Applied Structural Mechanics Fundamentals of Elasticity Part 1 pot

Principle of virtual displacements Pvd Principle of virtual forces Pvf Reciprocity theorems and Unit—Load —-Method Treatment of a variational problem Approximation methods for contin

H Eschenauer, N Olhoff, W Schnell

Prof Dr.-Ing H Eschenauer TA

Research Center for Multidisciplinary Analyses ot ry

and Applied Structural Optimization FOMAAS kX#1!5

Institute of Mechanics and Control Engineering i g {7

D - 57068 Siegen / Germany

Prof Dr techn N Olhoff

Aalborg University

Institute of Mechanical Engineering

DK - 9220 Aalborg East / Denmark

Prof Dr Dr.-Ing E h W Schnell

Technical University of Darmstadt

Institute of Mechanics

D - 64289 Darmstadt / Germany

ISBN 3-540-61232-7 Springer-Verlag Berlin Heidelberg New York

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Eschenauer, Hans A.: Applied structural mechanics: fundamentals of elasticity load bearing structures, structural optimization; including exercises / H Eschenauer, N Olhoff; W Schnell - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris;

Santa Clara; Singapur; Tokyo: Springer, 1997

ISBN 3-$40-61232-7

NE: Olhoff, Niels; Schnell, Walter

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ral—analytical computation software (eg Finite Element Methods) The

importance of computer-tools, may this be supercomputers, parallel compu- ters, or workstations, is beyond discussion, however, the responsible engineer

in research, development, computation, design, and planning should always

be aware of the fact that a sensible use of computer—systems requires a re- alistic modeling and simulation and hence respective knowledge in solid mechanics, thermo- and fluiddynamics, materials science, and in further disciplines of engineering and natural sciences Thus, this book provides the basic tools from the field of the theory of elasticity for students of natural sciences and engineering; besides that, it aims at assisting the engineer in

an industrial environment in solving current problems and thus avoid a mere black-box thinking In view of the growing importance of product lia- bility as well as the fulfilment of extreme specification requirements for new products, this practice-relevant approach plays a decisive role Apart from a firm handling of software systems, the engineer must be capable of both the generation of realistic computational models and of evaluating the

Following an outline of the fundamentals of the theory of elasticity and the

illustrates the transition and interrelation between Structural Mechanics and Structural

Optimization As mentioned before, a realistic modeling is the basis of

every structural analysis and optimization computation, and therefore nu-

merous exercises are attached to each main chapter

By using tensor notation, it is attempted to offer a more general insight into

the theory of elasticity in order to move away from a mere Cartesian view

An "arbitrarily shaped” solid described by generally valid equations shall

be made the object of our investigations (Main Chapter A ) Both the condi-

tions of equilibrium and the strain—displacement relations are presented for large deformations (nonlinear theory ); this knowledge is of vital import- ance for the treatment of stability problems of thin-walled load—bearing

structures When deriving the augmented equations as well as the corre-

sponding solution procedures, we limit our considerations to the most essen-

tial aspects All solution methods are based on the HOOKEAN concept of

the the As examples of load-be: \s examples of load-bearing str

plates and shells will be treated in more detail (Main Chapters B,C)

nally, an introduction into Structural Optimization is given in order to illus- trate ways of determining optimal layouts gf load-bearing structures (Main

as comprehensively as necessary However, it is highly recommended for the

reader to test his own knowledge by solving the tasks independently When treating structural optimization problems a large numerical effort generally occurs that cannot be handled without improved programming skills Thus,

at corresponding tasks, we restrict ourselves to giving hints and we have

consciously avoided presenting details of the programming

The authors would like to express their gratitude to all those who have as-

sisted in preparing the camera—ready pages, in translating and proofreading

as well as in drawing the figures At this point, we would like to thank Mrs

A Wächter-Freudenberg, Mr K Gesenhues, and Mr M Wengenroth who

fulfilled these tasks with perseverance and great patience We further ac-

knowledge the help of Mrs Dipl.-Ing P Neuser and Mr Dipl.~Ing M Seibel

in proofreading

Finally, we would also like to express our thanks to the publisher, and in

particular to Mrs E Raufelder, for excellent cooperation

April 1996

53 Material law for plane states 35

54 Material law for a unidirectional layer (UD-layer ) of a 37

fibre reinforced composite

Principle of virtual displacements ( Pvd )

Principle of virtual forces ( Pvf)

Reciprocity theorems and Unit—Load —-Method

Treatment of a variational problem

Approximation methods for continua

Basic equations and boundary-value problems

Solution of basic equations

Special equations for three-dimensional problems

Special equations for plane problems

Comparison of state of plane stress and state of plane

Tensor rules in oblique base

Analytical vector expressions for a parallelogram disk

Analytical vector expressions for an elliptical hole in

elliptical-hyperbolical coordinates

MOHR's circle for a state of plane stress

Principal stresses and axes of a three-dimensional state

of stress

Equilibrium conditions in elliptical—hyperbolical coordi-

nates (continued from A-2-3)

Displacements and compatibility of a rectangular disk

Principal strains from strain gauge measurements

Strain tensor, principal strains and volume dilatation of

a three—dimensional state of displacements

Strain-displacement relation and material law in ellipti-

cal—hyperbolical coordinates (continued from A-2-3)

Steel ingot in a rigid concrete base

Differential equation and boundary conditions for a

BERNOULLI beam from a variational principle

Basic equations of linear thermoelasticity by HELLIN-

GER /REISSNER's variational functional

Application of the principle of virtual displacements for

establishing the relations of a triangular, finite element

Hollow sphere under constant inner pressure

Single load acting on an elastic half-space — Applica-

tion of LOVE's displacement function

82 — Analytical solutions to the homogeneous bipotential 95

equation

9.2 Analytical solutions for shear-rigid plates 107

10.1 Isotropic plane structures with large displacements 1138 10.2 Load-bearing structures made of composite materials 118

B-8-1: Simply supported rectangular disk under constant load 123 B-8-2: Circular annular disk subjected to a stationary tempera- 128

ture field

B-8-4: Clamped quarter-—circle disk under a single load 133 B-8-5: Semi-infinite disk subjected to a concentrated moment 137 B-8-6: Circular annular CFRP-disk under several loads 139 B-8-7: Infinite disk with an elliptical hole under tension 145

B-8-8: Infinite disk with a crack under tension ˆ 151

B-9-1: Shear-rigid, rectangular plate subjected to a triangular 153

load

B-9-2; Shear-stiff, semi-infinite plate strip under a boundary 155

moment

B-9-3: Rectangular plate with two elastically supported bound- 157

aries subjected to a temperature gradient field

B-9-4: Overall clamped rectangular plate under a constantly 167

distributed load

B-9-5: Rectangular plate with mixed boundary conditions un- 170

der distributed load °

B-9-6: Clamped circular plate with a constant circular line load 172

B-9-7: Clamped circular ring plate with a line load at the outer 177

boundary

B-9-8: Circular plate under a distributed load rested on an ela- 179

stic foundation

x Contents

B-9-9: Centre-supported circular plate with variable thickness

under constant pressure load

B-10-1: Buckling of a rectangular plate with one stiffener

B-10-2: Clamped circular plate under constant pressure consi-

dered as a coupled disk-plate problem

C Curved load-bearing structures

— Chapter 11 to 14 —

C.1 Definitions — Formulas — Concepts

11 General fundamentals of shells

111 Surface theory — description of shells

113 Shear-rigid shells with small curvature

12_ Membrane theory of shells

121 General basic equations

122 Equilibrium conditions of shells of revolution

123 Equilibrium conditions of translation shells

12.4 Deformations of shells of revolution

125 Constitutive equations - material law

12.6 Specific deformation energy

13 Bending theory of shells of revolution

13.1 Basic equations for arbitrary loads

132 Shells of revolution with arbitrary meridional shape -

Transfer Matrix Method

13.3 Bending theory of a circular cylindrical shell

14 Theory of shallow shells

41 Characteristics of shallow shells

14.2 Basic equations and boundary conditions

143 Shallow shell over a rectangular base with constant

principal curvatures

C.2 Exercises

11-1: Fundamental quantities and equilibrium conditions of

the membrane theory of a circular conical shell

C-12-1: Shell of revolution with elliptical meridional shape sub-

jected to constant internal pressure

C-12-2: Dung boiler under internal pressure and centrifugal

Contents XI

C-12-3: Spherical shell under wind pressure

C-12-4: Hanging circular conical shell filled with liquid

C-12-5: Circular toroidal ring shell subjected to a uniformly dis-

tributed boundary load

C-12-6: Circular cylindrical cantilever shell subjected to a trans-

verse load at the end

C-12-7: Skew hyperbolical paraboloid (hypar shell ) subjected to

C-13-4: Circular cylindrical shell horizontally clamped at both

ends subjected to deadweight

C-13-5: Buckling of a cylindrical shell under external pressure

C-13-6: Free vibrations of a circular cylindrical shell

C-14-1: Spherical cap under a concentrated force at the vertex

C-14-2: Eigenfrequencies of a hypar shell

D_ Structural optimization

— Chapter 15 to 18 —

D.1 Definitions — Formulas — Concepts

15 Fundamentals of structural optimization

15.1 Motivation — aim — development

15.2 Single problems in a design procedure

15.3 Design variables — constraints — objective function

15.4 Problem formulation - task of structural optimization

15.5 Definitions in mathematical optimization

15.6 Treatment of a Structural Optimization Problem ( SOP)

16 Algorithms of Mathematical Programming (MP)

16.1 Problems without constraints

16.2 Problems with constraints

17 Sensitivity analysis of structures

171 Purpose of sensitivity analysis

17.2 Overall Finite Difference ( OFD) sensitivity analysis

173 Analytical and semi-analytical sensitivity analyses

problem

D-15/16-3: | Optimum design of a part of a long circular cylin-

drical boiler with a ring stiffener — sizing problem

D-18-1: Mathematical treatment of a Vector Optimization

Problem D-18-2: Simply supported column - shape optimization pro-

blem by means of calculus of variations

D-18-3 Optimal design of a conveyor belt drum — use of

shape functions D-18-4: Optimal shape design of a satellite tank — treat-

ment as a multicriteria optimization problem D-18-5: Optimal layout of a point-supported sandwich pa-

nel made of CFRP-material — geometry optimiza- tion

References

A Fundamentals of elasticity

B Plane load-bearing structures

Cc Curved load-bearing structures

List of symbols

Note: The following list is restricted to the most important subscripts, notations

and letters in the book

Scalar quantities are printed in roman letters, vectors in boldface, tensors

or matrices in capital letters and in boldface

1 Indices and notations

The classification is limited to the most important indices and notations Further

terms are given in the text and in corresponding literature, respectively

Index for a layer of a laminate

subscripts for covariant components superscripts for contravariant components

indices in brackets denote no summation

prime after index denotes rotated coordinate system eg 0,.,,

comma denotes partial differentiation with respect to the

quantity appearing after the comma, eg u,x

superscript prime before symbol denotes deviator, e.g

vertical line after a symbol denotes covariant derivative rela- ting to curvilinear coordinates E', eg v,

bar over a symbol denotes virtual value, eg F;

roof over a symbol denotes the reference to a deformed body tilde denotes approximation

asterisk right hand of a small letter denotes physical compo-

nent of a tensor, eg a¥

asterisk right hand of a letter denotes extremal point

asterisk right hapd of a capital letter denotes the comple- mentary of work or energy, eg U*

nabla-operator differential operator

intersection of A and B

A is a subset of B for all

co— and contravariant components of a surface tensor

semiaxis of an ellipse

determinant of the covariant curvature tensor

co-, contravariant and mixed curvature tensor volume dilatation

orthonormalized base (Cartesian coordinates )

permutation symbol volume force vector

objective function, ~ vector

weight per area unit

determinant of the metric tensor

inequality constraint function, — vector

co— and contravariant base (arbitrary coordinates )

co— and contravariant metric components, metric tensor

distance to axis of rotation along the curvature radius

load vector position vector to an arbitrary point of the mid-surface or a

body

vector of search direction

wall thickness, layer thickness (k = 1, n) stress vector

components of a stress vector

state vector of a cylindrical shell, state variable vector displacements in meridional and in circumferential direction displacement vector

displacements tangential to the mid—surface displacement perpendicular to the mid-surface weighting factors, penalty terms

approximation for deflection design variable vector

Cartesian coordinate system, EUCLIDIAN space

shape parameter

transformed variables complex variable state vector at point i of a shell of revolution

area, surface; concentrated force at a corner

strain-stiffness matrix; matrix of A—conjugate directions

B-spline base functions, BERNSTEIN- polynomials

rotation matrix; coupled stiffness matrix

transfer matrix of a shell element, total transfer matrix

elasticity tensor of fourth order

elasticity matrix

flexibility matrix

tension stiffness of an isotropic shell strain— or shear stiffnesses of an orthotropic shell flexibility tensor

YOUNG's modulus, elasticity matrix

plane elasticity tensor objective functionals concentrated forces; load vector implicit representation of a surface symmetrical flexibility matrix - mixed transformation tensor

— system matrix

operator of inequality constraints

elasticity tensor of a shell

pseudo-resultant moment tensor

Tnemi

normal and shear components of membrane forces

effective inplane shear force

polynomials

transverse shear forces effective transverse shear force boundary force

penalty parameter

polynomial approximations

radii of principal curvatures

shape function of a shell surface n-dimensional set of real numbers

stress tensor

List of symbols — XVII

specific deformation energy

specific complementary energy potential for field of conservative forces volume

tensor of deformation derivatives, deformation gradient

strain tensor (symmetrical part of V)

tensor of infinitesimal rotations (antisymmetrical part of V)

weight

external work, complementary work

feasible design space, subset

semi-—angle of a cone optimal step length

coefficient of thermal expansion

strains of the mid—surface of a shell LAGRANGE multipliers

distortions of the mid—surface of a shell components of a rotation matrix

thermal -elastic tensor

KRONECKER's tensor in curvilinear and Cartesian coordi- nates

MAXWELL 's infQuence coeffients

factor of the step length, strain vector permutation tensors

strains in Cartesian and spherical coordinates

vector of free thermal strains

coordinate perpendicular to mid—surface