LARGE DISPLACEMENT ANALYSIS OF AXI-SYMMETRIC T:TIN

SHELL STRUCTURES UNDER ARBITRARY LOADING BY

THE FINITE ELEMENT METHOD

By

ANTHONY FIRMIN B.Sc.(Eng) A.C.G.I.

A THESIS SUBMITTED FOR THE DEGREE OF

THE DOCTOR OF PHILOSOPHY OF THE

UNIVERSITY OF LONDON

1971

In writing this thesis the author is particularly indebted to

Dr. A.S.L. Chan whose inspiration and constructive criticism have been

greatly appreciated throu3hout the course of the research. He would

also like to thank Professor Argyris for his support and encouragement

of the work.

The author is also grateful to the Central Electricity Generating

Board for financing the work and for making available their computing

services. In particular the patient and most helpful support of the

Board's computing branch staff, whose outstanding service is in the

author's opinion second to none. - Dr. Ewing of the. Board's Leatherhead

laboratories also provided help in the form of a number of fruitful

discussions.

Many members of the aeronautics department have also lent a hand

and given advice when needed. In particular Dick Henrywood who pro-

vided valuable support in the early stages. Among others the author

Must thank are Dr. Davies, Miriam Pook, Jill Mair and Julie Bartley

who was responsible for the line drawings.

11

SUMMARY

An incremental, linearised theory is developed for the large dis-

placement small strain analysis of axi-symmetric thin shells. The

theory is developed through the finite-element displacement method

applied to an axi-symmetric thin shell element under arbitrary loading.

The small displacement theory for the element is derived and subse-

quently demonstrated to be an excellent basis for the analysis of thin

shell structures. The non-linear problem is solved via an incremental

method, for each step of which the displacements remain small.

Examples demonstrate the importance of such a theory in predicting the

distribution of shell stresses and show its equivalence with other

existing methods of solution. The approach is shown to give a good

theoretical estimation of the buckling load and to be capable, under

suitable circumstances, of following the load/displacement curve into

the post-buckled region.

A computer program has been written to set up and solve the equa-

tions derived in the theory. A precis of the program is given in

the form of flow charts, together with the more important computational

procedures used in it.

1

2

CONTENTS

Acknowledgements

Summary 1

Contents 2

Notation 4

Introduction 6

PART 1 - Small Displacements 12

1.1 Introduction 12

1.2 Displacement Functions 14

1.3 Description of Meridian Curve 19

1.4 Strain-Displacement Relations 22

1.5 Rigid Body Displacements 22

1.6 Shell Stresses 24

1.7 Generalised Force and Element Stiffness Matrix 27

1.8 Formation of the Structural Stiffness Matrix 29

1.9 Kinematically Equivalent Loads (for a distributed

loading) 32

1.10 Initial Loads 34

1.11 Application of the SABA Element to the Linear

Analysis of Shells 35

1.12 Conclusion to the Work on Small Displacements 46

PART 2 - Large Displacements 56

2.1 Introduction 56

2.2 General Outline of the Linearised Incremental

Approach

59

2.3 The Geometric Stiffness for an Infinitesimal

Shell Element 62

3

2.4 The Additional Geometric Stiffness Due to the Applied

Pressure 66

2.5 Derivation of the Geometric Stiffness Matrix for the

SABA Element 67

2.6 The Coupling between Harmonics in the Large Displacement

Problems 70

2.7 Formation of the Large Displacement Structural

Stiffness 76

2.8 Iterative Procedure for the Incremental Steps 79

2.9 The Displacement Increment Method of Solution 81

2.10 Application of the SABA Element to the Non-Linear

Analysis of Shells of Revolution 83

2.11 Conclusion to the SABA Analysis 114

APPENDIX

3.1 Introduction 116

3.2 Evaluation of the Normal Shear Terms Q6

and Q 117 4)

3.3 Calculation of Terms in Band B 120 an

3.4 Numerical Integration Procedure 121

3.5 Suppression of Nodal Freedoms 122

3.6 Algorithm for the Cholesky Inversion 122

3.7 Solution of the Large Displacement Equation 124

3.8 Calculation of rI and zx for the Hyperbolic Shell 127

3.9 Flow Diagrams 129

References 150

NOTATION

2, Meridional half length

True meridional distance

Non dimensional meridional distance

Matrix of polynomial terms in n

G

Hermitian coefficients

u, v, w Displacements in the local system of coordinates

u, v, w Displacements in the global system of coordinates

6,8

Vectors of displacements

T

Cartesian transformation matrix

Column matrix of nodal displacements

F

Diagonal matrix of Fourier terms

H

Diagonal super-matrix of

A Diagonal super-matrix of G

r

Radial coordinate

z

Axial coordinate

Nodal geometry data for r direction

zI Nodal geometry data for z direction

D Strain/displacement operator

Vector of strains

se co, the Membrane strains

Ke K0, Kee Bending strains

S Stress vector

Ne N0, S Membrane stresses

Me M0, T Bending stresses

K Material stiffness matrix relating S and E

Kinematically equivalent load vector for an element

B Matrix product D T H F

k Element stiffness matrix

C Transformation matrix

a

Boolean transformation matrix for adding element g

stiffnesses to global stiffness

Structural load vector

r Structural displacement vector

K Structural stiffness matrix

Vector of distributed applied loads

s

Nodal data for ingerpolation of applied loads

N

Initial stress vector

T Initial strain vector

J Initial nodal load vector

a Transforms nodal displacements into strains

Cr

Matrix of internal stresses used in large displacement

theory

.Matrix of surface rotations, strains and displacements

L Transformation matrix to produce W from 6

L1 1J2 etc. Rows of L

M Matrix product LTHF

A

Magnitude of load step

Y11Y12 etc.

BM

9111m Terms in

an

Lower half of B matrix

All other symbols are defined in the text.

5

INTRODUCTION

The relatively low strength of wood and stone has for centuries led

man to use a very solid made of construction. Stout structural members

have meant that deformations have generally been small and as a result

it has been possible to assume a linear relationship between loads and

displacements. However with the use of iron and steel during the last

century it became clear that deformations need not be small and that

members in compression could fail through lack of stability. Since

then the engineer has discovered that to optimise a structure by mini-

mising its weight and cost it may be advantageoUs to permit large dis-

placements and even in the extreme loading case to allow plastic

deformation if he can be sure this will not lead to collapse. Under

these circumstances the response of the structure will be far from

linear. Assuming then, that the best design will emerge from the

most accurate analysis such non-linearities should be accounted for

when considering a structure's behaviour. Love [1] in his famous

treatise on elasticity derived the compatibility relations for carte-

sian strain which were complete in this respect. It is unfortunate

however that the description of non-linear phenomena leads to non-

linear equations for which solutions are considerably more difficult

to obtain. Consequently it has been more usual to ignore the non-

linear terms, apart from their use in the calculation of buckling

loads. The difficulties of non-linear analysis has led to a barrier

which engineers have been reluctant to cross. But where the classi-

cal approch fails it is possible to employ numerical methods to

obtain an approximate solution. As in all branches of engineering

science such analysis has been spurred on in the last decade by the

development of high speed digital computers. By the discretisation

6

of a complex structure into a number of finite elements and by the use

of numerical methods to solve the non-linear equations an analysis may

now be made which hitherto would have been impossibly long.

A Review of the Use of Finite Elements in Solving Geometrically

Non-linear Problems

Since the pioneering work of Argyris [2-5] the finite element dis-

placement method has become a widely used and very successful approach

to the analysis of complex structures. The method is dependent on

representing the structure by a number of elements each of which has a

defined stiffness. The element can be seen as a physical entity

being an isolated part of the continuum. The generalised displace-

ments may be obtained by solving the complete structural stiffness,

formed by an assembly of these elements, with the generalised loads.

It is more usual to assume a linear relationship between the loads and

displacements; however considerable effort has been put into extend-

ing the theory to include geometric-non-linearity and the method is

well established in connection with beam and plate elements [3-12].

A linearised incremental approach is usually adopted whereby the

force-displacement relationship is constructed in a series of small

increments each of which is a linear step. The non-linear terms

being included in the finite element method by an additional geometric

stiffness matrix [4-12] which is added to the elastic stiffness at the

beginning of each step. Such matrices were derived from a considera-

tion of equilibrium of the nodal forces. However with the intro-

duction of more complicated elements and in particular shell elements,

the geometric stiffness has more usually been derived from the non-

linear equations of continuum mechanics [7,12-16]. In the classical

theory of elasticity the non-linear terms arise from the inclusion of

second order terms in the strain displacement relations. An account

7

8

of this theory may be found in references [17-19.]. A notable exception

to this approach is the work of Argyris et al. where the geometric

stiffness has consistently been formulated in terms of the 'natural

modes' of the element [20]. References [10,11] derive the non-linear

terms for triangular plate bending elements and curved tetrahedronal

and triangular elements. Whilst references [20-22] provide a general

triangular shell element for small and large displacement analysis.

Fundamental to the derivation of the geometric stiffness matrix is

the frame of reference within which the theory is developed. Connor

[12] and Hibbitt et al. [23] use a Lagrangian systems where the strains

are related to the undeformed geometry. Whereas Argyris [4] by up-

dating the geometry at the end of each increment of load is using a

mixed Lagrangian-Eulerian system where the equations depend on a fixed

coordinate system only during the increment of load. The latter

approach simplifies the derivation of the geometric stiffness and is

correct, providing one assumes the rotations during a step remain

small.

The effect of finite displacements may be extended to include a

consideration of finite strain. Little work has been done that in-

cludes such terms but notable contributions have been made by Oden

[24,25] and more recently Hibbitt et al. [23]. The latter reference

develops the general finite element formulation for the large strain,

large displacement problem including the effect of finite displace-

ments on the applied loading. It is however sufficient for the most

part to restrict oneself to small strains.

Alternative techniques for the solution of the non-linear equations

Instead of forming an incremental relationship to solve the non-

linear equations it is possible to use an iterative technique. Oden

[13,14] has developed constant strain triangular and tetrahedron ele-

ments and his solution using these elements employs a Newton-Raphson

iteration. However perhaps the most successful approach is a hybrid of

the two methods used by Connor [12] and Stricklin et al. [26].

Connor's method is to take a number of relatively large load steps and

then pause to correct the solution using a Newton-Raphson iteration.

He demonstrates the technique on a shallow shell element and obtains a

favourable comparison with other methods of solution. Stricklin et

al. employ successive substitution of each load step to obtain a solu-

tion for an axisymmetric shell element. Successive substitution only

has first order convergence and Archer [27] has shown this method may

fail to converge even when the equilibrium position is stable. However

when used in conjunction with the load increment method it provides a

valuable correction. Schmit, Bogner et al. [9,28] have developed a

direct search technique for minimising the total potential energy func-

tional and have used this in connection with geometrically non-linear

rectangular plate and cylindrical shell elements. Their approximate

method of solution is however restricted as it will only converge for

stable branches of the load/displacement curve. A transition from

stable to unstable equilibrium is indicated by a jump to another stable

branch. It is preferable to have the capability for tracking the

complete load/deflection curve as this gives a clearer insight into the

behaviour of the structure.

The Non-linear Analysis of Axisymmetric Shells

) By Finite Elements

The application of the matrix displacement method to shells of

revolution,using conical frusta as the finite elements was first des-

cribed by Grafton and Strome [29]. Subsequently, Percy, Pian et al.

10

[30] developed the SABOR III program using frusta with a curved meridian

and incorporating asymmetrical deformations. Navaratna et al. [31] have

used this element to investigate linear buckling [where the behaviour

prior to buckling is assumed linear] of shells of revolution, finding

close agreement with other theoretical results. They employed an

eigenvalue approach to obtain the critical load. Stricklin et al.

[26] extended this element to include arbitrary distributed loading.

The non-linear terms were however incorporated in a pseudo load vector

which was used to modify the applied loads at each increment. The

remainder of the solution being identical to the linear problem. This

is a most convenient method but is limited in that one can only esti-

mate the buckling load. The post-buckling behaviour cannot be deter-

mined and hence failure to converge is taken as the stability criterion.

In addition because the tangent stiffness matrix is not produced one

cannot solve the buckling problem by an eigenvalue approach.

b) By Alternative Methods

Kalnins [32,33] has analysed shells of revolution using a multi-

segment method of integration to solve the non-linear differential

equations derived by Reissner. The method involves iterations of

trial solutions with respect to the non-linearity of the equations.

Although restricted to axisymmetric loading the method is capable of

giving solutions with uniform prescribed accuracy. Ball [34] uses

the method of finite differences to solve Sanders' non-linear equa-

tions for axisymmetric shells for the case of a general distributed

loading. He also groups the non-linear terms in a pseudo load vec-

tor which precludes post-buckling analysis.

11

Objectives of This Paper

Although there now exists a general triangular shell element [21,

22] which could be used in a large displacement analysis of axisymmetric

shells such a solution would be inefficient as it would fail to take

account of the symmetry of the problem. As far as the author is

aware, there does not seem to have been any attempt at a large dis-

placement non-linear analysis of axisymmetric shells using a geometric

stiffness matrix. This paper therefore develops the geometric stiff-

ness of an axisymmetric shell element under arbitrary distributed load-

ing. The first part gives, as a necessary first step the small dis-

placement theory of a suitable finite element and compares the results

obtained using this method with other methods of solution. The ele-

ment is shown to form a very good basis for the extension to the large

displacement problem. The second part derives the geometric stiff-

ness for thin shells in general terms and applies it in particular to

the axisymmetric element. A number of examples are chosen to illus-

trate different aspects of the method, in particular the efficiency

of the incremental approach in predicting the. buckling load. Finally

an outline of‘the computing techniques used in programming the theory

are given in an appendix.

PART 1 - Small Displacements

1,1 Introduction

The finite element displacement method is based upon the discre-

tisation of a structure into a number of elements. The stiffness of

each element can be calculated from the assumption of certain displace-

ment patterns within these elements and from the principle of virtual

work. The displacement functions contain a number of generalised

displacements which have corresponding to them an equal number of gen-

eralised forces which may be calculated from the applied loading.

The stiffness matrix linking the vectors of displacement parameters

and generalised forces for the whole structure is assembled by an

addition of the element stiffness matrices. A solution of the result-

ing stiffness matrix with the generalised load vector produces the

required displacement parameters. The finite eleMents considered

here are frusta taken from shells of revolution and employ displace-

ment. and force parameters at the nodal circles to characterise their

behaviour.

The accuracy of the small displacement solution is of great

importance as it will reflect directly on the large displacement

analysis each step of which contains a small displacement stiffness

matrix. An element must therefore be chosen which is capable of

giving a reliable result. In addition the large displacement analy-

12

sis usually requires an iterative procedure for which the computing

time required is many times that of the small displacement solution.

It is therefore desirable that the element should give an accurate

description of the structural behaviour while using as few unknown

variables as possible. It is also important that the theory should

ensure that a rigid body movement will not give rise to any strains.

13

Otherwise the equilibrium conditions will not be satisfied and may lead

to a serious error.

The SABA family of elements described in this paper are designed

with these particular objectives in mind. The elements are developed

for axisymmetric shell structures under an arbitrary distributed load-

ing and take full advantage of the axisymmetrical nature of the thin

shell structure. Each element is generated from a segment of the

Meridian curve revolving around the axis, forming two nodal circles

at the ends. Displacements around the circumference are assumed to

vary as a Fourier series, where each harmonic is associated with a

polynomial in the meridian direction. The latter variation is, in

practice, interpolated by Hermitian functions in terms of the displace-

ment and their derivatives at the nodal circles. The order of the

polynomial can be varied according to the degree of sophistication

one desires. The geometry of the meridian curve is also interpolated

from similar parameters at the nodes using the same Hermitian poly-

nomials. Thus, with the strains given in terms of the displacements

by Novozhilov's shell theory [35], a rigid body movement will not in-

duce strain anywhere in the element, and equilibrium between the gen-

eralised nodal forces and the internal stresses is completely satisfied.

It has been found sufficient to interpolate the meridional varia-

tion of a variable by a fifth order polynomial in terms of its value

and its first and second derivatives at the nodes. The SABA5 element

so derived has not only continuity of the displacement, slope and

curvature across the element boundaries but also of the membrane and

bending stresses. However, the normal shear, which is dependent on

the third derivative of the displacements, will not necessarily be so.

If continuity of normal shear were considered important it would only

be necessary to change the meridional functions to a seventh order

polynomial.

14

The applied loading on the structure is decomposed into a number

of Fourier variations around the circumference.. Each component is

then converted into a kinematically equivalent generalised load•vector

at the nodal circles. For the small displacement analysis, each

harmonic component of the applied load is related only to the corres-

ponding nodal displacements of the same harmonic in the stiffness

matrix. In other words there is no coupling between different har-

monics. As a result there are only as many harmonics terms in the

displacements as it is necessary for an accurate description of the

applied load. The advantage of this is that each harmonic can be

solved separately, requiring little storage space and giving rapid

and accurate numerical results. It will be shown in Part II that,

for the large displacement analysis, coupling will generally occur

between the harmonics.

In the following sections, the steps leading to the formation of

the stiffness matrix are described. The derivation of the kinematic-

ally equivalent loads and the calculation of the stress resultants

in the shell are also given. Finally a number of examples are chosen

to illustrate some of the many applications for this element. Where

possible the results are compared with other methods of solution.

1.2 Displacement Functions

Consider a frustum of an axisymmetrical thin shell in the

cylindrical polar coordinate system. The displacements u, v and w

at a point on the middle surface are expressed as a Fourier series:

A = Ao + E(A. sin j0 + A. cos j0)

Js jc

where A represents u, v or w, positive in the r, 0 and z directions

respectively. Each coefficient of the series describes the variation

in the meridian direction, which is interpolated in terms of the para-

meters at each end by a Hermitian polynomial which for the SABA5 ele-

ment is fifth order and is of the form

En'(n)li = (f1(1)°1 f2 (1)°1 "3(1)°Y f4(n)A2 f5(n)62

f6 (11)APj (1.2)

where n is a non-dimensional coordinate along the meridian (Fig 1);

2 A A

an a A A', , a .. denote , ... respectively, and subscripts 1 and

011 @n2

2 refer to values at nodes 1 (n = +1) and 2 (n = -1). For the purpose

ofthispaperthefunctionsf.(n) (i = 1, 2 .. 6) are taken to be

fifth order Hermitian polynomials whose values at the ends return the

nodal parameters

e.g. A(+1) = Al; requiring

fl (+l) = 1, f.(4.1) = 0 (j # 1) etc.

To obtain higher order members of the SABA family higher derivatives

will be required in (1.2). In general two extra terms will appear on

the R.H.S. for every two increases in order made to the Hermitian

functions.

This equation may be written as a matrix expression.

A. = n G A.

where

iij = {.51 Al Al A2 A2 Apj

= [1. n n2 n3 n4 n5]

(Note: { } indicates a column matrix)

and

11.3)

(1.4)

15

16

= -1< 11 < +1

DERIVATIVES t it ARE a AND a2 RESPECTIVELY zrq 2

FIG. 1 SABA ELEMENT

AXIS

FIG. 2 SHELL DISPLACEMENTS

17

G 1 - 16

8 -5 1 8 5 1

15 -7 1 -15 -7 -1

O 6 -2 0 -6 -2

-10 10 -2 10 10 2

O -1 1 0 1 1

3 -3 1 -3 -3 -1

(1.5)

is the matrix of the constant coefficients of the Hermitian functions.

It is more convenient to obtain the strains, not in terms of the

global displacements u, v and w, but as functions of local displace-

ments — _

u, v and w (see Fig 2), where u is a tangential displacement in

the meridional direction and w is normal to the surface. Both systems

follow a consistent right-handed convention.

For the jth immonictwodisplacementvectors. aij and • are

defined:

8. fu v wl.

• {u v w}. J

they are related to each other by a transformation matrix T giving:

T 8 • (1.6)

where

cos (V 0 =sin_ tp_ --

T 0 1 0 (1.7)

sin 4 0 cos (I)

In order to ensure that the harmonics do not couple in the small

displacement analysis it is necessary to group symmetric and anti-

symmetric displacements separately. This means that the u and w sets

are multiplied by cosine terms while the v displacements are multiplied

by a sine term and vice versa.

The displacements oi for

terms of the parameters at: the

th. again for the 3 harmonic,

th the j harmonic may be expressed in

two nodal circles, defined by the vector,

= {Au A Awl1j (1.8)

where

Au

(u1 1 1 ul un u2 2 2 ul u"}

(1.9)

and similarly for Avand A with v and w in place of u. The generalised

nodel displacements are the sum of all the terms of the Fourier series

as specified in eqn. (1.1) with the pi defined in (1.8) as their co-

efficients. The series may be written as

Po ± FiPPiP

F OP • - rjP (1.10)

3

where the IP's are diagonal matrices

li jf) 1—cos je 16 sin je 16 cos je Ij

3p = rsin J O 16 cos je 16 sin jeIg

The subscript p or p is used to distinguish between the symmetrical and

antisymmetrical deformations.

Therelationshipbetweenojand pa now follow from eqn (1.3)

which can be expressed as

5.3 = 11 A F j pi

H FiA P 3 (1.12)

where .11 and A are diagonal super-matrices

18

19

H 1-1-1

A = (1.13)

The position of F) and A can be interchanged since they are both dia-

tonal super-matrices.

The subscripts p or p have been omitted from eqn (1.12) so that

this equation now applies for either case, implied by the harmonic

subscript j.

1.3 Description of the Meridian Curve

A point on the surface is defined in cylindrical polar coordinates.

by r, 0 and z (see Fig 3). However, instead of using the exact geo-

metry of the structure, the meridian curve of this axisymmetrical

element is constructed approximately from the nodal parameters

1. 2 =

z1 =

art a 2r, ar2

a2r2

) r, L an

aZ1 z

an2

a2Z1

r2 an

az2

a n2

.a2z2

1 an ant z2

an ant

(where subscripts 1 and 2 are again values at the two nodal circles)

in exactly the same way as the displacements of eqn (1.3), giving

r = q G I* /

q G z (1.14)

where 1) is the row matrix of the polynomials (1.4) and G the 6 x 6

matrix of constants for the Hermitian functions given in eqn (1.5).

N09

distadt'

Me Mel

9

Me

20

FIG. 3 SHELL GEOMETRY

FIG. 4 ELEMENT STRESSES

a ra0

1 — sing) r 1 — cos 4) r

a kan

0 • r

a 1 Dr kan r2. an ra6

1 a 1 ar 4, (— — ) krcl, - r n 0 1 1 or al a _ 32

R2 - 4) -11 an an2

1 ar

rr an sin 3 r2 De

a2 1 Dr 3 r2 302 r2,2 an an

1 a rr 30

a _ sin 4 ar r Qan kr2 3n

- a2 kr 363n

Zara r an 36

FIG: 5a

1.4 Strain-Displacement Relations

The linear strain displacement relations for the axisymmetric case

as derived by Novozhilov [35] can be expressed in the matrix form

• D S (1.15)

where

• {e$ E 0 0

E._ K. K K0 (P01

K__ (1.16)

The differential matrix D appears in Fig 5a. The three strains E

c0 and c.

0 are membrane strains describing the stretching of the mid

surface and Ke Ke, Kee are bending strains related to the rotations of

the tangents of this surface.

1.5 Rigid Body Displacements

It is well established that, in designing a curved element, care

must be taken to ensure that no strain is produced in a rigid body move-

ment anywhere within the element. Otherwise equilibrium between the

nodal forces and internal stresses will not be truly satisfied.

It has been shown in reference [21] that this can be achieved by inter-

polating both the displacements and the geometry within the element

in terms of the corresponding nodal parameters with the same interpola-

tion functions, providing that the strain-displacement relations are

not deficient in the first place. The equations from Novozhilov's

thin shell theory are used here, which are basically correct in this

respect.

It is important to show that the interpolation scheme used in

1.2 and 1.3 is capable of giving a true rigid body movement to the

eleinent in this parti6ular case, as follows. From the axisymmetrical

22

23

nature of the element geometry and of the displacement functions eqn

(1.1) it is evident that the behaviour of the element in a rigid-body

motion can be studied from the behaviour of any chosen meridian line.

It is only necessary to show that, in a rigid-body movement, the dis-

placements at any point on a given meridian line are given by

u u0+ 41XX

(1.17)

where X is the three-dimensional cartesian position vector

u is the three-dimensional cartesian displacement vector

and 4i is the vector of rotations about the axes.

The subscript 0 denotes a constant displacement.

The cartesian position vector of any point on the meridian line is

effectively given by eqn 1.14 as a fifth order Hermitian function of

.the form

X = f1X1 + f2 X1 + f

3 X1

+ f4

xa f5X2

+ f X2'6

Hence

‘,1

1 8X= f1

SXI + f

2 dA +

(1.18)

(1.19)

If the nodal displacement parameters are incremented in a rigid body

manner, then, at the end points 1 and 2, the new nodal parameters are

by definition:

/ = sx ; x = sx' 1 1

X 2 -1— ax a ; x;= ax /2

X"= x"—I—sx" 1 1 1

it ti X 2 = X 2 + bX 2

(1.20)

For a given rigid body movement, the increments at the end points are

given by:

g X = 0(0+ q)><X1

sX = 4X X

5X 1̀ = 4'X X III

SX =

=

11 6X a =

24

Hence from (1.19) and (1.21)

8 X = fis X0 + f4,5 X0 + 40( (fi f2 ;K1 f3 4.

X0-1- X X

(1.21)

(1.22)

Thus providing the displacements are interpolated by the same function,

i.e.

SX = = f1u1 f2ul

+ f3u1

+ • • •

then the displacements satisfy the relation (1.17) and the whole line

will move as a rigid body.

1.6 Shell Stresses

It is more convenient, in the case of thin shells, to formulate

the theory in terms of the stress resultants instead of the stresses.

The state of stress in the shell is completely described by the follow—

ing 6 statically equivalent resultant forces and 4 bending moments,

acting per unit length in the appropriate direction (see Fig 4):

r

(1 + C/r0 ) dC N40

44)

ao

aOr

a 0

0

(1 + E/r ) E dE

25

(1 + t/r0) t dE

(1.23)

where r and r(I) are the principal radii and r = resin cp. All the inte-

grations are performed through the thickness from E = -t/2 to +t/2.

According to the theory given by Novozhilov, the 6 equations of

equilibrium involving these ten quantities can be satisfied, to a

degree of accuracy consistent with the small displacement assumption,

by the approximation

(1.24) MO = M = T

Ocp

when the shell is loaded only at the edges. Although as one can see

from (1.23) this is only identically true for a spherical shell or a

flat plate. This simplification can very conveniently be adopted by

the present finite element analysis, since the applied loads are here

represented as equivalent forces acting on the nodal circles (see

Section 1.9).

Introducing the notation

MAsn N = N -

eib re r4)

(1.25)

which satisfies one of the equilibrium conditions identically, and

utilizing two other equilibrium equations

1 raM (1) DT ,

r + - M0) cos

a [ rn me + + 2T cos 4)] (1.26)

The ten stress resultants (1.23) may then be obtained in terms of the

following 6 quantities, collected into a column vector

S = s m4) me T1

Its relations with the strain in 1.16 is given by:

S = IC E

(1.27)

(1.28)

where for an isotropic material,,the stiffness matrix of the infinitesi-

mal element is

1

26

1 0 1— V 2

=

1 t2

0 12 1

1-v

The anistropic or orthotropic material stiffness may be substituted at

Et

1—v2

this point.

1.7 Generalised Force and the Element Stiffness Matrix

For every nodal displacement vector pi there is a strain Ej and th

hence a stress resultant Si , varying as a j harmonic in the 0-co-

ordinate direction. The stress distribution can be represented by a

generalised nodal force Pi , written as an 18x1 column vector in which

each item is associated with the corresponding item in pi. This load

vector can be determined by using the principle of virtual work which

ensures the equilibrium of the nodal forces Pj with the stresses Sj

but, as in all displacement methods, finite element analysis, makes no

provision for the equilibrium conditions within the element to be

satisfied in detail. Thus, for a system of virtual displacements !Spi

and compatible strain gEj of the same harmonic variation the P.V.W.

gives

6 pliPi bEI Si r deldrt (1.29)

Using equations (1.6, 1.12, 1.15, 1.28), it can be written as

6 RI pj Opt At.131K BS A rl &$ cll p j (1.30)

where

Bi = D T H Fj (1.31)

The expansion of this matrix is given in Fig .51)-(overleaf).

Noting that Sp is arbitrary, eqn 1.29 becomes

Pi = ki pi (1.32)

where

k j 1 At K BjT de dry A (1.33)

27

cos 4) sin 4) r Pj 0 cos

2cp

P • q 2, sin cp ii

P • Ti J kr4 Pj

1 cosy pj11 • 0 . — 1 s — incl)PJ .11

1 1 — r P 3 • ri r P • ri 0

cos 4,t r P • '1Ji

1 ► cos cl) — sin 4) ,

(•k r TDPi r P J • 11

sin cp ..,... _1 sin 4 cos 4) i sin 4) —, cos cp , c0s2S i _ _ •T'l P •11 _ P • q P • r2 v3 1 kr PJ r2 J r2 J kr j

1 I I

1 2 A sin 4) cos 4) , sin 4) sin 4) coscP — sin 4) — i cos cos cp , i

Pi ?I — kr Pill i _ • + P • 11 I. P t• I P • 11 r2 PJ kr j kr 3

r2 r2 •3

eitherPJ = cos je

PJ sin JO •

orP.) . = sin Pi T. = cos JO

q' = an

FIG: 5b

It can be• shown mathematically that the integration .of

B. K B•rdedn

is zero since B.1 will contain F1 which is a different harmonic from

the n in Eli . This is equivalent to saying that there is no virtual

workdonebYthestressdistributionSion a strain BE: of a differ-

ent harmonic variation. Hence eqn 1.29 defines the generalised force

P.3 which is equivalent to (i.e. in equilibrium with) the jth harmonic

stress distribution Si. Equation 1.32 relates this generalised

force to the corresponding generalised nodal displacement and the

matrix ki in 1.33 is the element stiffness matrix for the harmonic.

There is no other relationship between pi and other generalised forces

Pi where i # j. In other words the stiffness matrix for each har-

monic is uncoupled.

Evaluation of the integral proves simple as the integration in

the 0 direction is exact. The remaining integration in the 11 direction

is obtained numerically using the Gaussian quadrature method, the

result being exact for polynomials up to the ninth order if five

Gaussian points are taken. Further details of the evaluation of the

individual terms in B and a description of the Gaussian quadrature

method may be found in the Appendix on computing procedures.

1.8 Formation of the Structural Stiffness Matrix

As the element stiffness matrices are uncoupled between the har-

monies, it is natural to assemble the stiffness matrix of the complete

structure for each harmonic and find the solution separately. There-

. fore there will only be as many terms in the Fourier series for the

displacements as there are in the loading. To facilitate the compact

29

assembly of the complete stiffness matrix of the whole structure for a

particular harmonic, it is necessary to reorganise the column matrix

pj of eqns (1.8) into a new form in which all the parameters at each

end are grouped together. In addition the nodal parameters of slope

and curvature are no longer derivatives with respect to the non-

dimensional meridian length n but are derived with respect to s, the

true meridian length. Thus a new column ps is defined where

I I II I II I II I II I II I pj = / u1 u1 u/ v1 171 vl w1 w/ wl u2 u2 u2 172 172 17.2 W2 W2 T.,72 1j

(1.34)

where uI, u

II etc. refer to 211 and 1 respectively.

as Ds2

I Then pj is related to pj by

pi = C

where

(1.35)

Z3 •

Z3

30

C ,s-

Z3 •

• Z3 • • Z3

•

and Z3

£2

The stiffness matrix corresponding to the new definition is given by

1

n

31

k = C k• 1C

(1.36)

th If the displacement vector of the complete structure for the j harmonic

is defined as

T.3 rk • • • el

(1.37)

where each rk is the displacement vector at node k

r k = u uI uII v vI vII w wI wII (1.38)

Then for the gth

element,

1 • • • g

[] Pg = P = [ 0 0 . 1 0

0 0 . 0 1 •

a gr (1.39)

th th . . where the subscript j for the 3 harmonic has been omitted. The 3

harmonic of the applied force.

Ri • • . Rk • . R t • (1.40)

is then related to• by ra

13, = Kirj (1.41)

th. where the complete stiffness matrix for the 3 harmonic is obtained

E K3 = agkgiai (1.42)

from

which has the following form for an unbranched structure.

32

1

KN

1.9 Kinematically Equivalent Loads (for a distributed loading)

In general the distributed loading on the axisymmetrical structure

can be decomposed into a series of harmonic functions around the circum-

ference of the shell. Its variation in the axial direction may be

de"scribed by a polynomial function in n. In this case it is conven-

ient to use the same Hermitian polynomial function as in eqn (1.3).

Arranging the three components in the global co-ordinate directions as

a column vector

q = (1.43)

its harmonic variation may be expressed concisely as

q ' = cio E Fipqi E

(1.44)

where n?and FOP are matrices of harmonit-ftnictioh§—as defined in eqn

(1.11).

The variation of qi and qj with n are now described in exactly

the same manner as for the displacements, namely:

qi= HAsj (1.45)

33

where

c4.1cilit oir2q;2c2(101qhqhcle2c1;2(fLcizoLqii icizziLce;2 }

the elements of which are the values of q and its 1st and 2nd derivatives

with respect to n at the nodal points.

The kinematically equivalent load vector may now be found from the

principle of virtual work

6Ptj 8i5 3 cl.rldecin (1.46)

and hence from equations (1.12 and 1.45), noting alo that Fj is dia-

gonal,

Pi (HAF) IHAFi rldecln si (1.47)

For the particular case of a pressure p, constant along the meridian

and symmetric around the circumference, and normal to the shell locally

everywhere, eqn (1.47) is simplified to

Pi t= RH A .F3 ) Ttcos je { 0 0 1 } rd0 dn (1.48)

th where p. is the pressure for the j harmonic.

circumference first

21T

Integrating around the

Flops je d0 = r 1'6 06 ij j > 0

o = 416 06 I6.1 j = 0

then

+1.

Pa t t . Gib =-(2) j.071p. { G q P 06 cos ¢)}rdn (1.49)

This can easily be evaluated knowing the meridian geometry. The

sequenceoftheelementsin.may again be rearranged by the C matrix

34

from eqn 1.35 before being assembled into the applied load matrix,

giving

R t ag C

t P • (1.50)

1.10 Initial Loads

When there is initial stress INTi as a result of the initial strains

th j (due to, e.g. thermal expansion) Tj (varying as a harmonic)

Ni = - K T. (1.51)

an initial nodel load Jj arranged as an 18x1 column matrix may be in-

cluded in the calculation. This column is given by the virtual dis-

placement principle

Spy Jj = f SE A NJ dA (1.52)

with the virtual strain given by eqns (1.15, 1.12, 1.31) as

&E.3 BiALt ip

t IL Bt PC1A.A. (1.53)

Assembling this column for all the elements, the equilibrium condition

of eqn (1.41) is modified into

Ri = Kiri + z atg CtJgi • (1.54)

from which rj can be found after the kinematic freedoms are suppressed.

35

1.11 Application of the SABA Element to the Linear Analysis of Shells

A program has been written which will apply the SABA element to

the linear analysis of shells of revolution. Computational procedures

used in the program will be described in the Appendix. From a des-

cription of the meridional geometry, the material properties and the

element divisions, stiffness matrices are calculated for each element

. and assembled into the structural stiffness matrix. The nodal dis-

placements are computed from a solution of the stiffness matrix with

the applied load vector, for each harmonic present in the applied load-

ing. The shell stress resultants are then produced from the nodal

displacement for a number of chosen points along the meridian.

Finally all the harmonics of displacement and stress are summed, to

give the complete solution for the shell. The program which is written

in FORTRAN IV has been run on an IBM 360/75 computer.

. Analysis of two types of axisymmetrical structure are presented

here as representative use of the SABA element. The first is a torus

under constant internal pressure and the second a cooling tower under

a distributed wind load. Results for both structures have been obtained

with the SABA 5 element (using fifth order Hermitian functions for the

meridional interpolations) as this was deemed the best element to ful-

fil the original objectives. Analysis has also been made using SABA 7

which shows there is little advantage to be gained in the accuracy of

the results by using a higher order polynomial.

a) The Torus

This problem was originally chosen as a test case by virtue of

its analytical simplicity, since only a constant circumferential load-

ing is involved. However, it also proves to be a good test of the

element's ability to handle rapid changes in bending moment, which is

an important consideration when one wishes to use the minimum number of

elements.

The dimensions of the torus analysed and the material properties

are shown in Fig 6. There is no exact solution to this problem and so

the results are compared with those of Kalnins [32] who used a multi-

segment (predictor-corrector) integration method to solve the relevant

shell equations. Six equal elements were used over half of the sym-

metrical shell to obtain the results shown in Figs 7,8. The radial

displacements Fig 7 and membrane stresses Fig 8a are in perfect agree-

ment while the bending stresses Figs 8b and c show a minor disagreement

at the outside edge. The integration for Kalnin's method commences

at this edge and hence the difference may possibly be explained by an

error in the starting procedure, although this cannot be confirmed as

details are not given. Figs 9 and 10 investigate the convergence of

the element. A good result may be obtained using as few as three

elements, convergence of the solution being very rapid as the number of

elements is increased. . Consideration of the bending moment diagrams

shows that three elements would be sufficient for one per major turn-

ing point, which is the minimum number one might expect to give a

reasonable solution. This suggests a principle for selecting the

optimum number of elements to be used in subsequent examples.

b) The Cooling Tower

The cooling tower analysed is shown in Fig 11 and represents a

typical example of those presently in use, with the restriction that

the materials' properties are considered isotropic and no provision

is made for the stiffness of the supporting legs. The tower has

however been analysed previously by Albasiny and Martin [36] using

the finite difference method, as well as by the finite element method

36

Internal Pressure = 103 Ibif. / in? E = 107 LW. / in? v = 0.3

FIG. 6 PRESSURISED TORUS

1

4

z

3

c 0 U it 2

Finite element solution X X Kalnins's solution

311. 0 20 40 . 60 80 100 120 140 160 180

se

FIG. 7 RADIAL DEFLECTION OF TORUS

x

FINITE ELEMENT SOLUTION

X KALNINS'S SOLUTION

20

40 • 60

80

100

120

140 160

180

FIG.8a MEMBRANE STRESSES N®

0-2

0-1

0

-0-2

X ...................00.0X-",..,... x

X

X 200 400 800 100° 120 1 60 180° X

tp0

FIG. 8b BENDING STRESSES Mg)

01

X X

20 40 60 80 100 120 140 180 ce)

0 CD

X 0

X i.

- 0.1

FIG. 8c BENDING STRESSES Me

ux lOa ins.

2

RADIAL DEFLECTION AT POINT A vs N'D• ELEMENTS.

42

KALNINS'S SOLUTION

I I I I I

4 6 8 10 12 NO. OF EL.

FIG. 9

01

20

0

02

-

x

60 80 100 120 140 x'-o160 180 tip

A

ax = 2 Elements

x = 3 Elements o = 4 Elements

= 6 Elements

FIG. 10 CONVERGENCE OF Mo STRESSES FOR INCREASING NO. OF ELEMENTS

44

with a variety of other elements. The former results are chosen here

for comparison purposes, since they seem to be the only ones so far

that give a reliable estimate of the stress distribution.

All vertical dimensions are measured from the throat of the tower,

which is assumed to have been generated by a hyperbola. For the pur-

pose of this analysis the base is taken to be 'built-in', although in f

practice the support legs would allow rotation about the circumferential

- axis. This, condition is simulated by suppressing the u uI, v v

I w and

I w displacements. The distributed wind loading is assumed to be sym-

metrical around the circumference and constant along the meridian of

the tower. The circumferential variation is shown non-dimensionally

in Fig 12, and may be decomposed into the first ten harmonics (includ-

ing the constant term) of a Fourier series as follows

10

pressure coefficient = Ep.cos i 0

= 0

where pi (i = 1 - 10) are

.22892

.27779

.59821

.47010

.06269

-.12010

-.02678

.04443

.00180

-.01981

The results are for a value of ipv2 = 1 lb/ft2, corresponding roughly

to a wind speed of 29 ft/sec. Five elements were used to produce the

graphs shown in Figs 13 - 18. The nodal spacing of the elements which

appears in Fig 15 corresponds to fractional division of the meridian

line which are, reading from the top of the tower,

.06 .24 .6 .07 .03

The SABA 5 results, marked by crosses, are compared with the

solid line representing Albasiny and Martin's finite difference solu-

tion. Agreement is very good in every case, even where there is a

violent variation of bending moment at the bottom of the tower (Fig 17).

It appears from the calculations that, apart from the built-in end

where the edge conditions are dominant, the meridional bending moment

is largely induced by the effect of Poisson's ratio as a consequence

of the circumferential bending moment Me. For this reason M

0 is

chosen as an instrument for the comparison of results from solutions

with a different number of elements and different element spacings.

A tolerable result has been obtained with as few as four elements

although convergence to the true solution is not reached everywhere

until five or more elements are used. The elements are spaced so as

to give an optimum coverage of the major turning points in the Me dis-

tribution. It can be seen in Fig 15 that at some of the element

boundaries, although the bending moment is continuous there are small

'kinks' indicating that the slopes of the bending moment are not con-

tinuous, as would be expected. This can be seen most clearly at the

-240' position. An analysis with SABA 7 which has continuity of the

normal shear resultants (and hence the first derivative of the bending

moment) eradicated these discontinuities) In addition)by virtue of

the closeness in the results proved the sufficiency of the SABA 5 ele-

ment in this context. It is evident that the use of only five ele-

ments is quite adequate for the representation of the behaviour of the

structure under the given loading.

45

46

Other finite element solutions of the same problem have been obtained,

using such elements as TRIB 3C [37] and the axisymmetrical element SABOR

[30]. Both these elements give good predictions for the displacements

but by their very nature fail to give a satisfactory solution for the

bending moments. The SABA element with 9 unknowns at each node has 54

variables per harmonic. Compared to the SABOR program, which required

something like 80 variables per harmonic to give the same accuracy in

the displacements but with an inferior stress distribution. The tri-

angular TRIB 3C element required 1200 unknowns, and by comparison pro-

duced a most inaccurate result. The SABA element is thus a great

improvement on both of these.

1.12 Conclusion to the Work on Small Displacements

The analysis of the cooling tower using five SABA 5 element takes

only 0.9 minutes of computer time on the IBM 360/75. This solution

time, which is all important in considering an extension to large dis-

placement problems, compares very favourably with the other existing

methods of solution. SABA was deliberately conceived as a more sophis-

ticated element than existing.alternatives, so that fewer would be

needed for an accurate solution. This approach seems to have been

amply justified in that the complete linear solution has been obtained

at no extra expense compared with earlier attempts. Difficulty has

been found in calculating the stresses for previous elements which have

assumed at most cubic displacement functions, as continuity of curva-

ture is not obtained and considerable effort has been expended in inter-

preting the bending moment results. The ease with which a solution

is obtained using SABA once again supports the philosophy of using a

11144*-number of higher order elements. The SHEBA 6 [21] triangular shell element which is similar in derivation to SABA 5 would also have

874'

THROAT RADIUS 84'

47

E= 4- 32 x108 lbs/ft2

V= 015

12

1 b2

0 (NJ

0.583'

-BUILT IN ALONG THIS LINE

FIG. 11 HYPERBOLIC COOLING TOWER

1:6

- 0.4

1.2

1

COEF

F IC

IEN

T

0.8

0.4

• 1

30 60 90 120 150 180 , - 0-8

0

8 IN DEGREES

FIG.12 CIRCUMFERENTIAL DISTRIBUTION OF WIND PRESSURE

= • 001 ft.

FINITE DIFFERENCE SABA 5

49

FIG. 13 HORIZONTAL DEFLECTION AT TOP OF TOWER

601-

30

0

H -30 w

-60

cr z -90

cr -120 u_

0 G-J -150 I

-180

-210

50

X

x

-240 - ix x I

xxi x 1 1 1 t 1 i 1 I ,....

0 • 05 01 015 0.2 025 03 035 0.4 u x102 Ft.

FIG. 14 RADIAL DEFLECTION AT e = 0°

z 0 0

0 z

POSI

TIO N

OF

11••••••

••••••••

w

z

-90 0

-120

-150

-180

7)2

1 0

E x240 X

60 x 6

30 -

0

-30 -

- 60 -

x = SABA 5 A = SA BA 7

X

0 2

4

51

-4 -2 x A X J to'

6

8 10 mo lbf.

FIG. 15 M0 AT e=o°

I

52

-60-

-90-

-120-

-150-

-180-i

-210

-240- x

x x x

I I t IC I I I -3 -2 -1 0 1 2 3 4 5

Ibt

FIG. 16 Mt, AT e =90°

MEW

IMO

Om,

t 20

NNW

PIM

24

16

8

0

-8

4dx 60 80 100 ,0„.-----x 1-1„t0_)1.60 1t0 e. . I I . . . 1 .....

x

Is x -16

-24

FIG. 17 M 0 AT -267 FT (NEAR BASE)

60 54

30

-30

-60

z -90

-120 0 Li.

-150

. -180

-210

-240

=10 ELEMENTS

a =8

x-5

-10- 0

10

20 Me lbf

FIG. 18 CONVERGENCE OF RESULTS FOR M e

55

the above advantages but would involve a longer solution time, as no

simplification can be made which would take advantage of the symmetry

of the problem. The SABA element is therefore considered an excellent

element for the solution of axisymmetric thin shell problems and a

good foundation for the extension to the large displacement analysis.

The SABA 5 element, in particular, is used extensively in the develop-

ment of the large displacement theory.

56

PART 2 - Large Displacements

2.1 Introduction

Having established a reliable finite element, which fulfils the

objectives originally suggested as a sound basis for the large dis-

placement problem, it is now necessary to extend the theory to include

the effect of finite displacements. Such a consideration leads to

a non-linear analysis which is generally termed geometric non-linearity

to distinguish it from other non-linear effects, e.g. material behaviour,

which may be included if one wishes. In the classical theory of con-

tinuum mechanics, the non-linear terms arise from the inclusion of second

order terms in the strain-displacement relations. A different approach

is adopted here however as a linearised incremental method is used to

solve the non-linear equations. The force-displacement relationship

is constructed in a series of small increments, for each of which the

displacement is small and hence the linearised theory can be applied.

However, the usual elastic stiffness at the beginning of each step is

based on the changed geometry of the structure rather than the unde-

deformed geometry, and in addition, the effect of the existing stress on

the subsequent deformation is taken into account. This latter con-

sideration gives rise to an additional stiffness which is often termed

the 'geometric' stiffness. This is entirely equivalent to the ortho-

dox approach.

In the derivation of the stiffness matrices for the following

analysis, one further assumption has been made. It is that the radial

displacement of the shell is, even in a large displacement analysis,

only of the same order as the thickness of the shell, which is small

when compared with its radius. Hence the radius of the shell may be

assumed constant. This enables the strain-displacement relationship

57

for the axisymmetrical case, as derived by Novozhilov and adopted in

Part 1, to continue to be used. It is a perfectly adequate assumption

for the majority of thin shell problems up to a consideration of the

initial buckling load. It will also be suitable for cases where the

structure remains axisymmetrical when deformed, such as a spherical

cap or a torus under uniform pressure; and will be able to predict

the buckling load of any initially axisymmetrical shell structure.

However it will not be capable of giving the deformation right up to

the point of collapse, when the geometry would have completely changed.

To do that it would be necessary to use a more general form of the

strain-displacement relationship for the shell. The complexity of

the theory and the programming would be considerably increased which

in turn would mean the cost of computing such a problem would be pro-

hibitive with the present day computers.

The geometric stiffness of the SABA element is obtained by examin-

ing the virtual work done due to each item of the existing stress

resultants and applied loading at the beginning of an incremental step,

on the subsequently imposed virtual deformation. This is an applica-

tion of the basic principle first suggested by Argyris and Scharpf

in reference [21], where it was illustrated on an alternative formula-

tion for the geometric stiffness of the beam element. The method is

general and may be applied to any other element in a similar manner, •

depending only upon a fundamental, mechanical argument. It has the

advantage that the retention or rejection of any particular term is

clearly indicated by its relative physical importance in given circum-

stances, and does not depend on a purely mathematical assessment. In

particular, the effect of the applied pressure acting on the shell at

the beginning of the incremental step, may be readily included. This

contribution could easily be overlooked otherwise. As a result the

58

geometric stiffness matrix of the SABA element is made up of a series

of matrices each one arising from a membrane stress or normal shear

stress resultant, or existing pressure force, although not all are of

the same degree of importance at the same time. The bending stresses

give no contribution to the geometric stiffness, since the state of

equilibrium of a moment remains undisturbed in any displaced position.

In the small displacement analysis of Part 1, the displacement

of each harmonic depends only on the loading of the same harmonic

variation, and the solutions for them can therefore be obtained sep-

arately. This is no longer possible in the non-liriear case, as

coupling in general occurs between all the harmonics. The solution

of the equations becomes a much more complex matter and leads to a

regrouping of the displacement parameters to procure a more efficient

reduction.

The displacement increment corresponding to an increment of load

may be obtained with a stiffness matrix calculated from the geometry • and the existing stresses at the beginning of the incremental step.

The relationship between the load and displacement increments is then

simply linear, based on the initial tangent stiffness. An improve-

ment to this approximation can be achieved by evaluating the stiffness

at the end of the incremental step also, and then repeating the cal-

culation using the average stiffness between the beginning and the

end. Thus an iterative loop is executed for each incremental step

and can be continued until the result converges, It will be shown

that this procedure is equivalent to the inclusion of second order

terms in the equation between load and displacement increments, i.e.

a quadratic approximation for each step. The calculations given in

this paper were all performed with only two iterations. The reason

for doing this is merely to save on computing time. It should not ser-

59

iously affect the accuracy of the results and gives an enormous improve-

ment on a purely linear calculation.

The following sections describe in detail the derivation of the

geometric stiffness matrix and the development of the structural stiff-

ness matrix. The incremental methods used in producing the load/

displacement curves are explained including a method for entering the

post-buckled region of the curve. Finally a number of examples are

used to illustrate the theory, the results being compared with existing

solutions.

2.2 General Outline of the Linearised Incremental Approach

The incremental approach to large displacement finite element

analysis has been well established. Reference may be made to the work

of Argyris [4, 5 and 20] for a formal development of the method applied

to beam and plate elements. However, to facilitate the following

derivation of the geometric stiffness matrix for the SABA elements an

outline of the general theory is given at element level.

For any equilibrating set of applied nodal forces p and internal

stresses a, and any compatible system of stresses, displacements

and strains E, the principle of virtual work gives

t P E t a dv (2.1)

This is the equation used in obtaining the equilibrium equations in

the displacement method. In the case of small displacements the

strain is assumed to be a known function of the displacements in the

form of

a p (2.2)

Iwhere a is, based on the undeformed geometry of the elements.

When the displacements are large, the relationship between the

strain and displacement becomes non-linear and eqn (2.2) is no longer

valid. However, the loading may be conceived as being applied in

stages, and if the increments are sufficiently small, the corresponding

displacements for each step will also be small. In this case, the

strain increment dE may still be taken as linearly proportional to the

displacement increment dp as in eqn (2.2) giving

d E and P

(2.3)

where ali o is now based on the geometry at the beginning of the loading

step. The strain obtained is therefore not Lagrangian as it is

derived at each step from a new geometry.

Its corresponding stress is a as in the small displacement

analysis.

The equilibrium equation between the applied forces Po and the

internal stresses Cr o at the beginning of the step is obtained from

the principle of virtual work by the use of a virtual system of com-

patible strains and displacements satisfying eqn (2.3), resulting in

Po aot uo dv (2.4)

The corresponding strain and displacement are Eo and porespectively.

After the application of a force increment di', the quantities

at the end of the step become

applied force = Po d P

stress = ao + da

displacement = P6 d p

strain = Eo

d E

The relationship between the strain and displacement increments changes

also from eqn (2.3) to

60

61

d E

(ao cia)d P (2.5)

since the structure has deformed further during the step, which must be

taken into account. Equilibrium between the force and stress at this

instance is again expressed by the virtual work principle, where the

compatibility of the virtual strain/displacement system must be in

accordance with eqn (2.5).

Thus

Sp (P0 +dP ) 8 Et( a + do' ) dv 0

8 Pt ( aot+ d at) (cro +d a ) dv (2.6)

Then using eqn (2.4)

dp

f (at dcr + d atao + d atda ) dv o

(2.7)

where the final term is second order and may be ignored for the time

being.

It is evident that after the substitution of

da K dE = Kadp 0 (2.8)

where K is the material stiffness, the first term relates part of dP

to dp by the usual elastic stiffness

d Pe - ke d p (2.9)

where

k e =f ae K ao dv (2.10)

with o. °based on the geometry at the beginning of the step. The remain-

ing part of . dP arises from the effect of the existing stress cro on the

where

th. due to the change of the 3 element of di). The matrix k g is known

dai is .th a square matrix. Its 3 column being the change dai dp

62

change of da, which is a consequence of the deformation during the step.

th If e. is the i element of the column cr , corresponding to the i

th

column (dal.) of the dat matrix, the second term in (2.7) may be re-

written in the form

d Pg = f dat CTo d V = Eai da. dV

(_. dV) d p dp elp (2.11)

as the geometric stiffness matrix.

Eqn 2.7 now becomes

P = dP dP = (ke + kg) dP

ko (2.12)

and the assembly of this equation for all elements in the usual way

leads to

dR = (Ke + K ) =Kdr (2.13)

for the complete structure. This is the incremental load/displacement

relationship in its simplest form. Increments of the initial load

dJ(see eqn (1.54)), may be added when appropriate.

2.3 The Geometric Stiffness for an Infinitesimal Shell Element

The 'geometric stiffness given in eqn 2.11 may be obtained in some

cases by an evaluation of the symmetric matrix 1111. dp

tation of this matrix is however difficult and care must be taken if

this approach is used as may be seen from Ref. 20. An alternative 1

The interpre-

63

method is used in this paper which is also more convenient. The

principle involved was first given at the very end of Ref. 20, where it

was applied to the case of a beam in space. The geometric stiffness

matrix is obtained by considering the equilibrium of the existing stress

system pro and how this is altered by an increment of displacement.

The additional force required to restore the equilibrium gives rise to

the additional stiffness as in 2.11. This approach may be applied to

the shell problem as follows.

Consider an infinitesimal shell element (Fig 19) in a state of

equilibrium under an existing system of stresses (represented by stress

resultants) acting on its edges. During an incremental step, the

element will suffer further deformations (extensions and rotations), as

a result of which the original equilibrium state of the stresses is

disturbed. Extra forces are then required to sustain the state of

eqpilibrium in the new position. Thus the structure acquires an addi-

tional stiffness whose influence is most conveniently calculated as a

virtual work. Take the typical forces on the meridional face:

a] Membrane force N0 rde (Fig 19a)

Due to an increment of rotation dw 0'

duces a bending moment about the 8-axis.

tained by an extra moment dM0 = N

0 dw0 r de

imposition of a virtual displacement dwrp ,

07a =(I) N

(I) dw ) dA

where dA*= area of element = ds r de.

this membrane force intro-

Equilibrium can only be main-

ds being applied. On the

this moment gives virtual work

(2.14)

N. I (c) S

n

ds

dM 55% dwods d NO-6- .% WO

(a)

dWO QodWoc/NC:::" ".

\ I

.0)

dNo-oe- dWgi_\#.1

1Q0

64

d(A) 0 ds

"ro dwo -

pdAdWo

'4 d pdA

I ic 1/4 ..

dM0 dEds-- du

(d)

FIG. 19 FORCES ACTING ON AN ELEMENT

b] Normal shear force Q rd0 (Fig 19b)

This time the equilibrium can be disturbed in two ways. Firstly,

as a result of the strain increment deb, the length of the element

changes by deeds, inducing an unbalanced bending moment dM, = %deg) dsrd0

which would rotate the element unless an opposite moment is imposed.

Secondly, an increment of rotation dw(1) produces a component of force

Q rdedw along the plane of the element in the displaced position,

which would extend the element unless equal and opposite force dii(1, is

applied. The virtual work done by these additional forces is:

(2.15) alb = Ow4)(14)deq) -1-dc(1) yydA

This also shows the symmetric nature of the stiffness matrix as these

two terms separately give rise to unsymmetric matrices, one of which is

the transpose of the other.

c] In-plane shear force Srd8 (Fig 19c)

As a result of an incremental rotation dw1, this force moves through

a distance dsdw requiring an additional bending moment dM0 = Sdw dA

which does work on the subsequently imposed virtual rotation dwe. A

similar term arises from the complementary shear force so that the total

work from the in-plane shear is

c. =

0 Sdw(I) + dw(I) Sdw

0 )dA

(2.16)

Similar terms to those produced in a] and b] may be constructed with

N0 and Q

8 respectively. Clearly the resultant moments never enter into

the calculation of the additional stiffness since equilibrium of the

element is not altered by the displacement of their positions of appli-

cation. Contributions from terms involving increments of rotation in

the plane of the element may be ignored since the effect of such a de-

formation is clearly negligible. The total virtual work on the element

65

66

due to the effect of the existing stresses is then

SW = Owel(pdw(1) + dwoNodwo 6wsyscp

• 6c(pylw(1) + Swoyso + Scoywo

▪ (Sy dwo + (Sy dA

pt d pg (2.17)

where the last equal sign equates it to the virtual work of the extra,

geometric force. It only remains to express the virtual deformation

Sw(1) 000 etc. in terms of the virtual displacement•&p, and similarly

d(1) etc., for the particular element, to obtain the geometric stiffness

relating the force increment dl?g to the displacement incrementAip.

The detailed development for the SABA element follows later.

2.4 The Additional Geometric Stiffness Due to the Applied Pressure

If the shell structure is loaded by a pressure force distributed

over its surface, then in the original equilibrium of an infinitesimal

element (Fig 1), this pressure should also be. included. Hence the

presence of this pressure force will in general contribute to the geo-

metric stiffness in a subsequent deformation of the element as well as

the other forces although the contribution is more usually small as

will be shown.

Referring to Fig 19d, consider first an increment of rotation

dw • The pressure on the element will produce a_component pdw4 in

the meridional direction, and the equilibrating force then does work

over the virtual strain dc(1) (Su = SE(1) ds) equal to ipdw(I) dAdu. Alter-

• natively, due to a strain increment dce the resultant force pdA moves

parallel through a distance of idu, requiring a balancing moment

dM = ipdAdu which does work on an imposed virtual rotation dw . (I)

67

Similar consideration in the 0-direction gives the total contribution to

the virtual work by the existing pressure as

SW = ip(dm u + Sudw + dmdv + Svdmo) dA (2.18)

(I)

which should be added to the SW of eqn 2.17.

Hence in general an additional geometric stiffness arises due to

the applied loading. This is similar to the so-called initial load

stiffness matrix mentioned by Oden and Hibbitt in references [15] and

[23]. They illustrate the same principle by the example of centri-

fugal body force loading. The present derivation shows how it may be

derived from a simple physical consideration. The importance of the

stiffness may also be evaluated by consideration of the virtual work

terms. When.,.,,the-shell4lis loaded by ,a essurt,p,i the meMbra,ne440esS ,. .- • , - - - -••%'••:,':,C•,4 - ; 4".• .' . ),

' .*It resultants 4re offhe'',brder of p times' radius of,iffe:Sheet r:.;; 'From,

- '.. • ,,,,:. .,... , 47he end term o n 2.17, theVirUai'Wpri due to the meMbrane. ' '

. •

••,•:.,, • .,

4s4'is proportional to prdw2, while tha.:last/tw9 terms of eqn

- — ile::the - virtual4Ordue to preiifurecas gdw0471iei sn, when the order of

. .

lailusion pr othetwise;af'thd.-pressure - •

114Dme•casehowev6il aSIvii4d4e"

in the examples of application, the result could be affected quite

appreciably.

2.5 Derivation of the Geometric Stiffness Matrix for the SABA Element

The following analysis deals exclusively with formulation of the

geometric stiffness matrix for the SABA element and will refer to the

notation used in Part 1.

The deformations required for this development may conveniently

be written into the column matrix

dviiA much smaller than fdto.

tefm litfiei'difference.

68

GO. = {to(I) we

(I) c0 u

). 0 3

(2.19)

th. which may be related to the 3 harmonic of the local displacement

vector

{ u v w }.

by

a

where

1Ir

0

L 8i

cosr r30

0

sin (I)/r

0

a

3 L 1

L 2

L 3

L 14-

L 5

L 6

(2.20)

(2.21)

req)

r30

1/r

sin (1)/r

0

0

r e(1)

1

0

0

1

In writing eqn (2.18), it is tacitly assumed that the radial displace-

ment is very much smaller in magnitude when compared with the radius of

the shell, so that the structure remains practically a shell of revo-

lution for the purpose of calculating the strain increments. The prin-

cipal radii are still taken to be r0 and re and the expressions for the

axi-symmetrical shell theory as given in Ref. 18 are then valid. This

assumption applies also to the evaluation of the elastic stiffness

efor the incremental calculation. This allows the use of the small

69

displacement theory given in Part 1. The following theory is there-

fore applicable only if this condition is satisfied. It may be used,

for example, in the initial buckling stages of a cylinder, but will

not be true in the post-buckling stages when the cylinder is near to

the state of final collapse. This assumption greatly simplifies the

subsequent mathematics and computation and is perfectly adequate for

most thin shell problems.

th. Employing the definitions from eqns 1.6 to 1.13, the 3 harmonic

th. of the local displacement increments is given in terms of the 3

nodal displacement increments d p' by

d6. = T H F. dpj (2.22)

Hence

d wj = L THFA dp J . (2.23)

th Likewise, a virtual deformation in the

.harmoniC

= LTHFiASpi (2.24)

th The

.harmonic of the force increment can now be related drgi

th to the j harmonic of the displacement increment dpi by the applica-

tionofavirtualdisplacement8—Then, according to eqns 2.17

and 2.18, and after the elimination of Sp. from both sides,

dp. ( ) . . dp g J

(2.25)

where

( k At {.1Filit,rt[No • NottaLz+ LtiL3 g ) l j

+ Q041141- go 44+ go L4tL2 + S LILz S LtzLi

+ 1/2p(LiLs LtL1 + 44+ • LtiLz)] THF jdA}A

(2.26)

70

and Li etc. are the rows of the L matrix in eqn 2.21. It is most con-

venient to define the matrix

114i

. j . L T TIP.

{ M1 M2 M-5 MF M5 M6 }

(2.27)

whose 6 rows correspond to the 6 rows of the operator L . The expan-

sion of this matrix appears in full in Fig 20 overleaf. Using eqn

2.27 with 2.26 produces

( kg = At {.1. [*i.NoM1 j M2i.NeM 2 j yla3i

Rift n MK Wfft nff t n

'2ineLNi4j 11/14c% e M 2j

• MUS M 2j M2i SMij '12 -1" ( M tii M 3j

• 111.5i 1W ij + 2i NI 6 j hlta AI 2J.)]dA}A

(2.28)

th This is the geometric stiffness coupling the j harmonic of displace-

ment to the harmonic of the load increment, and is obviously

symmetrical with respect to i and j.

2.6 The Coupling between Harmonics in the Large Displacement Problem

In eqn 2.28 the stress resultant or the applied pressure involved

in each term is also made up from a series of harmonics. In general,

for example, we express Nri, as

N = N(po + N

Ot cos kO + N

Ot sin k0

(2.29)

This means that the integrands always contain the multiplication of

3 harmonic functions. The four possible combinations are

(a) sin 2,0 sin me sin nO

MATRIX M J

sin 4) Q.

13 • J ll

sin ¢ ' r P3 • 11

0

sin 4 —p n r '

_ coQ P • i/ • J

cos cp. r P •J ri

cos 2 PJ •

0 sin 4) P • q

0 I — r

, 'i ll

0

cos ¢ p j T) 0 sin q) P jTl

0

FIG. 20

(b) sin tO sin m0 cos nO

(c) sin StO cos m0 cos nO

(d) cos P.O cos m0 cos nO

where k, m and n can be any one of i j or k (32, # m # n). It may be

easily proved that the integration of these functions around the cir-

cumference of the shell is non-zero only if the relationship

++ - m - n = 0 (2.30)

can be satisfied. If one of the harmonic numbers is zero (the con-

stant term in the harmonic series), then the other two must be equal

and must both belong to the same harmonic function (either sine or

cosine) to give a non-zero integral. In any case since (a) and (c)

contain an odd number of sine functions, they are anti-symmetrical

about 8 = 11' and their integral between 0 and 2n must identically

vanish. Hence only types (b) and (d) with harmonic number satisfying

eqn (2.30) need to be considered. Furthermore, as in the calculation

of 'e (1.33), all integrals involving non-zero harmonics have the

value Tr and those involving only the constant terms have the value

2n. Thus the number of relevant terms in the calculation is greatly

reduced.

Consider the example of a symmetrically loaded structure. In

the first incremental step there is no, geometric stiffness, the dis-

placements and stresses are given by the linear analysis and hence

all are symmetrical. This means that the direct stress resultants

N(i)' N0 and the normal shear Q

(I) will have a constant term and cosine

variations, while the normal shear Q0 and the in-plane shear S can

only have sine functions. As the load increment dPi. is symmetrical

72

73

and associated only with symmetrical virtual displacements, the matrices

Fl will be given as in eqn 1.11. After the operation by L in eqn

2.28, the 1st, 3rd and 5th rows of Nniyill only contain cosine func-

tions, which include the constant term as a special case; and the 2nd

and 6th rows of Micontain only sine functions. The non-zero terms

in the integration of (kg)ij can only be obtained from nAjwith rows

as given in the following table: (where c = cosine functions; s =

sine functions)

Terms

a. Fk M.

mt 1

Mt N O 2

Mt

1

Mt

Q 3

Mt

2 go

Mk go

Mtn S

.Mt

S 2 mt 1

Mt

p 5

mt P 2

Mt

P 6

M1

M2

M1

M4

M2

M2

M1

M5

1

N6

M2

Function Variations

M. Fk M.

c c c

s c s

c c c

c c c

s s c

c s s

c s s

s c

c c c

c c c

s c s

s c s

+ - with harmonic numbers satisfying i - j - k = 0. From this table

it may be concluded thatfik has only symmetric functions in the pre-

sence of symmetric stresses and virtual displacements and is of the

same form as IVl• This agrees with the physical principle that sym-

metrical loading can only produce symmetrical displacements, and no

anti-symmetrical terms can arise in the calculation.

In the case of an anti-symmetrical loadingfrthe displacement incre-

ments are not confined to purely anti-symmetrical ones. To see this,

it is sufficient to consider the geometric stiffness given by, say,

-the term nAlli mAii alone.

11 N(Pic An M 1 i

s s c

s c s

c c c

For an anti-symmetrical loading, the columnMt~will contain sine

functions, and so will the stress resultant No. Non-zero stiffness

can then be obtained with RAlicontaining cosine functions, which is

produced by symmetrical displacements. Hence the.anti-symmetrical

load increments are coupled not only to anti-symmetrical displacement

increments, but also to symmetrical ones. This may be explained

physically by the fact that the existence of 'the anti-symmetrical

stresses destroyed the symmetrical property of the structure, so that

it is no longer axi-symmetrical after the first loading increment.

It should be noted that this may include the bifurcation phenomenon

of anti-symmetric displacements occurring under a symmetric loading.

It is shown later that problems involving bifurcation in which a har-

monic displacement occurs under, for example, a constant loading

require the harmonic to be present in the form of an eccentricity

before a buckling solution may be obtained. Hence an anti-symmetric

buckled mode merely assumes that an anti-symmetric harmonic is pre-

sent in the original analysis.

74

0 1 2 3 I -1 -2. -3

0,2 1,3 2,4 1 -2 -1,-3 -2,-4

0,4 1,5 1-1,-3 -4 -1,-5

kc

0,6 1-2,-4 -1,-5 -6

cosine

Symmetric

(2)t o,6

(3) 1

ks (2)

1

10,2 1,3 2,4

0,4 1,5 sine

As a result of the requirement that i - j ± k = 0, the stress

harmonics k which enable the loading harmonic i to couple with the dis-

placement harmonic j in the geometric stiffness matrix for a symmetri-

cal loading case are given as follows:

0 1 2 3 If 5

0 0 1 2 3 4 5

1 1 0,2 1,3 2,4 3,5 4,6

2 0,4 1,5 2,6 3,7

3 0,6 1,7 2,8

0,8 1,9

5 0,10

•

The pattern may be easily extended.

Similar tables can be constructed for the antisymmetrical case

which will include the coupling between sine and cosine terms

cosine sine 0 1 2 3 j -1 - 2 -3 ... -j

( - indicates sine harmonic )

75

0

2

3

•

_2

-3

76

Coupling between anti-symmetric harmonics of load and displacement

[3] is through the presence of symmetric stresses only (kc). The anti-

symmetric stresses (ks) lead to the coupling terms between the sym-

metric and anti-symmetric loads and displacements [2]. Block [2] is

itself in general unsymmetric.

2.7 Formation of the Large Displacement Structural Stiffness

As has been shown the large displacement problem involves coupling,

in general, between all harmonics. This means that the simple method

of solution used for the linear theory is no longer the most efficient

as the displacements cannot be solved for each harmonic separately.

The solution of the equations becomes a much more complex matter. It

is now advisable to group all the displacement parameters of all har-

monics at a nodal point together, rather than assembling all nodal

displacements of the same harmonic as for the linear analysis. Since

coupling occurs only with adjacent nodes, the stiffness matrix acquires

a 3-banded block structure, where each block is a square matrix of the

order of the number of harmonics times the number of displacement

parameters at a node.

As before a new column vector is defined [eqn 1.35]so that the para-

meters for one node and one harmonic are grouped together

Ai = c ps,I

and of course pi I

c pi (2.31)

The stiffness matrix corresponding to this new definition is then

kii ctl .(ga. .c (2.32)

when i = j the linear elastic stiffness corresponding to this defini-

I tion i.e. (from eqn 1.36) must be added to the geometric stiffness

1

016 = r2 0 0

. . g . .

1 0

0 1

[0] [ 0 0

_

g

.k 1. .j. . k

. 0 0 . . 0 . . 0

. 0 0 . . 1 . . 0 [

/ pc

1]

2

13i iP

77

th. hence the element stiffness matrix for the 1 harmonics of load and

displacement is

k.. = Ct ( kg + C xi (2.33)

Now if the displacement vector for the complete structure is defined

for N nodes as

1 r1 r2.. r

k. . r

N 1". (2.34)

where each rk is the displacement vector at node k for all L harmonics

1 r 1 ' r2

r r L[

j (2.35)

and each rj

is in turn the displacement vector at node k for the har-

monic j

rk = {u uI uII

v vI vII

w wI wII). (2.36)

As for the small displacement case there is a matrix a for the gth

element such that

. a g (2.37)

I However in addition there is a new relationship linking p gj for the

th gth

element and .

harmonic to Og

Pg (2.38)

C7. io

iii

1,1

i,J

•

]

1 j,i

L,L L D ,

1,1

SYMMETRIC

ilj

.,.

._ .. _

ti

..

ILI

1

L

1

L

Node 1

Node 2

_ — Similarly of course. la

I a. n = rga r g

Hence the stiffness for the gth

element including all harmonics is

given by

L L t

aikijaj

(2.39)

which has the following structure (in general fully populated)

Node

Node 2

Harmonics Harmonics

• L 1

j

78

The complete stiffness for the whole structure may be assembled as

before by

- E eka g g g

(2.40)

the structure for this being

1 2 gq- 1 ....

k --

g

2.8 Iterative Procedure for the Incremental Steps

The linearised incremental theory given in Section 2.2 may be

improved by a simple iterative technique. This non-linear incremental

step theory may be derived as follows.

After the application of the load incremental' , let the force

P +dP be equilibrated by a stress aro+ dcr+ id2a That is,

instead of just the linear increment as in Section 2.2, the second

order term in the Taylor series expansion is also taken into account.

Likewise, the strain/displacement relationship at the end of the step

may also be made more accurate than that of eqn (2.5) by writing

dE (a +d a + 2-d2a )dn (2.41)

where the second order change of a is included. Equilibrium between

the force and the stresses at the end of the step is once again

expressed by the virtual work principle, giving

79

80

Spt( PO + dP ) = J SEt (a+ + ) dv (2.42)

which on substitution of eqn (2.41) and taking into consideration the

initial equilibrium of eqn 2.4, becomes

dP .f[ 2 t t p coda + datl

go4-del do ( d a + a era] dv o 0 (2.43)

where only terms of the second order are retained. The first two

linear terms give the elastic and the geometric stiffness matrices

respectively, based on conditions at the beginning of the step.

Adding together they give the total stiffness !co of eqn (2.12)

kodp aotda + da o) dv (2.44)

It only remains to include the three second order terms, which may be

achieved by seeking the first order change of the stiffness. Taking

differentials of eqn (2.44):

2 dkdp = (clatda + ao

t da + da 2 t + d ) dv

Hence eqn (2.43) is simply

dP (k +2cik)d. p

The change of stiffness for an incremental step may be put.as

dk k- k 1 0

where k1 is the total stiffness at the end of the step.

Then eqn (2.40_ finally becomes

(2.45)

(2.46)

(2.47)

dP k 0 + )cip

(2.48)

81

This is in fact a non-linear relationship since k1 is dependent on the

displacement increments ap. However, a solution can be obtained

iteratively. As a first approximation the linearised incremental

theory can be used to find the displacements at the end of the step.

(This is equivalent to the assumption that I I = ko ). The ensuing

k1 is then used in eqn (2.48) and the calculation repeated until the

result converges. This simple iterative procedure, as has been seen,

is entirely equivalent to the inclusion of second order terms in the

incremental equation of equilibrium and the incremental relationship

between the strains and displacements.

2.9 The Displacement Increment Method of Solution

Two methods of solution have been used. One is the orthodox

method of incrementing the loading at every step, solving for the dis-

placement increments by eqn (2.13) and thus constructing the complete

load/displacement curve step by step. This works well generally

except for the buckling problems, where the stiffness matrix becomes

singular when buckling occurs. This difficulty can be overcome by

the second method of solution, which is to increment the most repre-

sentative critical displacement. Thus, instead of solving for all the

unknown displacements, the magnitude of one displacement increment is

specified. . The choice of this critical displacement is sometimes

straightforward, (e.g. the central displacement of a spherical cap

under uniform pressure) or, in any case, can be provided by the largest

displacement from a load increment calculation before it breaks down.

The set of linear equations 2.13 may be partitioned so that the

displacement that is to be specified is separated from the remaining

displacements. The equations are therefore rewritten as

82

dR r mit LxdRj

kll k121 r drj L k22J L dr

(2.49)

where A is the unknown size of the incremental load vector, dr is the

displacement to be specified and dR is its corresponding kinematically

equivalent load. The remaining loads and displacements being dRR and

dik. The second line of eqn 2.49 may be rewritten as

, -1 drR = K2( X c113,2 k21 dr) (2.50)

By substituting for di in eqn 2.49 an equation is obtained in which

A is the only unknown

-1 AdR = k

11 dr

1/12k22(XdPIR. - k21 dr)

Rearranging gives

= k11dr

dR

-1 t kip k 24 kAr 1(12 k 22 cm, R

(2.51)

The magnitude of the load step is therefore determined and it only

remains to substitute A in eqn 2.50 to obtain the remaining displace-

ments.

By this method, it is possible to enter into the post-buckled

region of some of the snap-through problems. The method is however

dependent on there being a stable deformed shape which may be reached

by a continuous positive increment of displacement. The method

fails if there is a bifurcation from the equilibrium position (as in

the case of one of the spherical caps discussed later).

2.10 Application of the SABA Element to the Non-linear Analysis of

Shells

The program outlined in Section 1.11 of this paper has been

extended to include the theory of the preceding sections in Part 2. The

SABA 5 element has been used throughout, following the success achieved

with it in Part 1. Solutions have been obtained for a number of

different axi-symmetric shells, in an attempt to illustrate different

aspects of the theory and to compare the results with other methods of

solution. Examples include the torus under external pressure for which

a comparison is made between the linear and non-linear stress distribu-

tions. Buckling loads are obtained for a series of caps and cylinders

and are compared with those derived from other sources. Finally an

analysis of the cooling tower brings the work to an end.

The calculations for the following examples have all been per-

formed using the iterative technique described in Section 2.8.

However, the iteration does not carry on until the convergence of the

result, but is terminated at the end of the second step. This is

adopted merely to save computing time. The result is a great improve-

ment on the accuracy of the linear incremental theory (no iteration),

and seems to be good enough when compared with calculations from other

sources.

a) Torus under external pressure

As in the first part of this paper the torus proves an excellent

test for the element. It is fortunate that comparison may again be

made with the results of A. Kalnins [33] who has provided a non-

- linear analysis. The deformation, however large, will always be axi-

symmetrical and hence only the constant term need be considered. The

half structure of the torus is shown in Fig 21 together with the

83

84

material properties. The meridian line was divided into five elements

as shown, the points of division corresponding approximately with the

peaks in the curve for the meridional moment Mc. This arrangement was

chosen to give the optimum number of elements; fewer would have led to

a poor solution as was shown in Section 1.11a. The external load was

applied in ten equal increments up to a value of 100 psi. The results

are shown in Figs 22-24.

It is interesting to note that the maximum radial deflection is

only one third of the thickness of the shell. In spite of this 'small'

deflection the meridional bending moment M and the membrane stress N (I) 0

are entirely different from that predicted by linear theory. The

method employed by Kalnins uses an iterative technique to solve the

governing non-linear equations and should be capable of a very accurate

solution. It was found that the results obtained using ten increments

lay very close to Kalnins solution though earlier attempts using a

smaller number of increments failed to reach the same peaks. The non-

linear terms obviously play a very large part in the analysis of this

structure and demonstrate that a linear analysis would be quite inade-

quate in predicting the bending moment distribution. It is concluded

that this example provides a very good comparison with other methods

of approach and shows the equivalence between the 'geometric' stiffness

method and the more classical solution. At the same time it justi-

fies the restriction of the iteration to two steps in that a good solu-

tion has been obtained in a severe test within a reasonable number of

increments.

b) Spherical Cap under Uniform Pressure

Like the torus the shallow spherical cap under constant pressure

loading is another axi-symmetric problem even in the post-buckled state

and no harmonic deformation need be considered. The two caps chosen

85

YOUNGS MODULUS = 1 x 107 p.s.i. POISSONS RATIO = 0.3 APPLIED PRESSURE = -100. p.s.i.

ELEMENT MERIDIAN LENGTHS (AS FRACTIONS OF HALF CIRCUMFERENCE)

ELEMENT 1 2 3 4 5

MERIDIAN LENGTH .22 •18 •13. •18 .29 ,

FIG. 21 TORUS UNDER EXTERNAL PRESSURE

NON- LINEAR

1 / \ 1 1 1

\—LINEAR ',...

%. ••••

I I I I I I I I 1

/ ....• / ‘

/

3

RADIAL DISPLACEMENT

x10-2 INS.

2

1

0 20 40 60 80 100 120 140 160 180 te

F I G. 2 2 TORUS RADIAL DISPLACEMENT

KALNINS SOLUTION X SABA 5 — LINEAR 11

Ne x 102 1 • ..--, ‘

LBS/IN. I // ‘ ‘ / / \

5 / \

..... .... \ ... / xe .... . / . \ x/

/ . . ‘

00 90° 18o'

FIG. 23 TORUS CIRCUMFERENTIAL MEMBRANE STRESS Ne

40.

4

2 MO

LBS IN / IN X

.• .007/ \

0 ."."-...."-N°•

-x- --x----x-- •

X • •

X

-2

-4

-6

-8

KALNI NS SOLUTION X SABA 5

U

--- LI NEAR

x ----X-X-

/ /

• / / X / \

I/

••• .....0

o° 90°

180°

F 1G. 24 TORUS MERIDIONAL BEN DING MOMENT Mo

89

for the analysis were both tested experimentally by Kaplan and rung

[38]. The first one (No.21) was also analysed by Gallagher et al.

[39] who used an eigenvalue approach together with the finite element

method to obtain the buckling load. As this cap showed a character-

istic of increasing stiffness with increasing deflection, another

example from Kaplan and Fling's report was chosen, which exhibited a

different behaviour in the experiment.

The dimensions of these caps and the material properties are

given in Fig 25. The edge conditions are assumed to be fully clamped,

hence•u, u', w and w' were suppressed at the outside node. Three

elements were used in the analysis dividing the structure into equal

angular divisions between the edge and the centre, with the nodal

circle at the centre of the central element degenerating into a point.

No significant change in results was found when five elements were

tried.

The buckling load is indicated by an indeterminate solution when

the load increment method is used. However, the displacement incre-

ment approach gives a more accurate estimation since it is possible to

continue the calculation even after the maximum ldad is reached. In

practice, the continuation of the curve is meaningless as the structure

would have 'snapped-through' to an entirely different configuration

after the buckling load is exceeded. Nevertheless it enables the

complete load/displacement curve to be calculated, and the post-

buckling or snap-through behaviour of the structure to be examined.

The points marked on both graphs indicate the increments taken in pro-

ducing the curve. It may be seen from Fig 26 that regular displace-

ment step sizes are quite sufficient to produce a smooth curve in the

region of the buckling load.

h

90

SPECIMEN* t INS h INS R INS r INS

4 •1 •365 22.8 4 21 •054 •413 19.56 4

YOUNGS MODULUS = 6.5 x 106 p.s.i.

POISSONS RATIO = 0.32

*Ref. ( 9 ) Kaplan and Fung

FIG. 25 SPHERICAL CAPS UNDER UNIFORM PRESSURE

100 PRESSURE

p. s.i. 75

50

25

A

125

0 2 4 6 8 10 12 14 16 Central deflection x102ins.

FIG. 26 KAPLAN & FUNG CAP No. 4

1.0 1.5

40 PRESSURE

p.s.i. 30

Central deflection x 102 ins.

FIG.27 KAPLAN & FUNG CAP N°• 21

93

Figs 26 and 27 show that the response of the two caps is entirely

different. Cap No.4 has a load/displacement curve which shows a

decrease in the stiffness up to the buckling load of 100 psi. This

compares well with the value calculated from the work of Budiansky,

Ref [40] which gives 97.5 psi. There is, however, a large discrepancy

between these results and the experimental value of 60.5 psi, which is

most probably accounted for by the uncertainty of the boundary condi-

tions, and the presence of initial eccentricities. Cap No.21 behaves

differently in that it becomes stiffer as the load is increased up to

the critical value. The calculation reaches a load of approximately

46 psi (Fig 27), at which point both methods become very sensitive to

the slightest change of displacement, which accounts for the difficulty

in obtaining an accurate solution; This is lower than that normally

obtained from an eigenvalue solution: 64.5 psi calculated from the

linear theory of Ref. 39, or 63.1 psi from Ref. 40, but is much closer

to the test result of 33.6 psi. Unfortunately they [38] did not give

the load/displacement curve for this particular test specimen.

However, from the curves for the other caps with comparable geometrical

parameters (e.g. Nos. 8 and 9) it is not unreasonable for it to behave

in a similar manner.

c) Asymmetrically loaded cap

In order to obtain a direct comparison for a case involving bar-

monic loading, a cap loaded over one half of its span is considered. - - -

The dimensions and properties of the cap which appear in Fig 28 are

taken from a paper by Ball [34]. Three elements were used as before,

no significant improvement being found with the use of five elements.

The edge conditions were once again full clamped. The rather crude

loading pattern given in Fig 28, is the same as that used by Ball and

94

is restricted to the first four Fourier terms. It is supposed to re-

present a constant loading over half the shell if Pcr = -33.1 psi,

which is the classical buckling load of a uniformly loaded sphere with

similar dimensions. The displacements in the analysis are also con-

fined to four harmonics,which is sufficient for the purpose of the

comparison.

The results are compared in Figs 29 and 30, in which point A

refers to the centre of the cap and point B corresponds roughly to the

position of the maximum deflection. There is close agreement between

the results to a point near buckling where the SABA element predicts

a lower critical load of .64 compared to .66 from Ball's method. It

is interesting to note that coupling between the harmonics produces a

significant displacement in the cos 20 term although no loading occurs

in this harmonic.

b) Buckling of a cylinder under circumferential pressure

The buckling of a cylinder under constant lateral pressure in-

volves the deformation of the circumference of the cylinder into a sym-

metric harmonic waveform. A prediction of this waveform and the

critical pressure have been obtained, using linearised theory, by

FlUgge [41] and Von Mises et al. [42]. Gerrard [43] compares the

results of his theory with experimental results and shows that close

agreement is obtained for the range of cylinders tested. For short

(low aspect ratio) cylinders the circumference buckles into a large

number of waves but for long cylinders where boundary conditions have

little effect the buckled form is elliptic corresponding to the

cos 28 harmonic. In the latter class of cylinders it is important

to retain second order terms which account for the movement of the

loading with the surface. This is because buckling in the lower har-

St

R= 1000"

6.02° 2.01°

YOUNGS MODULUS = 27.3 x 106 p.s.i.

POISSONS RATIO = 0.3

PLOT OF P(8) = -16.55 - 21cos 8+7.04 cos 38

-30

-20 P(0) p.s.i.

-10

45° 90°0 180°

FIG. 28 ASYMMETRICALLY LOADED CAP

95

PiPcr .6

.4

0 R.E. BALL RESULTS

4- SABA 5

-8

-2 .4 .6 .8

1.0

DISPLACEMENT Wit

FIG. 29 ASYMMETRICALLY LOADED SPHERICAL CAP

0 0 0 _ 0e cos, 38 0 cos 2e o~const o cos 8

o R. E. BALL RESULTS

SABA 5

P/Pcr •6

.1 -2 .3 .4 .5 .6

HARMONIC DISPLACEMENT WB/t

FIG. 30 ASYMMETRICALLY LOADED CAP DEFLECTIONS AT B (8=0°)

98

monics involves a considerable stretching movement in the circumferential

direction, (see Section 2.4) compared with the higher harmonics where

deformation is mainly surface rotation. Inclusion of the pressure

terms considerably lowers the buckling load. It is usual to ignore

these terms for the higher wave number solutions where their effect is

small.

A range of cylinders has been analysed using the SABA 5 element,

the dimensionsand properties of which are shown in Fig 31. The boun-

dary conditions assume that the radial displacement (u) and the merid-

ional curvature (u") are both zero at the ends and the axial displace-

ment only is restricted at the mid point. Half the cylinder was

needed, covered by one element. In general the non-linear solution

of an instability problem by an incremental method is only possible if

an eccentricity is introduced. Hence in the analysis of the 100 in.

cylinder, loads of the order of 10-6 smaller than the constant pressure

are added in a sufficient number of harmonics. The only methods for

finding what is a sufficient number, being experience or trial and

error. In this case four harmonics were used. In the first instance

results are obtained by a displacement increment approach. Fig 32

shows that when the constant pressure load reaches 9.6 psi the dis-

placement corresponding to the 3rd harmonic increases more rapidly

than any others, indicating that the cylinder buckles into the cos 30

waveform. In fact, further calculations show that the other harmonics

have little effect on the solution. If the harmonic in which the

cylinder buckles is known, then it is only necessary to apply a per-

turbation load of that harmonic in order to obtain the buckling load.

In that case, the calculation will only produce displacements in the

zero and the specified harmonics, and also in the multiples of that

harmonic. However when a 600 in. cylinder was analysed, which

buckles into the shape of an elliptical (cos 20) form (Fig 33), the

10"

L

99

YOUNGS MODULUS = 10 p.s.i POI SSONS RATIO = 0.3 RADIAL DISPLACEMENT AT A -.77- WA

-COMPARISON OF BUCKLING LOADS .

L IN

BUCKLED WAVEFORM

BUCKLING LOAD (p.s.i.) FLOGGE SABA 5 SABA 5*

30 cos 58 30.5 31. 4 30.5 100 cos 38 8.9 9.6 8.9 200 cos 28 4.0 4.9 4.0 600 cos 28 2.7 3.5 2 .7

* INCLUDING NON-LINEAR PRESSURE TERMS

FIG. 3.1 CYLINDER UNDER EXTERNAL PRESSURE

ft It SS o INCLUDING "

10 .-t""

cos 28

cos 38 /1(

X EXCLUDING NON -.LINEAR PRESSURE TERMS

cos 48

I I

1 I 2

3 4 5 6

HARMONICS OF WA x 10 INS.

FIG. 32 LOAD DISPLACEMENT CURVE FOR 100" CYLINDER

CONSTANT APPLIED 6 PRESSURE

p.s.i. 4

16 18 0 2 4 6 8 10 12 14

2nd HARMONIC OF WA X 106 INS.

FIG. 33 LOAD DISPLACEMENT CURVES FOR 200" AND 600' CYLINDERS

5

4

CONSTANT APPLIED 3 PRESSURE

p. s.i. 2

1 0 Including to 1111

200.

600"

x Excluding non - linear pressure terms

Um.

•

1

5th Harmonic 3 2

of WA X 106 ins.

•••

""" .414' .••• 30

FIG. 34 LOAD- DISPLACEMENT CURVE FOR 30° CYLINDER

CONSTANT APPLIED 20 PRESSURE

p. s. i.

10

- x Excluding non-linear pressure terms

0 Including ro 41

103

displacements of the 4th and 6th harmonic were many orders of magnitude

smaller than that of the 2nd harmonic, and made absolutely no differ-

ence to the buckling load by their inclusion. It is clear, therefore,

that no further harmonic need be included than the one in which the

cylinder buckles. The 200 in. in Fig 33 and the 30 in. cylinder in

Fig 34 are analysed accordingly, on the same basis. The table in

Fig 31 shows a comparison between the critical loads thus obtained and

those from FlUgge's theory. There is complete agreement between the

results in all cases if the effect of the applied pressure is included

in the geometric stiffness. When this effect is excluded, the results

from the SABA element show a large discrepancy. with FlUgge's theory

for the longer cylinders. This is equivalent to assuming the load

remains normal to the undeformed geometry.

c) The cooling tower

The cooling tower analysis followed as a natural extension of the

previous example. Two shells were analysed, the first a cooling

tower model under constant all round pressure. This enabled a compari-

son with results obtained from Dr. Ewing [44] in a private communica-

tion. The shell dimensions and material properties appear in Fig 35.

Four elements were used in this analysis which was aimed solely at pro-

ducing the buckling load. The element fractions of the full height

starting from the top of the tower were

0.1 0.6

0.24 0.06.

Four load increment steps were used in producing the load/displacement

curve which is given for the cos 50 harmonic in Fig 36. The inclusion

of other harmonics showed this to be the critical one. The graph

indicates the buckling load as 1.7 psi. This compares well with the

theoretical results of Ewing [44] of 1.8 psi, but is higher than the

experimental result of 1.4 psi.

104

The details of the full size tower which was analysed under the

distributed wind loading also appear in Fig 35. It should be noted

that the shell thickness was 5 in. which is different to the shell

analysed in Part 1. The tower was divided into five elements as in

Part 1 and had the same built-in boundary conditions at the base.

The analysis was performed in two parts, the first evaluates the effect

of the non-linear terms on the stress distribution prior to buckling,

the second finds the buckling load. Both parts used the wind load-

ing distributed as shown in Fig 12 and described in Part 1, being

constructed from 10 harmonics. The dynamic head taken for the first

part is ipv2 = 100 psf corresponding to a wind speed of 198 mph which

is in excess of the worst case one might normally encounter, but is

convenient for the purpose of our present analysis. The full load

is applied in four equal increments. Figs 37 - 42 show a selection

of the results obtained. The displacements and membrane stresses

are hardly affected by the non-linear terms. The bending stresses

M and M0 are markedly different in the top half of the shell, espec-

ially near the throat. Their distribution around the circumference

is also found to have changed. By further incrementing the dynamic

head an estimate of the buckling load has been achieved. The load/

displacement curve so obtained is shown in Fig 43. The cooling

tower would buckle at a dynamic head of about 250 psf corresponding

to a wind speed of 314 mph, the largest displacements occurring in

the 5th, 6th and 7th harmonics. Experimental work carried out on

models [46] indicate that the corresponding full scale winds speeds

for buckling lie between 230 and 280 mph. The analytical result once

again appears high which may be explained by the difficulty in pro-

ducing an accurate representation of the real tower and also by the

effect of imperfections which have not been accounted for.

105

h2 THROAT RADIUS

a x2 Y2 —2- b2 a

V

L BUILT IN ALONG THIS LINE

a hi hz. t ' E V D 2 -/b2.

COOLING TOWER MODEL 4 IN 11.92 IN 3.67IN .038IN 5.5xIi 0

5 p.s..

0.4 .16

FULL SIZE TOWER 84 FT 270 FT 60 FT 5 IN .3 2 x1 08 4 p m. 0.15 -16

FIG. 35 HYPERBOLIC COOLING TOWER

P p.5.1.

1.6

1.2

-8

.4

•

I ' I

-2 0 .2 -4 -6 -8 1.0 12 1.4 RADIAL DISPLACEMENT AT TOP OF TOWER FOR 5th HARMONIC

x 10-7 INS.

FIG. 36 LOAD DISPLACEMENT CURVE FOR COOLING TOWER MODEL

1-1 =0.1 ft.

0 NON-LINEAR X LINEAR

FIG.37 HORIZONTAL DEFLECTION AT TOP OF TOWER

107

No

LINEAR x NON-LINEAR

0 1 2 3 4 5 6 7 N x10 4

LBF /FT. FIG. 38 Ne AND N0 AT 8 = 00 FOR 5" SHELL

8 9 10 '8 Go

60

0

-30

70

-190

-230

-270 -2

109

1 Me x10-3 FT LB/FT

FIG. 39 M® AT 8 = 0 5" SHELL

110

+•••.....".er...............a.........t.............

-240

2 4

6 M0 x10

-2 FTLB/ FT

, F1G..40 M 0 AT e = 0° FOR 5" SHELL

F

60

6, LINEAR

X NON- LINEAR

Mex 10 2 FTLB/FT

,FIG. 41 VARIATION OF MG WITH 8 AT+ 60'

4 M010

2

FTLB/FT 3

LINEAR x NON— LINEAR

/°16> / \

X\

1 frii y , 7 .0- ,.......,xA

, , , , x / i A 1

•

i .,, I

1\ 45 901 135 00 180 \\ t

k, /xThc 1

X ..xi , ‘ . \ A

‘ ,

.44 I

VARIATION OF May WITH 0 AT + 40'

2

1

0

—1

—3 FIG. 42

400- •

300

1 2 pv lbf/fe'

200

100

1

2

FT. RADIAL DISPLACEMENT AT TOP OF TOWER

FIG. 43 LOAD DISPLACEMENT CURVE FOR 5" COOLING TOWER

The cooling tower has provided a good example of the use of the

SABA element in a practical analysis. It has been shown that a linear

analysis is inadequate in producing the correct distribution of bending

moment near the throat. The increased bending moment, if not allowed

for, could cause cracking of the shell along a vertical axis. The

buckling dynamic head is well in excess of any normal wind speeds and

shows that the shell stiffness is quite sufficient to prevent buckling

under a static wind load. It may be of interest to consider, for the

future, how the stress build-up affects the normal frequencies of the

tower and also to consider the interaction of a tower with the real

wind loading, which is turbulent. In addition the effect of founda-

tion settlement might be included. A more general analysis including

non-axisymmetric defects such as vertical cracks could be made with a

general triangular shell element but would of course be considerably

more complex.

2.11 Conclusion to the SABA Analysis

It is of interest to consider the considerable increase in com-

puting time needed for the large displacement analysis. The worst

case is the cooling tower analysis in view of the large number of

harmonics. The small displacement results were obtained in 0.9

mins on the IBM 360/75 whereas the incremental step (with two itera-

tions) of the large displacement analysis takes 12 minutes. This is

an order of magnitude greater and only represents onestep of the solu-

tion. This extra time is largely due to the coupling of the harmonics

which makes the solution procedure much more complicated. It is

therefore imperative that the number of harmonics be kept to a minimum.

The importance placed upon finding a solution with the minimum number

of nodal freedoms has also proved to have been well founded.

114

115

Overall, the accuracy of the SABA results compared with other

methods of solution has proved very good. This must largely be due

to the choice of displacement functions which give full compatibility

between elements both in displacements and stresses. The iteration

used for each step of the large displacement solution must also have

contributed and still greater accuracy could be achieved if the itera-

tion were carried further and a measure taken of the convergence.

This would of course be considerably more time consuming. From the

evidence obtained in the examples the SABA element may confidently

be applied to the analysis of any axi-symmetrical thin shell structure

for which the loading may be described by a reasonable number of har-

monics.

The objectives laid down in Section 1.1 have been fulfilled. A

geometric stiffness matrix has been derived for the case of an axi-

symmetric shell element under an arbitrary distributed loading and has

been used together with an incremental method of solution to solve a

range of non-linear problems, including the prediction of buckling

loads. The SABA element has been demonstrated to be a flexible and

capable tool which should be a useful addition to the existing family

of finite elements.

APPENDIX

Computational Procedures

3.1 Introduction

The following sections seek to enlarge upon some aspects of the

theory of the SABA element in order that they may more easily be pro-

grammed. With this in mind the matrix from which the normal shear

components are calculated is expanded, and the method for evaluating

terms within it, is developed. In addition, certain numerical tech-

niques necessary for the analysis are explained. The integration of

terms in the stiffness matrix may be performed exactly around the

circumference but requires a numerical method in the meridian direc-

tion. Gaussian quadrature has been found to be most successful for

this purpose and is therefore outlined for use with the SABA 5 element.

Because of the difference in length between the small and large dis-

placement equations separate routines are used for their solution.

The first uses the Cholesky method and assumes that the full set

of equations for one harmonic may be contained in the core of the

computer. The second uses a method of elimination coupled with the

Cholesky method tailored to suit the form of the large displacement

.equations and requires that the equations for all the harmonics of

one element be stored together. Algorithms for both these methods

are given. For most shells the nodal parameters used in inter-

polating the meridian curve for the element may be obtained directly

from the exact equations; however when this is not possible it may be

necessary to generate them from an alternative interpolation using,

say, Lagrange polynomials. From the four examples in this paper the

derivation of the nodal parameters is only given for the hyperbola,

the other three being trivial in comparison. The final section gives

116

the more important aspects of the Fortran IV program which has been

written to analyse structures using the SABA element. Flow diagrams

give a precis of the program structure explaining how each step is

taken.

3.2 Evaluation of the Normal Shear Terms Q0 and Q.

For most thin shell structures the normal shears play a small

part unless the shell is loaded by a bending moment. However should

they be required (as in the large displacement problem) they may be

calculated from eqn 1.26. In forming Q0 and 04 the derivatives of

the bending moment are required. Whilst derivatives with respect to

are straightforward, the corresponding derivatives with respect to

11 are less easily obtained. An evaluation of these derivatives rests

on the differentiation of the B matrix (Fig 5b) the lower half of

which may be referred to as AA.

R41 _ Et3 8Bm

an 0] Dn Pj 12(1- v2) El

Co o I)

and DT.

Eta 3 am P- an 12(1+v) 9n J

(3.1)

where Y11

0 Y13

Y21 Y22 Y23

V31 V32 Y33

and each .submatrix is a 1x6 the terms of which are listed overleaf.

117

Y. 11. =

ar o ) _2 cosM- r

m[(sin +cs

rcp 00 r(f) TI

(I) sin 11

]Pi/2 22

r, 11

Y (-cos + 2cos sin - -4 ) p /r2 - sin0 r--=It=g /9, — (cos24) — sin2 4) + sing) cos24) -

r r-1) p. 11 /kr

r2

0 (I) - sin cos 4 q

• rk2 Pi

sin0 r

Y31 - (sin24) - cos24) + 2sin cos24) r / ) p! rl/r2 + (2.sin0 — - 1) cos0 3 C11 • '1 (I) r kr 22 Pjr

0

Y22 =

r, . (1 - 2sin y

cosr) + sin -1=- P.111

r20 Y 7: kr2

Yz3 = (sin24) - cos24 + 2sin cos24) r( )

r r2

2r (r

p r, sin cos - cos 0) T

rk 22r ' - 0 sing) pi n

Y31

I 52.

I33

=

=

=

. 4, ate) II/ //

cos 4) qui sin -1 2 pin 4) q [(cos 4) - — + - r, P•in

r, ;4) r4, 2,

40 T 2,2 J 4.

/ rA r, (sin 4) + 2cos24)r / ) pl: ri /r2 - cos.4) -4=- piA l + (2sin (to +

cos r4)) c

o pjq

cos2 4, qj p .1.1 II 4) r j r £r J r22, 3 2,2r

r 3 2, 1.4) A, ptqf cos (I) r p!ti n

4) -(sin' + cos24) —) p:q + (2 cos24) r`" + sin 4)) --ar 4) 3

r2 2,2r

3.3 Calculation of Terms in B and a n

ar, Since the functions sin 41, cos 4 and r4' appear in B and

an

in , it is necessary to calculate these at any position of the

meridian line, their values being constant in the circumferential.

sense. The first three functions may be expanded as follows

sin 4)

cos (I)

1 az _ St, an

1 Dr t an

120

r4)

3Dr2- /2 1+

(az) (3.1)

a2riaz2

The derivatives an z may be calculated from the interpolated meridian

curve by using eqn 1.14

az .)11 G

an an

an an r/

(3.2)

a 2 z D 2 r Similarly may be formed when required.

ant ant

Let an , a2z

ant etc. be z', z" .. and aL a2r be r', r"

an ' an2

then ar _ r' - az

and a2r rt, z" r'

(3.3) az2 (z')2 (zI)3

from which r(i) may be constructed by the use of eqn 3.1.

121

The evaluation of

of az D2r — — and az ' az2

ark

an air az 3

is more complicated but may be written in terms

as

[ 3 (1 + (3r)2 aZ

(1 4. (Darz)2)3/2

at 3r 33r az

az3

az an

an

(-)a2r 2

az2

(3.4)

where-pi a r

and ---- are given by eqn 3.3 and 12-1 is found from az az2 az3

a 3r 3(z1)21., (z" r' + 3z"' r") r" (3.5) az3 (z')5 (z')4 (z1 ) 3

3.4 Numerical Integration Procedure

The Gaussian quadrature method was used throughout for the merid-

- ional integration. The method is exact for polynomials of order

2n-1 if n Gauss points (n 2) are used. Five points were found to be

sufficient for the integration of the stiffness matrix of SABA 5 pro-

viding an exact result for all terms below the ninth power. (SABA 7

required seven Gauss pts. for the same accuracy.) The application of

.this method of integration was greatly simplified by the choice of +1

the origin of n. Evaluation of a typical term Jr f(n) do is then -1

simply

+1 1=5

ff(n) do 1=1

A.f(n.)

(3.6)

where ni and A. take the following values

122

1 2 3 4 5

n. 1 .90617985 .538/:b931 0 -.5384693 -.90617985

A.1 .23692689 .47862867 .568888 .47862867 .23692689

3.5 Suppression of Nodal Freedoms

The structural stiffness matrix as given in eqn 1.41 is in general

singular. However, by removing from the displacement vector sufficient

freedoms to suppress the possibility of rigid body motion and thus

removing their rows and columns from the stiffness matrix, the equa-

tions are made soluble. This may be achieved by extracting these

freedoms and coefficients and collapsing the stiffness matrix. However

in this case a different approach is adopted which maintains the regular

pattern of the equations.

The rows and columns of the stiffness matrix, corresponding to

those freedoms that are to be suppressed, are set to zero except for

the diagonal value which is set to 1. The corresponding load in the

load vector also being set to zero. This is equivalent to reducing

the equation for the nth freedom to

1 . rN

= 0.

By this method the pattern is retained whilst the nth freedom is

effectively removed.

3.6 Algorithm for the Cholesky Inversion

The solution of the small displacement equations is found by use

of the Cholesky method. This approach is particularly well suited to

sets of regular equations and has been shown to'be the most stable of

A11 . Alj . . . A- ln

A

Ai1 . A.

the direct methods of solution. It is also used, together with an

elimination process to solve the large displacement equations. The

method decomposes the stiffness matrix into a lower triangle and its

transpose,

A x

LLt x (3.7)

where A is a symmetric positive definite matrix.

The solution may then be obtained in two steps by

L 1 y

—it L z (3.8)

The following algorithms give the computational procedure required to

form the lower trianglar matrix and to invert it

123

11

The lower triangle L may be produced from:

L. = 13

i-1

A. L . L 13 4.4 1P 31)

p=1 (3.9)

L1..

where j = 1 N for each value of i = 1 ± N.

To produce LT1 the relation LL-1 = T is used

124

. L'. .

13 11 0 • •

The inverse of L is then computed in the same working space as L

is stored by the following algorithm:

= 1./ 11 L..

11

"where L represents a term in the inverse

L. = 13

p=i

E Lip Lpj

P=i i > j (3.10)

L.. 11

'where j = 1 N for each value of i = 1 N.

3.7 Solution of the Large Displacement Equations

It has been shown in Section 2.7 that by a careful choice of

the arrangement of nodal freedoms in the displacement vector it is

possible to obtain a three-banded stiffness matrix. The form of this

matrix is shown below, for an unbranched structure.

125

k22

N

\

kN kNM

k11

\ , ̀\ \

. \

• \ \ \ \ \

.\

The blocks contain stiffness coefficients equal to 9 x the number of

harmonics. Only one half of the symmetric matrix need be stored.

By combining the Cholesky and Gaussian elimination methods a most

efficient algorithm for the systematic solution of this stiffness matrix

Kwith a load vector R is obtained. The process requires as mini-

mum storage the equivalent of three blocks plus the load and displace-

ment vectors equivalent to two nodes. Instead of the solution pro-

ceeding equation by equation the unity of the blocks is retained and

the calculation uses these as coefficients of the equations in a

super-matrix. Hence the elimination of the first row in the super-

matrix, following the addition of the first element, gives an expres-

sion for ri the displacement vector associated with the set of free-

doms for the first node.

126

-1 1 r k11 ( R1-

12 r 2) (3.11)

where R1 is the load vector for node 1

and r2 is the displacement vector for node 2.

Following this elimination the second element may be added to the

stiffness matrix in the space available from the first row. The

elimination then continues with the second row by

- -1 _ r2 = k22( R2 k23 r3)

where k22 =

k22 "12k111k12

R2 = R 2 -

2 1 R'1 (3.12)

The evaluation of the inverse of the diagonal block need never be

explicitly completed in these expressions. Instead the lower tri-

1 angle is formed and its inverse used to produce 191"12 and L11R1 ,

both of which are retained for use in the back-substitution described

later. Eqns 3.12 are then

k t 2 = k - k 22 12 L11 L11 :L111 k 12 t vo R2 R2 k

= 12 11 L 11 ''1 (3.13)

This method minimises the number of matrix operations required. The

Nth step may now be written as

t- -1 M

= k k 4 k NM

t 1t_ -1 IL - k LL 14 iTM N (3.14)

The back substitution necessary to find r is then simply

1 t - -1- 1k r rN L (LN g R, - LN 1,111 m - ) -0 (3.15)

127

The factors in the brackets being available from the forward decompo-

sition ( r being obtained from the previous step).

3.8 Calculation of r and z for the Hyperbolic Shell

From the four different meridional geometries considered in the

examples the torus, spherical cap and cylinder are trivial and calcu-

lation of the vectors rI and z1 from eqns 1.14 is simple. The

expression for the nodal parameters of a hyperbolic shell through

straightforward are more complicated and are given here for reference

purposes

The equation of the hyperbola in cartesian coordinates is

r2 z2 -

a2 b2 = 1

this is more conveniently written for our analysis as

r2 - cz2 =

(3.16)

where c = a2/b2.

The derivatives with respect to the meridian length s which is

defined as +ve with increasing 4 (see Fig 3) are given as follows

dr -cz ds (r2 + (cz)2)14-

(3.17)

dz -r ds

(r2 + (cz)2)9

der _ crag

ds2 (r2 + (cz)2)2

(3.13)

d2z

-c2za2

ds2 (r2 + (cz)2)2

The third derivatives which are required for SABA 7 are

d3r- c2a2z

(3(r2 + (cz)2) + 4ca2)

ds3 (r2 + (az)2)31

(3.19)

128

d3z c2a2r (3(r

2 + (cz)

2) - 4a

2)

(r2 + (az)2)31 ds3

d do

d2 d3 and

dn2 dn3 may be obtained from these expressions by k ds

2,2d2

ds2 3 and 2.3 d respectively, where 2, is half the meridian length.

ds3

The meridian length is found from

+ (dz)2 dz

gl + (rZ)2)i

dz (3.20)

where the integration is performed numerically by Gaussian quadrature.

3.9 Flow Diazrams

It is intended that the following diagrams should show as descrip-

tively as possible the philosophy that may be used in programming the

theory of the preceding Sections. The details of such a program may

easily be filled in by the use of simple logic and the aid of a suit-

able programming language such as Fortran. The diagrams are based

upon a program that has been written in Fortran IV and has been run

Successfully on an IBM 360/75. In order to reduce the number of

words needed some of the variables that occur in the program are also

used here. Those that require definition are listed below.

List of undefined variables used in the flow diagrams

NOEL Number of elements.

NOEL1 Number of nodes.

NHARM Number of harmonics.

NLOADS Number of load steps.

NSTART The number - of the load step at which the increment

procedure commences.

Working space used for the assembly and inversion of the

nt h . small displacement structural stiffness for the j

harmonic.

129

fds

130

X Load vector of kinematically equivalent loads.

IX A counter controlling the iteration within each step.

KN, KNM Working space used for the assembly and inversion of

KM&P the large displacement equations and loads.

MAP

Array defining the pattern of the geometric stiffness

matrix (see Section 2.6).

LISTH List of harmonic numbers used.

LEN {L3 L3 L3 L3 L3 L3} where L

3 = {1 it, k2}

JUMP Flag signifying the exit point from AXIK used by CSTR

when computing B for a point on the meridian line.

NSUP Number of nodal freedoms to be suppressed.

NSPT Number of stress points.

PT Integer list of internal points used for numerical

integration and for calculation of stresses.

LTH, M Matrices defining rotations referred to in Section 2.5

eqn 2.27.

j j

1 A A

I I

I I r:' ,2

NAIN S~rEERING SEGHENT FOR LAHGE DISPLACEl·1m,!T PROORAH

r;;.:,\ \!!ATY /~l .

( wr.I~TE ' DATA

'" .-- --.-.-----.-..--~---

COMPUTE GEOMETRIC DATA FROM SUBROUTINE OPTIOHS HYPDAT" CYlOAT, TORDAT, SPHDAl

r--------INlftALiS"E-it';P-OR-UUT ] VAAl AOtES

-~-~----

/r-R-e-r-EA-T-rOA tI LO fR~;---) ------~START ~_~OAOS_____ -----~-.. ---~-- ... -, .. ~ ..

IIST~E:..:.S _____________ ._

?

SELECT LOAO tRACTION ----l 1..-________ 1

REPEAT FROM 1 TO NHARM

r---( REPEAT FROM 1 TO N~

COMPUTE LinEAR ELASTIC I STifFNESS AND K.E. LOADS ..

SU B. AX I K __ ... _J

ASSEMBLE ELEHEUTS AND LOADS I" KJJ AnO X.

f SU,.,ft£SS fREEDOMS IN 'J I KJJ AND X .. --.--~~--- .. ------.-------

COMPUTE ItJV!RS£ Of ~OWER TI''' ANiLE Of KJJ:& L­SUB.CHOlBI

I !

r I I j

I ! I I

.......... , \

! .-( ) \ ':-t- i

131

DISPLACEMENT VECTOR rt- 2 L- L- X

SUB. TRICOT

PRINT DISPLACEMENT VECTOR

STORE

DISPLACE-. MEFITS

SuM DISPLACEMENT HARMONIES AND PRINT

SUB. FSD

COMPUTE STRESSES

SUB. CSTR

NO

RETRIEVE DISPLACEMENTS AND STRESSES AT ENO OF PREVIOUS LOAD STEP

FORM STRUCTURAL STIFFNESS FOR STEP AND SOLVE FOR DISPLACEMENTS

SUB. GSOL

SELECT LOAD FRACTION

REPEAT IX .1 TO 2

132

•

NO

( PRINT DISPLACEMENTS

(

STORE 01 SPLACE-

IICHTS

N.

SUM 0! SPL AC EMENT HARMONIES AND PRINT

COMPUTE STRESSES

SUS. GSTR

IS THI S THE

LAST STEP

?

YES

133

Is

THIS THE YES

LAST NODE

NO

REPEAT FROM 1

RETRIEVE ELASTIC STIFFNESS AND K.E. LOADS FOR BEGINNING OF STEP

J FORM ELASTIC STIFFNESS

(AXIK) AND K.E. LOADS FOR END OF STEP.

AVERAGE STIFFNESS AND K.E. LOADS

FORM ELASTIC STIFFNESS (AXIK) AND K.E. LOADS FOR BEGINNING OF STEP

STORE ELASTIC

f STIFFNESS AND K.E. LOADS

ADD ELASTIC STIFFNESS AND K.E. LOADS To WORKING SPACE KN, Kni, KM, AND P

YES SUPPRESS NODAL FREERomS

SUB. SUPS

SUBROUTINE GSOL 134

Steering routine to form structural t;tiffness matrix for each step (>1) and ;valve:

--\ REPEAT FROM I TO NOEL1

COMPUTE GEOMETRIC STIFF-NESS ANO A00 TO ELASTIC STIFFNESS IN WORKING SPACE

SUB. GSTIF

COMPUTE INVERSE Of

LOWER TREARGLE OF

KN LW1

SUB. CHIN

Is

THE LAST NODE

NO

KS L

KM = KM — KS+ KS PS L4-1 P(A)

WHERE A REFERS TO TOP HALF OF VECTOR AND B TO LOWER HALF

YES

YES

RETRIEVE KS, PS

FOR RODE NOEL1—NI

135

YES

NO

PS PS - KS x DPNI-1

136

REORGANISE DISPLACEMENT VECTOR AND SUM WITH

PREVIOUS STEPS

R, z TjAZI R', 1I ri t Azi R", Z" 7. Tit IARI, AZI

P Z":

SIN cp:

cos To

DR ... R'/Z' DI -

5211 en' R" - (zip ri77

(14 e)2)3/2

02R/D;2

COMPUTE TERMS

IN 0

0/1.

SUBROUTINE AXIK 137

Com utes elastic element stiffness matrix and kinematically equivalent loads for one element and one harmonic

SELECT HARMONIC NUMBER AND ELEMENT DATA UPDATE NODAL GEOMETRY.

IS HARMONIC

\ HO. e 0

No

YES C = 2

C;1

AZI = 6z1

ARI SR I

REPEAT FOR 5 GAUSS PT% OR FOR 1 STRESS PT. IF ROUTINE CALLED FROM /

cm.

SIN (Q

/13 <' JUMP = 0

7

NO

YES

K 2 B+ K B

1 K2 . K2 + Ki x Gauss consr x L 3 c

0

G_ ÷+ cos q) if

P.

R = C Pi x ic I. C X PRESSURE COEFFICI EUT

KE r.. C+A +K 2 A C

END)

X GAUSS CONS,'

138

SUBROUTINE GSTIF

Computes all geometric stiffness matrices for one element

139

SELECT ELEMENT DATA AND

UPDATE NODAL GEOMETRY

SET UP LEN

COMPUTE ROWS OF LTH FOR 5 GAUSS POINTS

1_

)

( INCREMENT NSTRS FROM 1 To NHARM

I

)

INCREMENT N FROM 1 To NHARM _J SELECT ROW IR FROM MAP,

I OF GEOMETRIC STIFFNESS

i

CORRESPONDING TO STRESS MRS AND CDL. N

(INCREMENT II FROM 1 TO 5

SELECT STRESSES FOR HARMONIC NSTRS AND GAUSS PT. II

SELECT HARMONICS FROM LISTH CORRESPONDING TO IR AND N

COMPLETE ROWS OF MIR AND Ii

COMPUTE TERMS OF GEOMETRIC STIFF KG1

aA' K61

K62 C

Is

IX 1 YES STORE

KG

NO

RETRIEVE KG1( K=1) FROM

STORE AHD AVERAGE WITH

K G (IX r..2) 2

K G (KG1 KG2)/2

ADD KG TO WORKING SPACE

KM, KN, KMN ,

0 co 11+0

ENTER WITH DUMBER OF SUN-. RESSIONS AND LIST OF FREEDOMS TO BE SUPPRESSED

(......

REPEAT FROM 1 TO NHARM

I

REPEAT FROM 1 TO NSUP

YES NO

SUPPRESS ROW AND - COLUMN IN KN PLACING

1.0 ON DIAGONAL

SUPPRESS ROW IN KMN AND P

SUPPRESS ROW AND COLUMN IN KM PLACING 1.0 ON DIAGONAL

SUPPRESS COL. IN KMN AND now IN P

ENO

SELECT ROW/COL. NUMBER N OF FREEDOM TO DE SUPPRESSED

SUBROUTINE SUPS

Introduces suppressions to working space KN, KMN, KM rand P

REPEAT FROM 1 TO NSPT

SELECT STRESS POINTS FROM PT (foR THE NON-LINEAR STEPS THESE MUST BE THE GAUSS PTS0)

COMPUTE B SUB. AXIK

aM — K ae

5; -T

=r 1 Li art, ar R ao

(M(I)

-MO)cos cp]

Q 0 i[R ar R a—r1 a 0 +mos tp 3

SUBROUTINE CSTR

Steering routine for the calculation of stresses within the elements

( 1 TO NHARM INCREMENT NUH FROM

r INCREMENT NEC FROM 1 TO NHARM

C ( SELECT DISPLACEMENTS FOR ELEMENT NEC AND HARMONIC NUH, p

T X 32 A p

PRINT STRESSES

SUM MEMBRANE STRESS HARMONICS AND PRINT

SUB. FSS

SUM BENDING STRESS HARMONICS AND PRINT

SUB. FSS

( STORE STRESSES

1k3

IS THE TOTAL

NO. OF COLUMNS DIV. BY 8

145 SUBROUTNE PRINTE

PRINTS OUT DISPLACEMENTS (OP STRESSES) AT NODES

ENTER MATRIX TO BE WRITTEN OUT ANGLES IN DEGS. MARKER FOR DISP. OR STRESSES

PRINT TITLE )

DIVIDE TOTAL NO. COLS. BY 8 w L

REPEAT FROM 1 To L

PRINT 8 ANGLES

PRINT 8 COLUMNS IN GROUPS OF 'MARK' ROWS PER NODE

YES

NO

PRINT REMAINING ANGLES

PRINT REMAINING COLS.

1

!END

Z •• F x N+1 -N .N-- RN LN = FoR/2

Geometry Subroutines 146

SUBROUTINE CYLDAT

Computes nodal point data and meridian lenzths for a cylinder

fN = fracdon of height of cylinder

INCREMENT N FROM 1 TO NOEL

INCREMENT N FROM 1 TO NOEL1

1 zu = 1 Ri = 0 zri - 0 RN =

PRINT DATA

END

SUBROUTINE TORDAT

Computes nodal point data and meridian length for a torus

READ DATA

INCREMENT ,N FROM 1 I TO NOEL

fN = fraction of 180°

N 7.1 7E X FN

241 = A TN

INCREMENT K FROM 1 TO NOEL1

(PT 4- (PN R = acosrpT +R zN = RsiRcpr Rs N z Strupr z' cOS9 T R" cosipm

SINT T/A

▪ TOP CO—ORDINATE

✓ SA.SE COORDINATE

= ELEMENT FRACTIONS OF HEIENT OF CYLINDER

▪ A2/52

READ DATA

INCREMENT N FROM 1 To NOEL

No = ZN + 41(Z1+ z2)

11

12 F

( ..,—.)

INCREMENT N FROM 1 To NOEL1

RN .7. (R2 .1. czN)114 czN/K1 +c2z0

OARN/(RN (cz )N )

2 - 2 2

ROW .1• c2z4' c2ZNAI(Rti2 c2v

INCREMENT N TO NOEL

OR 2L = f k 1 -( 2) oz

IT;

zA

z"

FROM 1

'SUBROUTINE HYPDAT

Commutes nodal point data and meridian lenf0;11 for a hyperbolic shell

147

PRINT DATA

ENO

24-1 = -(PN-1)

_INCREMENT N FROM 1 ro NOEL1

▪ R cospN = fi simpt4 ▪ stuTN • cosy N ▪ cosyR/R

• " "CP N/R

~~ PRIhT DATA

(/

\ INCREMENT N FROM 2 TO NOEL1

RN ZN

RhRf N zN

ita

zN

148

SUBROUTINE SPHDAT

Computes 2.1212 oint data and meridian ten th .for a Shallow ~iericrrl cn.

READ

DATA

17, ANGULAR POSITION IF

NODAL POINTS

R = RADIUS OF SPHERE

SUBROUTINE CIIOLBI

Computes inverse of the lower triangle of a symmetric matrix stored in a two dimensional array. The routine assumes the matrix has a bandwidth of NB and restricts the calculation to this bandwidth.

A-1 L-1

SUBROUTINE TRICOL

Multiplies inverse of lower triangle by column vector, or if

X = L-1 Y

MARK=.1 multiplies by transpose of lower triangle

X = (L-1 )+ Y

SUBROUTINE CHIN

Computes inverse of the lower triangle of a symmetric matrix stored in one-dimensional packed symmetric form.

SUBROUTINE MBTFB

Computes congruent transformation on a symmetric matrix

D A+ B A

SUBROUTINES MUL1, MUL2, MATD

Perform various matrix multiplications.

SUBROUTINE REORG

Reorganise K matrix in accordance with the transformation .(see text)

K2 = C+K 1 C

149

150

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