for sufficiently small values of b/a, of a suitably stated problem with all boundary conditions of the local kind. Inasmuch as this aspect of the problem has not been discussed in the previous literature, we here formulate a system of local boundary conditions for an eventual use in conjunction with an asymptotic analysis of the two-dimensional problem. We replace the global conditions on Nx and Nt in (12) and (13) by the two local conditions:

Given that the expressions for Nx and Nt in (15) imply that w = W(y) 1/2kx 2Txy, we replace the global conditions on Mx, Px, and Mt in (12) and (13) by two local conditions:

It remains to be established that the boundary value problem with (31) and (32) in place of (12) and (13) does in fact have a solution which asymptotically coincides with the solution of the one-dimensional problem, except in zones of order b adjacent to the edges x = ±a. References Bisplinghoff, R. L., 1957, ''The Finite Twisting and Bending of Heated Elastic Lifting Surfaces," D.Sc. Dissertation, Zurich Institute of Technology. Chen, C. H. S., 1974, "Finite Twisting and Bending of Thin Rectangular Orthotropic Plates," J OURNAL OF APPLIED MECHANICS, Vol. 41, pp. 315316. Mansfield, E. H., 1959, "The Large-Deflection Behavior of a Thin Strip of Lenticular Section," Quarterly Journal of Mechanics and Applied Mathematics, Vol. 12, pp. 421430. Reissner, E., 1957, "Finite Twisting and Bending of Thin Rectangular Plates," J OURNAL OF APPLIED MECHANICS, Vol. 24, pp. 391396. Simmonds, J. G., 1958, "Finite Bending and Twisting of Thin Wings," B.S. and M.S. Thesis, Massachusetts Institute of Technology, Cambridge, MA.

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A Note on the Shear Center Problem for Shear-Deformable Plates [Int. J. Solids Structures, 32, 679682, 1995] Introduction Recent results for the shear center problem in the framework of Kirchhoff plate theory have left open the extent to which the effect of transverse shear-deformability becomes significant with increasing thickness-width ratio and with decreasing transverse shearing stiffness. In the following this problem is considered for an orthotropic plate, using the principle of minimum complementary energy in conjunction with Saint-Venant type stress assumptions. In an earlier application of this approach to non-shear-deformable plates (Reissner, 1991), the numerical consequences were found to be quite close to corresponding results obtained by a more accurate and more complex analysis in which account was taken of anti-clastic curvature constraints by Reissner (1989) and Gu and Wan (1993). There is no reason to suppose that the same would not be true when transverse shear-deformability is taken into account. The present approximate analysis reduces the problem to an ordinary second order differential equation. It is found that this equation can be solved explicitly for plates with linear widthwise-thickness variation, with a resultant closed-form expression for the shear center coordinate in terms of an appropriate dimensionless parameter. Formulation Consider a rectangular cantilever plate of span L and width a, with edges at y = 0, a and x = 0, L. The edge x = 0 is clamped and the edges y = 0, a are traction free. The edge x = L is stipulated to deflect uniformly by an amount W, in conjunction with two conditions of absent bending moments and edgewise rotational displacements. The minimum complementary energy formulation for a shear-deformable plate is, for this problem, given in terms of stress couples Mx, My, Mt and stress resultants Qx, Qy by the variational equation

The complementary energy density V is for a linearly elastic orthotropic plate, to which attention is restricted in what follows, of the form

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where

for a transversely isotropic homogeneous plate of thickness h = h(y). Equation (1) is associated with constraint differential equations

and with constraint boundary conditions, which in this case are

Equations (1) and (5) will be used in conjunction with the Saint-Venant type assumptions

for an approximate determination of a force Q and a torque T,

and a shear center coordinate

Reduction The three relations in eqn (4), in conjunction with eqns (5) and (6), give the following as expressions for Qx, Mt, and Mx:

with the prime indicating differentiation with respect to y. The introduction of eqns (6) and (9) into eqns (2) and (1) leads to the one-dimensional variational equation

with constraint boundary conditions

and with

.

The Euler differential equations of (10) are

While it would be possible to reduce eqn (12) to one second order equation for Mt, it is preferable to proceed as follows. Introduction of relation in (12) into the second gives the following as an expression for Mt in terms of Qx:

from the first

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With eqn (13) the first relation in eqn (12), together with eqn (11), leaves the boundary value problem

It is allowable, for simplicity's sake, to set 3W/L3 = 1. Furthermore, except for terms of relative order h2/L2, eqn (14) may be replaced by

With this and upon observation of eqns (15) and (13), the expressions for Q and T become

Upon setting Bx = 0, the non-shear-deformational plate theory result

becomes an immediate consequence. A Closed-Form Solution It is possible to obtain an explicit solution in closed form for plates for which

In view of eqn (3), this includes the case of a homogeneous orthotropic plate of linearily varying thickness h = h0K. Upon setting

eqns (14) and (15) assume the equi-dimensional form

with dots indicating differentiation with respect to K. An inspection reveals that the differential equation in (19) is explicitly solvable in terms of suitable powers of K. Upon satisfaction of the two boundary conditions the solution comes out to be

and

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The introduction of eqns (17) and (21) into eqn (16) gives, after some transformations,

and therewith, in accordance with eqn (8),

An impression of the significance of the effect of transverse shear deformation may be gained by considering a homogeneous orthotropic plate for which

with G = Gt = E/2(1 + Q) for the case of isotropy. As H increases, the value of ys/a decreases, at first approaching the cross-sectional centroid value yc/a = 2/3 from above. For sufficiently large H values, ys/a becomes smaller than yc/a. For example, when H = 1 then ys/a = 0.59. In this connection it is worth noting that for a "plate," for which the cross-section is an equilateral triangle with and for which a plate theoretical analysis is clearly not rational, eqn (23) gives ys/a = 0.647 when G/Gt = 1 and H 2 = 4/15, in place of the correct value ys/a = 2/3. It seems reasonable to limit the applicability of eqn (23) by the stipulation that h0/a d 1/2. This does not preclude the possibility of significant numerical effects, for sufficiently large values of G/Gt. References Gu, C. H. and Wan, F. Y. M. (1993). Approximate solutions for the shear center of orthotropic plates. Arch. Appl. Mech. 63, 513521. Reissner, E. (1989). The center of shear as a problem of the theory of plates of variable thickness. Ing.-Arch. 59, 325332. Reissner, E. (1991). Approximate determinations of the center of shear by use of the Saint Venant solution for the flexure problem of plates of variable thickness. Arch. Appl. Mech. 61, 555566.

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SHELLS My first contribution to the literature dealing with shells was motivated by the need to explain the subject to students of Applied Mathematics. Knowing threedimensional elasticity and having been a student in Dirk Struik's course on Elementary Differential Geometry, I found a relatively straightforward derivation of a twodimensional linear shell theory, involving the Kirchhoff-Love hypothesis [21,28]. While the equilibrium equations of this theory were unambiguous, the same could not be said about the associated strain displacement relations. It was only later that I learned to use the principle of virtual work to obtain consistent strain displacement relations, with the Kirchhoff-Love condition as a consequence of constitutive stipulations [136]. Participation in an industry-sponsored research project dealing with pre-stressed spherical domes next led to an analysis of shallow spherical shells. I derived a system of differential equations, without realizing that these could have been obtained by transforming Marguerre's cartesian system to polar coordinate form [41]. I then applied these equations to obtain quantitative data for a variety of axi-symmetric loading and support conditions [46]. This was followed much later by a study of certain unsymmetrical load problems, this time in connection with some guided missile problems [120]. Returning to the period 19451950, two efforts stand out. The first was a two-dimensional theory for the behavior of sandwich-type shells [58]. It was clear that the relative core softness required that allowance be made for the effect of transverse shear deformability, the same as for the sixth-order generalization of Kirchhoff plate theory. I found that for shells with soft cores it was also necessary to consider the effect of transverse normal stress deformability, with this being analogous in a physical sense to the cross-sectional flattening effect in the bending of curved tubes. A much later return to the transverse normal stress deformation effect [208] was to show its significance relative to Naghdi's considerations of "Cosserat-type" shells. The second effort was an attempt to deal with nonlinear finite deflection problems of shells. It came to me that it should be possible to generalize the well known H. Reissner-Meissner linear-theory equations for the symmetrical bending of shells of revolution so as to have equations for finite deformations by considering meridional slices of the shell as plane elastica resting on an elastic circular ring foundation [56, 68]. I later extended this work by a consideration of transverse shear deformation [141] and, as part of a final comprehensive account, of transverse normal stress in conjunction with anti-symmetric transverse shear, with this resulting in a symmetric system of three simultaneous second order equations in place of the usual two-equation system [262].

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After the first two non-linear shell-of-revolution papers, I soon considered a variety of linear and non-linear problems concerned with shallow helicoidal shells [85, 90], buckling of hyperbolic-paraboloidal shells [91], vibrations of shallow shells [94, 95, 107] and corrugation effects for circular cylindrical shells [97]. A membrane-theoretical treatment, in Wilhelm Flügge's pioneering text, of the problem of non-axisymmetric edge loads acting on spherical shells led me next [102] to a boundary layer analysis of this problem on the basis of the shallow shell equations [41]. I established the extent to which the elementary membrane solution came out as an 'interior' solution, and what the form of the 'contracted' boundary conditions for this interior solution would be. I furthermore discussed how the interior solution could be either a membrane solution or an inextensional bending solution, depending on the nature of the prescribed edge conditions. The seeds of the analysis in [102] would prove to be useful, later on, in a number of other plate and shell studies. Further work from this time period concerned symmetrical shell of revolution problems with intriguing solution properties. Bob Clark and I considered the problem of a pressurized ellipsoidal shell [109] to establish the range of validity of a membrane solution due to H. Lorenz, and to obtain information on bending stiffness effects. An analysis of the effect of polar orthotrophy showed unexpectedly strong consequences [110]. A study of the finite-deflection equations of shallow isotropic shells [121] brought out interesting connections between co-existing linear bending and non-linear membrane boundary layers. A General Lecture for the 3rd National Congress of Applied Mechanics [115] presented an opportunity to summarize these results, as well as the results for some other problems, in particular for toroidal shells. A project of a different nature which had been on my mind for a while was to derive asymptotically a system of two-dimensional shell equations from three dimensional elasticity, for the case of a symmetrically deforming transversely isotropic circular cylindrical shell. The basic thought was to non-dimensionalize the equations of the three-dimensional theory for a shell of radius a and wall thickness h by the introduction of a third length , based on a knowledge that the two-dimensional theory to be obtained could be expected to involve this length b. An outline of this procedure led to Millard Johnson's 1957 Ph.D. dissertation, and subsequently to a joint publication [117]. It fell to others, in particular to E. L. Reiss, to complement our expansion involving the small parameter h/b by a second expansion in terms of the smaller parameter h/a, for a more rational consideration of the boundary conditions problem than we were able to entertain without this second expansion. In a later attempt to derive shell theory from three-dimensional elasticity I pursued the thought that since the two-dimensional theory was concerned with forces and moments, there might be an advantage to start out from a 3D theory involving moment stresses in addition to force stresses [181]. From an analytical point of view one of the attractions of this approach was that now a system of first order equilibrium differential equations became associated with a system of first order compatibility equations, rather than one of the second order as in the theory without moment stresses. I found that this approach allowed a particularly simple derivation of 2D equilibrium and compatibility equations, and that the absence of second derivatives in the 3D theory made it appealing to attempt an asymptotic

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derivation of 2D constitutive relations from a given 3D system by transforming the 3D relations into a set of integro-differential equations through elimination of derivatives with respect to the shell thickness coordinate. I believe that my attempt was, in essence, successful but it left a feeling that I might not, in fact, have "dotted all the i's" in deducing the consequences of my integro-differential equation system. I am hoping that some day someone else will go back to this work for further progress. I would like to mention some other concerns with problems of shells, alone or jointly with Jim Simmonds, Fred Wan, and W. T. Tsai. 1. Asymptotic expansions for the solutions of the two-dimensional equations for circular cylindrical shells involving 'long and short' characteristic lengths [148, 154]. 2. Interior discontinuity aspects for rotating shells of revolution [150]. 3. An 'elementary' version of finite-deformation shell theory [157]. 4. An inextensional dislocation solution for finite bending and twisting of conical ring sector shells [162]. 5. Inversion problems in connection with the form of 2D constitutive relations [155, 158]. 6. A 'complete' formulation of the linear version of the symmetric shell of revolution problem, including the possibility of constitutive coupling of bending and twisting [166, 182]. 7. Studies of the behavior of laminated anisotropic cylindrical shells [183, 185, 195, 205]. There were finally two particularly meaningful efforts. The first of these was the formulation of a completely self consistent two dimensional finite deformation shell theory, including transverse shear deformation and drilling moments [196, 232]. In this I was influenced by related earlier work of Simmonds and Danielson. The essence of my analysis was to start with a readily established system of vectorial equilibrium equations, with the subsequent use of virtual work for the establishment of a vectorial system of virtual strain displacement relations. It was the step from virtual to actual strain displacement relations, by way of the introduction of a special triad of unit vectors, where I needed to know what Simmonds and Danielson had done (although I had earlier independently resolved an analogous task for the symmetric shell of revolution problem). The second final effort was a serendipitous consequence of wanting to show to the students in my course on shells a simple example of the use of linear shallow shell theory for an explicit solution of a problem where the step from plate to shell would be associated with qualitative consequences the nature of which could not be foreseen intuitively. I found two such examples by considering the effect of a small circular hole on states of otherwise uniform transverse twisting or membrane shearing of a spherical cap. While for the plate the corresponding classical problems of bending and stretching are no more difficult than the problems of twisting and shearing, the same turned out not to be the case for these shell problems, due to the terms in the shell equations responsible for the coupling of tangential and transverse action. The two circular-hole problems were solved first [217, 218] and these solutions were then supplemented by corresponding solutions for a rigid insert [219, 228]. Remarkably, two of these four problems showed a very substantial effect

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of shell curvature while for the other two the effect was relatively minor. An asymptotic analysis involving a boundary layer adjacent to hole or insert revealed the following circumstances. While the solutions for twisting and shearing were, in the absence of the hole or insert, of the inextensional bending or membrane type, respectively, the effect of an insert or of a hole, in contrast, induced a membrane state or an inextensional bending state, respectively, just outside the boundary layer. This meant a conflict for the combinations hole and membrane shear, and of insert and transverse twisting, with no such conflicts for the other two problems. It became apparent that the substantial effects of shell curvature on stress concentrations were associated with the conflict situations and not with the other two. Of particular interest was the observation that the indicated conflicts resulted in interior solution portions with inextensional bending or membrane far-field behavior, and opposite near-field behavior, and with a transition zone in which the two types of behavior were of equal importance. Given the long standing notions concerning mutually exclusive inextensional bending and membrane states it was intriguing to have come upon a physically meaningful situation with no such exclusivity.

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On the Theory of Thin Elastic Shells [Contr. Appl. Mech., Reissner Anniv. Vol., 231247, J. W. Edwards, 1949] Introduction The present paper is concerned with the subject of rotationally symmetric deformations of thin elastic shells of revolution. First a self-contained formulation of the problem of finite symmetrical deflections of shells of revolution is given. From this the equations of the small-deflection (linearized) theory are obtained by specialization. An essential step in the treatment of the small-deflection problems is their reduction to two simultaneous secondorder differential equations. This reduction was first given by H. Reissner [11] for spherical shells of constant thickness. Subsequently E. Meissner [6] published the corresponding reduction for general shells of revolution. Here this reduction is carried out in a slightly modified manner, which is believed to possess certain advantages, which will be indicated. From the general equations of the small-deflection theory a simplified system of equations is obtained in a systematic manner which applies for shallow shells. It is shown that the solution of this system of equations can be expressed in terms of Bessel functions for the entire class of paraboloidal shells of constant thickness. This generalizes known results for the case of a shallow spherical shell [9] for which the meridian curve is equivalent to a second-degree parabola. It is also shown that the solution can be given in terms of elementary functions for a class of shallow shells with varying thickness, such that the problem of conical shells with linearly varying thickness is included as a special case. Finally, some observations are added on the subject of the asymptotic integration of the differential equations of shell theory, which was first shown in H. Reissner's work [11] to be an appropriate method for this class of problems [5] These observations concern the effect of significant changes of thickness and curvature of the shell over distances of the same order of magnitude as those associated with the edge effect in shell theory. Formulation of the Problem of Finite Symmetrical Deflections of Shells of Revolution Geometrical Relations (Figure 1) The equation of the middle surface of the shell is written in the parametric form

so that [ together with the polar angle T in the x, y-plane are the coordinates on the middle surface. The sloping angle M of the tangents to a meridian curve is given by

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Fig. 1. M iddle surface of shell, showing coordinates [, T on middle surface and unit vectors associated with middle surface.

From Eq. (2) follows that

where primes denote differentiation with respect to [ and where D is given in the form

Let i, j, and k be unit vectors in the x, y, and z-directions, respectively. The radial and circumferential unit vectors jr and jT, are then defined by

and tangential and normal unit vectors j[ and n by

The radius vector R to any point of the shell may now be written in the following form:

where ] represents the distance of the point from the middle surface.

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The quantities [ , T, ] define a system of orthogonal curvilinear coordinates in space. The linear element for this coordinate system is of the following form:

Finally it is noted that

where R[ and RT are the principal radii of curvature of the middle surface of the shell. Analysis of Strain (Figure 2) The quantities referring to the undeformed middle surface are indicated by a subscript o, and the equation of the deformed middle surface is written in the form

The quantities u and w are then the components of displacement in the radial and axial directions, respectively. Moreover

Fig. 2. Side view of element of shell in undeformed and in deformed state. Also shown are visible stress resultants and couples and load intensity components.

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where E is the angle enclosed by the tangents to the deformed and the undeformed meridian, at one and the same material point. The customary assumption is now made that the normal to the undeformed middle surface is deformed without extension into the normal to the deformed middle surface.* Then

and dS2 as given by Eq. (8) refer to the same material element in the body. Comparison of Eqs. (12) and (8) gives for the components of strain H [ and H T in the tangential and circumferential directions the following expressions:

In what follows attention is restricted to thin shells in the sense that the thickness h is small compared with the magnitudes of the radii of curvature R[ and RT as defined by Eq. (9). The terms with ] may then be neglected in the denominators of Eq. (13) and instead of Eq. (13) may be written,

where

Note that H [ and H T as given do not involve the axial displacement component w. This component is obtained from the relation zc = D sin M in the form

Finally a relevant compatibility relation is set down which follows without calculation from a comparison of Eqs. (15) and (16) in the form

The present modification of the customary procedure consists up to this point in using radial and axial displacement components rather than normal and tangential displacement components. This gives the possibility of obtaining the formulas of the finite-deflection theory with no more difficulty than the formulas of the linearized small-deflection theory. It also permits a simpler derivation of the compatibility equation than is otherwise the case. Definition of Stress Resultants and Couples (Figure 3) Since rotational symmetry has been assumed, the non-vanishing components of stress are the four components V[, *It is recalled that this means that deformations due to transverse shear stress and transverse normal stress are neglected compared with the deformations due to the remaining stresses, and that this way may be justified for thin shells by a study of the equations of the three-dimensional theory of elasticity.

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Fig. 3. Element of shell showing stress resultants and stress couples.

VT, V], W[]. The stress resultants and couples are defined as follows:

In writing Eqs. (21) and (22) terms of the order h/R, compared with unity, have again been neglected, that is, attention has again been restricted to thin shells. Resultants and couples may be combined to resultant and couple vectors as follows:

In addition to this a load intensity vector p is introduced in the form

For what follows it is convenient to write N[ and p in the alternate form

The radial (horizontal) stress resultant H and axial (vertical) stress resultant V are related to N[ and Q as follows:

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Differential Equations of Equilibrium Force and moment equilibrium conditions for elements of the shell are, in vector form,

With N T from Eq. (23), M[ and MT from Eq. (24), and N[ and p from Eq. (26), Eqs. (28) and (29) imply the three scalar equations

Stress Strain Relations As the effect of transverse shear and transverse normal stress on the deformations is neglected, the relevant stress strain relations for an isotropic medium are

In Eq. (32) H [ and H T are taken from Eq. (14) and the result is introduced into Eq. (21) and (22). This leads to the system

where

Eqs. (30), (31), (32), (34), and (35) represent seven equations for the seven quantities u, E , H, V, N T, M[, MT and thus form a complete system of equations for the problem. The subject of the finite deflection theory will not be pursued further here, but rather from now on attention will be restricted to the linearized theory. The Equations of the Small-Deflection Theory The small-deflection (linearized) theory follows from the foregoing by referring the differential equations of equilibrium (30) to (32) together with Eq. (27) to the undeformed shell and by omitting in the expressions for the strains as given by Eqs. (16), (17), all non-linear terms. The resultant system of equations was shown in Refs. 6 and 11 to be reducible to two simultaneous second order differential equations for E and RT0Q of a remarkably symmetrical appearance. In what follows this derivation is modified slightly by choosing as one of the two basic variables the quantity r0H rather than RT0Q. In so doing it is possible to pass directly from the equations of shell theory to the equations of stretching and bending of circular plates, while with E and Q as variables the equation for stretching of plates is certainly no immediate consequence of the shell equations. While this advantage of ready transition to a special case might be thought to be of no practical consequence, it will be shown that it does permit a ready discussion of the problem of the shallow shell, with results which go beyond those known heretofore.

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It may further be stated that with the selection of E and r0H as basic variables the problem of finite deflections may be reduced in a corresponding manner to two simultaneous second order equations. Discussion of this latter aspect of the problem is, however, left for a future occasion. In terms of E and r0H and in terms of the components pV and pH of external load intensity the following expressions obtain for all other quantities:

The two simultaneous equations for E and r0H are obtained by introducing M[ and MT from Eqs. (41) and (42) into the moment equilibrium equation (32) and by introducing H [M and H TM from Eq. (34), with N[ and NT from Eqs. (38) and (40), into the compatibility equation (20c). The results can be written as follows:

If this is desirable Eqs. (45) and (46) may be reduced to one differential equation of fourth order for either E or r0H. It was found by Meissner [6] that in some cases this fourth order equation may be factorized into two independent second order equations. This factorization is of advantage if the solutions of the uncoupled second order equations can be expressed in terms of tabulated functions. It seems to the writer that for the purpose of obtaining solutions in series or asymptotic form the coupled equations (45) and (46) are as convenient as the uncoupled equations. A case in point is the problem of the spherical shell of constant thickness where the uncoupled second order equations are hypergeometric equations. The relative order of magnitude of the various terms in (45) and (46) becomes more readily apparent after a transformation which eliminates the first derivative terms in Eqs. (45) and (46). To this end two new functions X([ ) and Y([ ) are introduced,

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The differential equations for X and Y may be written in the form

The quantities O, ), T, and < are given by

In Eqs. (50) and (51) the subscript r indicates the value of instance, for the case of the spherical shell of radius a for which

at a suitably chosen reference station. In general the quantity O2 is a large number, as, for .

Stresses and Deflections of Shallow Shells The equation of the middle surface of the shell is written in the form

where a is a reference length, gc([ ) is of order unity, and P a number small compared with unity. From Eqs. (50) to (55) are obtained

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A shallow shell is defined by the stipulation that for it terms of order P2 may be neglected compared with terms of order unity. If this is done Eqs. (48) and (49) assume the following form:

In what follows the question of finding particular integrals for Eqs. (62) and (63) will not be considered but rather it will be shown that Eqs. (62) and (63) with right sides equal to zero can be solved in terms of elementary or tabulated functions for certain classes of shells. Shallow Shells of Uniform Thickness When hc = 0 and F = G = 0 Eqs. (62) and (63) reduce to the following form:

Eqs. (64) and (65) can be written as one complex equation

Eq. (66) is solvable in terms of Bessel functions, whenever

that is, for the case of a parabolic shell of nth degree. When n = 1 this case is contained in more general results for conical shells [7]. When n = 2 the case is that of the shallow spherical shell [9]. No previous solutions are known to the writer for other values of n. The solution of Eqs. (66) and (67) can be written in the form

where Z is the symbol for the general cylinder function. The application of this solution to the treatment of specific problems will not be considered here. Among such specific problems may be mentioned the problem of the shell with clamped edge as treated earlier for the case n = 2, and for n < 0 the problem of the infinite shell with circular hole with uniform radial tension at infinity. A still simpler case arises when

as then the solution of Eq. (66) is composed of powers of [ . A Class of Shallow Shells of Variable Thickness For shallow shells with thickness variation variation

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Eqs. (62) and (63), with F = G = 0, become

It is apparent that Eqs. (71) and (72) become a system of equidimensional equations and therewith have solutions which are composed of powers of [ when

The case m = 1, which gives a conical shell with linearly varying thickness, is included in E. Meissner's result [7] for the conical shell of any opening angle. Note that m = 2 gives a shallow spherical shell with quadratically increasing thickness away from the apex. Evidently, applicability of this solution for m = 2, and, in fact for all values of m, is restricted to ring shells only, so that the apex is not part of the actual shell. A problem of practical interest here may be the problem of a ring shell rotating about its axis with thickness being largest at the inner edge, which is the case when the exponent m is negative. Note on Asymptotic Solutions Reverting now to the general differential equations (48) and (49) it can be seen that in general the quantity O2 is a number large compared with unity, while the functions T,