Applied Mechanics and Mathematics

(1)

(2)

title: Selected Works in Applied Mechanics and Mathematics

author: Reissner, Eric.

publisher: Jones & Bartlett Publishers, Inc.

isbn10 | asin: 0867209682

print isbn13: 9780867209686

ebook isbn13: 9780585363530

language: English

subject Mechanics, Applied--Mathematics.

publication date: 1996

lcc: TA350.R45 1996eb

ddc: 624.1/7

(3)

Selected Works in Applied Mechanics and Mathematics

(4)

Editorial, Sales, and Customer Service Offices Jones and Bartlett Publishers

40 Tall Pine Drive Sudbury, MA 01776

Jones and Bartlett Publishers International 7 Melrose Terrace

London W6 7RL England

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.

Manufacturing Buyer Dana L. Cerrito

Typesetting Doyle Graphics Limited

Cover Design Hannus Designs

Printing and Binding Hamilton Printing Company

Library of Congress Cataloging-in-Publication Data (Not available at press time.)

Printed in the United States of America 99 98 97 96 95 10 9 8 7 6 5 4 3 2 1

(5)

CREDITS

The following publishers graciously gave permission to reprint the articles listed.

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

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.

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.

(6)

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.

(10)

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

(11)

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

(12)

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.

(13)

(14)

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

(15)

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 moment-deformational 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 pbuckling deformation terms in the 1904 paper. I re-examined 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].

(16)

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 angegeben1). 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. Metzner3). 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. Schnadel4).

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.

(17)

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. y-Richtung mit u bzw. v, die Winkeländerung mitJ bezeichnet, folgendermaßen:

(18)

Querkontraktionsver-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 2O 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)

(19)

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

(20)

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

(21)

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.

(22)

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

(23)

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

(24)

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 vx and vy are Poisson ratios, the virtual work per unit of sheet area, t(VxGHx + VyGHy + 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.

(25)

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.

(26)

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}

(27)

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 + x0), t = t0(x + x0)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 + x0₎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

(28)

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.

(29)

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

(30)

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

(31)

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_{/dx2-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

(32)

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, E., Die Formeln zur Berechnung der ''vollmitragenden Breite" dünner Gurt und Rippenplatten , Der Stahlbau, 9, 7378, 1936. 2Cox, H. L., Smith, H. E., and Conway, C. G., Diffusion of Concentrated Loads into Monocoque Structures, R. and M. No. 1780, 1937.

(33)

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

6Ebner, 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, G., Die Spannungsverteilung in den Flanschen dünnwandiger Kastenträger, Jahrb. d. Schiffbaut, Ges. 27, 207291, 1926. 20Sibert, H., Effect of Shear Lag upon Wing Strength , Journal of the Aeronautical Sciences, 6, 418, 1939.

21von Kármán, Th., Die mittragende Breite, August-Foeppl Festschrift, Berlin, pp. 114127, 1924. 22Winny, H. F., The Distribution of Stress in Monocoque Wings , R. and M. No. 1756, 1937.

(34)

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.

(35)

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).

(36)

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 Hx 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

(37)

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 x1 and x2 denoting the ends of the interval of integration,

(38)

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

(39)

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

(40)

Fig. 2.

Diagrammatic sketches of beams analyzed as examples of application of the theory. write the moment distribution in the form

(41)

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

(42)

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).

(43)

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