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31293 00882 6343

This is to certify that the

dissertation entitled

Static Analysis of Structural Thin

Flat Membranes

presented by

Abdelilah Elguennouni

has been accepted towards fulfillment
of the requirements for

Doctor of Philosophy degreein Civil Engineering

//4%...%

flMajor professou

 

 

Date August 5, 1992

 

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STATIC ANALYSIS OF STRUCTURAL THIN

FLAT MENIBRANES

BY

ABDELILAH ELGUENNOUNI

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of civil and Environmental Engineering

.1992

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ABSTRACT

STATIC ANALYSIS OF STRUCTURAL THIN
FLAT NIEMBRANES

BY
ABDELILAH ELGUENNOUNI

The static nonlinear behavior of structural thin flat
membranes which are subjected to transverse loading was
investigated. Using a: nondimensional formulation, two
geometrically nonlinear finite element models were considered.
First, a simplified model based on the von Karman strain-
displacement relationships was developed and validated by
comparison to previously developed models. Then, a general
model based on the exact strain-displacement relationships was
developed.

A comparison between.the two models was made to'determine
the limitations of the simplified model. An extension to
deformation-dependent loading was also studied. Several
parametric studies were.conducted.to investigate the.effect of
Poisson's ratio, initial prestressing, and membrane boundary
conditions on deflections and stresses. An incremental-
iterative procedure was used to solve the nonlinear finite
element equations. To overcome the difficulty associated with
the ill-conditioning encountered for the first several
increments, a new technique referred to as "initial virtual

prestressing" was developed. This was found to be effective

and convenient compared to the usual strategy of guessing the
initial deflected shape.

It was shown that a nondimensional coefficient k which is
a function of five parameters (load intensity, characteristic
length of the membrane, Young’s modulus, Poisson’s ratio and
membrane thickness) determines the state of straining in the
membrane. For values of 1: smaller than 0.01, the maximum
strain is less than 1.75%. The simplified model leads to
results that are sufficiently accurate for design purposes.
For values of k greater than 0.05, the maximum strain exceeds
5%, and the accuracy of the simplified model diminishes.

The use by previous investigators of second Piola-
Kirchhoff stresses is inappropriate because Cauchy stresses
are the more accurate representation of real stresses.
However, for values of k smaller than 0.01, the numerical
difference between the two stress measures was found to be
negligible. Also, it was shown that for values of k smaller
than 0.05, external pressure loading may be assumed to be
deformation—independent. Variationcanoisson’s ratio, initial
real prestressing and membrane boundary conditions were shown

to have significant effects on deflections and stresses.

To the memory of my father

MOHAMED ELGUENNOUNI

and to my mother

DAOURA AICHA

FOR THEIR LOVE AND SACRIFICE

iv

Acknowledgments

I am gratefully indebted to Dr. Frank Hatfield, my major
professor, for his invaluable advice and continuous support

throughout the course of this study.

Sincere appreciation is extended to the guidance
committee members: Dr. Fred Bakker-Arkema, Dr. John Masterson,

Dr. Parviz Soroushian, and the late Dr. William Bradley.

The Hassan II Agronomic and Veterinary Institute, Rabat,
Morocco, facilitated and encouraged my research; the United
States Agency for International Development provided financial
support; and the International Agricultural Programs of the
University of Minnesota handled administrative functions. This
research would not have been possible without the existence of

those three institutions, and I am deeply grateful to them.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF SYNIBOLS

1. Introduction andBackground 1
1.1 Introduction.......... ............. ... ........ ....1
1.2 Review of Technical Literature .................... 3
1.3 Scope of the Research...... ......... .. ........... 11

20The0ryand1rnplementation 0....0.0.0.0..........OOOOOOOIl3
2.1 Introduction.....................................13

2.2 Basic Assumptions. ................. . ........ .....15
2.3 Fundamental Equations Using a Nonlinear Finite
Element Formulation .................................. 16
2.3.1 Basic Problem. ............................ 16
2.3.2 Formulation of the Continuum Mechanics
Equations of Equilibrium ........................ 17

2.3.3 Description of the Curved Isoparametric
Finite Element Used and the Corresponding Shape

Functions............. .......... ...............18
2. 3.4 Reformulation of the Equilibrium

Equations... .................................... 21
2.3.5 Incremental Equilibrium Equations ......... 22

3. Fundamental Equations for the von Karman Model . . . . . . . . . 25

3.1 Derivation of Element Matrices...................25
3.1.1 Derivation of the Basic Element Matrices..25
3.1.2 Derivation of Element Stiffness Matrices..29

3.2 Rearrangement of Element Stiffness Matrices

for Computer Implementation ........ . ................. 33
3.2.1 Numerical Integration..... ..... ..... ...... 33
3.2.2 Rearrangement of Element Matrices.........35

3.3 Nondimensional Formulation of the Incremental

Equilibrium Equations ................................ 37

vi

 

4. Extension to the General Nonlinear Model . . . . . . . . . . . . . . . 46
4.1 Derivation of Element Matrices ................... 46
4.1.1 Derivation of the Basic Element Matrices..46

4.1.2 Derivation of Element Stiffness Matrices..48

L 2 Nondimensional Formulation of the Incremental
Equilibrium Equations ....................... ......50

4.3 Extension to the Case of External Pressure ...... .56

5.SolutionTechnique 66

5.1 Assemblage of Structure Matrices......... ........ 66
5.2 Solution of Equilibrium Equations.. ......... .....66
5.2.1 The Incremental Iterative Solution
Strategy ...... ............ .... ........... .....66
5.2.2 Convergence Criteria.. .......... ..........68
5.2. 3 The Initial Virtual Prestressing
Technique ....................................... 68
5.3 Evaluation of Strains and Stresses ............... 72
5.3.1 Evaluation of Strains ..................... 72
5.3.2 Evaluation of Second Piola-Kirchhoff
stresses ........................................ 72
5.3.3 Evaluation of Cauchy Stresses.... ......... 72

6. Model Validation and Parametric Studies . . . . . . . . . . . . . . . . 74

6.1 Model Validation ................................ 75
6.1.1 Convergence of the von Karman Model ....... 75
6.1.2 Comparison of the von Karman Model
with Previous Models .................... ........80
6.1.3 Discussion ................................ 82
6.1.4 Comparison of the von Karman Model with
the General Model ............................... 87

6. 2 Parametric Studies .............................. 110
6.2.1 Effect of Poisson’ 5 Ratio ................ 110
6. L 2 Effect of Initial Prestressing... ........ 112
6.2.3 Effect of the Boundary Conditions ........ 114
6.2.4 Effect of Nonconservative Loading: Case of
External Pressure .............................. 119
6.2.5 Comments About the Initial Virtual
Prestressing Technique ......................... 119

7. SummaryandConclusions...............................120

7.1 Summary ......................................... 120
7.2 Conclusions ..................................... 121
7.2.1 Comparison of the von Karman Model
to Previously Developed models ................. 121
7.2.2 Second Piola-Kirchhoff Stresses and
Cauchy Stresses ................................ 122
7.2.3 Limitation of the von Karman Model ....... 122
7.2.4 Effect of Poisson's Ratio ................ 123
7.2.5 Effect of Initial Prestressing ........... 123
7.2.6 Effect of Boundary Conditions ............ 124

vii

 

 

 

 

 

 

 

7.2.7 Effect of Nonconservative Loading........124
7.2.8 Use of the Initial Virtual Prestressing
Technique ...................................... 124

Bibliography 126

viii

Table

Table

Table

Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table

Table
Table

Table

Table

Table

13

14

15
16

17

18

19

LIST OF TABLES

Square Membrane Under Uniform Transverse

Loading Comparing Different Models.......... ..81
Rectangular Membrane Under Uniform

Transverse Loading (u=6/7) Comparing

Different Models............................ ..81
Rectangular Membrane Under Uniform

Transverse Loading (v=2/5) Comparing

Different Models.. ............................ .82
Stress Coefficients for a Square
Membrane............. ........ ..... ....... ... ..83
Stress Coefficients for a Rectangular

Membrane (v=5/7)............................ ..84
Stress Coefficients for a Rectangular

Membrane (v=2/5).................. ...... .... ..84
Relative error in Stress Coefficients....... ..85
Strains in a Square Membrane........... ..... ..86
Strains in a Rectangular Membrane (v=6/7)... ..87
Strains in a Rectangular Membrane (u=Q/5)... ..87
Central Deflection Coefficients ............. ..92
Cauchy Stress Coefficients in a Square
Membrane......... .......... . ....... ..... ....... 93
Cauchy Stress Coefficients in a Rectangular
Membrane (u=5/7).. ............. ... ..... . ....... 93
Cauchy Stress Coefficients in a Rectangular
Membrane (v=2/5)... ........ ................. ..94
Relative Error for a Square Membrane........ ..95
Relative Error for a Rectangular

Membrane (v=5/7)................ ........ . ...... 95
Relative Error for a Rectangular

Membrane (v=2/5) ......... ..... ................. 96
Comparison of Strains for a Square
Membrane......................... ......... .. ..97
Comparison of Strains for a Rectangular

Membrane (v=5/7) ............. ........ .......... 97

ix

Table 20 : Comparison of Strains for a Rectangular
Membrane (”=2/5).0.........0.0.00.000000000000098

Figure

Figure
Figure

Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

[0

10

11

12

13

14

.0 O. O.

O. O. O. O. .0 O. 0.

LIST OF FIGURES

Representation of the Eight-Node

Master Element................................19
Region for Rectangular Membrane........ ..... ..75
Convergence of Deflection and Stresses

for a Square Membrane.........................77
Convergence of Deflection and Stresses for

a Rectangular Membrane (u=6/7)................78
Convergence of Deflection and Stresses for

a Rectangular Membrane (u=2/5)................79
General Model Convergence of Deflection

and Stresses for a Square Membrane...... ...... 89
General Model Convergence of Deflection

and Stresses for a Rectangular Membrane

(u=5/7) .......... ........... . ........ .... ...... 90
General Model Convergence of Deflection

and Stresses for a Rectangular Membrane

(u=2/5) ...................... ...... ... ........ .91
Typical Quarter of a Membrane......... ........ 99
Deflection Coefficient Distribution for

a Square Membrane............................100
Deflection Coefficient Distribution Along

The Shorter Side of a Rectangular Membrane
(v=5/7)........... ............... ............1o1
Deflection Coefficient Distribution Along

The Longer Side of a Rectangular Membrane
(v=5/7)............. ....... . ...... ...........102
Deflection Coefficient Distribution Along

The Shorter Side of a Rectangular Membrane
(v=2/5) ...................................... 103
Deflection Coefficient Distribution Along

The Longer Side of a Rectangular Membrane

(u=2/5) ....................... ...... ...........104

xi

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure
Figure
Figure

Figure

15

16

17

18

19

20
21
22
23

24

25

O.

.0

Stress Coefficient Distribution for
a Square Membrane............................105

Stress Coefficient Distribution Along

The Shorter Side of a Rectangular Membrane
(u=5/7)......................................106
Stress Coefficient Distribution Along

The Longer Side of a Rectangular Membrane
(v=6/7).......................................107
Stress Coefficient Distribution Along

The Shorter Side of a Rectangular Membrane
(v=2/5) ..108
Stress Coefficient Distribution Along

The Longer Side of a Rectangular Membrane

(v=2/5) ........... ................... ........109
Effect of Poisson's Ratio. ................... 111
Effect of Initial Prestraining...............113
Comparison of Normal Deflections Along

the Center Line a-a..... ..... ....... ......... 115
Comparison of Normal Deflections Along

the Center Line d-d........ ...... .... ........ 116
Comparison of Cauchy Stresses Along

the Center Line a-a ...... ........ ...... ......117
Comparison of Cauchy Stresses Along

the Center Line d-d .......................... 118

xii

LIST OF SYNIBOLS

half-length of the longer side of a

rectangular membrane

— nodal displacement vector

on:

half-length of the shorter side of a
rectangular membrane

— strain-displacement matrix

elasticity matrix
Young’s modulus

- external load vector per unit area

itinerant:

W ‘4 D‘*fi E

vector of body forces per unit volume
vector of surface tractions per unit
area ‘

stress function

vector of external loads

membrane thickness

Jacobian matrix

nondimensional coefficient characterizing
The membrane problem

linear stiffness matrix

nonlinear stiffness matrix

load stiffness matrix

tangent stiffness matrix

stress stiffness matrix

a characteristic length of the membrane
in-plane stress resultant components
nodal shape functions

Matrix of shape functions

xiii

r,

U!

V;

vector of equivalent internal forces
normal load intensity

= natural coordinates

displacement components respectively

in the x- and y- directions

displacement component in the z-direction
also called normal deflection

volume of membrane element in the

initial configuration

surface area of the membrane element

in the initial configuration

displacement vector

Green-Lagrange strain vector

Green-Lagrange strain components

Poisson's ratio
second Piola-Kirchhoff stress

components

second Piola-Kirchhoff stress vector

Cauchy Stress components

xiv

1 . INTRODUCTION AND BACKGROUND

1.1 Introduction

The high capital cost.of glass greenhouses has stimulated
interest in film plastic clad alternatives which require only
20 to 45% of the capital required for glassk

In Morocco, there is a trend toward the construction of
plastic film greenhouses. More than 1000 hectares for banana
production have been covered by this type Bf structure during
the period 1981-19882.

Because: of ‘the favorable Jhorticultural qualities of
polyethylene film (particularly for the Mediterranean climate)
and its low price compared to other covering materials, it.has
been used extensively for greenhouses. Due to the fact that
the structural behavior of this covering material is not well
known, particularly under the random action of wind, the
reliability of these greenhouse structures remains uncertain.
Existing designs span a large range of reliability; some are
prone to structural failure while others appear to be
overdesigned to an uneconomic degree.

Greenhouse structures are hybrid systems in which plastic
membrane panels span between primary load carrying members
such as prestressed cables and rigid elements. Some of the

advantages of membrane structures are:

2

- they are lightweight and collapsible and therefore easy to
transport and erect;
- the environmental loads are efficiently carried by direct
stress without bending;
- they are load-adaptive in that the members change geometry
to better accommodate changes in load patterns and magnitudes.

The structural mechanics of tension structures such as

greenhouses is well described in references3 4 5 6 7.

Large
deformations due to wind pressure, the resulting tearing of
the plastic due to the high membrane tension, and the collapse
of other structural elements are critical problems. -

Since the typical cost of greenhouse structures designed
for Morocco is between 250,000 and 300,000 Dirhams‘ per
'hectare, failure of a greenhouse represents a significant
loss. Therefore, an efficient and safe structural design of
these buildings must be achieved. To do this, several factors
that influence the design must be considered. The main factors
that are specific for these structures are the design wind
load and the structural behavior of film plastic membranes.

The first factor has received considerable attention; only a

few investigations have been concerned with the latter.

1.2 Review of Technical Literature

A.membrane can only sustain tensile stresses. Therefore,

in order to be stable , i.e. have an equilibrium position, it

 

‘. In June 1989, 1 Dirham = $ 0.125.

3

must be prestretched. This can be effected by tensile forces
acting on the membrane edges, by selfweight,or, when a space
is completely enclosed, by pressurizing. Prestretching forces
stabilize the structure and provide stiffness against further
deflections. Membrane structures respond in a nonlinear
fashion to both prestretching and service loads, regardless of
linearity of materials, even if the loading is deformation-
independent.

The static analysis of prestretched structures comprises

two main problems:
1- Determination of the prestretching forces at equilibrium;
2— Establishment of the maximum tensile forces arising in the
system at the given load application. These, together with the
strength of the membrane, determine the maximum size of the
panel.

In order to determine the required prestretching forces
as well as the maximum stresses, it is necessary first to
determine.the internal forces. The ordinary membrane theory of
shells may be used, provided the material is only slightly
deformable so that the loads can be considered to act on the
undeformed system. It is far more difficult to determine the
state of stress for highly deformed membranes. In this case
the initial shape is incapable of supporting any load, and.the .
membrane undergoes finite deformations until an equilibrium
shape is reached. The state of stress depends markedly on the

final shape of the membrane. However, this shape is unknown,

4
as are the internal forces. When the equilibrium state of a
system must be determined for the deformed position, it is
necessary to apply the theory of finite deformations, using
nonlinear strain-displacement relationships. This leads to
nonlinear displacement equations.

According to Shaw and Perronéfi the governing nonlinear
equations of 'membranes were first derived by Foppl and
Teubner". These equations follow directly from the von Karman
large deflection flat-plate equations by setting the plate
stiffness identically zero. The von Karman equations for the
large deflection of a thin flat plate of uniform thickness

are”:

(1.1)

 

 

 

 

341.32 64F + a4F=E( 62w)2_ azwazw
6x4 Ehfldyz dy‘ 1%{3Y 8x269)r2

 

 

 

 

jtg4i3 éFW' +é¥wg=lg£g+i¥F’d%v+i¥F'6%v__2 in' Eyw
6x4 dxzdyz dy‘ .D h dyzéhfl axzc'iyr2 dxifirdxih/

(1.2)
where w is the normal deflection, q=g(x,y) is the applied
normal load intensity, h is the plate thickness, E is Young’s
modulus for the plate material, D its bending stiffness
defined as Eh3/12 (l-VZ) where V is Poisson’s ratio, and F is a
stress function related to the forces per unit length in the

plane of the plate by the formulas:

N=hazF N=hazF N =hazF

x ayz y 6x2 ’0’ ' —axay ”'3’

 

 

Making the bending stiffness zero results in two simultaneous
nonlinear equations relating the membrane deflection and the

stress function

 

+ 5bfaY' Iggidyz

2
64F+2 an: 35 = E[( 62w) _ 52.4933] (1,4,
3x4 Ehfiayz dy‘

szdzw+ 62wa_2 62F 62w

6y2 6x2 32:36—37; 6x 6y 6x 5y = (1.5)

 

 

 

.g +
h

The exact solution for the uniformly loaded rectangular
membrane has not been obtained.

In 1920, Foppl and Foppl” used the energy approach to
obtain an approximate solution for stress and deflection at
the center of a square membrane. They assumed a trigonometric
function for the membrane deformations. This function, which
contains a certain number of unknown coefficients, was chosen
so as to satisfy the boundary and symmetry conditions. The
unknown coefficients were evaluated by minimizing the total
energy of the membrane. A year later, Hencky12 achieved a
rather lengthy numerical finite difference solution for a
square membrane. His results differ slightly from Foppl and
Foppl’s.

According to Borg”, in 1940 Neubert and Sommer14 carried
through the Foppl's computations for the rectangular case and
drew curves for stresses and deflections. Additionally,

Neubert and Sommer obtained satisfactory experimental

6

verification for the Ffippl’s and Hencky solution for the
square membrane, but.did not test a rectangular membrane. Head
and Sechler" got similar results from their experimental
work, but, for rectangular membranes with high aspect ratios,
they noted.a discrepancy from.the Foppl and Fappl solution for
stresses. Borg13 obtained an exact solution for the semi-
infinite membrane by taking the limit of the semi-infinite
tied-platel6 as the plate bending stiffness approaches zero.
He estimated the deflections and stresses of rectangular
membranes by interpolating' results for square and semi-
infinite rectangular membranes. He drew curves for central
deflection and central stress between the two limits in such
a way as to satisfy known experimental requirements.
Differences of the order of 19 per cent appeared between the
Borg and F6ppl's results in semi-infinite membrane solutions.
Borg attributed this discrepancy to the fact that as the
aspect ratio decreases, the trigonometric function assumed by
Foppl and Foppl becomes less accurate.

In 1954 Shaw and Perrone8 employed a finite difference
approximation in conjunction with a nonlinear relaxation
technique to obtain a solution for an aspect ratio of 5/7.
Rather than using the Foppl’s formulation, the ‘membrane
problem was dealt with numerically in terms of displacement
components. Their results compared fairly'well with Borg’s. In
1972 Kao and Perronel7 extended the solution to other aspect

ratios.

7

In 1987, Allen and Al-Qarra18 used an incremental finite
element method in a total Lagrangian coordinate system. An
advantage of the method is that the problem of the large
deflection of thin flat membranes subjected to normal forces
is formulated in terms of simple physical concepts, and a
numerical solution is achieved without dealing directly with
the complex nonlinear differential equations.

All the investigations described above used Hooke’s law
to express the stress-strain relationships, which for an

elastic and isotropic material are

a=De (1.6)

where the stress vector 0, the stain vector 6, and the

elasticity matrix D are given by:

 

a={ax,ay,oxy}T (1.7)
e: {ewewexy}T (1.8)
1 v 0
D = E2 V 1 0 (1.9)
1-v 0 0 l-v

 

When large displacements are considered, Hooke’s law may still
be valid provided that second Piola-Kirchhoff stresses are
used in conjunction ‘with. Green-Lagrange strains and the
material straining is small”. Consequently, the previous
research mentioned above used implicitly second Piola-

Kirckhoff stress as a measure of stresses. There has been much

8

discussion about the physical nature of the second Piola-
Kirchhoff stress tensorwmh However, It should be recognized
that the second Piola -Kirchhoff stresses have little physical
meaning, and.in practice, Cauchy stresses should be the stress
quantities to be compared to available experimental work.

The:general definition.of strains which, is valid whether
displacements or strains are small or large, was introduced by
Green and St.Venant, and.is known as the Green’s strain tensor
or the Green-Lagrange strain tensor. In a fixed xyz-Cartesian

coordinate system, the strain components in terms of the

displacement components are:

 

__ au 1 duz 6V2 aw2

Qx- ax +I2<dx) +(6.x) +(BXH (1°10)
_ 8v _1 duz 6V2 6w2

8’" 6y + 2(6)!) +(3Y) +(3YH (1°11)

 

Ky...g_y_g§%gg_y%rg}g%g_; (1...,
The investigators mentioned above, except Allen.and Al-Qarra“,
utilized a simplified geometric nonlinear model in which it is
assumed that the squares and products of derivatives of the
in-plane displacement components u and v are small compared

with those of w and therefore may be neglected. Thus, the

following strain-displacement relationships were used

_ Bu 1 6w2
ex _ .3; + 3(3) (1.13)

6V 1 dwz
= __ _.__. 1. 4
8” 6y + 2(3)!) ( 1)
8 Ba + 6V + awdw (1.15)

”‘5 a as
From now on, the term "von.Karman.model" will refer to a model
using equations (1.13-14-15) as the strain-displacement
relationships, while the term "general model" will refer to a
model using equations (1.10-11-12).

Allen and Al-Qarra mentioned in their paper18 the use of
a general model and that their results compared well with the
previous von Karman models cited above, but did not draw any
conclusions regarding this point. This does not seem to be
coherent, since an obvious and important conclusion that
should be drawn from their work is that analysis of membranes
with large displacements does not require the use of’a general
model. Also, it is clear from Equations 13 of their paper18
that the values of the load intensity, length of the membrane,
membrane thickness and Young’s modulus might have a
significant effect on the results for deflections and
stresses. Therefore, the use of the general model by Allen and
Al-Qarra is questioned.

Thin flat structural membranes with negligible bending
stiffness rely on catenary action to support transverse
loading. Therefore, in the initial flat position, the problem
has no linear solution. An iterative procedure is then

necessary. The procedure employed by Allen and Al-Qarra18 is

10
based on the generalized Newton-Raphson method; a deflected
shape for the loaded membrane was assumed as an initial guess
to start the numerical procedure. In some applications of this
technique, ill-conditioned stiffness matrices may arise from
an improper choice of the deflected shape. This situation can
be circumvented by arbitrary prescribing a new set of initial
displacements. The convergence rate is sensitive to the
accuracy of the assumed deflected shape, which depends on the
problem atqhand,and on the analyst’s intuition and experience.
To improve significantly the rate of convergence, Oden and
Sato21 proposed to start the analysis with a coarse finite
element representation of the membrane, then use the results
obtained as starting values for a more refined representation,
the displacements of the added node points being obtained
through linear interpolation. Dealing with the complete set of
undetermined displacements when guessing the initial deflected
shape, is a cumbersome task, particularly when there are few

restraints on the membrane edges.

1.3 Scope of the Research

The present research will investigate the static behavior
of thin flat membranes which are subjected to transverse
loads. It will include the following parts:

1. Theoretical development and computer implementation of a
nondimensional incremental nonlinear finite element model

using von Karman’s strain-displacement relationships to

11
analyze thin flat membranes subjected to transverse loading.
A total Lagrangian formulation will be used.
2. Implementation of a new numerical technique which will
reduce the number of variables in the initial guess of the
deflected shape to at most three. These variables will be
referred to as "the initial virtual prestressing variables"
and the numerical technique as "the initial virtual
prestressing technique".
3. Investigation of several numerical examples in order to
test the validity of the model by comparison to the results of
previous research.
4. Evaluation of Cauchy stresses in the membrane to allow
meaningful comparisons with experimental results.
5. Theoretical development of a general incremental nonlinear
finite element model where the general strain-displacement
relationships will be used instead of those of von Karman.
6. Comparison of the von Karman model and the general model
and determination of the range of applicability of the von
Karman model.
7. Investigation of the effect of membrane aspect ratio on
deflections and stresses.
8. Investigation of the effect of Poisson’s ratio on stresses
and deflections.
9. Investigation of the effect of membrane initial

prestressing.

12
10. Investigation of the effect of varying the boundary
conditions.
11. Investigation of the effect of considering a
nonconservative loading ,that is, one that changes due to
deformation. This permits a more accurate representation of

differential air pressure.

2.Theory and Implementation

2.1 Introduction

Because of the lack. of bending stiffness, membrane
structures show large displacements. Prestressing forces
stabilize the structure and provide stiffness against further
displacements. The response to prestressing forces is always
nonlinear because the equilibrium configuration as well as the
state of stress are: dependent. on 'those forces. But. the
response to in-service loads may be either nonlinear or quasi-
linear depending on the directions and magnitudes of the in-
service forces compared to stresses and deformations of the
prestressed structure. Three phases may be distinguished in
the physical behavior of a membrane structure7:

Deployment phase

During this phase, the membrane unfolds from its compact
configuration into a state of incipient straining and is
stress-free. The membrane behavior is lightly nonlinear, but
the equations of statics and constitutive equations are not of
interest until the state of incipient straining is reached and
the prestressing phase begins.

Prestressing phase

During this phase the structure undergoes large

displacements until a static equilibrium configuration is

reached. Therefore, nonlinear strain-displacement

13

 

 

l4
relationships are needed. Strains are small but relative
rotations may be large and thus second-order terms of
displacement gradients may be significant. During the
prestressing phase displacements predominate over strain
effects.
In-service phase

The additional displacements due to in-service loads are
generally much smaller than the prestressing displacements.
This is due to the fact that the prestressing stiffens the
membrane structure. The membrane behavior during this phase is
either nonlinear or quasi-linear depending on the relative
magnitudes of the prestressing forces and the in-service
loads.

The state of stress in a membrane subjected to loads
normal to its plane depends markedly on the final shape of the
membrane. However, this shape is unknown, as are the in-plane
membrane sectional loads. Accordingly, the theory of finite
deformations has tot be applied, and therefore nonlinear
strain-displacement relationships are needed. These strain-
displacement relationships are given by Equations 1.6, 1.7,
and 1.8. Also, the equilibrium conditions of the membrane

should be considered in the deformed configuration.
2.2 Basic Assumptions
The present work is concerned with the static behavior of

thin flat membranes which are subjected to transverse loading.

The following are the basic assumptions:

15
- the initial state of the membrane is either a state of
incipient straining or is initially prestressed by in-plane
forces so that the initial configuration can be assumed to be
in a state of equilibrium prior to load application,
- The membrane is assumed to be flat in its initial
configuration, i.e. lying in the x-y plane of a fixed
Cartesian coordinate system xyz.
- the membrane material is homogeneous and isotropic and has
a linear elastic constitutive behavior.
- the membrane bending stiffness is generally negligible and
therefore may be discounted. Consequently, the membrane may be
considered as a two-dimensional material body in a biaxial
state of stress where only in-plane stresses occur.
- Due to the fact that generally the membrane thickness is
very small compared to the other dimensions of the membrane,
a uniform stress distribution across the membrane thickness
may be assumed. Therefore, the membrane stresses may be
replaced by membrane sectional loads.defined.as the statically

equivalent loads per unit length of section.

16

2.3 Fundamental Equations Using a Nonlinear Finite Element

Formulation

2.3.1 Basic problem

In the development to follow, a thin flat structural
membrane in a Cartesian coordinate system is considered. When
subjected to transverse loading,the membrane can experience
largeldisplacements and large strains, but exhibits a linearly
elastic constitutive response. However, there is a strong
geometric nonlinearity in the deformation process of the
membrane. The aim is to determine the configuration of the
membrane in its final state of equilibrium and the
corresponding state of stress.

In order to include the effect of large deformation for
this geometrically nonlinear problem, special treatment is
required. The underlying theories and their solutions can be
formulated by means of nonlinear equations or through
equivalent variational principles. The latter approach will be
used in conjunction with the finite element method. This has
the advantage of solving the problem without dealing directly
with the complex nonlinear equations, and allows an easier
computer implementation.

To develop a finite element strategy, the membrane
continuum will be approximated by a finite number of small
components called elements. These elements are assumed to be

connected on their boundaries at selected node points on the

17

membrane. Thus, the continuum is represented by a discrete
model at the onset. This way, the problem reduces to one of
evaluating a finite number of discrete variables which are the
displacement components of the node points.
2.3.2 Formulation of the Continuum Mechanics Equations of
Equilibrium

Using a total Lagrangian approach, the equilibrium of a
membrane finite element in the final configuration is
expressed by applying the principle of virtual displacements:
V09 53" a dVo" = [V00 6U” :3 dV0° + L. 6U3T £5 dse + ; aaf F1

(2.1)

where:

V0‘ is the volume of the membrane element in the initial
configuration,

5? is the surface area of the membrane element in the initial
configuration,

f” is the vector of body forces,

15 is the vector of surface tractions,

.E.are concentrated forces,

U is the displacement vector,

If is the surface displacement vector,

U. is the displacement vector corresponding to points of

application of concentrated forces F,

I

o is the second Piola-Kirchhoff stress vector,

6 is the Green-Lagrange strain vector, and

18
6 means "variation in".

In the development to follow, it will be assumed that no
point forces are applied.to the membrane element and that body
forces are either negligible compared to transverse loading or
will be embedded in the vertical component of the transverse

loading. Equation 2.1 may be written as:

as” a dvoe =f 50” f5 d5:e (2.2)
v; s'

For clarity, the superscripts "e" and "S" will be dropped, and

Equation 2.2 may be written as:

de’adVo =fs dUdeS (2.3)

V0

2.3.3 Description of the Curved Isoparametric Finite Element
and the Corresponding Shape Functions
2.3.3.1 Isoparametric Finite Element

There are many possible choices for the master finite
element. As mentioned earlier, the membrane problem may be
considered as a plane stress problem. In a Cartesian
coordinate system, the simplest element form is a rectangle.
An eight-node isoparametric element called "serendipity
element" was chosen because it allows more accurate modelling
than a four-node element (see Figure 1 below). To ensure that
a small number of elements can represent a relatively complex
form, the two-dimensional rectangular element will be mapped

into a distorted rectangular element”.

19

 

 

 

 

 

7
4: e :3
s
80 r 06
1: e :2
5

Figure 1: Representation of the Eight-Node Master Element

2.3.3.2 Shape Functions

The basic procedure in the isoparametric finite element
formulation is to express the element coordinates and element
displacements in the form of interpolations using the natural
coordinate system of the element. The coordinate

interpolations are:
8
x=2Nixi y=2Niyi (2.4)
' 1%

where x and y are the coordinates at any point of the element.
It should be mentioned that in a total Lagrangian formulation,
all computations are referred to the initial configuration of
the membrane. Therefore, the element nodal displacements will
be chosen aligned with the global assemblage nodal
displacements, and the element local coordinate systems

coincide with the global coordinate system.

20
The interpolation functions also called shape functions
are defined in the natural coordinate system of the element,
which has variables r and.s that each vary from -1 to +1. They
are defined as follows (see Fig.1 above):

- for corner nodes (i=1,2,3,4)

N1. = —:-(1+r0) (1+so) (r0+so-1) (2.5a)
where:
r0 =rir so=sis (2.5b)

riand siare the natural coordinates of node i, respectively,
in the r and 5 directions.

— for midside nodes (i=5,6,7,8)

s?
1 (1+sb)(1-I2) (Z'SC’

I?
1V-= 1 (1+r0)(1-sz) + 2

1 2

 

 

2.3.3.3 Degrees of Freedom

In. what follows, only the translational degrees of
freedom u, v, and w will be considered. Therefore, there will
be 3 degrees of freedom per node.

In the isoparametric formulation, the element
displacements are interpolated in the same way as the
geometry:

8 8
11: 23Agui tr: 23AQV3 1V: 23AQW3 (2-6)
=1 l=l

l

21

where u, v and w are the local element displacements at any
point of the element and u,, v,, and w,, i=1,...,8, are the
corresponding element displacement at its nodes.
2.3.4 Reformulation of the Equilibrium Equations
Let:

U={u,v,w}T a={u1,vl,w1, ...... ,u8,v8,w8}7 (2.7)
The displacement vector U may be written as:

U=Na (2-8)
where N is a 3x24 matrix defined as

N= [N11,: ...... :NiI3i ...... {NBI3] (2.9)

and I} is the 3x3 identity matrix. Also, if B is the strain-

displacement matrix (to be defined later), then

de=36a de=Bda (2.10)

Consequently, Equation 2.3 becomes
6aT(f B’adVo—fNdes)=o (2.11)
w 5

6a being an arbitrary variation of a, Equation 2.11 may be

rewritten as

B’adVo-F=0 (2.12)
V0
where
r=fNdes (2.13)
3

Equation 2.12 constitutes the nonlinear equilibrium equations

of the finite element model of the membrane problem. In the

22

solution procedure, a direct method does not guarantee
convergence to the solution, and consequently Equation 2.12
will not be used directly. Instead, an incremental version of
this equation will be derived.
2.3.5 Incremental Equilibrium Equations

The basic approach for deriving the incremental
equilibrium equations is to assume that the solution for load
g is known, and that the solution for load g+Ag is required,
where Ag is a suitably chosen load increment.

The equilibrium equations corresponding to g+Ag is:

w(q+Aq) = o (2.14a)
or

P(q+Aq) — F(q+Ag) = 0 (2.141))
where

P = B’adVo (2.15)

V0

and F is defined by Equation 2.13.

Since the solution corresponding to load g is known,

(2.16)
P(q+Aq) = PM +1”

where AP is the increment in nodal point forces corresponding
to the increment in element displacements and stresses from
load g to load q+Ag. AP can be approximated using a tangent
stiffness matrix KT(g) which corresponds to the geometric

conditions at load g as

23

AP = K,(q) As (2.17)

where Aa is a vector of incremental nodal displacements.

Substituting Equations 2.17 and 2.16 into Equation 2.14blgives

K,(q) An = F(q+Aq) -P(q) (2.18)

and solving for Aa.approximates the.displacements for the load

q+Aq as
a(q+Aq> = a<q> +Aa (2.19)

The exact displacements at load q+Aq are those that
correspond to the applied load F(g+Ag). Because Equation 2.17
was used, the displacements computed in Equation 2.19 are only
an approximation to the exact displacements. Having an
approximation for a(q+Ag), the strains, stresses and
corresponding nodal forces at load.g+Aquay be evaluated. Then
the next increment is started. Due to the approximation made
in Equation 2.17, and in order to avoid instability, an
iteration process is required in order to obtain a
sufficiently accurate solution of Equation 2.14b. By combining
the INewton-Raphson iterative :method. with. the incremental

method described above, Equations 2.18 and 2.19 become,

respectively,
K251“ (q+Aq) A8“) (q+Aq) = F‘“ (q+Aq) -P”’1’(q+Aq) (2-20)
a‘“(q+Aq) = ia(1‘1’(q+Aq) +Aa‘1’ (2.21)

where i is the iteration number.

24

The initial conditions are:

 

KI” (q+Aq) = x,<q)
8‘°’(q+Aq) = a(q) * 12:22)
P‘°’(q+Aq) = P(q) .

If an index In describing the mth increment is introduced,

the above equations may be written as

Kan“) A23" = Id" -P.f"“ (2.23)
a)" = a.”‘” +Aa.”’ (2.24)
Irina) = I’m-1

a)” - a“ > (2.25)
P.(o) Pfl-l ,

 

3. Fundamental Equations for the

von Karman Model

3.1 Derivation of Element Matrices
3.1.1 Derivation of the Basic Element Matrices
The von Karman strain-displacement relationships are

defined by Equations 1.13, 1.14, and 1.15. Rewriting these

three equations in a vectorial form gives

r w r

 

 

 

 

22 11v?
3x ax 2(dx)
:1: 2, =4 .31}: +++%(%’)2» (3.1)
a... 611.21 mm
3%! 6x; , ){dy‘
g1=¢L+gfim (302)
r du ‘ ' a '
‘1.” g}: r: 0 Tia} 0 v =LU (3-3)
.Qg+iflf .31 _Q.0
\dy' ax _dy'lix .

 

 

 

 

where the subscript 1 refers to the von Karman model, and

O (3.4)

 

 

25

From Equation 2.8,

where

With the help

 

 

 

 

 

 

 

 

 

26

of Equation 2.9,

an: (3:: ...... :35: ...... :33
where
'aN, ‘
ax 0 O
aN-
B:=L[Ni1'3] = O a; O
6N} EFL 0
_757 ax .
also,
{1222‘ "—61 o-
2(dx) 6x ‘2!
._1aw2_1 flax_l
"" 2(8)“? 35’ 91' 3410‘
my _a_w a; ay
_ibtdyy _éb’ ax
where
"aw .
5; 0
6w
= O ——-
A1 ay
1"! 9!
(hr ax

 

 

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

27

6w 6
——- 0 0 ——-ll
0: ax = ax v =L’U (3.11)
2‘." o o _6_
6y 6y

With the help of Equations 2.8 and 2.9,

 

 

01:01.: (3.12)
where
a1= [911: ...... : 1‘: ...... :a:] (3.13)
o 0 33f;
611:1/ [N113] = 6N (3.14)
o o 1'
65/

By taking the variation of Equation 3.2, and using Equations
3.5 and 3.9,

6.91 = deL+ 5811”,

(3.15)
68L=BL58 (3.16)
emf-gamefigraal (3.17)

but

61:10:41,661 (3.1a)
therefore

defi=3m16a (3-19)

BHL1=AIGI (3.20)

28

With the help of Equation 3.13,

em = [31,: ...... mg“: ...... :Bgm]
dwémfi
° 0 are;
1 - 22%
3mm 0 0 6y 6y
0 . flflflfl.
_ dy 6x 8x dy.

 

 

Using Equations 3.16 and 3.19,
621 = 816a
£H==BL+Bmu

Equation 3.2 becomes
:1 = 31a

2: ......m

or, in a partitioned form
31’ = [Ia/1: ...... : If: ...... :3’2]
where

1 1 1 1
3,1 = BI: 4.38an

(3.21)

(3.22)

(3.23a)

(3.23b)

(3.24a)

(3.24b)

(3.25a)

(3.25b)

 

29

 

 

 

 

 

 

.aNi o .lfyfilfif
ax> 2 fix ax
6N3 ].aN16w
31:: 0 6y 333'5 (3.25c)
6N, 6N1- 1( 6N1. aw+ 6N1- 6w)
fa? 8x 3 ‘6‘)?8—52 Tia—x,

3.1.2 Derivation of Element stiffness Matrices

In the following paragraphs, it is assumed that the
loading is deformation-independent and the dimenSions of each
finite element are sufficiently small that the load intensity

q is uniform over each element. Thus,

F=q N’e,d5’o (3.26)
so

where e, is the unit vector normal to the membrane in its

initial configuration.
The nonlinear equilibrium equations of the finite element

model of the membrane problem are given by (see Equations 2.12

 

and 2.13):

1H8) = VOBlraldVo-F=O (3.27)
By letting

01 = 1-Ev2 of a; = {afimfimfiyf (3.28)

and using
dV,J =hds0 (3.29)

Equation 3.27 becomes

 

 

3O

Eh
1—v2

 

fla) = [SB1TafidSo—F=o (3.30)

The tangent stiffness matrix is defined by
dt =Knda (3.31)

Differentiating Equation 3.30 with respect to a, and using the

assumption that F is independent of a, gives

Eh
1-v2

dt==

 

(f5 0113de dso +f331’daé dso)(3.32)

The stress-strain relationship is
01=D81+ao (3.33)

or equivalently

 

 

 

04=D’£1+0{, (3.34)
where
E E'
D: l-sz’ 00: 1-v20é (3.35)
1 v 0
pl: V 1 0 . (3.36)
O O l-v
2

Using Equations 3.23a and 3.34,

da§=D’del=D’31da (3.37)

then

1.303le0: 61.90 = [s BITDIB:l dS'o da = (K1,.+Kp;r.1) d3 (3.38)

 

 

31

where

K}, = BLTD’BL d5o
so

I
Kim‘=j;(3mtpflamn*“QnuTDflBi+1%marpaamu)dSo
0
Also, using Equation 3.20,

_Lwdafraidso=iflflF5Tdm1T°éd5b

 

 

 

.04.
13-1 yawn-(v.13?
0 (43") (1(5)?) LO/ny ax}, 0y d(—y)

where
/ /
/ a} an,
51:
a/ 0/
xy y

 

 

using Equation 3.12,

cflfl==qlda

“deli” afi d30 = fSOGITsl’ c;1 d30 da = Kg. da
and

K51 = f 1311's; 91 dso
50

(3.39)

(3.40)

(3.41)

(3.42)-

(3.43)

(3.44)

(3.45)

(3.46)

(3.47)

 

 

32

 

-mnf=xi+xén+3$1

or, alternatively

“n:=xi+xhu+3%1

.5” is the element tangential stiffness matrix

 

Ki==

K; is the linear element stiffness matrix

Eh

I

 

Khm:=

Eh” is the nonlinear element stiffness matrix,

the geometric stiffness matrix

Kfi.='JEE"Kg

1-v2
and.Ka_is the element stress matrix.

The components of 81’ are given by

(3.48)

(3.49)

(3.50)

(3.513)

(3.51b)

also termed

(3.51c)

 

 

33

of, = :.",‘+v(:y+af,x
/_ +, + / '
ay-vex 8y 0m, , (3.52)
_ 1-v
J

 

3.2 Rearrangement of Element Stiffness Matrices for Computer

Implementation

3.2.1 Numerical Integration

To be able to evaluate the element stiffness matrices, it
is necessary to compute the basic element matrices BL, Em,
and 6,. These matrices are obtained in terms of derivatives of
element displacements or element shape functions with respect
to the local coordinates. The derivatives should be expressed
in terms of derivatives with respect to the natural

coordinates r and s as follows

_a_ _3_
&x 1 61

= ‘ 3.53
a .7 a ( )
ay 63

where J is the Jacobian matrix defined as

6x ED:

J: Br Br (3.54)

&X‘QX
as 63

Also the element surface is expressed as

 

 

 

34

d3o = Jdrds (3.55)

where J is the determinant of the Jacobian matrix.
The general expression of element stiffness matrices can then

be written as

XI = [soddso = f_:f_:xg(r, s) J(I,S) drds (3.56)

To evaluate explicitly the surface integrals in the
expressions of element matrices is not effective. Therefore,
numerical integration is employed. In fact, numerical
integration is an integral part of the isoparametric element
matrix evaluations. Practically, when using this numerical
integration procedure a choice should be made concerning the
kind.of integration scheme and the order of integration. Gauss
quadrature was chosen because it requires fewer function
evaluations to get accurate results. However, it should be
mentioned that the Newton-Cotes formulas also may be
efficient”. According to Bathe”, a third order Gaussian-
quadrature is a reliable integration order for an eight-node
isoparametric element.

Using numerical integration, Equation 3.56 may be written

£18:

3 3
K’=ZZd<ri,sj)J(rl-,sj)wiwj (3-57)

i=1 j=l

where.n, and.§ are the natural coordinates of Gauss points,

and w“ and wiare the corresponding Gauss weights.

 

 

35

The expression of the load vector F can be written as

a -1

1r: qfsoro aso = qfdf’l 1P0(r,s) J(r,s) drds (3.58)
where

17,, =N’e, (3.59)

or, by using numerical integration

3 3

F: q: Zrfirflsj) J(ri,sj) wiwj (3-60)
i=1.j=1

The element internal force vector P appearing in the right-

hand side of Equation 2.23 may be written as

 

 

Eh I
p = B o/ds (3.61
_ Eh _ Eh ’1 '1
P1 — W soP°'1dS° - 1—v2 -1f-1 Po'1(r,s) J(r,s) drds (3.62)
where
pm 2.3170; (3.63)

or, using numerical integration

Eh
l-\’2 is

Eda

R1:

 

3
jfl

H

For clarity, the arguments r and s will be dropped in the
following paragraphs.
3.2.2 Rearrangement of Element Matrices

The derived expressions of element matrices are very

general. To allow easy implementation of these matrices in a

 

W

 

36

computer program, the partitioning properties of matrices will
be used. 7

Because an element has eight nodes with three degrees of
freedom each, the element stiffness matrices will be of order
24. By taking advantage of the general form of the elementary
matrices BL, G, , and Bzvu (Equations 3.7, 3.13, and 3.21), and
using partitioning, all the element stiffness matrices K'o may

be put in the following form:

K5: . [Ir/0U]3x3 . (3.65)

where i is a row index, and j is a column index, both taking
values from 1 to 8. K'is thus partitioned into 64 submatrices
K’f of order three. The expressions of the typical submatrix

for each element stiffness matrix are given by

KI§£=BLJTDIBIZ1 (3.66)

I I I I
K0351 = Kom1,1+xom1,2+xoum,3 (3'67)

where

 

 

 

37

1 _ 1T
JKQflnA.-1%;.D'Béu
1 T
Iflfihhz==‘mn.D’Bf
1 T
3:31am: = 11.1 9,331.1

1 _ 1T I
K’oi, - a, .91 of

The load 'vector integrand .E, may also

partitioned form as
1",, = [170“: ...... :FJ': ...... mg") T

where

O
PfE=Aglgeq==Nge = C)

i

(3.688)

(3.68b)

(3.68c)

be written

(3.69)

in a

(3.70)

(3.71)

3.3 Nondimensional Formulation of the Incremental Equilibrium

Equations

Instead of dealing with all the parameters involved in

membrane analysis in its original form, it is more interesting

to recast the above formulation.in a nondimensional form. This

has the advantage of reducing the number of parameters

involved by grouping them into a smaller number. In addition,

solving one problem gives the solution of a whole class of

problems.

38
The variables L1, L2, h, E, u, q and W defining the

membrane problem may be related by a function:

f1(L1,L2,h,E,v,W) =0 (3.72)

where

L1, L2 are two orthogonal representative dimensions of the
membrane in its plane, and W is a representative displacement
of an arbitrary membrane point e.g. one of the displacement
components u, v, w.

Using the Pi Theorem”, a relation equivalent to Equation
3.72, but expressed in terms of:nondimensional.parameters, can

be found:

f3(n1,n2,n3,n4,n5)==0 (3.73)
where
L2 L1 q W 4
1t =v; n =-——; n3=-——; 'n4=-—; fl5=-—— (3.74)
1 2 1 h E L1

or, alternatively, by combining dimensionless parameters

L2 3 qu) (3.75)

W==L f v,-—n ,———
1 2[ L1 L1 Eh
It is clear that the nondimensionless parameter gin/Eh is the
most important among the four dimensionless parameters
included in Equation 3.75.

To permit comparison with previous work, the following

relationships are used%

 

 

39

= - 2 9E:
k 2(1 v ) Eh (3.76)
x=Lf{; y=Lj7 (3.77a)
3 3 ' l 3 b
U=Lk3fi v=Lk3V w=Lk3W (~77)
i={fi1,x71,w1, ...... ,fi8,\78,v78}7' (3.776)

where L replaces L1 and the factor 2(1-22) is introduced.

Then, it may be shown that

a = Lk'i'ué’ (3-788)

where M'is a 24 by 24 diagonal matrix written in a partitioned

 

 

form as
"Mo 0 o o o o o o‘
0 Mo 0 o o o o o
o 0 Mo 0 o o o o
o o o o o o o
M: M° (3.7813)
0 o o 0 Mo 0 o o
o o o o 0 Mo 0 o
o o o o o 0 Mo 0
o o o o o o 0 Mt
3
k3 0 o
M = _1' 3.786
° 0 k3 0 ( )
o o 1

and 0 is a 3 by 3 null matrix.

 

 

40

If the variables with the tilde symbol are the "nondimensinal

variables", then

 

 

 

ds0 =L2d§o J=LZJ
au 361'} 6v 36)?
_.=k3 ’0 _=k3_;
6x a2 ax' a"
figzkéafi; flzkéafi-
6y 657 BY 557'
el=.k3é;
Equating Equations 3.24a, and
following relationships:
- 1- - .
ELM: k3BL 3mm: 3,,“
5'.= "15
where

Then
3
01:1(351/
3-
s{=k3 ’

The nondimensional stress-strain

aw- :22
E’c‘k a~
aw- 31-32
E k 6*
3.80,

relationships are

(3.79a)

(3.79b)

(3.79c)

(3.80)

and using the two

(3.81)

(3.82a)

(3.82b)

(3.83a)

(3.83b)

 

 

41

 

1?
II
NH
b
m
E

l
h
m:
5

hp

ll
NH
P:

wlw
l

51 =

$"
+
k'

E

Therefore

u __ 1 "Hi
‘Koi"E;HK0L

u _ k
57.3.... -

 

 

Klgrmm _ k E,”

M! - k
-Komna

 

Then

#1 _ k
Irum1‘

 

where

(3.84)

(3.85)

(3.86a)

(3.86b)

(3.87)

(3.883)

(3.88b)

(3.88c)

(3.89a)

42

 

 

1
"11.1 _ ”I11 "I11 "' "I11
Kama ‘ Kama. +Kon1,3 +k 3 Korma
Also
2
111 _ k 3 ~11:
K 001 - 2 K 001
L
Then, with
[1 ._ [1 [1 I1
.Kqél-erihKogn+lde
[1.1 _ l ~I11
K on ‘ —3 Ken
L
where
1 2
"11:! _ "I11 - "I11 " “11.1
Kan "Kon+k3Koum+k3Kon
Also
1
I11 _ k 3 “71.10
' K on ' 2 K on
L
where
"l-U’ _ ~I11 ~11: -‘- "I11 ~11: ~I11
Kan " K0L+K0NLL1 +k 3(K 03114.2 +Koum,3+xon)

and

= flflq. l—V 6N1 5N]-
a)? a)? 2 637' 657

 

...1 .
K’.¥~.<1.1>

= v 8N1. 6N]. + 1-. 61v, em].
as: as; 2 357 352

 

E’éilum

(3.89b)

(3.90)

(3.91)

(3.92)

(3.93)

(3.94)

(3.95)

(3.963)

(3.96b)

 

 

 

 

 

           

 

 

 

K’i"(1 3)_ LN1(%§E+V_3LVZQQ) + 1- v6N(aN, a_w+6N _Jaw)
011 a? a3 a? a? a? 2 ay' 3? fly a? ax
(3.960)
, aN aN 1- BN- 6N-
K’ij = 1 j V 1 J 3.96d
1<,m(2 1) V 657' a)? 2 62 a5"; ( )
. 31v. 31v 1- 6N aN.

Iij = 1 j V 1 .7 3.96
Izmn(2I2) a? a??? 2 a? 3X ( e)
R/ij‘ - 6NW( flaw+a—1N4_a_~w) + l—v 6Ni(aNj—_ aw+ aN' 46—147)

0 T1 ay 62 ax ay ay 2 65? 6x 6y 63’ 31?

(3.96f)

Kara, 1) = k%[a__Nj(aNi_ 3W LIE-fir), 1‘” aN(aN aw+ 6N 64)]

an ax ax 6x 637 617 2 35’ 5X aY ‘97 ax

K/lj "

(3,2) =ki[a_N1(v

 

 

 

 

8_N1 6w _a_N1 6w

(3.969)

 

5N. 669+ 3N. 6:21)]

—)+ l—V6MG( __. _____.
2 a)? a)? 337 a)"; a}?

 

K011 a? 6x 6x 3y 6y
(3.96h)
21-. .1. 6N.a~ 6N.a~ 6N.a~)
I] = 3 ll 1 W .__-7__E
K°Tl(3'3) 1‘ as; a~(ax a7+v as; 657
. 61537 211.93 31193)
6" ay 657 a" a)? 6"
(3.961)
. 1-v 911v “1217) (flflaflzfl
2 a" a" a? a? 62 ~ ~ ~
+ aNi(§/_a_1!1+a“=’ % 6Ni(5 flJra/EIZZH
6* "6" "y 837 3" "y as: y 657
Us1ng Equations 3 57 3 65, 3 78, 3.79a, 3.92 and 3.94,

I _~I
Kim-Kn

(3.97a)

 

 

 

44

 

 

 

 

l-~.
x’,,u=k3x’n (3.9713)
Also, with the help of Equations 3.58 and 3.59,
F0 =fI‘o F=L2f' (3.93)
where
f'= q)?"
3 3 (3.99)
P" = 2: EFo(ri,sj)J(ri,sj)w1-wj
i=1 j-1
Also, using Equations 3.63 and 3.64,
3
k3 2 .2- .
Po,1 = L 130,1 p1 =Lk3 1 (3 100)
Eh W
1‘5 = P"
1 l-VZ 1
P1’= 2 z Po'1(ri,sj) J(ri,sj) Win
1-1 j=1 ,
Using Equation 3.76,
P 2qL2jy
1 _ 1 1 (30102)
k?
The

incremental equilibrium equations are given by
Equation 2.23.

If the indices m and i are dropped, this
equation may be written as

KilAa==Fb£3

(3.103)
Using Equations 3.48, 3.76, 3.78, 3.97, 3.98, 3.99, and 3.102,

Equation 3.103 becomes

45
If Aqiis the incremental load at the jth increment where

jsm, the load intensity at the mth increment is

11)
gr; qu (3.105)
'1
Letting
ACI- "' quL
p =__.J. a = p k =___(1—v2) (3.106)
j g m 1.1 j m Eh
gives
qm=amq km=amk (3.107)

If k is replaced by km, the incremental equilibrium equation

becomes

 

~n (b1) <')_ a -1~n) ~M a) 3.108
[K 15]”) {Aa}ml _ 2m k 3 F ml _P1m1 ( )

Equation 3.108 has been derived for an arbitrary element.
However, it still represents the structure's fundamental
equation, provided each term contained in this equation be
regarded as the equivalent termicorresponding'tO'the assembled
structure. The assemblage procedure will be discussed in

Chapter 5.

4. Extension to the General

Nonlinear Model

4.1 Derivation of Element Matrices

4.1.1 Derivation of the Basic Element Matrices
The general strain-displacement relationships are defined
by Equations 1.10-11-12. Rewriting these three equations in

matrix form gives
82=31+Ae (4.1)

where the subscripts 1 and 2 refer to the von Karman model and

to the general model,respective1y, and

 

 

’ 3.2.922 i
49x (ax) (6x)
= A =_1_ 21.2 i” (4.2)
A: A8’ 2) (3Y)+(3Y) }
‘3 gamma);
k(6x895! axébJ,
or
Ae=%A,0, (4.3)
where
au 6V au 6V T
= __ _.__. _.__. .... 4.
0 {6x” 3x“ ay' ay} ‘ 4a)

46

 

 

_ay (hr 5; (be

 

 

 

also
Q,=(%£l
where
G-[Gl' 'G" "
a-- 3) ...... :3) ...... I3
1
6N3 0 0
dx
6N3
0 ——— 0
Gf- ax
6N1' 0 o
6y
0 6N10
6y .

 

 

Differentiating Equation 4.1,

dz, = d81+dA8

dAe =%d1l, 02%;}; d0,
Using
dAaa3=Wfiid03

dAe=Azd02=A2Gada

then

47

there results

(4.4b)

(4.5)

(4.6a)

(4.6b)

(4.7)

(4.8)

(4.9)

(4.10)

 

 

48

dAe = AB da (4.11)
where

AB=A263 (4.12)
Therefore

de2=33da (4.13)

B,=31+AB (4.14)

Also using Equations 3.25a, 4.1 and 4.13, gives

22:132.: (4.15)

where Egzhs the strain-displacement matrix corresponding to

the general model and is given by

132=1§1+—AB (4.16)

4.1.2 Derivation of Element Stiffness Matrices
Similarly to the the von Kérmén model, the tangential

stiffness for the general model may be written as

 

Eh I
K72: 21:22 (4.17)
l-v
where
K42=K£+K§22+K52 (4.18)

With the help of the equations derived in the previous

paragraph together with those derived in Chapter 3, and using

49

the same matrix techniques, the following equations may be

 

written
Kim = Kfm+AKQ (4.19)
where
Ax}... = Axgmxg (4.21)
A191,,1 = SOB1TDIABdS° (4.22)
4321:. = SOABTD’33 d3o (4.23)
Letting
0:: 1-Ev2 0; Ué={0,/(, 05/" 0:0,}1" 09°24)
Ax5=fs c;,T.s,’c-:.'2 ozs0 (4.25a)
0

/ /
I 02L, gal;

33 = (4.2513)

I /
QMJ; 03%;;

where I} is the 2x2 identity matrix.

Then

K42=K1£1+AK4 (4.26)

where

Ax; = Ax’m nut},re +Ax;

Equations

4.2 Nondimensional Formulation of the Incremental Equilibrium

 

.£~ 1 - k%
1 ~ ~ ~ 3 -
L
Ae=lA§a=545u§
2 2

But employing

1

A§M= k3A§

ulc-

k
A =
8 2

 

455

Then, with the help of Equation 3.73,

82

50

(4.27)

Using Equations 3.67-68-69 defined in Chapter 3,

(4.28)

 

(4.29)

(4.30)

(4.31)

Equation 4.1 becomes

(4.32)

(4.33)

51
and

 

. (4.34)
1 2

The stress-strain relationships are given by

0§=D’8z+0€ (4.35)
Then
3 3-
aé-kJér’,’ s,’=k3 3’ (4.35)
where
3
6',’=D’£’,+6'0/ a{,=k350/ (4.37)
Using Equations 4.22, 4.23, 4.25a, and 4.27,
’ - 3 "' 4.38
ARfini-—k3AR;,n1 ( )
’ — 3 ~’ 40 9
ARQQ-I:FAR%% ( 3 )
AK,’,=1<%A1?; (4.40)
1.. 3 ~/ ~/_ ~ / ~ / ~/ .
AKT-ksAK, AK, - Arm +41%: +419, (4 41)
The

incremental equilibrium equations are given by

Equation 2.23. If the indices m and i are dropped,

this
equation may be written as

52
x” Aa=F-P3 (‘0‘2’

Using Equations 4.17, 4.26, 4.41 and 3.96, it is easy to show

that

3 ...
K” = 17132 K,’.1+k3AK,’.) (4.43)

 

Using Equations 3.76, 3.78, and 3.97, the left hand-side of

Equation 4.42 becomes

 

Kr: A8 = zqu fill-243 (4.44)
k7
where
i2, = f’;1+Afi; (4.45a)
and
A~;=k%A~;.M ' (4.45b)

Let the following quatities be defined

 

 

 

y Y' ) (4.463)
aNiaN3 aNiaNz
A3= ~-—1: A4=-—1r—<§
ax 6y 6y ax J
y y) (4.4613)
B=flfl Bzflgg
3 are 657' 4 637 65?,

 

53

 

 

 

 

 

 

 

 

 

 

 

c1 =A1 6'," +A2 ay’ + (A3 +114) axy’ (4.46c)
2 x ’7 y ) (4.46:!)
8N aa aN-aa
E =_7j_ E =_-7_~.
3 8x 37 4 57 a J
2 22 y y ) (4.469)
6N av 3N av
F =—=11—= F =—~1—.:
3 a a 4 a a ,
G _ 8N1 6!? G _fl 6!:
1 as? a}? 2 657 657
l (4.46f)
BNng G _ 6N1. aa
3 a}? 657 4 657 62?,
6N1 61‘? 6Ni 6“?"
H1 — H ~—-— ..
ax as: 2 6 6
57 y , (4.46g)
3N1 av 6N.- av
Hs‘ are—65"; H4‘a—ya2
_ 6N. aw 0N. aw
Tl‘ 652$? T2 857 ay
l (4.46h)
T = aNii‘? = aNi 6W
3 3" 6" 4 as; are,
Zl=A1+ 1_VA2 Z2=VA3+ EVA,
) (4.46i)
Z _ 1—v _ 1—v
3"VA4+ A3 Z4‘A2+ A1

 

Z5=EQ+VEQ
z,=E3+vFE
z9 = 131 +sz
Z11 = G3 + G4
Z14 = E3 +E4
an
Z17 = Z1???
av
Zzo = Z2};
69'
Z23 = 23:9}
61'}
Z26 — Zz‘a}
l-V
Z29 = "—2_Zn

Z32 = (ZS Gl + 26 62 + Z14 Z29)k
Z33 = (Z7 G'1 +Z8 G2 +215 Z29)k
Z34 =(Z5H1 +Z6 H2 + Z14 Z3o)k

Zas = (Z7 H1 +Z3H2 + 215 Z3o)k

54

 

 

z6 =E2+vE1
Z8 = F2+vF1
Zlo==Bz+VB1
Z12 = H3 +H4 Z13 = T3 + T4
ZlS=F3+F4 Z16=BB+B4
613 av
Z18 = Zz'a—f, Z19 = Z15}
_ ail _ 3131'
221 ' 235}: 222 ‘ Z433;
6i? afi
Z24 = Z4533; Zzs = 23-6—35
_ av _ av
227 ‘ Z353"; Z28 ‘ 2253'?
l-v 1—v
Z30 _ 2 Z12 Z31 — 2 213

who

ulu

 

who

The coefficients of the 3x3

to 41%;, are given by

submatrix

 

(4.46j)

(4.46k)

(4.461)

(4.46m)

corresponding

 

55

DIN

41.65241!“ (2Z17+Z18+Z25+Z32+C1)k

who

430341: 2) = (Z19 +Z20 +222 +Z26 +Z33)k

uIN

AKJT<1:3> = (Z9G1 + Z10G2 + Z16Z29)k

UIN

41.60.7(2'1) = (Z19+Z21+Z22+Z2.,+Z34)k
l
AKJTQIZ) = (Z23+2Z24+Z28+Z35+C1)k3
l
AKo.r(2I3) ‘—'(Z9H1+2710H2+Z16Z3o)k3 '

uIH

Affirm, 1) = (2521'1 + z5 T2 + z14z31)k

UIH

ARO‘T(2,3) = (z.,:l'1 +231"2 +zlsz31)1<

ARO‘T(3,3) 0

Equation 4.42 then becomes

 

~/2 ~_ k ~I "I
IrflrAa- 2 P’— 2
where

By using Equation 3.107,
expressions of element matrices,

equation becomes

U'H

1
(i) _ ' _ ~[(1-1)
m " 3 P

[Kan {Aa} 7mk F; ,m

(4.46n)

 

(4.47)

(4.48)

and substituting km for k in the

the incremental equilibrium

(4.49)

56

4.3 Extension to the Case of External Pressure

The tangential stiffness matrix developed in the previous
paragraphs is valid only for membranes subjected to loads
which do not change in.direction as the membrane deforms. This
might be a severe restriction, particularly when the
structural membrane exibits large displacements and large
deformations, since the loading applied to membrane structures
is, in general, external pressure, which exerts force normal
to the deformed surface. In this section, element stiffness
matrices will be developed to account for changes in the
external loading due to deformation.

The element load vector is given by Equation 2.13. The
term de in Equation 2.13 is the load vector acting on the
deformed membrane element dS.

As the membrane deforms, there are changes in both the
direction of the external pressure force and the area on which
it acts. It is reasonable to assume that the intensity of the
external pressure does not change with deformation and that
the dimensions of the finite elements are sufficiently small
that the external pressure is essentially uniform over the

surfaces of each of them. Therefore,
f=qE', (4-50)

where E2 is the outward unit vector normal to S at an

arbitrary point of the deformed membrane.

 

 

57
Expressing the product E2 dS as a function of the unit vectors
in the global fixed Cartesian system of coordinates and of the

area dSo of the undeformed membrane element gives

__ a a
E, dS - (93+a—g)®(ey+a—;7) d3o (4.51)
where the symbol ® stands for the cross product operation, and
the displacement vector U is given by
U = ue,+ve,+we, I (4.52)

Writing Equation 4.51 in vector form,

 

f flfl-fl(1+_aif) ‘
axay 6x 6y
_ 2291-31 .62 4.53
2:, d3 —( 6y ax ay(1+ ax) >dSo ( )
au 6V_§y_fl/
. (“Exhfifl 6y 5X .

 

Decomposing E; into a difference of a constant vector and a

vector depending on the displacement components gives

 

' -fixfi’.§_¥1+§z’ ‘
0 axdy 6x( 8y)
_ _flfl Q E
2:,ds— o dso—( 6y6x+ay(1+ax) Hiso (4.54)
1 -21.-_a_x_»-_a_mz.22a<
. 3X 33’ 5X 6y 8y 8X J

 

The first vector on the right-hand-side of Equation 4.54 is

ezdSO. The second vector may also be decomposed into two

components:

13de = (9,-Eq1—Eq2)dso (4.55)

 

 

 

 

 

 

where
f g}! l
ax
_ .91
'En -< aY }
-224
. ax ay
and
’ _alfljzfly
ayébc axébr
E _) fiefirjgi’
9". axébr (firax
2222-222
L (bray axébr

 

J

58

Thus, Equation 2.13 may be written as

F:
or
F==FQ,-.fin-PE,
where

qu = qfsoN’ e, ds0

En
'En

F“ is the

conservative loading.

1'
quON qu d30

1'
quON sq, d3o

load

 

A

vector

1' __ T ... 1'
q SON e, dSo quON E“ :130 qfsoN z“ dso

corresponding to

the

(4.56)

(4.57)

(4.58)

(4.59)

(4.60)

case of

59

With the help of Equation 2.8, the following expressions may
be written

sq, = AaN a

 

 

 

 

(4.61)
where
6
O 0 -EE
01 6y
_ a -i
ax ay 0
, 0 -£&( it? 0 SW -u§Y
dy 6y ax ax
6w __6u _jfig .23
A?" 53—, 0 6y 6x 0 5X (4.621))
-_3X E o if _Q 0
. ay ay 6x 6x
and
Bu 6v 8w Bu 8V 6w T
= —I_l—I_l—I— 4'62
0 {ax' ax ax ay ay ay} ( c)
Differentiating Equation 4.59 gives
d? = -qu1-qu2 (4.63)
Substitution into Equations 4.60 and 4.61, leads to
1",, = (stoN’quN dSo a qu, = quda (4.64)

where

_ 1'
Kg: - qfsoN AqlNdso (4.65)

 

 

60

Also

div-q, = qfsaN’quzdSo

dz“ = gauge», $44,010
but, noting that
dAafi=Aq2d0
Equation 4.67 becomes
cfl%,==Am,d0

Noting also that

 

 

 

 

 

9:03 o=[ol:...:o,:...:oa]
where
'aNi
—— o 0
6x
0 6N1 0
6x
aN.
O 0 a;
G _.
1 6N1- o o
0y
o 6N1 0
0y
aN.
o o 1
6Y1

 

 

(4.66)

(4.67)

(4.68)

(4.69)

(4.70)

(4.71)

 

61

Therefore

d0=Gda dEa=Aq2Gda (4.72)
and

(11",,2 = Kg, da (4.73)
where

Kg, = quONTquGdSo , (4.74)

Thus Equation 4.63 may be rewritten as

d? = _qua (4075)
1Q may be called the "load stiffness matrix", and is given by
re = K¢+Ka (4.76)

Introducing nondimensional variables,

 

1 ~ 1 ~
A“ = EA“ G = 3C: (4.77)
letting
Eh I
qu 1_v21rq1 (4.78)

and using Equation 3.76, gives
I
Kg, = —x1 k1 =fSNTAq1NdSO (4.79)
0

~ ~

K1 = LK1 K1 =f§N75q1Nd§o (4.30)

 

 

62

 

I k.-
'K&I=Ԥ1Q
Also, if
Eh
K = x’
«2 l-v2 ‘2
then
K’ =ix, k,=fN’A ads
“7 2L % 9‘ o

By writing

 

 

 

 

Ac: = qu.1+Aq2.3
where
6w 6w
0 -—— 0 O ——-0
6y 6
A == 6w 6w
«2.1 — o o -— o 0
3y 6x
0 O 0 O O 0
6V avl
O O -—— 0 O -——
3y 6x
au Bu
= O 0 -—— 0 0 .__
Ac“ ay ax
_jbf.§2 o .QZ [EH 0
. (hr 3y dx ax
then
33==321+REJ
where

(4.81)

(4.82)

(4.83)

(4.84)

(4.85a)

(4.85b)

(4.86)

(4.87)

 

 

N°ting that

 

 

 

 

 

 

 

(4.88)

(4.89)

(4.90)

(4.91)

(4.92)

(4.93b)

 

64

it may be shown that

 

 

2 1L2 "3
3'44“ q; ‘43 (4.94)
k3
where
~. _ k‘i' ~. %~. 4.95
0 - 2 [ +k ] ( )

“8

 

 

The coefficients of the 3x3 submatrix corresponding to q are
given by
K3] (1,1) = 0
1
~..‘ 3
K33 (1,2) = 1; Ni(B4-B3)
3 cm 2
~..‘ 3 . _
K37 (1,3) =1; Nj[—a§7+k’(F3-F4)]
K;J'(2,1) =-K;3'(1,2)
(4.96)

~13. _ )
Kg (2,2) - 0

Kg] (2,3)

 

% dNfi 3
k2 “(av-M" (EVE-3’]

1
”ij‘ _ "" ”13'.
Kg (3,1) 1(3Kq (1,3)

1
~13. _ "~ij4
Kg (3,2) k3 g (2,3)

~

ijt _
Kq (3,3) — 0

 

 

65
The nondimensional incremental equilibrium equations are then

given by Equation 4.49, provided that
~14 _ a". ..
K,,—K,1+AK,+K (4.97)

and that k is replaced by km in the expressions of element

matrices.

The extension to external pressure loading is also valid

for the von Karman model.

 

 

5. Solution Technique

5.1 Assemblage of Structure Matrices

Since the nondimensional element stiffness matrices and
the nondimensional load vector were derived for a typical
membrane finite element, they hold for any element in the
finite element mesh.

The assemblage process to obtain the nondimensional
tangent stiffness matrix may be written in symbolic form as

E=Zflm 64)

D

where the term under the summation symbol is the
nondimensional tangent stiffness matrix of the typical
element, and the summation goes over all elements in the
finite:e1ement mesh. Similarly, the nondimensional load vector
is assembled from the element nondimensional load vectors.
It should be mentioned that the nondimensional tangent
stiffness matrix is not symmetric (see Equations 3.96, 4.46,
and 4.96). The loss of the symmetry property is due to the

form of the matrix H (see Equations 3.78b and 3.78c).

5.2 Solution of Equilibrium Equations

5.2.1 The Incremental Iterative Solution strategy
The nonlinear incremental equations to be solved are

given by Equations 3.108 and 4.49 and may be written as

66

67

5 HIP

a
2

ulu

~/(i) "1(1-1) (5.2)
ka-Pm

[i;];1-1) {A5},‘,,“ =

where the full Newton-Raphson iteration method in conjunction
with an incremental loading procedure is implied. In this
method, each load step consists of the application of an
increment of external load and subsequent iterations to
restore equilibrium. This implies that the tangent stiffness
matrix has to be updated at each iteration. Since the major
computational cost per iteration lies in the evaluation and
factorization of the tangent stiffness matrix, it is in
general more effective to use some modification of the full
Newton-Raphson algorithm”. In this work, the modified Newton-
Raphson iteration procedure was used. Therefore, the tangent
stiffness matrix is assumed to be constant. within each
increment, and is updated only at the start of the next
increment. This is clearly more economical at each step but
convergence is slower. The modified nonlinear equations to be

solved are then

 

. _1_ ~ . ~ ._
[It/HS) {A5}? = kJF/(1)_P/u 1) (5.3)

whereIfiznP is the value of the tangent stiffness matrix at

the beginning of the mth increment based upon the nodal
displacement solution vector obtained in the last

iteration of the (m-1)th increment.

68
5.2.2 Convergence Criteria
A problem associated with iterative techniques is the
decision as to whether the current iterate is sufficiently
close to the solution which is unknown. To overcome this
problem, a convergence criterion must be set. A displacement
criterion based on the maximum norm will be used to measure

24

the size of the error . The maximum norm is defined by

Aai

i.ref

(5.4)

 

N
“ell, = max
1

 

 

where N is the total number of unknown displacement
components; Aai is the change in displacement component 1'.
during a given iteration; and avg-is a reference displacement
quantity equal to the absolute value of the largest
displacement component of the corresponding type (u, v, or w) .

The convergence criterion is then

ll6|l..<# (5.5)

The value of u is usually between 10‘2 and 10‘, depending on

the accuracy desired.

5.2.3 The Initial Virtual Prestressing Technique
Iklnonlinear finite element analysis, the linear solution
is often used as the initial guess in the iteration procedure.
For initially thin flat membranes which are subjected to
transverse loading, there is no linear solution due to the

lack of bending stiffness. Therefore, the analyst has to make

69

an initial guess to start the iteration procedure. As
mentioned earlier in the literature review, the initial guess
used in previous work involved all of the unknown.displacement
components. To avoid dealing with all of the components of the
displacement vector in formulating an initial guess, a new
technique referred to as the "initial virtual prestressing
technique" is introduced.

Unless the membrane is very strongly prestressed prior to
service loading, the tangent stiffness matrix in the first
stages of the incremental iterative procedure is either
singular or close to singular. Therefore, an ill-conditioned
problem arise. The basic idea of the initial virtual
prestressing technique is to circumvent this ill-conditioning
problem by introducing an imaginary initial prestressing so
that the tangent stiffness matrix is far from singularity. The
membrane problem is then modified. Since the main purpose in
the analysis is the evaluation of the final equilibrium
configuration of the membrane and its state of stress under
the true service loading, it is necessary to remove this
virtual prestressing before the final configuration is
reached. This can be done by removing part of the virtual
prestressing at the beginning of each load increment so that
by the end of the procedure the virtual prestressing is

removed completely.

 

70

By letting the following vectors

f 1 f

 

 

 

 

v I:
00x 00x
4:40;” 03205),» (5.6)
v t
(”0’02 90ny

be, respectively, the initial virtual prestressing vector and
the initial true stress vector, the initial stress vector at

the first load increment becomes

051): 034-0: (5.7)
To reduce the initial virtual prestressing in subsequent

increments, a reduction factor A," is introduced so that at the

mth increment

05m = 03+4ma: (5.8)
and at the last increment

affl==¢fi (5.9)

where M is the total number of increments, and km is a

function of m. The initial prestressing may be suitably

expressed in terms of an equivalent initial prestraining as

a§=Ee§ aZ=E8§ (5-10)
in which
r . r
I:
80x 83x]
80=<sgy> 8::(8Zy) (5.11)
1:
Font (3310’,

 

 

 

 

 

 

71
By using Equations 4.23, 4.32 and 4.37, the nondimensional

stress at the mth increment may be written

6’,’ = (1—v2)(é',*+1m€,') (5.12)
The reduction factor A“ should be a decreasing function of m,
and should satisfy the following conditions

,1 =71 (m=1) =1

1 m (5.13)
1M=lm(m=M) =0

Many possible choices for A," can be used. The initial choice

for this work was

1 for m=1
I1 = l (5014)

-W for m22
where B is a constant between 0 and 1. From preliminary
computations, it was found that a small value of B (0<B<0.3)
ensures numerical stability when a small number of increments
is considered. However, with five increments, B=1 was used
successfully. To speed up convergence, the following

alternative reduction factor was considered

1 for m=1

1m== 1 [ 1 ] (5.15)
1——-——— for m22
(1+B)M-m

 

With the latter reduction factor, the average number of

iterations per increment required for convergence was about

 

 

72
one third of that obtained with the reduction factor given by
Equation 5.14.

By using the new method, all that is needed to start the
incremental iterative procedure is an initial guess of the
three components of the initial virtual stress vector. An
"optimum" guess of these values may be obtained by trying
different values with a coarse finite element mesh, and then

use this optimum with finer finite element meshes.

5.3 Evaluation of Strains and Stresses

5.3.1 Evaluation of Strains
Once the element nodal displacements are obtained, the
strains at any point can be evaluated as (see Equations 3.80,

3.82a, 4.32 and 4.33)

ulu

e=k é" €=§’&' (5-15’

5.3.2 Evaluation of Second Piola-Kirchhoff Stresses
The nondimensional second Piola-Kirchhoff stresses are
obtained using the following stress-strain relationships (see

Equations 3.84 and 4.37)
€=D’é’+6’0’ (5.17)

Using Equations 3.28, 3.83, 4.24, and 4.36, the second Piola-

Kirchhoff stresses are given by

 

5/ (5.18)

73
5.3.3 Evaluation of Cauchy Stresses
The Cauchy stress tensor is defined as19
r = 1110:" (5.19)
pa
where p is the mass density in the final configurations, p0 is
the mass density in the initial configuration, a is the second
Piola-Kirchhoff stress tensor, and X is the deformation

gradient given by

Bu Ba
1+... .__

x: 6" 65' (5.20)
.61 1.31» -
6x 6y

and
p0 = pdetx
(5.21)
- 31. 3: -6112
detx — (1+ ax)(1+ 6y) 6x By

The components of the Cauchy stress tensor in terms of the
components of the second Piola-Kirchhoff stress tensor are

then given by

 

 

= 1 222 $2 3.4 :23 .
x detx[ax(l+6x) ”Y(ay) +20” 8Y(1+3X) (5 225)
ry = .__..dgtx [ay(1+.g}‘f)2 + 0x(%—;)2 + zawgk‘fhhgfl] (5.221))

= .__1__ _.__. __ .__.
1"” detx[ " 8x( 6x 3’ 6y 6y 6y 6x 6x 8y
(5.22c)

0 6V 1+§E)+a 611(14-av)+axy[i§fl+(1+2‘5)(1+§_¥)u

 

6. Model Validation and

Parametric Studies

A computer program was developed to test and validate the
theoretical models described in the previous chapters, and was
used to undertake some parametric studies. The initial virtual
prestressing technique was implemented in the computer
program.

To allow for an adequate comparison of the results of
this work with previous research, the membrane deflections and
stresses are conveniently presented in the form:

= 3.12%
W7 aL(Eh)

2 2 %
0:3(th2 E) } (6.1)

_ qZLZEi

 

The coefficients a,fl and y are, respectively, the normal
deflection coefficient, the second Piola-Kirchhoff stress

coefficient, and the Cauchy stress coefficient.

74

 

 

75

6.1 Model Validation

6.1.1 Convergence of the von Karmén Model

To study the convergence characteristics of the von
Karman model, a square membrane and a rectangular membrane
with two different aspect ratios were analyzed (see Figure 2).
The aspect ratio n is defined as the length of the membrane's

shorter side over the length of its longer side:

0 = (6.2,

mlo

 

 

 

 

 

 

 

 

Figure 2: Region for Rectangular Membrane

 

 

76

Letting L=b, Equation 5.1 becomes

= qb-i- ‘
w (Lb(§5)
2 2 3
a=p(.Q_'_13b2_.E_:)3 ( (6.3)
2 2 %
”7(qu E)

 

In all cases, the membrane was assumed to be fixed along
its four edges (u=v=w=0), and the transverse loading acting on
the membrane was considered uniform and deformation-
independent.

Due to double symmetry, only a quarter of the membrane
was required in the analysis. Different mesh sizes were
considered. The number of subdivisions in the x-direction N1‘
and in the y-direction Ny were chosen to be identical (Nx=Ny=N) .
Figures 3, 4 and 5 show the variation of the central
deflection and central and maximum principal second Piola-
Kirchhoff stresses with mesh refinement. It is evident from
the results that the von Karman -model exhibits good
convergence characteristics. The value of the nondimensional
coefficient k used to plot Figures 3, 4, and 5 was taken to be

equal to 1, and Poisson's ratio was taken equal to 0.3.

Deflection and stress coefficients

‘77

 

1.00

0.90 '!

0.80

0.70,

0.60

(150

(140

 

 

(130

-'—- Moxhnurn deflecfion

0
4)

"4" Central stress

“.2, Moxknuw13Uess

0

 

 

Number of subdivisions N

”V..-” ~ (M d
. .. ... -1 ...._... .....T
‘‘‘‘‘‘‘‘‘‘ "'--“““‘—~~—v~——--__-_---——v——-————-——-—-—:(
1 l L l L l 1 L l
2 3 4 5 6

Figure 3: Convergence of Deflection and Stresses for a

Square Membrane

 

Deflection and stress coefficients

78

 

1.1 l I l I l I l I I

1.0‘ -—*—- Moxknunwdeflecfion
‘ "“Lm Centrol stress

09 “.._l\ ------ l“- Moxlmum stress

 

 

 

 

 

O3 1 i l l 1 J 1 l 1

 

Number of subdivisions N

Figure 4: Convergence of Deflection and Stresses for a

Rectangular Membrane (17:5/7)

Deflection ond stress coefficients

79

 

1 3 l I T I fl I I I I
1.2(7’ I . . _
1‘ + Moxnmum deflection -
\
1.1 —\ ‘-*-' Centrol stress I
\
“ \ ------ I ------ Moximum stress ‘
1.0 — \ _

 

 

0.4 — _

 

 

 

03 1 l 1 l L l 1 l 1
l 2 3 4 5 6

Number of subdivisions N

Figure 5: Convergence of Deflection and Stresses for a

Rectangular Membrane (n=2/s)

80
6.1.2 Comparison of the von Kirmfin Model with Previous Models

To test the validity of the numerical implementation of
the von Karman model, a square membrane and a rectangular
membrane with two different aspect ratios (n=5/7 and n=2/5)
were analyzed.

Because the nondimensional coefficient k enters the
fundamental incremental equilibrium equation to be solved (see
Equation 3.108), the analysis was carried out with seven
different values of k ranging from 0.0001 to 1, i.e. values of
0.0001, 0.001, 0.01, 0.05, 0.1, 0.5 and 1. However, it was
found that the values of the central deflection and the
central and maximum second Piola-Kirchhoff stresses were not
affected by the variation of k. Then, a comparison was made of
the results for the central deflection and central and maximum
principal second Piola-Kirchhoff stresses with the results of
Fopplu, Hencky”, Borg”, Shaw and Perronez, Kao and Perrone”,
and Allen and Al-Qarra”. The results given in terms of
deflection and stress coefficients are shown in Tables 1, 2

and 3.

 

 

81

Table 1: Square Membrane Under Uniform Transverse Loading
Comparing different Models

 

 

 

 

 

 

 

Maximum Central Maximum
Model deflection stress stress
Foppl 0.8 0.409 0.56
Hencky 0.716 0.436 0.458
Kao & Perrone 0.723 0.439 0.497
Allen & Al-Qarra 0.722 0.435 0.518
Present work
von Karman model 0.722 0.436 0.518

 

 

 

 

Loading (n=6/7) Comparing different Models

Table 2: Rectangular Membrane Under Uniform Transverse

 

 

 

 

 

 

 

 

Maximum Central Maximum
Model deflection stress stress
Foppl 0.943 0.543 0.735
Borg 0.864 0.504 -
Shaw and Perrone 0.824 0.520 0.591
Kao & Perrone 0.841 0.522 0.574
Allen & Al-Qarra 0.836 0.536 0.598
Present work
von Karmén model 0.836 0.533 0.594

 

 

 

 

 

 

 

82
Table 3: Rectangular Membrane Under Uniform Transverse

Loading (n=2/5) Comparing different Models

 

 

 

 

 

 

 

 

 

 

Maximum Central Maximum
Model deflection stress stress
F6pp1 1.069 0.642 0.889
Kao & Perrone 0.895 0.551 0.582
Allen & Al-Qarra 0.877 0.581 0.605
Present work
von Karman model 0.877 0.572 0.596

 

 

It is evident from the comparison, that the numerical
implementation of the von Karman model is consistent with the
implementation done by other investigators. In the author’s
model, the second Piola-Kirchhoff stresses were evaluated at
the element nodes and the center of each element. This may
account for some of the discrepancy between the tabulated
values and those obtained by Borg and by Kao and Perrone,
since those investigators do not specify where stresses were
evaluated. Also, it is important to note that the maximum
second Piola-Kirchhoff stress occurs at the middle of the
longest edge of the rectangular membrane (see point A in
Figure 2).

6.1.3 Discussion

As was mentioned in the literature review, second Piola-

Kirchhoff stresses do not represent "real" membrane stresses

when large displacements are considered, and in practice

 

 

83
Cauchy stresses should be computed. A comparison between
second Piola-Kirchoff stress coefficients and Cauchy stress
coefficients, using the von Karman model, is presented in

Tables 4, 5, and 6.

Table 4: Stress Coefficients for a Square Membrane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2nd Piola-Kirchhoff Cauchy Stresses
Stresses

k Central Maximum Central Maximum

' Stress Stress Stress Stress
0.0001 0.436 0.518 0.436 0.522
0.001 0.436 0.518 0.436 0.520
0.01 0.436 0.518 0.436 0.509
0.05 0.436 0.518 0.436 0.484
0.1 0.436 0.518 0.436 0.461
0.5 0.436 0.518 0.436 0.342
1.0 0.436 0.518 0.436 0.346

 

 

 

84

Table 5: Stress Coefficients for a Rectangular Membrane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(n=S/7)
2nd Piola-Kirchhoff Cauchy Stresses
Stresses
k Central Maximum Central Maximum
Stress Stress Stress Stress
0.0001 0.533 0.594 0.536 0.597
0.001 0.533 0.594 0.536 0.594
0.01 0.533 0.594 0.539 0.581
0.05 0.533 0.594 0.546 0.550
0.1 0.533 0.594 0.552 0.522
0.5 0.533 0.594 0.582 0.376
1.0 0.533 0.594 0.606 0.436

 

Table 6: Stress Coefficients for a Rectangular Membrane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(n=2/5)
2nd Piola-Kirchhoff Cauchy Stresses
k Stresses

Central Maximum Central Maximum

Stress Stress Stress Stress
0.0001 0.572 0.596 0.575 0.599
0.001 0.572 0.596 0.577 0.595
0.01 0.572 0.596 0.583 0.581
0.05 0.572 0.596 0.598 0.547
0.1 0.572 0.596 0.612 0.515
0.5 0.572 0.596 0.681 0.354
1.0 0.572 0.596 0.741 0.513

 

 

 

 

 

85
Table 7 shows the relative error when second Piola-Kirchhoff
stresses are used instead of Cauchy stresses. The relative

error in the stress coefficients is defined as:

At: 0"?
T

(6.4)

 

where a and r are, respectively, the second Piola-Kirchhoff

stress and the Cauchy stress.

Table 7: Relative Error in Stress Coefficients

 

 

 

 

 

 

 

 

 

 

 

Relative Error %
Central Stress Maximum Stress

k n=1 n=5/7 n=2/5 17:1 n=5/7 17:2/5
0.0001 0 -0.56 -0.52 -0.77 -0.50 -0.50
0.001 0 -0.56 -0.87 -0.38 0 0.17
0.01 0 -1.11 -l.89 1.77 2.24 2.58
0.05 0 -2.38 -4.35 7.02 8.00 8.96
0.1 0 -3.44 -6.54 12.4 13.8 15.7
0.5 0 -8.42 -16.0 51.5 58.0 68.4
1.0 0 -12.0 -22.8 49.7 36.2 16.2

 

 

 

 

 

 

 

It is evident from the above results that,

in general,

 

the use of second Piola-Kirchhoff stresses instead of Cauchy
stresses underestimates the central stresses but overestimates
the maximum stresses. For values of k smaller than 0.01, the
use of second Piola-Kirchhoff stresses leads to results that
but Cauchy

are sufficiently accurate for ‘most. purposes,

stresses need to be used for values of k greater than 0.05.

 

 

86

However, for high values of strains, the membrane may exhibit
an inelastic behavior. When this is the case, the assumption
of elastic behavior may no longer valid, and new constitutive
relationships modelling the material behavior should be
considered.

To check.the state of strain in the membrane, the central
and maximum actual strains were computed, and are listed in

Tables 8, 9, and 10.

Table 8: Strains in a Square Membrane

 

 

 

 

 

 

 

 

k Central Strain % Maximum Strain %
0.0001 0.04 0.07
0.001 0.21 0.32
0.01 0.95 1.48
0.05 2.79 4.33
0.1 4.42 6.87
0.5 12.9 20.1
1.0 20.5 31.9

 

 

 

 

 

 

87

Table 9: Strains in a Rectangular Membrane (n=6/7)

 

 

 

 

 

 

 

 

 

 

 

k Central Strain % Maximum Strain %
0.0001 0.06 0.08
0.001 0.29 0.36
0.01 1.33 1.69
0.05 3.88 4.95
0.1 6.16 7.86
0.5 18.0 23.0
1.0 28.6 36.5

 

Table 10: Strains in a Rectangular Membrane (n=2/5)

 

 

 

 

 

 

 

 

 

 

 

k Central Strain % Maximum Strain %
0.0001. 0.07 0.08
0.001 0.34 0.37
0.01 1.57 1.70
0.05 4.60 4.97
0.1 7.31 7.88
0.5 21.4 23.1
1.0 33.9 36.6

 

 

 

6.1.4 Comparison of the von Karman Model with the General
Model

To determine the limitations of the von Karman model and
therefore its rangeiof applicability, the three membrane cases
studied above were analyzed using the general model. Figures

6, 7 and 8, which are plotted for k=1, demonstrate that

 

 

 

88
convergence for the general model is comparable to that of the

von Karman model (Figures 3, 4 and S).

 

 

 

Deflection ond stress coefficients

89

 

 

 

 

 

 

 

 

 

1 O; I I T I I I I I I
g _
(19.9t -—*—- MoxanWideflecfion —
_ \fl "4'“- Centrol stress _
‘ __.-..._... Maximum stress
0.8 — _
,I”"'” s-
’ “I~.—— ---“ 1.1-1...——-- -. _—.
,_ \\ _
\
\\
Ofb— \\ ‘
\
l \
\x\
0.4 — “\\\ _
______ ,_____‘_-__
‘—‘“" ———————————— -V ————————————— _v
J l l L l I 1 l l
l 2 3 4 5 6

Number of subdivisions

Figure 6: General Model Convergence of Deflection and

Stresses for a Square Membrane

Deflection and stress coefficients

1.1

1.0

0.9

0.7

0.6

0.5

0.4

0.3

90

 

A

 

 

 

 

 

 

 

 

I I I 7 T l I I
\3 n
_ + Maximum deflection _
_ "4’"- Centrol stress .
_ ------ I ------- Moximum stress _
r —l
\\ ...- -...
L \ I. /._’_. ". ‘\ ~ -.. .__ __§ .. _‘_ -.. .__..-
‘1‘ I
r- . __.,,.. A
— \ .—
\
.. \\ “
_ ‘\ —
\\
.— ‘ ~~~~~~~ y __________ “
“““V‘ ------------ + ———————————— —v
1 1 J 1 l 1 1 1
2 3 4 5 6

Number of subdivisions

Figure 7: General Model Convergence of Deflection and

Stresses for a Rectangular Membrane (17:5/7)

 

 

 

Deflection and stress coefficients

91

 

1.1

0.71

0.6

0.5

0.4

 

0.3

 

 

 

 

 

l ' l ‘ ' i T 1
__¢—- Moxnnurn deflecfion
-_—v—-- Central stress
...... I Maximum stress
_l
' 4
1
\\ —+
\\' —‘
\\
\ _________ V ............. .4
4L _____________ v ------------ 'V
l 1 l L 1 1 l 1
2 5 4 5 6

Nunfixy ofsubdnngons

Figure 8: General Model Convergence of Deflection and

Stresses for a Rectangular Membrane (17:2/5)

92
A comparison was made of the general model with the von

Karman model for the central deflection, and the central and

maximum

deflection and stress coefficients,

12, 13 and 14.

Cauchy' stresses.

The results ,

Central Deflection Coefficients

given in terms of

are shown in Tables 11,

 

 

 

 

 

 

 

 

 

 

Table 11:

Von Karman Model General model

Aspect ratio Aspect ratio
k 1 5/7 2/5 1 5/7 2/5
0.0001 0.722 0.836 0.877 0.722 0.836 0.877
0.001 0.722 0.836 0.877 0.722 0.836 0.876
0.01 0.722 0.836 0.877 0.722 0.836 0.876
0.05 0.722 0.836 0.877 0.722 0.835 0.874
0.1 0.722 0.836 0.877 0.722 0.835 0.873
0.5 0.722 0.836 0.877 0.724 0.836 0.872
1.0 0.722 0.836 0.877 0.726 0.838 0.873

 

 

 

 

 

 

 

 

 

Table 12: Cauchy Stress Coefficients in a Square Membrane

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Von Karman Model General Model
Central Maximum Central Maximum
k Stress Stress Stress Stress
0.0001 0.436 0.522 0.437 0.522
0.001 0.436 0.520 0.436 0.521
0.01 0.436 0.509 0.432 0.517
0.05 0.436 0.484 0.422 0.507
0.1 0.436 0.461 0.415 0.499
0.5 0.436 0.342 0.383 0.466
1.0 0.436 0.346 0.362 0.445

 

Table 13: Cauchy Stress Coefficients in a Rectangular Membrane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(n=5/7)
Von Karmén Model General Model
Central Maximum Central Maximum
k Stress Stress Stress Stress
0.0001 0.536 0.597 0.535 0.597
0.001 0.536 0.594 0.534 0.596
0.01 0.539 0.581 0.531 0.592
0.05 0.546 0.550 0.522 0.582
0.1 0.552. 0.522 0.514 0.574
0.5 0.582 0.376 0.484 0.543
1.0 0.606 0.436 0.465 0.523

 

 

 

 

94

Table 14: Cauchy Stress Coefficients in a Rectangular Membrane

 

 

 

 

 

 

 

 

 

("=2/5)
Von Karman Model General Model
Central Maximum Central Maximum
k Stress Stress Stress Stress
0.0001 0.575 0.599 0.575 0.599
0.001 0.577 0.595 0.574 0.598
0.01 0.583 0.581 0.571 0.594
0.05 0.598 0.547 0.565 0.585
0.1 0.612 0.515 0.559 0.577
0.5 0.681 0.354 0.539 0.549
1.0 0.741 0.513 0.527 0.531

 

 

 

 

 

 

 

Tables 15, 16 and 17 show the relative error in deflection and
stress coefficients when the von Karman model is used instead
of the general model. The relative error in a variable Q

(deflection coefficient or stress coefficient) is defined as:

 

(6.5)

where Q, is the corresponding variable in the von Kérma’m
model, and Q: is the corresponding quantity in the general

model.

 

 

Table 15: Relative Error for a Square Membrane

95

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Error %
k Central Central Maximum
Deflection Stress Stress
0.0001 0 -0.23 0
0.001 0 0 -0.19
0.01 0 0.93 -1.55
0.05 0 3.32 -4.54
0.1 0 5.06 -9.62
0.5 -0.28 13.8 -26.6
1 -0.55 20.4 -22.2

 

Table 16: Relative Error for a Rectangular Membrane (n=5/7)

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Error %
k Central Central Maximum
Deflection Stress~ Stress
0.0001 0 0.19 0
0.001 0 0.37 -0.34
0.01 0 1.50 -1.86
0.05 0.12 4.60 -5.50
0.1 0.12 7.39 -9.06
0.5 0 20.2 -30.8
1 -0.23 30.3 -16.6

 

 

 

 

 

Table 17: Relative Error for a Rectangular Membrane (n=Q/5)

96

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Error %
k Central Central Maximum
Deflection Stress Stress
0.0001 0 0 0
0.001 0.11 0.52 -0.50
0.01 0.11 2.10 -2.19
0.05 0.34 5.84 -6.50
0.1 0.46 9.48 -10.7
0.5 0.57 26.3 -35.5
1 0.46 40.6 -3.39

 

 

It appears from the above results that the von Karmén model
overestimates the central stress but underestimates the
maximum stress. The accuracy of the von Karman model declines
sharply as k increases beyond 0.01.

In addition, a comparison between the two models for the
central and maximum actual strains is made. The corresponding

results are shown in Tables 18, 19 and 20.

97

Table 18: Comparison of Strains for a Square Membrane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Von Karman Model General Model
k Central Maximum Central Maximum
Strain % Strain % Strain % Strain %
0.0001 0.04 0.07 0.04 0.07
0.001 0.21 0.32 0.20 0.32
0.01 0.95 1.48 0.94 1.50
0.05 2.79 4.33 2.69 4.49
0.1 4.42 6.87 4.20 7.25
0.5 12.9 20.1 11.3 22.5.
1.0 20.5 31.9 17.0 36.8

 

 

Table 19: Comparison of Strains for a Rectangular Membrane

 

 

 

 

 

 

 

 

 

(n=5/7)
Von Karman Model General Model
k Central Maximum Central Maximum
Strain % Strain % Strain % Strain %
0.0001 0.06 0.08 0.06 0.08
0.001. 0.29 0.36 0.28 0.37
0.01 1.33 1.69 1.31 1.72
0.05 3.88 4.95 3.71 5.17
0.1 6.16 7.86 5.74 8.38
0.5 18.0 23.0 15.2 26.2
1.0 28.6 36.5 22.7 42.8

 

 

 

 

 

 

 

 

 

98

Table 20: Comparison of Strains for a Rectangular Membrane

 

 

 

 

 

 

 

 

 

 

(n=2/5)

Von Karman Model General Model

Central Maximum Central Maximum

k Strain % Strain % Strain % Strain %
0.0001 0.07 0.08 0.07 0.08
0.001 0.34 0.37 0.34 0.37
0.01 1.57 1.70 1.54 1.73
0.05 4.60 4.97 4.36 5.23
0.1 7.31 7.88 6.73 8.49
0.5 21.4 23.1 17.6 26.8
1.0 33.9 36.6 26.0 44.0

 

 

 

 

 

 

 

The von Karman model overestimates the central strain but
underestimates the maximum strain. Again, it should be
mentioned that for higher values of strains the membrane may
exhibit an anelastic behavior. When this is the case, the
assumption of elastic behavior is no longer valid, and new
constitutive relationships modelling the material behavior
should be considered.

To illustrate the distribution of deflections and
stresses within. the 'membrane, the. deflection and stress
coefficients for the three membrane.cases studied were plotted
along the center lines a-a and d-d, quarter lines b-b and e-e,
and edge lines c-c and f—f as shown in Figure 9. Figures 10 to
19 illustrate the results. The nondimensional coefficient k

was taken equal to 0.01.

 

 

99

yl

 

 

 

 

 

Figure 9: Typical Quarter of a Membrane

 

 

 

Normol deflection coefficient

(19

(18

0K7

(16

(15

04-

0L5

(12

04

(10

100

 

 

1 l 1 l 1 L 1 l

 

0(1 (12 (14 (16 (18

Distonce from center

Figure 10: Deflection Coefficient Distribution

for a Square Membrane

 

Normol deflection coefficient

101

 

 

 

 

0.9 l I I I I I I ‘I I

1* ‘
0.8 —- _
0.7"; ------------ k‘\ _
0.6 '_ \\'\\\\ _1

— \‘\\\\ -I
(15-— \\\ _

‘\

l ‘
(1.41 —- \\\\ __
0.3 — _ \ .1

W0 O \\ .
0.2 e '"*" W j

l. \x. 5
0.1 '_ \\\\ —1
00 L J 1 I 1 i 1 l 1

0.0 0.2 0.4 0.6 0.8 1.0
Distance from center
Figure 11: Deflection Coefficient Distribution Along the

Shorter Side of a Rectangular Membrane (17:5/7)

 

 

 

Normol deflection coefficient

 

 

 

 

 

102
0.9 I I I I I I fir I I I I I I
i“ -
0.8 — —
0.7 —- _
v _____________ _,_ ......
0.6 _ ~~~~~ *\\ _
0.5 — \‘1.\ _
0.4 — —
_ _ \\ ‘
0.3 ~— d—d \\ ~—
+ w \
* e—e \\ A
0.2 F “Jr-_- W i\ ~—
_ l
0.1 —- \\ _
— \\\ —1
OO 1 i 1 i 1 i 1 i 1 i 1 l 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Distance from center

Figure 12: Deflection Coefficient Distribution Along the

the Longer Side in a Rectangular Membrane (1;:5/7)

   

103

 

Normol deflection coefficient

 

OO 1 ‘I 1 l 1 L 1 l 1

 

 

 

01) (12 04- (16 (18

Distance from center

Figure 13: Deflection Coefficient Distribution Along the

Shorter Side of a Rectangular Membrane (n=Q/5)

Normol deflection coefficient

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

104

 

 

 

 

 

 

I
.
1..
V»— ———————————— -V ------------- 4— .....
L-
l—
+ W
' ‘ e
-—-v-—— W
1 1 1 L
o l 2

Distance from center

Figure 14: Deflection Coefficient Distribution Along the

the Longer Side in a Rectangular Membrane (17:2/5)

 

Cauchy stress coefficient

105

 

0.60 T I I I I I I I I

 

 

 

 

 

0.20 — o-o —
‘—+—'1' \
1- ___'___ T — \.\.\"
c—c \
0.10 - " T 1 —
0'00 1 i 1 L 1 L 1 i 1
(10 (12 (14 (16 (18

Distonce from center

Figure 15: Stress Coefficient Distribution for a Square

Membrane

 

 

 

106

 

0'5 ' 1 ' 1 ' 1 ' 1

 

 

Cauchy stress coefficient

 

 

 

(12-— _ _
+ 7
_ —v—— 7- _
0. 1 e --------l- C _ C —
O .0 1 1 1 1 1 1 1 1 ii"
01) (12 (14 (16 (18

Distance from center

Figure 16: Stress Coefficient Distribution Along the

Shorter Side of a Rectangular Membrane (n=5/7)

 

Cauchy stress coefficient

107

 

 

0.3 — _.

0.2 — d—d ' —

 

 

 

+ T X
_e ‘1‘
F -_.'__- .1- x
O 1 ,_ ...... I ....... T _ —
l'
00 1 i 1 l 1 i 1 i 1 i 1 l 1 4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.

Distance from center

Figure 17: Stress Coefficient Distribution Along the

the Longer Side in a Rectangular Membrane (17:5/7)

 

 

Cauchy stress coefficient

108

 

 

 

 

 

 

 

0.6 I I 1 i 1 l r é I
A A I -------- v _____________ 4r
' -..--; ............. ..1 _____
‘=-=:.:f_-:: —————— -V— .........
(15
0.4
(13
0.2
+ T
_ -—+—.- 7 _ ‘
C—C \\
0.1 _ ---. -
\
0'0 1 l 1 L 1 1 1 1 1
OOO 0'2 0-4 0.6 0.8

Distance from center

Figure 18: Stress Coefficient Distribution Along the

Shorter Side of a Rectangular Membrane (17:2/5)

109

 

 

 

 

 

Cauchy stress coefficient

 

 

 

0fl3—- —
— \ ..
“1"
0 . 2 — d _ d "1‘ ——
+ 7' “1,,
__ — e i \ '7
- _ _v_ _ - ,T.
f—f
01 — """ ' """" T x 1
\a
O . O 1 L L i
0 l 2

Distance from center

Figure 19: Stress Coefficient Distribution Along the

the Longer Side in a Rectangular Membrane (n=2flfl

 

110

6.2 Parametric Studies

6.2.1 Effect of Poisson's Ratio

To investigate the effect of Poisson's ratio on
deflections and stresses, a square membrane was analyzed. Nine
different values of Poisson’s ratio ranging from 0.2 to 0.48
‘were considered. These values are: 0.2, 0.25, 0.3, 0.33, 0.37,
0.4, 0.43, 0.45, and 0.48. Figure 20 shows the variation of
the central deflection, the central stress, and the maximum
stress with Poisson’s ratio.

It is evident from the results that as Poisson’s ratio
increases, the deflections decrease and the stresses increase.
From v=0.2 to 1=0.48, the variation in the central deflection
coefficient, the central stress coefficient, and the maximum

stress coefficient are, respectively, 10.7%, 10.5%, and 3.9%.

 

 

Deflection and stress coefficients

(19

(18

0X7

(16

(13

 

111

 

__+_ w .4
C
| T _
C
_.__..____ T
m 0 X n
D
1
.1 - 1-1.
' I I I-—-- »--—-—---I-——--—»—-»—--—l—---~ I

 

 

 

 

(13 (14

Poisson's ratio

Figure 20: Effect of Poisson’s Ratio

 

 

112

6.2.2 Effect of Initial Prestressing

The membrane cases studied above were assumed to be
stress-free before the application of normal loading. In
actual use, it is common to prestress membranes in order to
compensate for the lack of bending stiffness. To investigate
the effect of initial prestressing on membrane deflections and
stresses, a square membrane initially prestressed along the x-
direction was analyzed. For the purpose of generality, initial
prestressing is expressed in terms of initial prestraining.
Different values of initial prestraining up to 1.4% were
considered. Poisson’s ratio was taken equal to 0.4. Figure 21
shows the variation of the central deflection coefficient, and
the central and maximum.stress coefficients with these values.

As expected, the central deflection decreases as the
initial prestressing increases. The membrane behaves as if it
has an initial bending stiffness. Nevertheless, the stresses
increase ‘with. the increase: of the initial prestressing.
Therefore the latter may contribute to premature tearing of
the membrane. The variation of the central deflection
coefficient, and the central and maximum stress coefficients
are, respectively, 10.8%, 36.6%, and 29% when the value of the

initial prestraining varies from 0 to 1.4%.

 

 

Deflection and stress coefficients

113

 

(18

 

 

 

4 J i l L l i 1 1 1 i '1 1 1 i 1 l L L L 1 J J 1 1 l

 

(12 (14 (16 (18 l.0 1.2

Initial prestraining along ><—o><is

Figure 21: Effect of Initial Prestraining

114
6.2.3 Effect of the Boundary Conditions

In the examples studied in the previous sections the
membrane was assumed to be fixed along its four edges. In this
section, the membrane is assumed to have the two edges
parallel to the x-axis free, while the other two edges remain
fixed. This type of membrane boundary conditions is common in
plastic greenhouses.

Deflections and stresses of a square membrane, for this
type of boundary conditions and the one considered previously,
are compared in Figures 22, 23, 24 and 25. The superscripts
bcl and bc2 refer, respectively, to the previous case and the
present case. The values of the nondimensional coefficient k
and Poisson's ratio V considered were, respectively, 0.01 and
0.3.

Deflections and stress coefficients corresponding to the
present case are higher than in the previous case ( 23.1% for

the central deflection, and 12.4% for the maximum stress).

Normal deflection coefficient

1.0

115

 

OES—

0f3e

04-—

012—

 

0(1

 

 

011

Figure 22: Comparison of Normal Deflections Along

1 . 1 ' T
*‘x
\\'\\
‘x
\\
\\
\\
\
bcl
+ w
bc2
-__'__- w
l 1 1 1 l
(12 (14 (16

Distance from center

the Center Line a-a

 

 

Normal deflection coefficient

116

 

 

 

 

1 .0 T , l r l 1 T
’ __________ .1,
if ————————————— -v ————————————— 41- ————————————— V- ------------ "" A
(18- _
i— ‘1
(16-— 1
04-— _
’ bcl 1
w —.
_ ch
0.2 - '-"'L" w s
0.0 l L l l I 1 l
(10 (12 (14 (16 (18 1.0

Distance from center

Figure 23: Comparison of Normal Deflections Along

the Center Line d-d

Cauchy stress coefficient

117

 

0.6 . 1 . 1 .

 

 

0'4 1 L 1 1 4 1 1 l

 

 

01) (12 (14 (16 (18

Distance from center

Figure 24: Comparison of Cauchy Stresses Along

the Center Line a-a

Couchy stress coefficient

118

 

0‘5 T I ' I I 1 I I

 

 

 

 

O4 1 l 1 l 1 J 1 l

(10 CL2 011 (16 (18

Distance from center

Figure 25: Comparison of Cauchy Stresses Along

the Center Line d-d

LO

119
6.2.4 Effect of Nonconservative Loading: Case of External
Pressure

So far, the normal loading acting on the membrane was
assumed to be deformation-independent. In this section, a
deformation-dependent loading is considered. A square membrane
subjected to this type of loading was analyzed.

For values of the nondimensional coefficient k ranging
from 0.0001 to 0.05, the maximum deflection coefficients for
the two types of loading were identical to three significant
digits, as were the maximum stress coefficients. Apparently,
the assumption that the loading is deformation-independent is

quite accurate in the range of straining considered.

6.2.5 Comments About the Initial Virtual Prestressing
Technique

In all the numerical cases studied, the "optimum" initial
guess of the initial virtual prestressing vector was obtained
with a 3x3 finite element mesh, using the reduction factor
defined by Equation 5.15. Then, this optimum guess was used
with a 5x5 finite element mesh.

For the case of membranes fixed along their four edges,
the minimum number of increments needed was five, and the
average number of iterations per increment required for
convergence was 5.6. However, in the free-fixed square
membrane studied in section 6.2.3, 15 increments were required
while the average number of iterations per increment was only

2.4.

 

 

 

7. Summary and Conclusions

7.1 Summary

This research investigated the static nonlinear behavior
of structural thin flat membranes which are subjected to
transverse loading. The membrane material was assumed to have
a linear elastic constitutive behavior. Only geometric
nonlinearities were considered. An incremental nonlinear
finite, element. method. using' nondimensional variables 'was
employed.

First, a geometric nonlinear model referred to as the von
Karman model was developedd This model was based on simplified
strain-displacement relationships in which the squares and
products of derivatives of the in-plane displacement
components were neglected. An incremental-iterative procedure
was used to solve the equilibrium equations for the nodal
displacements. To overcome the difficulty associated with the
ill-conditioning arising at the start of this procedure, a new
technique referred to as "initial virtual prestressing" was
developed.

To validate this model, a square membrane and rectangular
membranes with two different aspect ratios were analyzed.
Then, results for the central deflection and central and
maximum second Piola-Kirchhoff stresses were compared with the

results of previous investigators who also had used the von

120

 

 

 

121
Karman simplification of the strain-displacement relation-
ships. Cauchy stresses were also computed and compared to
second Piola-Kirchhoff stresses to determine whether the use
of the latter was appropriate. To the best of the investi-
gator’s knowledge the comparison of the two types of stresses
has never been published.

Next, aigeneral geometric.nonlinear'model using the exact
strain-displacement relationships was developed. This model is
referred to as the general model. It was used to analyze the
three membrane cases mentioned above. A comparison of its
results with those of the von Karman model was made to assess
the limitations of the latter model. To the best of the
investigator’ 5 knowledge the comparison of the comparison
between the two models has never been published.

Finally, some parametric studies were conducted to
investigate the effect of varying Poisson’s ratio, introducing
initial real prestressing and changing the boundary
conditions. The effect of nonconservative loading also was

investigated.

7 .2 Conclusions

7.2.1 Comparison of the von Karman Model to previously
developed models

Comparison of the results of the von Karman model with
those obtained by previous investigators shows overall

agreement, particularly’ for' deflections. Because ‘the

122

nondimensional coefficient k enters the fundamental
incremental equilibrium equations, the analysis was carried
out for different values of k ranging from 0.0001 to 1. It was
shown that the central deflection coefficient and the central
and the maximum second Piola-Kirchhoff stresses were not
affected by the variation of k.
7.2.2 Second Piola-Kirchhoff Stresses and Cauchy stresses

Unlike the second Piola-Kirchhoff stresses, Cauchy
stresses vary with k. From the theoretical point of view, the
Cauchy stress represents the real membrane stress. A
comparison of these stresses with second Piola-Kirchhoff
stresses for different values of the nondimensional
coefficient k shows that the use of second Piola-Kirchhoff
stress leads to an underestimation of the central stress, and
an overestimation of the maximum stress. Nevertheless, for
values of k smaller than 0.01, the differences are negligible
for design purposes.
7.2.3 Limitation of the von Karman Model

A comparison of the results for the central deflection
coefficient and the central and maximum Cauchy stress
coefficients with those.obtained using the general model shows
that the von Karmén model overestimates the central stress,
but underestimates the maximum stress. However, for values of
k smaller than 0.01, the von Karmén model gives results that

are sufficiently accurate for design.

123

The nondimensional coefficient k (a function of load
intensity, characteristic. length of the membrane, Young’s
modulus, Poisson’s ratio and membrane thickness) determines
the state of straining in a membrane. For values of k smaller
than 0.01, the maximum strain is less than 1.75%. For values
of k greater than 0.05, the maximum strain exceeds 4.5%. For
high values of strains, the membrane may exhibit inelastic
behavior; in which.case, new'constitutive relationships should
be considered.
7.2.4 Effect of Poisson's Ratio

To investigate the effect of Poisson's ratio on
deflections and stresses, a square membrane was analyzed with
different values of Poisson’s ratio. The results shoW'that.the
variations in the values of the central deflection coeffi-
cient, and the central and the maximum stress coefficients
are, respectively, 10.7%, 10.5%, and 3.9% when the value of
Poisson's ratio is varied from 0.2 to 0.48.
7.2.5 Effect of Initial Prestressing

Initial prestressing was expressed in terms of initial
prestraining to allow for a general interpretation of the
results. The results of the analysis of a square membrane
initially prestressed along one direction show that when the
initial prestraining increases from 0 to 1.4%, the central
deflection coefficient decreases by 10.8%, while the central
and maximum stress coefficients increase, respectively, by

36.6%, and.29%. Consequently; in practical situations, initial

 

 

124

prestressing might be introduced when small deflections are
required, but the ‘magnitude should be limited to avoid
premature tearing of the membrane.
7.2.6 Effect of Boundary Conditions

A square membrane with the two edges parallel to the x-
axis free, and the two other edges fixed, was analyzed.
Comparison of results with those obtained in a membrane with
four fixed edges shows that the deflections and stress
coefficients corresponding to the former case are higher than
in ‘the latter' case. Variations. of 23.1% and 12.4% 'were
obtained, respectively, for'theicentral deflection and maximum
stress.
7.2.7 Effect of Nonconservative Loading

External pressure on a flexible membrane is a deforma-
tion-dependent loading. To investigate the effect.of this type
of loading on membrane deflections and stresses, a square
membrane was analyzed. The results obtained for the maximum
deflection and stress coefficients, for the range of values of
k between 0.0001 and 0.05, were nearly the same as those
obtained when considering a conservative type of loading.
7.2.8 Use of The Initial Virtual Prestressing Technique

Newly developed for this study, the initial virtual
prestressing technique was used in the incremental-iterative
procedure to avoid ill-conditioning and assure convergence.
The advantage of ‘this technique is ‘that. only’ the ‘three

components of the initial virtual prestressing vector need to

 

 

125

be guessed, while in previous work, an initial guess of the
deflected shape of the membrane was required. The "optimum"
values corresponding to the initial guess may be obtained by
trial and error on a coarse finite element mesh.
7.3 Further study

For plastic greenhouses, the practical value of the
nondimensional coefficient k ranges between 0.005 and 0.05.
Therefore, the models and the results for nondimensional
parameters developed in this study are potentially useful in
design. In. particular, the assumption of linear elastic
behavior might be very critical. To get a better insight of
the structural behavior of membranes, a further study is
needed to include more accurate constitutive relationships,
particularly for low density polyethylene which exhibits a

nonlinear viscoelstic behavior.

 

 

 

BIBLIOGRAPHY

 

 

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