A theory of latticed plates and shells

A THEORY of LATTICED PLATES mt SHELLS

This page is intentionally left blank

Series on Advances in Mathematics for Applied Sciences - Vol. 5

AlHEORYtf LATTICED PLATES «NI SHELLS

G. I. Pshenichnov Computer Center Russian Academy of Sciences Russia

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Torteridge, London N20 8DH

Library of Congress Cataloging-in-Publication Data

Pshenichnov, G. I. A theory of latticed plates and shells /G.I. Pshenichnov. p. cm. - (Series on advances in mathematics for applied sciences ; vol. 5) Includes bibliographical references. ISBN 9810210493 1. Elastic plates and shells. I. Title II. Title: Latticed plates and shells. III. Series. QA935.P75 1993 624.1'776,0151-dc20 92-33782 CIP

Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyanymeans, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Printed in Singapore by JBW Printers & Binders Pte. Ltd.

PREFACE Latticed shells are sophisticated spatial constructions consisting of a large number of rods. These shells are the most progressive elements of engineering constructions and are widely used in various spheres of modern technology (aircraft, industrial engineering, shipbuilding, municipal and civil engineering). The book expounds the reticulated shell theory as definite continual systems. This approach to the investigation into latticed shells made it possible to effectively apply the deformed solid body mechanical methods and the equation apparatus of mathematical physics. Main accent (Chapters 1, 3, 4, 5) is placed on the single-layer reticulated shell theory. A single-layer reticulated shell is specific as the axes of all its rods form smooth families of curves on the medial surface. In general the book applies the classical theory to the deformation of the rods but in certain sections refined theories which consider transverse shear deformation, cross section warping, geometric and physical nonlinearity are employed. This refinement is of particular importance for solving nonlinear shell theory problems as it substantially simplifies the existing numerical algorithms. Chapter 2 elucidating a decomposition method [49] which is a new effective way of solving equations and general boundary value problems is of special significance. This method is used to obtain numerical and analytical solutions of certain mathematicalphysical problems. In other sections of the book it forms the basis of simple highly accurate analytical solutions of some statical, dynamic and stability problems of reticulated plates and shells with elastic contours. Chapter 6 discusses a multilayer system theory. This theory made it possible to undertake a qualitative analysis of reticulated shells including assurance of their geometrical stability and differentiation of a general stressed state into elementary. The book contains the solutions and results of many actual problems arising in the theory of reticulated shells and plates. Much attention is given to optimum design. For this in certain instances methods of the optimal control theory were applied. A detailed analysis of the results obtained is given. Graphs and tables convenient for practical use are also presented. As a continual system is taken for the reticulated shell's calculation model the results given in the book are of great importance for the research of anisotropic

vi

Preface

shells. The obtained constitutive equations can be used to compose similar equations for reinforced composite shells by adding terms referring to the binding material. The book contains a substantial part of the work [48] on the classical theory of the deformation of rods, results of researches undertaken on reticulated plates and shells using non-classical theories and a description and application of the decomposition method. G. I. Pshenichnov Moscow, 1992

CONTENTS Preface

v

Consistently Used Symbols

xi

1

1 1 2 3 7 8 8 9

Reticulated Shell Theory: Equations 1.1 Anisotropic Shell Theory: Basic Equations . . . 1.1.1 Static equations 1.1.2 Geometric equations . . 1.1.3 Constitutive equations for anisotropic shells 1.2 Constitutive Equations in the Reticulated Shell Theory . . . 1.2.1 Constitutive equations for the rods of reticulated shells . . . . 1.2.2 Constitutive equations for a calculation model . . 1.2.3 Assessment of the deformation components and forces in the rods using the forces and moments of the calculation model . 1.2.4 Constitutive equations for an oblique-angled system of coordinates 1.2.5 More complex version of the constitutive equations 1.2.6 Study of the geometrical stability of the reticulated shell's calculation model. Deformation energy 1.2.7 Boundary conditions 1.3 More Precise Constitutive Equations in the Reticulated Shell Theory 1.3.1 Allowance for transverse shear, cross-section warping and transverse deformation of rods . 1.3.2 Allowance for the rods' non-linear-elastic deformation

2 Decomposition Method 2.1 Solution of Equations and Boundary Value Problems by the Decomposition Method 2.1.1 Decomposition method 2.1.2 Merits of the method 2.2 Application of the Decomposition Method for Particular Problems 2.2.1 Analytical solutions . . 2.2.2 Numerical solutions vii

15 22 24 25 31 33 33 37 39 39 39 40 41 41 47

Contents

viii

3 Statics 3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Plane Problem 3.1.1 A plate with more than two families of rods 3.1.2 A plate with two families of rods Bending of Plates 3.2.1 Differential equation for bending . 3.2.2 A plate with a rhombic lattice 3.2.3 A plate with more than two families of rods 3.2.4 Plates with an elastic contour 3.2.5 Plates made from composite material .... 3.2.6 Plates made from nonlinear elastic material 3.2.7 Bending of plate subjected to large deflections Shallow Shells 3.3.1 Various differential equation systems for shallow shells subjected to medium bending 3.3.2 Shallow shells with constant lattice parameters 3.3.3 Shallow spherical shells Small Parameter Method in the Shallow Shell Theory 3.4.1 Constitutive equations 3.4.2 Differential equation system . . . 3.4.3 Small parameter method . 3.4.4 Numerical method for solving boundary iteration process problems 3.4.5 Shallow non-circular cylindrical shells Circular Cylindrical Shells 3.5.1 Differential equation system 3.5.2 Cylindrical shell with a rhombic lattice 3.5.3 Cylindrical shell with a square lattice 3.5.4 Calculation tables for reticulated cylindrical shells Optimum Design of a Shell with an Orthogonal Lattice 3.6.1 Statement of problem 3.6.2 Solution using the optimal control theory Shells of Rotation 3.7.1 Basic relationships and equations 3.7.2 Axisymmetrical deformation 3.7.3 Non-axisymmetrical deformation 3.7.4 Cylindrical shell made from composite material 3.7.5 Shell of rotation made from nonlinear elastic material Momentless Theory 3.8.1 Basic equations and relationships 3.8.2 Shallow shells 3.8.3 Shells of rotation

51 51 51 54 59 60 62 64 76 81 83 85 92 92 99 103 118 119 121 122 124 143 150 150 152 154 162 166 166 168 173 173 175 182 188 190 192 193 194 197

Contents 3.9

Simple Edge Effect in the Reticulated Shell Theory 3.9.1 Simple edge effect equation .... 3.9.2 Integration of simple edge effect equation 3.9.3 Simple edge effect during axisymmetrical stress state in shallow shells of rotation 3.10 A New Method for Solving Nonlinear Problems

4 Stability 4.1 Stability of Plates 4.1.1 Stability equation 4.1.2 Stability of plates hinged along the contour 4.1.3 Stability of plates with an elastic contour 4.2 Stability of Cylindrical Shells and Shells of Rotation 4.2.1 Closed circular cylindrical shells 4.2.2 Shallow non-circular cylindrical shells 4.2.3 Large deflections of shallow cylindrical shells . 4.2.4 Shells of rotation

ix 199 199 202 203 207 209 209 209 210 216 219 219 232 235 240

5

Vibration 245 5.1 Free and Parametric Vibrations of Plates 245 5.1.1 Free transverse vibrations of plates 245 5.1.2 Free transverse vibrations of a plate with an elastic contour 251 5.1.3 Parametric vibrations of plates 255 5.2 Free and Forced Vibrations of Shallow Shells 257 5.2.1 Free vibrations of shallow shells with three families of rods . . 257 5.2.2 Solution of shallow shell vibration problems by the small parameter method 259 5.2.3 Free vibrations of a shallow spherical shell with an elastic contour266 5.3 Free Vibrations of Closed Cylindrical Shells 269 5.3.1 Cylindrical shells with a rhombic lattice 269 5.3.2 Cylindrical shells with three families of rods 270 5.4 Vibrations of Shells of Rotation 274 5.4.1 Free and forced vibrations . . 274 5.4.2 Study of axisymmetrical free vibrations by the asymptotic method279

6

Multilayer s y s t e m s 6.1 Structural Coatings 6.1.1 Differential equations 6.1.2 Plane girder structural coating 6.2 Ribbed and Multilayer Reticulated Shells and Plates 6.2.1 Constitutive equations 6.2.2 Bending of ribbed plates 6.2.3 Shallow ribbed cylindrical shell

287 287 287 292 296 296 297 299

X

Bibliography

Contents 303

CONSISTENTLY USED SYMBOLS The following symbols are used to describe the lattice's structure and characterize its stressed state: a, 0'—c a, /? 0 and a', a', /?'—curvilinear orthogonal and oblique-angled coordinates of the middle surface points; n—the number of rods' families in a lattice; -area, main central inertia moments corresponding to bending in Fi,J\i,J2i,J$i—area, Ei,J\i-, Jn, Jai— planes normal and tangent to the middle surface and the rod's cross section torsional inertia moment; a,—distance between adjacent rods' axes; Pi—angle between axis a and the rod's axis (counted from axis a towards axis

0);

lf 1/2 pji — ^jil^i) (Jji/Fi) of inertia of the rod's cross section; P#-{Jfi/Fi) 'U ■ U(j == 1,2)—radii l,2H. fji — 1,2) — dimensionless radii of inertia of the rod's cross section; rr,,— di it (j = 1,2) c. n. vYoune's '„ Bt. G. E{,Gi Cji, Lri — — Young's roung s modulus and shear modulus of elasticity for the material; N',Q',S'.** — longitudinal and transverse (in the direction of the normal and N-,Q;,S; - ion tangent to the middle surface) forces in a rod; M",G',H' — bending (in the normal and tangent planes to the middle surface) M*,G-„H:-\ and torsional moments in a rod;

jv , rii = ri{

,o EiJuK E{F{ EiJzi EjJ , _ EiJu EiJu 0 . li , Ai = •. /,- = a, a, a, a. a. a. a-i _ EjJu EjJii GjJsi , _ EjFi EjFj GjJzi &i ii H EiJu ' E\Fi E\F\ E.F,' E\J\\ ErJn " I 1 Ipi Si pi, c,?i == COS cosyj, s, == sin ipi, 1

nn

GiJzi GiJzi , a, GjJsi _ GtJsi e,- = G\Jii G\Jz\

o, =

—geometrical and physical parameters of a lattice;

V,

:

c , dd c, A]da

s, Si dd_ •SI B d/3'

A,

_ Si d0 4Ada

c, d 3~\90' 0'

A, B—coefficients of the first quadratic form of the shell's middle surface. In the said symbols the bottom index i denotes values relating to the i-th family of rods. If this index is unity (i = 1) it can be neglected. xi

Chapter 1 RETICULATED SHELL THEORY: EQUATIONS In the first part of the book equations of one layer reticulated shell theory are obtained, their analysis is carried out and problem of statement of boundary conditions, which are necessary for formulation of boundary problems is examined on the basis of continuum design model. We take some continuous shell for design model of reticulated shell. Its constitutive equations will be obtained proceeding from satisfaction of the following conditions: a) the reticulated shell middle surfaces and those of its calculation model coincide; b) deformations of the reticulated shell's rods coincide with those of the calculation model; c) forces and moments in one and the same cross-section of the reticulated shell (after their averaging) and its calculation model are statically equivalent. Three groups of equations were used when considering the calculation model: static equations or kinetic equations in the forces and moments, geometric equations linking deformations with displacements and constitutive equations. The first two groups of equations coincide with their corresponding equations of the continuous shell theory. The latter group of equations, depending on the lattice structure and material, is more complicated and in a particular case is similar to equations used in the anisotropic shell theory. As this is so we present equations and relationships for anisotropic shells as they will be required later.

1.1

Anisotropic Shell Theory: Basic Equations

Let us introduce the middle surface of the shell to the curvilinear system of the orthogonal coordinates a, /? and use the following notations: A, B—coefficients of the first quadratic form of the middle surface; &i = lfR\, k2 = I/R2—curvatures of the middle surface normal cross sections drawn along the coordinates; ki2 = I/R12— twist of the coordinates. 1

2

Chapter 1. Reticulated Shell Theory:

Equations

Figure 1.1: Functions A, B, k\, k2, k^2 are connected by three Gaussian and Kodazzi differential relationships:

a

d (\dB\ B d_ d (\dA\ A ,_„, , , . d_ = AD AB(k2n-k,k2), L + da) dp ( s dp) ) ) da dBk12 d/U, dA dB_ B , dA dAh , dB i212 _ dAkt = 0, + k2 + kn da da dp d/3 da dp da dP dp dAk1212 8Bk22 dBk BA^.dB dB dA dA^.BB 12 dAk dBk = 0. + kl7 + + kl ~bl3 dpd/3^ ^fa fada = °~dl da~ dl da Let us assume that u, u, w are projections of the displacement vectors at a point on the middle surface towards the unit vectors of coordinates a, P and the normal to this surface (Fig. 1.1). Let e%, c2, w, K\, K2, T be deformation components of the shell's middle surface. We assume that during the deformation, extension and shears are small compared to unity.

1.1.1

Static equations

Positive directions of linear forces and moments are shown in Fig. 1.2 (here Mis = M2, = 0 unlike reticulated shells). In this figure it is assumed that the moment tends to turn clockwise looking from the positive side of its corresponding vector. Functions Nj, Sj, Qj, Mj, Hj (j = 1,2) must satisfy the following static equations [33, p. 38]: dB'Ni

dA'S2

dA

dB'

dB'

dA'

*da£ *+^ ^ *£* * **g*- *dB' ££-♦*[^-2{"-s')w-Awh0 +{h +

-

(the third equation is derived from the second by substituting a *-* /?, A «-» B, ki fc2,*i «-» K 2 ) . Now we shall consider three possible cases of the deformed states of a shell's middle surface. 1.1.2a. Strong bending. If the displacements and curvature changes are random the components of the shell's middle surface deformation are determined through the components of the displacement vector u, v, w according to the formulae [33, pp. 19, 23]: 2 2 £i 7?), £1 = e u + 0.5(e*, 0 . 5 ( e n + e2l2l2 ++ 7, ), 2 2 2 £2 = e22 + 0.5{e\ + e + 7 ), 0.5(e 7 ) 2 22 , 22 221 1 u> + e + e e + e u> = = e12 i i 7i72, n 21 22 12 221 2 2 e 12 J 2 + 7172,

K,

Kl Kl

=

-~

K2

=

T

=

*2 -K,

r =

u U ut kie22 kue21 kie En - kue21 ~A\En~da~

"-

-A\ ^

l1

A

de de u d9d-fi" aa2Q2 dA de dA de _L IP ftp (TP IP dede ™ " _L _L cC fA fA * dA E:u+E33 E» ""u ++_L £22 1 E22 + E33 AB dp E22 da da da, 'da~ 'da')''ABW ~fa fa)-ABdp'

1 de d dj2 de 7i 8B j. L ( El2 IP ™2 2 , IP dede ™21 , IP ~tA Ti 9B E33 33 33 En k2e E22 U ++ k2e k k2e 2 E22 AB da' -B\ W WB dp WdP »-^-> ^»-B{ "~dj ~dj)~ABda-' dP.Wj~ABd^' lX de de22 ddj2 ai dAdA de2l _L JP de l 2 9 A t U uU (tp ( IP Se™ JP 9 t 2222 ,, IP J? 9~ + e22sm'ip)

+ (el2 + e 21 )sin2.AB AU) dP {BV)

(1.5) (L5)

-2IBW -1

= -26 + (e 2y>. (e,122 + e 2 i)Icos cos2

J6KI + D26K2 + D^r,

Mxx M

= =

- ((£D> ,I iI K K 22 ++ DD1166TT ) , / cI , + £> D1212/c

M22

=

-(D 122K\/c, + D22K2 —(D\

+ D26DT). 26T).

(1.13)

If the material is orthotropic and the main directions of elasticity coincide with those of the coordinates, Eqs. (1.13) become simpler owing to the condition: Cie = C16 — CC2626=—-Die D\e=—DD262e=—0. 0.

(1.14)

For shells made from isotropic material, besides (1.14) the following conditions should be fulfilled: C n„ C

2

= C22 C22 = Cutv 2C«6/(1 - v) ») = Ehf(l Eh/(1 - vv%), — —C 12/v = 2C««/(1

Dnn11 = D22 = D D,u2/v /v = Dee/(l-v)— D

V

= Eh /\2(l-v Eh /\2(l-v ), 2), 3

32

(1.15)

where E and t; are respectively Young's modulus and Poisson's ratio for the material.

8

Chapter 1. Reticulated Shell Theory: Equations

1.2

C o n s t i t u t i v e E q u a t i o n s in t h e Shell T h e o r y

Reticulated

1.2.1

Constitutive equations for the rods of reticulated shells

1.2.1a. Deformation of a reticulated shell's rods. We assume that a rod's deformation is equal to that of the line coinciding with this rod's axis in the calculation model. We fix one of the families (the i-th family, 1 < i < n) of the shell's rods. The position of the axes of this family of rods on the shell's middle surface is characterized by angle (in Fig. 1.3