Stresses in Shells. Wilhelm Fliigge. Dr. lng. Professor of Engineering Mechanics Stanford University. With 244 Figures

Transcription

1 Stresses in Shells By Wilhelm Fliigge Dr. lng. Professor of Engineering Mechanics Stanford University With 244 Figures Springer-Verlag Berlin Heidelberg GmbH

2 ISBN DOI / ISBN (ebook) AlIe Rechte, insbesondere das der "Obersetzung in fremde Sprachen, vorbehalten Ohne ausdriickliche Genehmigung des Verlages 1st es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) zu verviewut1gen All rights reserved_ This bock, or parts thereof, may not be reproduced in any form without written permission of the publisher by Springer-Verlag Berlin Heidelberg 1960 Originally published by Springer-Verlag OHG_, BerliniGottingenIHeidelberg 1960 Softcover reprint of the hardcover 1st edition 1960

3 PREFACE There are many ways to write a book on shells. The author might, for example, devote his attention exclusively to a special type, such as shell roofs or pressure vessels, and consider all the minor details of stress calculations and even the design. On the other hand, he might stress the mathematical side of the subject to such an extent that he virtually writes a book on differential equations under the guise of the mechanical subject. The present hook has been kept away from these extremes. At first sight it may look to many people like a mathematics book, but it is hoped that the serious reader will soon see that it has been written by an engineer and for engineers. In a theoretical subject such as this one, it is, of course, not possible to get very far with the multiplication table and elementary trigonometry alone. The ma,thematical prerequisites vary widely in different parts of the book, depending on the subject. In some parts ordinary differential equations with constant coefficients are all that is needed. In other sections ordinary equations with variable coefficients, product solutions of partial djfferential equations, the theory of complex variables, or numerical analysis ",ill be en countered. However, the author wishes to assure his readers that nowhere in this book has an advanced mathematical tool been used just for the sake of displaying it. No matter which mathematical tool has been used, it had to be used to solve the problem at hand. The book may be divided into four parts. Chapter 1 contains preliminary matter, and a reader sufficient,ly familiar with the basic definitions may omit. this chapter until he finds that a real need for studying it arises. Chapters 2 to 4 contain the membraue theory, i. e. the theory of shells whose bending rigidity may be neglected. The spectacular simplification thus obtained makes it possible to examine a wide variety of shapes and support conditions. In particular, the stress problems of tanks and shell roofs have benefited from this fact, and many examples of these applications have been included. There is, of course, a heavy penalty to be paid for the simplification, and the short.comings of the membrane theory are pointed out at many places

4 iv PREFACE in these chapters. It has been considered important to show that the inadequacies of the membrane theory can be discovered by a critical inspection of the membrane Rolutions, without any need for first solvirig the bending problem-a task which often enough is out of reach of the practical engineer and even of the research worker. Chapters 5 and 6 are devoted to the bending theory of the two mort important types of shells, the circular cylinder and the general shell of revolution. It is in this field that most of the development of the last two decades has taken pjace. Since the solution of most problems of this category requires a rather elaboratc preparation, a careful choice of subject matter had to be made; otherwise the proper balance between the simple and the complicated would have been lost. In these two chapters an attempt has been made to cover a wide variety of quertions and to carry every theory to a definite end, viz. to a set of formulas giving all the stress resultants and the displacements in terms of the constants of integration and the coordinates. In many cases it has been possible to present these results in the form of a table. It has, however, mostly been left to the reader to adapt a solution t,o his particular case of boundary conditions. Chapter 7 is concerned with the stability of shells. From a research man's point of view this is a rather unrewarding subject. A long struggle through the mechanic's and mathematics of a problem and a tedious numerical evaluation ultimately yield a curve or only a single numerical factor in a simple formula. And, after all, there is only a rather 100Re correlation between the actual collapse of a shell and the buckling load obtained from a linear theory. While in some cases a large-deformation analysis has thrown light into a dark corner of our understanding, the numerical labor involved is so prohibitive that the designer cannot expect too much help from this side. In this book a choice of stability problems has been made which is considered representative of the present state of knowledge. Some material contained in this book has been used in courses on shell theory and on shell design, which the author has been giving for many years at Stanford University. However, much of what is found in this book goes beyond the possibilities even of an elaborate university course. It has been wtitten essentially for graduate engineers. Among these, the author has been thinking principally of two groups, namely, design engineers and stress analysts who need shell theory for their work, and research workers entering the field or working in it. For the first group, the book offers a body of well-established knowledge that will help them in most cases or may show them what can be expected of the services of a special consultant. The second group may use the book as a reference work and as a starting point

5 l'r.klfavj<.; v for their own endeavors. The almotate~ bibliography should be particularly helpful in locating additional information on a specific subject. The author owes many thanks to colleagues and former students. Professor S. TIMOSHENKO gave the first encouragement to undertake the big task, and he has followed its progress through the years with steady interest. Professor J. N. GOODIER read substantial parts of Chapters 1 and 2 at a very early stage and gave the author much good advice during his first steps of writing a book in English. After completion of the manuscript, Dr. H. L. ENGEL read it painstakingly and made many helpful suggestions. Parts of the manuscript have been read at different stages by Drs. R. E. PAULSEN, F. T. GEYLING, R. H. STIVERS, H. V. HAllNE, D. A. CONRAD, P. M. RIrLOG, F. A. LECKIE, and R. A. EISENTRAUT, and smaller parts by many more of the author's students. All of them deserve the author's sincere thanks for constructive suggestions and for checking formulas. Los Altos, Calif., February 1960 w. Flugge

6 CONTENTS Chapter 1 GENERAL PROPERTIES OF STRESS SYSTEMS IN SHELLS Definitions Definition of a Shell Stress Resultants Membrane Forces Membrane Forces in Arbitrary Direction~ Rectangular Coordinates MOHR'S Circle Oblique Coordinates and Skew Forces Transformation of Moments Chapter 2 DIRECT STRESSES IN SHELLS OF REVOLUTION General Differential Equations Geometrical Relations Equilibrium of the Shell Element Loads Having Axial Symmetry Differential Equations Solution for some Typical Cases Spherical Dome..., Boiler End Pointed Shells Toroidal Shell Tanks Conical Shell Shells of Constant Strength Drop-shaped Tank Dome of Constant Strength Loads without Axial Symmetry General Equations Spherical Shell General Solution Distributed Load Edge Load... 53

7 CONTENTS vii Concentrated Forces and Couples Conical Shell General Solution Homogeneous Solution Solution for Shells of Arbitrary Shape Solution by an Auxiliary Variable Solution by Numerical Integration Shell Formed as a One shl'et Hyperboloid Deformations Strains and Displacements Inextensional Deformation Differential Equation Finite Solution for the Spherical Shell Solution for Arbitrary Shape of the Meridian Inhomogeneous Solution General Solution Axially Symmetric Deformation Toroidal Shell Strain Energy Statically Indeterminate Structures 105 Chapter 3 DIRECT STRESSES IN CYLINDRICAL SHELLS Statically Determinate Problems General Theory Differential Equations General Solution Homogeneous Solution... U Tubes and Pipes Circular Cylinder U Elliptic Cylinder U Inclined Cylinder Fourier Solutions for the Circular Cylinder Barrel-Vaults Circula.r Cylinder Elliptic Cylinder Cycloidal Cylinder and Related Shapes Special Cylinders' Critical Remark~ DefoJ'Jllations General Theory Differential Equations Solution in General Terms

8 viii CONTENTS Application to Pipes and Barrel Vaults Circular Shell Barrel Vaults of Other than Circular Profile Fourier Series Solutions for the Circular Cylinder Statically Indeterminate Structures Polygonal Domes Regular Dome under Regular Load Regular Dome under Arbitrary Load Non-regular Domes Folded Structures Uniform Load Fourier Series Form of Solution Examples Limitations of the Theory Chapter 4 DIRECT STRESSES IN SHELLS OF ARBITRARY SHAPE Conditions of Equilibrium Elliptic Problems Paraboloid of Revolution, Triangular Shell Elliptic Paraboloid Solution by Relaxation Method Hyperbolic Problems Hyperbolic Paraboloid, Edges Parallel to Generators Hyperbolic Paraboloid, Edges Bisecting the Directions of the Generators Membrane Forces in Affine Shells General Theory Applications Vertical Stretching of a Shell of Revolution Horizontal Stretching of a Shell of Revolution The General Ellipsoid Polygonal Domes Cylindrical Shells Chapter 5 BENDING OF CIRCULAR CYLINDRICAL SHELLS Differential Equations Equilibrium Deformation

9 CONTENTS ix Exact Relations Approximate Relations Secondary Stresses in Membrane Theory Differential Equations for the Displacements Solution of the Inhomogeneous Problem Loads Applied to the Edges x = const General Solution Semi-infinite Cylinder Cooling Tower Simplified Theory Loads Applied to the Edges cj> = const Exact Solution... ' General Theory One Boundary Only Symmetric Stress System Barrel Vaults..., The Differential Equation and its Solution Isolated Boundary Symmetric Case Simplified Barrel-vault Theory Isolated Boundary Symmetric Case Examples Q Half-filled Pipe Barrel-vault Roof Cylindrical Tanks and Related Problems Differential Equation Solution for Constant Thickness Homogeneous Solution Water Tanks Cylinder Subjected to a Ring of Radial Forces Cylinder with many Rings Shell of Variable Thickness Anisotropic Shells Elastic Law Plywood Shell Double-wa,lled Shell Gridwork Shell Shell wi.th Rings and Stringers Differential Equations for the Shell with Ribs Folded Structures 307 b

10 x CONTENTS Chapter 6 BENDING STRESSES IN SHELLS OF REVOLUTION Differential Equations Conditions of Equilibrium Deformations Axially Symmetric Case Axially Symmetric Loads Spherical Shell Differential Equation~ Solution Using Hypergeometric Series Asymptotic Solution for Thin-walled Shells Simplified Asymptotic Solution Bending Stresses in the y'icinity of the Apex Concentrated Load at the Apex Surface Loads Shells Having a Meridian of Arbitrary Shape Elastic Law... ' Differential Equations Approximate Theory for Thin Shells Conical Shell Constant Wall Thickness Example: Sludge Digestion Tanle Wall Thickness Proportional to Distance from Apex Solution for the Higher Harmonics Spherical Shell Differential Equations Membrane Forces and Inextensional Bending Oscillatory Solutions Conical Shell Differential Equations Solution Chapter 7 BUCKLING OF SHELLS Introduction Adjacent Equilibrium Energy Method Cylindrical Shell... '" Differential Equations for Compression and Shear Basic Concepts Differential Equations

11 CONTENTS Xl Solution for Shells without Shear Load Two-way Compression Axial Compression Only External Pressure Only Solution for Shells with Shear Load Torsion of a Long Tube Shear and Axial Compression in a Cylinder of Finite Length Nonuniform Axial Compression The Beam-Column Problem The Axisymmetric Problem Imperfections of Shape Nonlinear Theory of Shell Buckling Spherical Shell Appendix FORCES AND DEFORMATIONS IN CIRCULAR RINGS Radial Load Tangential Load Load Normal to the Plane of the Ring External Moments, Turning about the Ring Axis BIBLIOGRAPHY INDEX....' TABLES Table 1. Semi-infinite Cylinder Table 2. Cylinder Loaded along a Generator Table 3. Barrel Vault Table 4. Barrel Vault, Isolated Boundary Table 5. Barrel Vault, Symmetric Case Table 6. Functions y,. and y~ Table 7. Coefficients for Spherical Shells Table 8a, b. Values of ~1 and ~2" 390 Tllble 9. Coefficients for Conical Shells