l-1a'frix analysis of thin shells using finite...
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l-1A'fRIX ANALYSIS OF THIN SHELLS USING FINITE ELEMENTS
Aftab A. ~1ufti, to'i. Eng.
Department of Civil Engineering and Applied Mechanics '
ABSTRACT
A procedure using the displacement method of
analysis is presented which is applicable to the
solution of problems involving ~hin plates and shells
of arbitrary shapes, boundary conditions, elastic
properties and loading, including thermal effects.
The shell is idealized by a system of triangular
elements 'l7hose nodal points lie on th~ middle surface
of the actual curved shell. The solution requires
the evaluation, of the individual element characteristics,
the global stiffness rnatrix, and thermal load vector.
The analysis is accomplished by the simultaneous
solution of the nodal point equilibrium equations for
nodal displacements and from these thestres'ses and
strains at the centroids of the individual elements
'are determined.
Several examples are given to show comparisons
with ~esults obtained experirnentally and using other
methods.
Model tests were conducted for the different
support and Loading conditions.
v·
HATRIX ANALYSIS OF THIN SHELLS USING FINITE ELEHENTS 1
Nufti
• MATRIX ANALYSIS OF THIN SHELLS USING FINITE ELEMENTS
A thesis
by
Aftab A. Mufti, M. Eng.
Submitted to the Facu1ty of Graduate Studies and Research in partial fu1~ fL1lment of the requirement for ~he degree of Doctor of Phi1osophy.
McGi11 University Ju1Yi 1969
1 (ë) Aftab A. Mufti 1969
To my mother and father
•
e,
ABSTRACT
A procedure using the displacement method of
analysis is presented which is applicable to the
solution of problems involving thin plates and shells
of arbitrary shapes, boundary conditions~ elastic
properties and loading~ including thermal effects o
The shell is idealized by a system of triangular
elements whose nodal points lie on the Middle surface
of the actual curved shell o The solution requires
i
the evaluation of the individual element characteristics~
the global stiffness matrix and thermal load vectoro
The analysis is accomplished by the simultaneous
solution of the nodal point equilibrium equations for
nodal dis placements and from these the stresses and
strains at the centroids of the individual elements
are determinedo
Severai examples are given to show comparisons
with results obtained experimentally and using other
methods o
Model tests were conducted for the different
support and loading conditions o
ii
ACKNOWLEDGEMENTS
The author wishes to express his sincere thanks tOI
Professor P.Jo Harris~ who acted as research
director and suggested the study of shells o The author
is indebted to him for his help and advice o
Professor LoG. Jaeger~ who has taught, discussed,
advised and encouraged the author in his researeh, for
this the author is deeply grateful.
Dro R.G. Redwood p who suggested the finite element
method.
Dr. M.So Mirza for proofreading the final script.
BoL. Mehrotra, S.Z. Burney and J.C. Mamet, whose
discussions and helpful suggestions were greatly
appreciated.
Bo Cockayne 9 N. Ahmed and To Bowen, technical staff~
for their assistance in the laboratory.
My wife for typinS 9 proofreading and for her
encouragement 0
Acknowledgement is also made here of some of the
subroutines used from the book "The Finite Element Method
in Structural and Continuum Mechanics" by Professors
Zienkiewicz and Cheung.
The National Research Couneil of Canada, who awarded
a seholarship to the author and provided the funds for
this researcho
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF PLATES
LIST OF TABLES
TABLE OF CONTENTS
LIST OF SYMBOLS
INTRODUCTION
MATH~TIC4L P~V~L9PMENT
iii
i
ii
iii
vi
viii
ix
x
l
6
10 Development of Element Characteristics 6
a)
b)
c)
d)
e)
f)
g)
h)
Local coordinate system and displacement fields
Stress-strain relations for orthotropic material
Distribution of strains across the thickness
Thermal effects
Element characteristics in plane stress
Element characteristics in bending (cubic displacement field)
Element characteristics in bending (linear displacement field)
Transformation matrix
20 Formation of Global Stiffness Matrix and Thermal Load Vector and Solution
6
9
11
12
14
20
23
31
of the Equilibrl~ Equations 33
30 Stresses and Strains 35
EXPERIMENTAL STUDY 37
NUMERJ-CAL ~YAL~ATJ.QN AND RESU~,!,~
iv
Page
52
10 Convergence of Deflections and Moments 52
a} SLmply supported plate with a unifor.mly distributed load
b} Clamped square plate with a concentrated load at the center
c} Spherical shell with central circular load
52
54
55
20 Comparison with Experimental Results 57
a} Isotropie spherical shell with a central circular hole 57
b} Isotropie spherical shell, triangular in plan 57
c} Isotropie spherical shell under a narrow ring load 59
3 0 Application to Arch Dams
40 Thermal Stress Analysis
a} Thermal stresses in a plate of
59
60
non-uniform thickness 60
b} Thermal stresses and deflections in an axisymmetric cylindrical shell 61
50 Application to Orthotropic Shells 61
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH 83
REFERENCES 85
APPENDICES 92
10 Element Characteristics and Equivalent Nodal Forces 92
20 Qb and Kb Matrices 94 ,..., ,..., 3 0 Equi1ibrium Algorithm 96
40 Computer Program 99
v
Page
5. P~ograms to Generate Some of the Input Data for the Computer Program of
129 Appendix 4
6. Typical Strain and Dial Gauge Readings 137
10 Spherical Shell with Hole at the Crowno Load (Total) 200 lb. on Meridian Passing Through Center of the Loaded Arc 137
2 Q Spherical Shell with Hole at the Crown. Load (Total) 200 lb. on Meridian Passing Through Center of the Unloaded Arc 138
3. Spherical Shell Triangular in Plano Load 20 lbo at the Crown 140
40 Spherical Shell Triangular in Plano Load 20 lb. (~= 50 42') 141
LIST OF FIGURES
Figure
l~ Triangular Element 7
2. Layered Orthotropic Material 10
3 0 Membrane Stiffness Matrix!p Corresponding to ~p 19
40 Bending Stiffness Matrix !e Corresponding to ~e 27
50 Shearing Stiffness Matrix!s Corresponding to wl~ exl~ eyl~ etc. 30
6. OrthogonalCartesian Axes 31
70 Positions at which Thickness was Measured 37
80 Spherical Shell Triangular in Plan 38
90 Position of Dial Gauges 41
100 Positions of Loads 41
110 Cylindrical Loading Ring 45
120 Spherical Shel.l wi th Central Circular Hole 49
130 Meridians on which Strains were Obtained by Rotating the Oylindrical Loading Ring 50
140 Idealization of a Square Plate 63
150 SLmply Supported Plate with Uniformly Distributed Load 64
160 Clamped Plate with Concentrated Load 65
170 Spherical Shell with Central Circular Load 66
180 Meridional Strains in Spherical Shell Under Central Circular Load of 107625 lbo 67
19 0 Meridional Strains in Spherical Shell Under Intermittent Ring Load 68
200 Meridional Strains in Spherical Shell Under Intermittent Ring Load 69
vi
vii
Figure
210 Radial Deflections of Spherical Shell Under Intermittent Ring Load 70
22. Meridional Strains in Spherical Shell Triangular in Plan Under Concentrated Load 71
23. Meridional Strains in Spherical Shell Triangular in Plan Under Concentrated Load 72
24. Vertical Deflections of Spherical Shell Triangular in Plan Under Concentrated Load 73
25. Vertical Deflections of Spherical Shell Triangular in Plan Under Concentrated Load 74
26. Spherical Shell Under Narrow Ring Load 75
27. Meridional Strains in Spherical Shell Under Ring Load 76
28 0 Radial Deflections in Spherical Shell Under Ring Load 77
29. Cylindrical Arch Dam 78
30. Stresses and Deflections in Arch Dam 79
31. Stresses in Heated Plate 80
32. Uniformly Heated Cylinder 81
33. Orthotropic Cylindrical Shell 82
34. Forces and Deformations 92
35. Equilibrium Algorithm for Beam 96
36. Equilibrium Algorithm for Triangular Element 96
37 0 Flow Chart 100
38. Idealization of Shell 101
viii
LIST OF PLATES
Plate Page
10 Apparatus for Testing Model Shell 43
20 Close-up of Model Shell 44
3 0 Apparatus for Testing Model Shell 47
40 Close-up of Model Shell 48
viii
LIST OF PLATES
. Plate Page
1. Apparatus for Testing Model Shell 43
2. Close-up of Model Shell 44
3. Apparatus for Testing Model Shell 47
4. Close-up of Model Shell 48
ix
LIST OF TABLES
Table
10 Thickness Measurements 39
20 Showing Convergence of Deflection x 105 for Simply Supported Plate with Uniformly Distributed Load 53
30 Showing Convergence of Deflection x 104 for a Clamped Plate with a Concentrated Load at Center 55
40 Showing Convergence of Deflection x 103 for Spherical Shell with a Central Circular Load 56
50 Radial Deflection x 102 for an Orthotropic Cylindri.cal Shell under Internal Pressure 62
ex' ey
Yxyll YXZIl Yyz
0Xll 0yll Oz
0XYIl 0xzll 0yz
EXil Ey
E
G
v
t
LIST OF SYMBOLS
Direct Strains in x and y Directions
Shearing Strains
Direct Stresses in XII y and z Directions
Shearing Stresses
Moduli of Elasticity in x and y Directions for an Orthotropic Material
Modulus of Elasticity for an Isotropie Material
Moduli of Elasticity in Shear for an Orthotropic Material
Modulus of Elasticity in Shear for an Isotropie Material
Poissonls Ratios for an Orthotropic Material
Poisson's Ratio for an Isotropie Material
Coefficients of Thermal Expansion in x and y Directions for an Orthotropic Material Coefficient of Thermal Expansion for an Isotropie Material
Linear Displacements in X9 Y and z Directions
Rotations in x~ y and z Directions
Direct Forces in x ll Y and z Directions
Moments in x, y and z Directions
Flexural Rigidity
Thickness
Average Temperature
Temperature Difference
Length
xi
e R Radius of Curvature for a Spherical Shell
a Radius of Curvature for a Cylindrical Shell
Ji [3(1 -V 2>t a2t 2
x2' x3' Y3 Element Dimensions
[JT Transpose of a Matrix
[J-l Inverse of a Matrix
Other symbols are defined in the text where they appear o
The notations for the computer program are defined in
Appendix 4.
INTRODUCTION
Initially, shell roofs were designed by civil engineers using only past experience and many dames so constructed are still existing todayo These shells were all relatively thick in cross section and hence the intrinsic economic advantages of thin shells were not realized until as recently as the nineteen twenties o
A mathematical formulation of the problem was first given by Lamé and Clapeyron [lJ*, but it was restricted to membrane action onlYi Aron [2J, developed a theory based on bending behaviour, and the general theory was first advinced by Love [3J in 1892 0
However, design did not follow immediately after the development of the the ory 0 The rapid progress in reinforced concrete design made it possible to ;~tilize the differential equations of shells by Loveo The first engineer to use the theory for design purposes was Carl Zeiss of the Zeiss Optical works in Jena» Germany in 1924 0
1
The use of shells in a variety of engineering fields has attracted the attention of many engineers and mathematicians o Numerous texts, notably Gol@denveizer [4J~ Flügge [5J9 and a relatively recent one by Kraus [6J, giving theoretical treatments, are availablei while others like those by Billington [7J, Lundgren [8J» and Chronowicz [9Jp are of more interest to designerso
*Numbers in square brackets refer to references at the end of the thesiso
2
ln these texts it is made apparent that solutions of the classical differential equations are extremely complex and only possible in a restricted class of shell problems o ln order to analyze real shell structures, recourse is made to approximate methods such as the shallow shell theory given by Reissner [lOJ or the "beam" method of Lundgren [8Jo Tables and design charts have been published by the American Society of Civil Engineers for cylindrical shell roofs [llJ, and by the Portland Cement Association for domes [12] and hyperbolic paraboloids [13Jo
Despite the vast amount of effort that has been devoted to the deve10pment of analytical solutions to shell problems p many relatively stmple thin ahell forms still cannot be analyzed by classical methods o
The recent development of electronic computers has made the analysis of shells by numerical methods possible. Southwell [14J systematized the finite difference method and using this procedure he solved elasticity problem~ in torsion and plane stresso Since then investigators have used the finite difference method in the analysis of shells o Harris [15J has given a solution to axis.ymmetric spherical shells p and Kalnins [16]9 Budiansky and Radkowsky [17J have given a finite difference solution to unsymmetrical bending of she11s of revolution. A review of computer
1 programs for the analysis of stresses in pressure vessels, MOst of which were based on the finite difference approach, is given by Kraus [18Jo Soare 9 in his text [19J gives
3
s~ver~l solutions of _she~ls by the samemethod. Onc~
again~ some of the difficulties of the classical approach . are encountered in the finite differenc~ methodl that is,
when the problem is trea~ed in gene~alized form the governing differential equations, as approximated by finite differences, become extremely cumbersome due to their great length and the boundary conditions are often difficult to satisfYD Also, any variation in material properties or thickness and thermal effects is not easily taken into account o
The above difficulties are not only encountered while analyzing shells but also pertain to problems in elasticityo This led to the development of the finite element method by Turner et al [20J and Argyris [21J for the analysis of two d~ensional elasticity problems. The method is essentially the sarne as the well known stiffness method except that, for the first time, the concept of two dimensional elements was introduced and a continuum was divided into finite elementso
A detailed investigation of two dimensional structures was given by Wilson [22J using the triangula~ element developed by Turner et al [20Jo A text by Gallagher [23J also deals with plane problemso
Application of the finite element method to bending of plates~ axisymmetric shells and shallow shells is given in references [24J to [27J and [36Jo The general principles of the finite element method are to be found in the text by Zienkiewicz and Cheung [28Jo
4
The ~po~e .of thi~ thesis is t9 giye.a cqmplete ~~~~iç _l~e~:r;~naly~i_~.qf __ ~hin _sh~~J,s _\1~~ng_fl~t triangular finite elements. The analysis takes i'lltq account arbitrary geometry and boundary conditions as well as variations in loa~ing and material properties including thermal effects. The shell need not be shallow or axisyminetric.
ln the technical literature MOSt of the problems analyzed by the finite element method have dealt with Simple shella for which classiçal solutions are available. Shells with complex geometries, loading and boundary _ conditions have not been analyzed to any great extent. The autho:t' has investigated the behaviO\;lr of real shell structures by the fini~~ element m~thodo Since no classical solutions were available~ experiments were conducted on model shells to check the results obtained by the finite element method~
Two triangular eleme~ts9 one fully conformable and the other non-confo~ble9 are used to investig~te the complete deflection 9 stress and strain analysiso Several examples are solved to show orthotropic behaviour and thermal effects in shells:
• To analyze a real shell ~tructure a general three dimensional program is needed o Such a program incorporating all the effects mentioned above with a conformable triangle is developed by the author and the listing cf the program l.s included in this thesis 0 This is based on a prograDl previously reported by Mehrotra, the
author, and Redwo~d [29J in which a non-conformable triangle was usedo The details are reported in references [30J and [31Jo The author has expanded the pro gram to handle thermal effects which may be needed for shell analysis, and the application to isotropie and orthotropic shells and thermal stress analysis of plates and shells using a non-conformable triangle was presented by theauthor and Harris at the Second
5
Canadian Congress of Applied Mechanics at Waterloo [32J. The conformable triangle was first proposed by Melosh
[33J and used by Utku [27J, [34J. Only the displacement analysis of isotropie shallow shells was dealt with.
Mathematical details, excluding thermal effects, were rederived by the author [35J,. but in this thesis thermal effects are also included o All matrices are given in explicit formo
An explanation of input data for the program and supplementary programs to help generate this data are included in the appendices o
•
MATHEMATICAL DEVELOPMENT
The shell is idealized by a system of triangular flat plate elements whose nodal points lio on the Middle surface of the actual curved shello
6
The solution requires the evaluation of the stiffness and thermal properties of the individual elementso The global (overall) stiffness matrix and the thermal load vector are formed by suitable transformation of coordinates and superposition of the individual element stiffnesses and thermal loads o The analysis of the shell is accomplished by the simultaneous solution of the nodal point equilibrium equations for the nodal displacements and from these the stresses and strains at the centroids of the individual elements are determined o
1 2 Development of Element Characteristics
a) Local coordinate system and displacement fields
A local right-handed orthogonal Cartesian coordinate system x 9 Y9 z is chosen for each element with the origin at one apex (node) of the element and with the x-axis along one edge, the y=axis lies in the plane of the element and the three nodes are numbered as SllOwn in Figure 10 The x and y axes must coincide with the principal axes of the orthotropic materialo
Linear functions of x and y are assumed for inplane displacementa u and v in the x and y directions respectivelyo
y 3
1 /L-____ --"_2 ___ -x
z
Figure l, Triangular Element
For the bending action two alternate triangles are used. A cubic polynomial is assumed for transverse displacements w in the z direction for the first triangular element. A set of linear functions for transverse displacements and rotations are assumed for the second triangular elemento
Plane displacement field
* u =
v =
where
f -u
f -v
fu = i..v = ~
[1 x y]
A straight bar underlining or column), square brackets matrix and braces L 1 denote
(loa)
ego i denotes a vector row ] denote a row vector or a
a column vector o
8
(l.d)
(I.e)
Bending displacement field
Cubic function
w = (l.f)
(l.g)
(l.h)
Linear functions
w f l = CIb (l.i)
of: 2 e = ... S!b x (l.j)
3 ey = f CIb (l.k)
f = [1 x y]
l [ CI7 CIg 1 S!b = CIa
2 [ CI IO CIll CI12] ~b =
3 [ al3 CI l 4 aIS] S!b =
----------------...... Œl' Œ2~ 00 ••• , Œ1S are unknown parameters (sometimes called "generalized di:-splacementstl) which depend upon the nodal displacements, while P and Q are parameters which may be chosen somewhat arbitrarilYD By an appropriate choice of P and Q, !w reduces to either Tocher's function (p = Q = 1) [36J, or Gallagher's function (p = l~ Q = 0) [37J.
It should be noted that the cubic function (l.g) is not completely conformable; that is, it will not, in general, satisfy the requirement of compatibility of slopes of adjoining elements perpendicular to their common sides o Another form of discontinuity results between non co-planar adjacent elements when bending
9
and in=plane de format ion occurs. The bending of one results in cubic displacements in the plane of the other and~ with the assumed linear membrane function, these displacements clearly cannot be compatible. This effect reduces with the decreasing sL:;€'! of element. The linear displacement assumption in bending does not have the above mentioned discontinuities~ but a different ttifficulty arises in that the linear w field and linear ex and ey fields are not compatible. This will be further elaborated while deriving element characteristics in bending o
hl Stress~strain relations for orthotropic material For a linearly elastic orthotropic material with three
planes of symmetry coincident with the coordinate axes,
10
nine independent, elastic constants are required [38J. If a material is layered or is transversely
isotropie (planes parallel to xz as shown in Figure 2)
then such a material has five elastic constants.
x ..
/ y
Figure 2, Layered Orthotropic Material
Moreover, by Kirchoff's first hypothesis, vi~ Oz = 0, the independent elastic constants required reduce to
four, These are Ex Sl V xyl) Eyll Gxy ' Thus
l2:p ~1- [D* ~ li ~ ~s1 L - -p
Q .Es
where
~p = [ °x 0y °xy J r = ) ° Oyz} L xz
*The curly bar underlining (ego ~p) denotes a matrixo
(2,a)
(2.b)
(2.c)
$!p = [
~s = [
Ep III
Es al
Gxz =
G = yz
ex ey YXy]
Yxz Yyz ]
Ex Vyx Ex (1 .. V xy V > yx (1- VXy Vyx>
VXy Ey Ey
(1- Vxy Vyx> (1- V V) xy yx
0 0
Gxz 0
0 G yz
E x 2(1+ Vxy)
E ~ E 9 Gare modu1i of e1asticity x y xy
VXy ' Vyx are Poisson's ratios
where
11
(2 o d)
(2 o e)
0
0 (2.f)
G xy
(2.g)
(2 oh)
c) Distribution of strains across the thickness It ls assumed that plane stress causes on1y
12
stretching of the element and bending causes linear variation of strains across the thickness. Moreover, the displacements are small quantities compared to the thickness of the element and hence there is no coupling between bending and plane stress effects. For the linear displacement field in bending it is further assumed that the shear deformations ~are constant across the thickness.
dl Thermal effects
A steady state is assumed and the coefficients of expansion in x and y directions are Œx and Œy respectively. The temperature may vary in any fashion fram element to element but within a given element it must vary linearly across the thickness of the element, i.e. top and bottom faces of the element must each be at a uniform temperature.
T T = T +~ Jo a t 1:;,
Let
where T ls the temperature at a distance , fram the middle surface.
Td ls the temperature difference between the top and bottom surfaces.
Ta is the average temperature
t is the thickness of an element
The strains due to temperature rlse Tare
~o oJ = (3.b)
1
e -pO
~bO
QO
where
~O
D
D -p
=
=
=
=
=
=
i ttxTa ttyTa 0]
[ tt;Td ttyTd 0] 2
D ~O
Therefore
2.po ;: ExTa [ tt +a. V (l- V V) ;X y yx yx xy
13
(3.c)
(3.d)
0] (3.f)
t atC = ± l ' i.e o top and bottom surface of an element.
~bO ~(3.g)
The thermal stress analysis is accomplished in two stages 0 First, assuming all nodal points are restrained, the stresses in all elements due to temperature changes are calculated o Second p these restraining nodal forces are ~quilibratèd by a system of forces (thermal loads)
14
equal in magnitude but opposite in Signe
The final thermal stress distribution is the sum of the stresses due to these thermal loads and initial stresses in the restrained system.
e) Element characteristics in plane stress
Six equations expressing the two dis placement components in plane stress at each node can be written
= (1=1,2,3) (4.a)
where
where the subscript p denotes in-plane displacement o
ô = C ~p (4.b) -p -p
where
ôp = r .§.lp 6 -2p .§.3p 1 ~p = l ~ . Clv ]
15
~p = f ul 0
0 f:vl
!u2 0
0 f v2 (4.c)
!u3 0
0 !v3
Note that C --p is a 6 x 6 matrix.
Strain displacement relations are given in reference [39J.
= [~u Ox (4.d)
Substituting (l.a) and (l.b) in (4.d)
= (4.e)
from (4 o b)
-1 ~p = [ Cp] ~p
Substituting in (4.e)
-1 ~ = [ QpJ [ CpJ ~p
Define a strain matrix by
then
e == B -p -p .§.p
The complete expression for Bp ls
B = 1 -p lA
where
Using (2.a)
where
~p = Ep ~p
-Sp= ~ 2A
-Y3
o
o
-y 3
o
is a stress matrix
Vyx(x3·x2) Y3
'" VyxY3 ~(X3-X2) VyxY3 xy
- I1IlY3
16
(4.f)
o o o
o
o
(4.h)
- Vyxx3
-~ Y3 xy 4.1)
mx2 0
where
-E = Ex -----(1- Vxy Vyx)
lt can be shown, (see Appendix 1) that
where K is a membrane stiffness matrix as shown in -p
17
(4.j)
Figure 3 and~O is a thermal load vector acting at the corner of the element.
Lpo = [FIPO F2pO F3PO]
wheve
Using (3oc) it can be shown that
(4.1)
F 0 = EtTa -p 2
-(a. + V Cl) Y3 x yx y
o
18
(4.m)
Et 4A
e e
Y3+ m(x3-x2) 1 2 2
Symmetric
=(m+ Vyx)Y3(x3-x2) 2 V . 2 mY3+ ..:..n (x3-x2)
Vxy
2 mx3Y3+ Vyx (x3-x2)Y3 y2+ mx2 -y3-mx3(x3'"'x2)
3 3
--
Vyxx3Y3+m{x3-x2)Y3 -my~-~ (x3-x2)x3 -(m+ Vyx)x3Y3 2 V 2 mY3+-.::n x3 Vxy VXY
m(x3-x2)x2 -mY3x2 -mx3x2 2 my3x2 mx2
- Vyxx2Y3 ~ (x3-x2)x2 Vyxx2Y3 -~ x2x3 0 .J!.n. x2 2 VXY VXY l)XY
Figure 3. Membrane Stiffness Matrix!p Corresponding to ~
.... \0
fl Element Characteristics in Bendin! (çubic disp1acement field)
Nine equatiQn~ expressing the three disp1acements w, ex' .ey at each node can be written in terms of a7'
Wi = .!.wi J!.b
exi = = [~]
20
-[~;l c>y i (i=1,2,3) (S.a)
eyi = [-:L = [~] ab ()x i
l.wi = [ fw ] i
Positive directions for exi and eyi are opposite to those of the x and y axes respective1y using the right hand ru1e o
(i=1,2,3) (5.b)
where the subscript b denotes the bending disp1acements.
-[~ ], [~] .!w2 ••••• [~] l(s.c) ()Y 1 C>X 1 ()x J
Note that ~b is a 9 x 9 square matrix. 1
The strain displacement relationship is given in reference [40J for the top and bottom fibres.
e b == î
using (l.f)
!!b == t. Qb ~b 2 -
21
(5.d)
Fo11owing the same steps as in the case of plane stress
.!ob == t !b ~b 2 (S.e)
.2:b == Eb~ (5.f)
~ == fi JI BT _b ,E.b ~b dA (5.g)
!:bO == _t2 J~ BT ~b ~bO dA '6 . .::J) (5.h)
Where
Eb == D -p (5.i)
~b == 6 [H,. My H,.y ] ;2
(5.j)
[ BbJ · == [ QbJ [ CbJ -1 (5.k)
!b and FbO are the bending stiffness matrix (9 x 9) and thermal load vector (9 x 1) respectively.
22
Substituting (S.k) in (S.g) and S.h)
(S.t)
FbO = [ F1bO F2bO F3bO}
where
EibO = [WiO MxiO MyiO] (i=1,2,3)
o
o
o
o
-K and Qb were derived in reference [31J and are given ,.., ,..,
in Appendix 2 for comp1eteness. If Tocher's function is
23
used then Eb will be singular for an e1ement with a
certain nodal geometry. In this case the computer
program is arranged to use the average stiffness matrices
resu1tblg from putting P = 0 and Q = 1 and then P = l
and Q = O.
gl Element characteristics in bending (linear
displacement field)
Once again nine equations expressing three
disp1acements w, ex' ey at each node can be written in
terms of ~7' ~8o •• o.~150 (These ~'s differ from those
of the cubic disp1acement function since in bending there
is a choice of using this triangu1ar element or the
previous one. The subscripts and notations, unless
redefined, are retained to give continuity with plane
stress formulation.).
1 wi = fi gb
exi !i 2
:: ~b (i=1,2,3) (6.a)
9xi :: 3 !i ~b
f. :: [f]i -1
The positive directions of 9xi and 9yi are the sarne as
those of the x and y axes respectively using the right
hand ruleo
24
Define
.2.is = f wiJ
and (1=1,2,3) (6.b)
ô- e = [eXi 9yi] -1
Qs = Es ~s
where
js = [ 6 18 Ô2s Ô3s ] ~s = i f l f 2 f3]
~s = [Cl~ 1 Note that Es is a 3 x 3 matrix
and
.2.e = ~9 0:9
where
Qe • [ Ôle Ô29 ô391 Ea = il Q
Q il
i2 Q
Q Î 2
t3 0
Q !3
25
!!e 2 [tt~ tt~} Note that ~ is a 6 x 6 matrix.
Strain displacement relations are given in reference [41J.
eb=î[-t! :!r t [~ 2:
thence
.!b = t l!b ~9 2"
Qb = ~b .2.9
where
2 - 0 w ~
_ 09x Ty
~e = [6 1e 62e 63e]
2~ } ~y
09 . x -Tx
The strain matrix!b = ~e fë l
~b = 1 0 -Y3 0 Y3 lA
(x2-x3) 0 x3
0
"'Y3 (x2-x3) Y3 x3
and ~b t is a stress matrix. = Db- Bb - 2-
~} (6.c)
(6.d)
(6.e)
0 0
-x 2
0 (6.f)
0 -x2
26
Sb= Ët r- 4A Vyx(x2-x3} -Y3 Vyxx3 Y3 - Vyxx2
~(X2-X3) - VyxY3 ~X3 VyxY3 -~ x2 xy Vxy xy
o
(6.g) The thermal load vector and stiffness matrix ar~
formed in a similar way as previously mentioned. They are given below.
- 2 = - ~d 24
o ( lJ a + ~ ay }(x
2- x
3)
yx x lJxy
o
(V a + V vx ex ) (x3 ) yxx v=: y
xy
o -:
o
(6.h)
0
0
Ët3
48A
e
2 2 ~ (x3~ x2) + mY3 Vxy
(x3",x2 )( V yx+ m)Y3
œ~ (x3-x2)x3-my~ Vxy
- Vyx(x3-x2)Y3-mY3x3
v iC: (x3-x2)x2 xy
mx2Y3
Symmetric 2 2 Y3+ m(x3,",x2)
œ Vyxx3Y3-m(x3-x2)Y3 ~ x~+ my~ VQ
. 2 -Y3-m(X3œX2)X3
Vyxx2Y3
mx2(x3-x2 )
Vyxx3y3+ mx3Y3
- .J!.n x3x2 Vxy
- mx2Y3
.2 2 Y 3+1DX 3
- Vyxx2Y3
-mx2x3
Figure 4. Bending Stiffness Matrix~9 Corresponding to ~
V~ x2 - 2 Vxy
0
e
2 mx2
N
"
28
The strain displacement relations which take into account the shear deformations for the case of the linear displacement field are given in reference [41J.
~s = ~w - e ] x ~ (6.i)
As discussed before the linear ex and ey fields are not compatible with the linear w field. Linear ex ànd ey
"--di"splacements imply a constant moment, i. e. zero transverse shear. A constant shear means linearly varying moments. This inconsistent behaviour of linear functions leads to an undesirable stiffness matrix. To get around this difficulty practical models, called Equilibrium Algorithms, have been suggested by Melosh [33J and Utku [34J. Mëlosh's model gives poor convergence when compared to that of Utku. The Equilibrium Algorithm as suggested by Utku is used here to derive explicitly the shearing stiffness matrix o
The portion of the transverse shear stiffness"matrix associated with ~w and ~w, using a similar procedure as Tx Ty employed previously, is
Kll = t - s -4A Symmetric
2 G x2 yz
(6.j)
29
Now since columns of !lls represent nodal transverse forces because of unit nodal transverse displacements imposed individually on the kinematically determinate base system, it is possible to compute the unbalanced bending moment for each unit displacement and suitably distribute it to the nodes with a sign change. By so doing, the contributions of ex and ey are obtained.
The elements of !lls May be identified as f ij • Further details are given in Appendix 3. The shearing stiffness matrix is given in Figure 5. The total stiffness matrix in bending and shearing behaviour
then is ~ =!s +!s which is a 9 x 9 matrix.
t 4-A
e
t ll
-J...fal ':la 1 t 2. -4" 3\ 'ja
2
~(+21~+.fa,~) tCfal :l:3 ~3) -*(t21~2+~:X:~)
f21 0 _.L f21 :X::2. 2
0 0 0
; fZI ::Cl. 2.
0 _L ~ Xl. +
f31 t {~I 'j3 _1. -tô \ X3 2
-t f 31 ~3 -~ +31 { ~ +31 X,3 ':::13
~ ..fa, X 3 tfal x 3 ~a '.f 2. - 4 al"X3
-t2 2.
_.1-.f. .... 2. :~A:..J3
- 1 f -~ "+ 31::la
•
Symmetric
t(.ç3~~~i"t,:x.2) - ~-t.J1~u~a -~(f21:l~Tf3~1)
t32 t +32. 'Ja ! +31 ~2a f?>~
- '2 f~2. '.:la _l.. tô 2. ..J: 4- 3 - ~ +!1. X 1'b ~2> -k.(~;~ +fafl "j~) -~ (fô,~ +~l.~)
l t32~31 1 of l. -!(~:a\~T~~~) i.(~31'X3t\r~')~.3 "i(~I~+~~~ 4 ~l. 'X,31 'j~ -~ ~!12. X3~ 2
Figure 5. Shearing Stiffness Matrix~ Corresponding to w1 ' ex1 ' 9y1 ' etc. l.rJ o
31
h) Transformation matrix
The two sets of axes shown in Figure 6 are Cartesian
orthogonal systems, and if at a node six degrees of
x
Figure 6 0 Orthogonal Cartesian Axes
freedom are considered then there are 18 degrees of
freedom for an e1ement. Therefore, an 18 x 18
transformation matrix is needed to transform from
local axes to the global and vice versa.
T = À 0 0 0 0 0 ,..., ...., ,..., ,..., N ,.,
0 À 0 g 0 0 -.J ,.J ,.., ,.,
0 0 À 0 0 0 (7.a) ,.... -' ,-J ,.., rJ rJ
0 0 0 À 0 0 ,.J ""
,..., rJ r/ -'
0 0 0 9- À
~J -- rJ ,..., ,.., Q 0 0 ~ 0
-.J ,..., ,..,
where À is a matrix of the direction cosines. ,-
32
(7. b)
where Àll is defined as the cosine of the angle between
local x-axis and a common global x-axis. The text by
Jaeger [42J gives the dependence of elements of the 1
matrix. Out of these nine equations only three are
independent. Of the remaining six, three are the
orthogonality conditions and three the normality conditions.
Considerable simplification is obtained by taking
the local x-axis in a plane which is parallel to the xy
global axis as suggested by Zienkiewicz and Cheung [28J.
À <= Bl Al 0 - b b
... A1Cl _B1Cl A2 + B2 (7.c) 1 1
ab ab ab
~ a' a
*Superscript G refers to global coordinates.
33
= (7.d)
=
a =
b =
lt may be noted that if the local xy plane is
parallel to the global xy plane then Al = Bl = 0 and
the À matrix as given by (7.c) becomes indeterminate. f"-/
The program automatically takes an appropriate identity
matrix for such cases.
2, Formation of Global Stiffness Matrix and Thermal Load
Vector and Solution of the Equilibrium Equations
To form the thermal load vector and the stiffness
matrix with respect to a common global coordinate system
the following transformations [42J are used for each
element (n = 1,2,30000)0
= [Tn? [FJ = [Tn r [ K ] [ Tn ]
where
(B.c)
e· 34
K III !ll !12 !13 ,...,
!21 !S22 !23 (8.d)
,!31 1532 !33
FiO III [ FiPO FibO} (i=1,2,3)
where FO and ! correspond with which is
lt should be noted that FO given by (8.c) is a column vector with 15 elements. However, in order to apply the transformation to global coordinates (8.a)~
Eo must have 18 elements. The three extra elements are all zero, representing the moment with respect to the local z-axis and these are inserted in their correct locations by the computer before carrying out the transformation. The sarne procedure applies to K which ,.... as given by (8.d) is a 15 x 15 matrix while in (8 ob) it is an 18 x 18 matrix.
The overall thermal load vector and the overall stiffness matrix are obtained by adding together the approp~~ate components for each elemento After formation of these matrices the equation of equilibrium for all nodes in the global coordinate system is written as
(8.e)
35
For anode with a prescribed component of displacement,
the corresponding diagonal term of the stiffness matrix
KG is multiplied by 1011 while the corresponding ,..., force term is replaced by the newly formed diagonal
term multiplied by the prescribed displacement value [28J.
This procedure, however, is not appropriate trom the point
of view of computer memory but it simplifies the
programming t.chnique.
Equations (8.e) are solved using the method of
ttidiagonalization where inversion· of the individual
submatrices is performea using Cholesky's square root
method.
3 8 Stresses and Strains
The solution of the equilibrium equation yields the
displacements of the nodal points of the finite element
system o These displacements are then transformed back
to element coordinates to obtain the strains and stresses
at centroids of elements.
where
36
~o = L .!!.Op .!!.Ob ]
B =- B Q, --' ,...,p
j} t 2 ~b
Q • [~p âb ]
D = D Q, ..., ,...,p
It may be noted that ê and â are final strains and stresses -
37
S;XPERIMENTAL STUDY
lt was considered desirable to have finite element solutions for some of the shells for which classical solutions are not available. ln order to check the validity of the theoretical results based on the finite element method p deflections and strains were measured in model shells experimentally.
The experimental investigation was carried out on shells with different geometries and various loading positions 0 One of the models tested was a spherical shell which was triangular in plan. The shell was eut (see Figure ~) to the required geometry in the laboratory from a spherical cap made of aluminum alloy 3S-0. The thickness of the model shell was measured by a dial gauge using the same device described in reference [15J. The thickness was measured on three meridians joining the pole with the corner supports as shown in Figure ~.
Pole
D
• Location details are given in Table l
~~ ____ ~~~ ________ ~ .. Corner Support 15.688"
Figure 7. 'p Positions at which Thickness was Measpred
\
~
--0\ ..... o
" N
e 1 5"
~ 0 • Position of Strain Gauges (See Figure 22)
~A
600
Plan
R =
Section A-A
1 1 1 1 1 1 1 1 1 1
."..,.,.,"""'"
e
.......
."..,.,.,"" ."..,.,., .......
.,., ,.....,~
.", ....... ,."....."""""
."..,.,.,""""
Idealization of Shell
E = 9.76 x 106 psi
V = 0.322
,... ....... ~
w 00
Figure 8. Spherical Shell Triangular in Plan
'e 39
Distance along Meridian Meridian 1 Meridian 2 Meridian 3 from fixed edge • (inches) (inches) (inches) 3" 0.0773 000774 0.0779
6" 0.0773 000781 000786 9" 0.0789 0.0789 0.0792
12" 0.0819 000811 0.0810
pole 0.0819 000819 000819
Table l, Thickness Measurements
Measurements were taken several ttmes for each meridian and averages are shown in Table 10 The average of all these readings was approximately 0008 inches. This was used for computer input data.
The radius of curvature and the co-latitude of the points of support were calculated from the known geometry i.e. from the span of the spherical cap and the height of the shell~ and were found to be 30.5 inches and 31 degrees reapectively.
The top and bottom surfaces of the shell were thoroughly cleaned and prepared for the installation of electric foil strain gauges. The strain gauges were very carefully mounted along three lines of symmetry from the pole to the edge of the shell on both the top and bottom surfaces 0 These meridians were labelled "A"~ "B" and "C" for purposes of identification and were at angles of 120 degrees to each other. Near the load and end supports, the strain gauges were closely spaced. Along each meridian 12 strain gauges (lntertechnology EA~13=125AD-120 OPT~W)
1
40
were placed, using Eastman 910 cement, six at the top and six at the bottom; a total of 36 strain gauges thus being used. Each strain gauge was numbered according to the meridian and the· position at 'to7hich it was installed. For example, the three strain gauges near the pole on
. the top surface ~Tere numbered lAT, IBT, lCT. Si..rnilarly, for thebottom surface near the pole these were numbered as lAB, lBB, leB. A short gauge length (0.125 inches) was selected for aIL the gauges since uüder the regions of stress concentration high strain gradients were
expècted. Seven dial gauges were located on the shell, as shown in Figure 9, to measure the deflections.
The positions of the strain gauges are shotYD in Figures 8 and 220 Full use of symmetry ~vas made to check whether, during the experiments, the behaviour of the shell was similar along the meridians, and to check that the strain gauges were placed properly.
The four arm bridge circuit "tvas cornpleted by means of a three lead wire system. A Datran automatic digital strain indicator was used to record strains.
The edges of the shell were fixed against rotation and translation by pouring a loto}' m~lting point alloy ("Cerrosafe" supplied by Canada lietal Company Limited) into rigidly fixed metal boxes surrounding the corners. The metal boxes ware welded to the stand lmich consisted of a frame made up of 2:" ~ 2" x 3/16" angle irons. As desired, the whole set-up was very rigide
41
tJ is Oo-latitude
Figure 9. Position of Dial Gauges
Figure 10p Positions of Loads
The experiment was run for two different load
positions, Pl and P2. Loads were hung from a wire
which passed through a 1/16 inch diameter hole and
was secured at the top by means of a small knot.
Positions of the load are shown in Figure 100
The complete set-up is shown in Plate 10 The
second plate shows a close-up view of the shell.
The deflections and strains were measured at
42
10 Ib o intervals, with a datum load of l lb. The
maximum load applied was 50 lb. for both the load
positions. Typical test results are given in Appendix
6 0
The second shell tested was a spherical shell cap
with a hole at the crown. The shell was spun from the
same material as the previous shell. The load was
applied in the form of an intermittent ring load to
give the effect of a discontinuous load o This was
achieved by cutting a neoprene pad attached to a heavy
cylindrical ring [15J as shown in Figure tlo The width
of the cylindrical ring was 0 0 46 inch and was machined
to the slope of the shell o The diameter of the ring
was 19 0 152 inches and its weight was 161 lbo
The edge of the shell was secured between two
steel rings which were bolted togethero An annular
space between the rings was filled by maans of a low
melting point alloy (Cerrosafe). This a~angement gave
satisfactory fixed edge conditions.
43
<
43
• ~ ~ .
• .-1 .-1 Q)
.r:: (/)
.-1 Q) 'd 0 l::
c .... lJ II)
m ~
1-1 0
4-1
II)
::1 lJ Cl! 1-1 Cl!
4!
.-1
Q) lJ Cl!
.-1 0...
..
44
t Q) en o ~ u
44
4-l o
;j 1
Q) U)
o r-l U
1/4"
45
2.0 1/2" Dia. ·~--------------------~·t ~ I~ ~
~====~11=======rr~~~ 1 1 1 1 1 1
20" 0.0. 1 1 ~ 1 1
/t, 1
,/ Elevation
UOO:. \.18,693"
1.19.612"
Detail
/ Neoprane Gasket8 l/S" Thick
Plan
Figure_ll, Cylipdrical Loading RinS
The load was applied by means of a 150 gallon
domestic fuel oil tank which could be filled with
water. A glass manometer was mounted on the side
of the tank and calibrated in increments of 50 lb.
The smallest total load which could be read on the
manometer was 282 lb.
46
The full setœup showing shell, supporting ring$
loading ring, tank and stand can be ~een in Plate 3.
A view of the underside of the shell is shown in Plate
4. For further details of each component for this
setœup see"references [15J and [43J.
ln this case a180, three meridians were chosen, as
shown in Figure 129 on which 9 strain gauges on the top
and 9 strain gauges on the bottom were glued uSing
Eastman 910 cement. The strain gauges were supplied by
Budd l~struments Limited (C12-121A) and a total number
of 54 was used. A Baldwin Strain lndicator was employed
to measure the strains. The procedure for the labelling
of the meridians and the strain gauges was similar to
that adopted in the previous test.
Extensions made of auitable lengths of welding rods
were added to the shafts of the dial gauges and placed
under the loads to obtain radial deflectionso The dial
gauges were set perpendicular to the shel1 surface by
mounting a dexion angle so that its surface was at the
center of curvature of the shell which was marked on it.
An extension to the lower end of the dial gauge shaft,
a1so made of welding rod 9 was adjusted to point directly
47
.... .... Q) .a CIl
.... Q) '0 o X
47
.-1
...... Q) .c CI)
...... Q) '0 o ~
48
1 CI) al o ..... u
• /'fi \
t-I ~:'-4 ~ •
"J.~~-" • 'If 06? '" , . ~ ,", __ i , , ./ ,",'
48
...-1
...-1 Q)
.c rJ)
~ o
::l 1
Q) en o
...-1 U
--------------------............ . e • Position of Strain Gauges (See Figure 19)
5.01" x 0 0 46"
r
'-A
) "
~"-
~" \
200 lb o (Total)
ldealization of Shell 6 E = 9.76 x 10 psi
V = 0.322
Figure 12. Spherical Shell with Central Cirçular Hole
-
~ \0
50
at the center of curvature of the shell.
Two sets of dial gauges were employed to measure
the deflections of the model shell. One of the sets
measured the deformations of the frame, whilst the
other measured the deformations of the shell relative
to the frame.
The maximum load used in this test was 917 lb.
Though strain gauges were mounted only along the
meridians Al, Bl and Cl it was possible to measure
strains along A2 to AS, B2 to BS and C2 to CS by
appropria te rotation of the cylindrical loading ring
(see Figure 13)0 For example, to measure strains along
AS g BS and CS the ring was rotated by 30 degrees o Typical
test results are given in Appendix 6.
C4
CS
B4 B3
B2 Bl
Figure 13. Meridians on which Strains were obtained by rotating the cy1indrica1 loading ring.
51
The e1astic properties were obtained experimenta11y
from tension test specimens (A.S.T.M. E8-54T (1954»,
and the average values thus obtained were
E
V
=
=
9 0 76 x 106 psi
0.322
For detai1s see reference [15J.
52
NUMERICAL EVALUATION AND RESULTS
It is instructive to demonstrate the convergence
of deflections and moments for plates and shells by
increasing the number of triangular elements used.
Three examples are chosen. two in flat plate bending
and one from spherical shells. This study gives an
insight into the behaviour of plate and shell structures
so that the analysis of more complex forms, reported
later in this thesis, can be achieved.
10 Convergence of Deflections and Moments
L = t = E = V =
a) Simply supported square plate with a uniformly
distributed load
The dimensions and elastic properties are as follows.
10"
0 0 5"
30 x 106 psi
0 0 3
q = 0.04 psi
The four idealizations are shown in Figure 14 in
which numbers of nodes along the half sides are n = 2,3,6
and 11. The convergence of the deflections at the center
is shown in Figure 15 and Table 2 gives the numerical
results. The results for a triangular element with linear
displacement functions in bending increase monotonically
with n and are always less th an those obtained by the
classical method. This is due to the fact that a triangular
53
element with linear displacement functions satisfies the
convergence criteria, first laid down by Melosh [44J.
For monotonie convergence an element should have conformable
boundary displacements, permit rigid body motion without
straining, and admit a constant strain condition. The
convergence of results to the correct answer by a non-
conformable triangular element permitting rigid body
displacement and a constant strain criteria has been
disputed [45Jo However, it may be observed (see Figure
15) that when a sufficient number of nodes is taken, then
the non-conformable element a180 tends towards the correct
solutiono This was also pointed out by Zienkiewicz and
Cheung [28J po 22 and Gallagher [53J.
ln Figure 15, convergence of bending along with the
classical solution is plotted for n = 6 and 11 on the
center line of the plate for linear and cubic functionso
The convergence of bending moment and deflections by
linear functions is faster. The.difference between moments
obtained by linear functions for n = 6 and n = 11 is very
slight as can be observed in Figure 150
No. of nodes Cubic Linear Classical without on the side Function Functions shear [40J
n -2 006201 0.3414
3 0 05688 0.4445 004729
6 0.5132 0.4724
11 004972 0.4785
Table 20 Showing Convergence of Deflectiofi x 105 for Simply Supported Plate with Uniformly Distributed Load
54
b) Clamped square plate with a concentrated load
at the center
The following dimensions and elastic properties are
assumed. The four idealizations of the plate are shown
in Figure 14.
L :: 48"
t :: 1"
E :: 10.6 x 106
V :: 1/3
W :: l lb.
The convergence of the central deflections are shown
in Figure 16 and Table 3. The convergence of the moments
is shawn in Figure 16. Once again convergence using
linear displacement functions is faster despite its initial
"stiff" behaviour. Even though the deflection (for n :: 6)
is less than the deflection for n = 11, the moments 9 as
may be seen in Figure 16, have almost negligible variation.
The cubic function shows a tendency to be flexible but
has the desired behaviour of converging to the correct
limite Both the triangular elements (for n :: 6~ 11) gave
a bending moment of O.lZ lb. in/in. at the fixed end
(Figure 16), compared with 0.lZ57 lb. in/in. obtained
from the classical solution [40Jo The linear functions,
being conformablël) have a tendency to be "stiff" but the
rate of convergence is better than for many other
conformable elements whose stiffness properties are
derived by the element subdivision technique [Z8Jj [46J»
[54Jo For n :: ZI) the linear functions give negligible
e
55
deflections owing to the fact that no rotations are
possible in this idealization. Flexural behaviour is,
therefore p absent and the negligible deflections observed
are due only to shear deformations.
No. of nodes Cubic Linear Classical without on the side Function Functions shear [27J
n
2 0.1208 0 0 0006
3 0.1737 0.1007 0.13
6 0.1556 0 0 1236
11 0.1428 0.1288
Table 3. Showing Convergence of Deflection x 104 for a Clamped Plate with a Concentrated Load at Center
c) Spherical shell with central circular load.
The geometry of the shell and the finite element
idealization is the same as that used for the spherical
shell with a narrow ring load (Figure 26). The base
edge is completely fixed. The shell is loaded by means
of a circular load of 1.7625 lb. at the crown which is
applied on a circular area of 1.466 inch in diameter
centered on the pole. The theoretical investigation
based on the finite difference method is given in
reference [47J and the experimental investigation in
reference [48J.These results are compared to the finite
element solution. The deflections and strains at the
pole of the shell were calculated using Reissner's theory
of shallow spherical shells. Deflections thus obtained
56
were taken as a classical solution for the convergence investigation.
The convergence of deflection at the pole by both triangular elements is shown in Figure 17 and Table 4.
lt can be observed that the triangular element using linear functions shows a stiffer behaviour for the same number of nodes than the triangular element using a cubic function. The former gives a lower bound and the latter an upper bound after sufficient refinement of the idealized shell. This is a desirable guide-line as it plac~s the correct solution within the two bounds. Strains given by both triangular elements when compared with the results obtained by the finite difference method and the experimental tests compare favourably, as shown in Figure 18. This characteristic of triangular elements using linear displacement functions giving more accurate results for moments than for deflections was previously pointed out by the author [35J.
Total No. of Cubic Linear Classical [15J Nodes Function Functions
7 00532 .0081
21 .2290 .0797
31 .2495 .197 .2503
71 .2784 .212
85 .2810 .215
Table 4. Showing Convergence of Deflection x 103 for Spherical Shell with a Central Circular Load
2. Comparison with Experimental Resu1ts
a) Isotropic spherica1 she11 with a central
circu1ar ho1e
Taking advantage of symmetry, on1y one quarter of
57
the she11 was idea1ized. The geometry, e1astic properties
and the idea1ization are shown in Figure 12. The
experimenta1 procedure has been described in the previous
chapter. Figure 19 shows a plot of the strains on a
meridian which passes through the center of a 10aded arc,
and Figure 20 gives the strains a10ng a meridian which
passes through the center of an un10aded arc. Simi1ar1y,
Figure 21 gives radial def1ections for both the meridians.
The values p10tted in each case are the average of the
readi~lgs on the three radial 1ines of symmetry under a
total 10ad of 200 lb. As was intuitive1y expected, the
un10aded portion does bu1ge out and the max~ bending
strains are confined to the 10aded arc portion. The edge
of the ho1e~ as indicated in Figure 21, lifts uniform1y
but its affect on the strain distribution is not very
significant. It is again observed that 1inear functions
give 1ess deflection under the load o
b) Isotropic spherica1 she11, triangu1ar in plan
The geometry and e1astic properties are given in
Figure 8. On1y one ha1f of the she11 was ana1yzed
becauae of symmetry. The averages of strains and
def1ections measured on three radial 1ines of symmetry
58
are plotted and compared with the finite element solution
in Figure 22 and Figure 24 respectively for a 20 lh load
at the crown. The strains obtained e:x:.perimentally agree
quite well with those given by the two triangular
elements o
Investigation of the deflections shows that the
triangular element using linear functions is too stiffo
This stiff behaviour might be explained as follows o
Both the triangular elements used have the same
characteristics and are conformable in membrane action,
thereby giving identical results for plane stress problemso
The deflections obtained in this action will be smaller
than the correct results because of its monotonic
convergent property. In bending action the triangular
element using linear functions is conformable and,
therefore~ the deflections obtained in the bending
problem will be smaller than the correct values o The
triangular element using a cubic displacement function
is non~conformable and gives generally higher deflections
in bending action than the correct valueso (See Figures
15 and 16)0 When these two characteristics are combined,
that is p the membrane action and bending action~ the
triangular element uSing iinear functions becomes stiff
and the triangle using a cubic function for bending
displacements seems to compensate for the stiff behaviour
in the membrane action o (See Figure 17)0 Since spherical
shells 9 because of their double curvature, support loads
mostly by membrane action with bending localized at the
59
edges and under the load, the triangular element using
a cubie function for bending displaeements should give
better results for the same finite element size. This
has been the experience df the author in solving spherical
shells. However, sinee the primary interest of the
analyst and designer is to find stresses, the triangular
element uSing linear displacement funetions could be used
for the stress analysis with satisfactory results.
Similar observations are also made for a load of
20 lb. at the position 0 = 5°42' (see Figures 23 and 25).
ln this ease, although the strain plot is more complicated
than the previous one, the strains obtained by both
triangular elements eompare favourably with the experimental
resultso
e) Isotropie spherieal shell under a narrow ring
!.su!&
The theoretieal analysis based on the finite
differenee method is given in reference [47J and the
experimental. results were reported in referenee [48Jo
The plot of strains and deflections for a meridian is
eompared with the finite element method in Figures 27
and 28 respeetivelyo The finite element idealization
geometry and elastie properties are shown in Figure 26 0
3 0 Application to Areh Dams
The eylindrieal arch dam whose dimensions are shown
in Figure 29 was analyzed by Zienkiewiez and Cheung [49J
using rectangular elAments. This dam was also
investigated by others in the same reference [49J.
60
The foundation of the dam was assumed to be very rigid
so that fixed edge conditions were assumed o Since the
rectangular element could not idealize the exact
foundation configuration, Zienkiewicz and Cheung
approximated the actual foundation by an imaginary one o
Although the actual foundation configuration can be
taken into account by the triangular element, for
comparison ~~poses Zienkiewicz's idealization was
adopted o
The arch dam was analyzed assuming certain elastic
properties for the concrete and perhaps it was a
coincidence that nope of the investigators gave the
elastic properties which they used to obtain their
theoretical results. The results using the elastic
properties shown in Figure ~9 gave a reasonably good
comparison and subsequently Professor Zienkiewicz [50J
confirmed that he had a1so used these same e1astic
propertieso Figure 30 shows comparison of the results.
4e Thermal Stress Analysis
a) Thermal stresses in a plate of non-uniform
thickness
This example is of an isotropie plate with thickness
variation given by t = (1-Oo9y2) to and temperature
variation of T = (y2~ 1/3) TO along a chorde The
61
thickness approximation and the finite element idealization
are shown in Figure 31.
The the~l stresses are compared with the results
given by Gallagher et al [51J who used rectangular finite
elements and Mendelson et al [52J who obtained a classical
solution assuming the plate to be of infinite length o
bl Thermal stresses and deflections in an
axisymmetric cylindrical shell
The shell material is assumed to be isotropie, the
ends are fixed against linear or angular mov~ment and
the temperature is increased by TaoF above its initial
temperature. Figure 32 shows a comparison of deflections
and moments between the triangular elements and the
classical solution [6J.
52 Application to Orthotropic Shells
Figure 33 shows an orthotropic cylindrical shell
under uniform internal pressure. The ends are assumed
to be fixed o The idealization for this shell is exactly
the same as that previously taken for the cylindrical
shell under a uniform temperature increase. The elastic
properties and the geometry qf the shell are shown in
Figure 330 A computer program was written to do an
analysis based on classical methods [6J and Table 5
shows the deflections obtainedby the different methods.
Figure 33 shows a comparison of the cieflections and
moments 0
62
Node Classical [6J Cubic Function Linear Functions
l .2063 .2059 .2075
2 .2077 .2078 .2084
3 .2079 .2086 02074
4 .1945 .1953 .1915
5 .1486 .1467 01414
6 .0639 .0579 .0536
7 0 0 0
Table 5. Radial Deflection x 102 for an Orthotropic Cylindrical Shell under Internal Pressure
63
n = 2 n = 3
// 1// :/ 1/1/ 1/ !L '/ // 1// j VI/ /:/ / // [/1/ V VI/ V V '/ l/ 1/./ :/ 1/1/ IL IL 1/ 1/ 1// j 1/1/ V V V 1/ 1// :/ /1/ 1/ 'L / 1/ 1 ..... 1/ 1/1/ IL IL 1/ 1/ !/:/ 1/1/ 1/ 1/ 1/
1/1/ 1/./ V 1/1/ V V 1/ 1./1/ 1./1/ 1/ 1/1/ LI V 1/
N = 6 n = 11
Figure 148 ldealization of a Square Plate
• !i
'" 0 ~
~
~ .... .a.J U" CD ~ .... CD Q
~ cu u .... .a.J ~
~ ~ cu ~ .a.J ~ CD
CJ
.. .0 ~
N o ~
6,0
4,0
2.0
0
20
10
2
--Convergence of Def1ection
Classical ~CubiC Function
---- Linear Fupctions
4
o
• c •
6 8 10 12
Number of Nodes on Side "n"
Convergence of Bending Moment
- Classica1 [40J
Cubic Function • n. 11 c n = 6
Linear Functions • n lai 11 o n = 6
~ Center of Plate
64
~ Simple ~ ~~ ~
o 1" 21 3" 4" 5" Distance from Center Line
Figure 15. Simp1y Supported Plate with Uniformly Distributed Load
----~
U"\ 0 .... ~
Q 0 .... ~ u Q) ....
'1-4 Q)
0
.... CG u .... ~
'" ~ ....
Cd
'" ~ Q Q) u
N 0
! .... X
0
s:: ..-1
'" s:: ..-1
0
.0 ..... ~ s:: m 0 ~
105
0.5
0
2
-20
-10
0
10
Convergence of Deflection
____ Cubic Function
C1assical
----Linear Functions
4 6
Number of Nodes
Convergence
Center of Plate
4 0 8 11
8 10 1
on Side "n"
of Bending Moment
19.211 24 1
Distance from Center Line
Cubic Function ----- n = 11
B n = 6
Linear Functions 20 _____ n = 11
--e--n = 6
Figure 16 9 Clamped Plate with Concentrated Load
65
e
Convergence of Deflection
0
s:: ,.-f
4.0 "'"0 M
>< s:: 0
'" 3.0 Cubic Function
,I.l
~;e 0
.---Q)
~ ~Classical
.... --0
.... ---Q)
Finite Difference
Q
M cu
2.0-1 '" - --Linear Functions 'Ô / 9-cu D:: M cu ~ ,I.l s::
1.0 Q) u
o 7 î-"- --------47 . --.- ----67--- -----87- 107 127
Total Number of Nodes Figure 17. Spherical Shell with Central Circular Load ............ ----------------
0\ 0\
\0 o ~
~
CIl s::
...f CU J.I ~ CI)
e
Center Line
8 •• Bottom
4
-4
-8
Load
1
Bottom Surface - • Reference [15J[ 47J[ 48J
• Cubic Function • Linear Functions
Top -v
a o
Surface Refer~nce[15J[47J[48J Cubic Function Linear Functions
20 25 Number of Equal Divisions Along Meridian
e
Edge of Shell
Figure 180 Meridional Strains in Spherical Shell Under Central Circular Load of 1~625 lb o
J
0\ ~
\0 o .-4
>< CI)
r:: '" ~ ~ CIl
--120
80
40 Plan
Center Line
~eridian on which Strains were Measured
o
e
- Cubic Function
--- Linear Functions
e Top Fibre Strains (Exp.)
o Bottom Fibre Strains (Exp. )
Position of Load
01 E- - - ......,z "\ 1 ~'i \\ -p-c ='-" f ..... :Sb &7 l '\ ~ 5 - 1.0ç
-40
-80
-12
--. ---
Number of Equal Divisions Along Meridian
Edge of Hole "
el
/ / -_ ......
Fixed Edge
Figure 19, Meridional Strains in Spherical Shell Under ...... lntermitt:ent Ring Load
0\ 00
\0 0 .-f
~
fi)
s:: • .-1 qj ~ ~ CIl
e e 120
Cubic Function Meridian on which Strains were Measured Linear Functions 80
--1 1 1 1 1
• Top Fibre Straüls (Exp.)
0 Bottom Fibre Strains 401 (Exp. ) Plan
- ----<'JI. 0 ~osition of Load
1 Center Line
01 ...:::-=--=- -- -- .-
1 5 10 15 20 25 30
-40 -i
Number of Equal Divisions Along Meridian
Fixed Edge -801 L Edge of Hole
~120 Figure 20 0 Meridional Strains in Spherical Shell Under Intermittent Ring Load 0\ \0
s::: ,r-4
CO') o .....
e
3
2
e
Cubic Function
Linear Functions
o Experimental
Center Line
~ 1
5 '" -I-J o Q) .....
lt-4 Q)
Cl
..... cu
,r-4 '0 :
Position of Load Def1ection Ha1fway Between Loads
1 -.". ~ ~ / if - -~ " -== O:::J ~ 7f(' ~
.. 1
-2
-3
10 l5'\. 20
Number of Equa1 Divisions Along Meridian
I-I Def1ection Under ~ Load Center1ines
Edge of Ro1e Fixed Edge
Figure 210 Radial Def1ections of Spherica1 She11 Under Intermittent Ring Load "'-J o
e
----_ ....
l Free Edge of Shell
.... , 1 , 1
" 1 \ -401 \ 1 \\ Il
\ Il \ '1
\ 1 \ 1
\ ,
e
-- Cubic Function
--- Linear Function
o Top Fibre Strains (Exp.)
• Bottom Fibre Strains (Exp. )
~ (Deg.) Along Meridian
Edge of Shell
Figure 22. Meridional Strains in Spherical Shell Triangular in Plan Under Concentrated Load .....
1-'
e
15 1.0
Lree Edge of Shell
120
\0
~ 80 >< fi)
s:::: .... ~ ~ 40
-80
-120
o
\.1 1 1 1
\ \ \
e Cubic Function
Linear Functions
o Top Fibre Strains (Exp.)
• Bottom Fibre Strains (Exp.)
\ Position of Load
\ \ \ \
t
1. 1
~ (Deg.) Along Meridian
1 Edge of Shell
1
1
Figure 23. Meridional Strains in Spherical Shell Triangular in Plan Under Concentrated Load
""-J N
e
1
s:: 12 .,..f -N ) -
'b ~ 8 >C
s:: 0
o,..f ~ U Q) ~ 4 If.! Q)
t::I
o 1
----, -- -- " - '\. ../--/ ~ \.-4 / '-
o
• Position of Load
/
../ ,/
5 --
o
~ (Deg.) Along Meridian
Cubic Function
Linear Functions
Experimental
20 25
L Free Edge Fixed Edge
-1
Figure 24. Vertical Def1ections of Spherica1 She11 Triangular in Plan Under Concentrated Load
e
30
....... w
e
o
s:: ,ri -N :l --
12
('t') 8 o ~
>< s:: o
orf ~
u 4 Q) ~ \lof Q)
Q
e
Cubic Function
Linear Functions
o Experimental
Position of Load Free Edge of Shell
~ 15
/" /'
o
o 10 .-------------- --
-4
-8
-12
110
""" ...-"'-" /' " ,/ ,_/
2'5 /30 Fixed Edge of Shell
~ (Deg.) Along Meridian
Figure-250 Vertical Deflections of Spherical Shell Triangular in Plan Under Concentrated Load
....,
.c::-
e
• Position of Strain Gauges (See Figure 27)
lb. (Total)
.08"
Idea1ization of She11 6 E • 9.76 x 10 psi
V .; 0.322
Figure 26. Spherica1 Shell Under Narrow Ring Load
e
..... VI
e
12
8
\0 1 Center Line 04'" .... ~
Gl s:: "" cu
--o R~f~~ence_[47J[48J
• 0 Cubic Function
• 9 Linear Functions
Bottom.
~Position of Load Fixed Edge ~
< ~ ~ 0 < ~ ,,> > 3 1 ra ~ 1 :::::> _-.
CI) 1 .......... 7 , i • 7 c::::::::
25
o -4
-8
-12
o
2
o
Number of Equal Divisions Along Meridian
Figure 27. Meridional Strains in Spherical Shell Under Ring Load
...., 0\
\0 o ....
e
~ ,
4
" Center Line ~ ........ :t Il 2
l:t "-'
s:: o .... 4J U Q) ....
lf.I Q) Cl
.... Cd .... 'tI
~
0
.. 2
-4
e
• R~fe:r;ence. [47J[ 48J
o Cubi~Function
D Linear Functions
Fixed Edge J Position of Load ~ •
~ "l>' 7·t 2b 25 30
Number of Equal Divisions Along Meridian
-6 J \ " 1 ::j
Figure 28 0 Radial Deflections in Spherical Shell Under Rin~ Load
e
o
a 1t"I
o .....
Elevation
Plan
Figure 29, Cylindrical Arch Dam
/ .\ Actual Foundation Line
e
-il 3 m~
~/7,
Section
5 2 E = 2 x 10 Kg./cm.
'lJ= 0.15
• a o M
"'-J 00
30 \.
25 i\~
20 "
79
Downstream Face Vertical 15 Stresses in Central Cantilever
Upstream Face Vertical Stresses 25 in Central Cantilever
20
1 15
1 \ 10
10
- Reference [49J
- - - Reference [49J (U.S.BoRo) 0
v
Cubic Function
Linear Functions
Center Line Horizontal Deflection (mm.)
o ~~ __ ~ __ ~ __ __ Figure 30 9 Stresses and Deflections in Arch Dam
1 .. Sc (
lo35c Dlc .lc
Plan
0 ,d
• \0 O· " \Q
,d .. C"'l
" 00 "
-1 ,do 1.-\0 00 0\--
a
Section-AA
o 11) N
D
80
0 0 • ....
Temperature Profile
0.~----------------------~-----------------------1 Stresses
=-
-o.
o
Stresses
Stresses
- Reference [51J
--- Reference [52J
____ Triangular Elements
1 =o.~~--~--~----~--~ ____ ~ __ ~ ____ ~ __ ~ ____ ~ __ ~ Figure 31 p Stresses in Heated Plate
co E-t d co ....... ~
+
L Fixed Edge
1.
0 0 8
Oc
0.4
0.2
o
81
ldealization of Shell
Radius = 20 Thickness
Length Radius = 3
Nodal Deflections
--Reference [6J • Cubic Function • Linear Functions •
Longitudinal Moments
x/2L
o 1 0 0 °r 4 =l.O~--~--~----~--~--~----~--~--~----~---Figure Uniformly Heated Cylinder
o
!i
0
s:: .... .........
s:: .... ..
,0 ..... ':1:.'-<
o
10 psi Fixed dge
-+ ________ ~~~x
82
-ë N
- R~fe~ençe .[6J
• Cubic Function
• Linear Functions
For Idea1ization of Shel1 (See Figure 32)
Ex = 16 x 106 psi ..,60 Ey = 106 psi
Vxy = 0 0 667
Vyx = 0 0 042
G =: 1 0 7143 x 106 psi
o
x..,axis
20 Figure 33 9 Orthotropic Cylindrical S~ell
15
83
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH
The method of deflection and stress analysis
presented in this thesis has been proven to be applicable
to plates and shells of uniform or non=uniform thickness,
isotropie or orthotropic material properties, including
thermal effects o The preceding numerical examples
indicate the wide variety of thin plate and shell
problems that can be solved o The results compare
favourably with both experimental and theoretical
resultso One of the great advantages of the finite
element method is that the governing differeD~ial
equations need not be known, end complex forms are
handled in the same manner as simpler shapes o
The procedure 9 which is thoroughly tested, is
provided in the form of a computer program for designers
to use o The input data for this program is easy to set
up and supplementary programs are provided to generate
this data. The output is in the form of a detailed
deflection and stress analysis of a plate or shell
structure 0
The use of the finite element method for non-linear
theory and problems of stability and vibrations is of
interest to researcho The stability and vibrations
investigation of shells should be approached with a
precision element. Precision can be of two types~ one
ia to take the shape of an element auch that it fits in
as close to the CO~Lèct geometry as possible and the
other to use such a displacement field that higher orders of conformability May be achieved.
84
Much effort is being put into both these refinements by researchers. Complete success with a curved element for a general shell has not yet been achieved. ln reference [27J a quadratic polynomial was fitted to approximate the curvatures of the element but it was abandoned without testing any examples. Since it will take little effort to expand this program, it should be tried again especially from the stress analysis point of viewo The doubly=curved element developed by Gallagher [53J is suitable for shells of translation onlyo The element seems to be refined and the results obtained by him even with a coarse mesh are accurate enough compared with the curved element of Wempner et al [55J. This was pointed out in reference [56J. lt would be desirable to use such elements in order ta obtain the eigen-values accurately for stability and vibration analysis of sheils. If flat elements 9
which are satisfactory for the static stress analysis of shells are used for eigen=value problems 9 difficulties would arise because of the very large size of the matrices required for sufficient accuracy.
85
REFERENCES
10 Lamé, Go» Clapeyron» Eo, "Mémoire sur l'Equilibre Interieur4es Corps Solides Homogènes"p Memoire Présenté h 18Académie des Sciences de l'Institut de Franceg Second Series 9 Vol o 49 1828 0
20 Aron 9 Hog "Das Gleichgewicht und die Bewegung Einer Umendlich Dunnen» Beliebig Gekrummteng Elastischen Schale"» Journal for Reine and Angeo Matho 9 18740
3. Love» AoEoH0 9 "A Treatise on the Mathematical Theory of ElasticitY"g Dover Publications, New York (4th Edo 1926)0
40 Go18 denveizer~l AoLo »"Theory of Elastic Thin Shells", Pergamon Press~ New York 19610
50 Flügge g Wo 9 "Stresses in Shella"» 3rd Printing, Sprin~er' Verlag g New York 19600
60 Kraus 9 Hop "Thin Elastic Shells"9 Wiley 1967 0 70 'Billington p DoPog "Thin Shell Concrete Structures"»
McGrawoHill 1965 0
80 Lundgren p Hop "Cylindrical Shells" Volo 19 The Danish Technical Press p The Institution of Danish Civil Engineers» Copenhagen 19600
90 Chronowicz g Ao p "The Design of Shells» A Practical Approach"p Crosby Lockwood and Sons 1960.
100 Reissner g Eo g "Stresses and Small Displacements of Shallow Spherical Shells"» Jo Matho Physicso 25 9 pp080 0 85 9 19460
110 Anon og "Design of Cylindrical Concrete Shell Roofs"9 AoSoCoEo Manual of Engineering Practice p Noo 31 9 1952 0
86
12. Anon., "Design of Circular Dames", Portland Cement
Association Publication, Chicago, Illinois.
13. Anon., "Elementary Analysis of Hyperbolic Paraboloid
Shells", Portland Cement Association Publication,
Chicago, Illinois, 19600
140 Southwell, RoVo, "Relaxation Methods in Theoretical
Physics"$! Clarendon Press 9 Oxford 1946 0
15. Harris p P.Jo, "The Analysis ofAxially Symmetric
Spherical Shells by Means of Finite Differences"$!
PhoDo thesis, McGill UniversitY$! July 19640
160 Kalnins, A09 "Analysis of Shells of Revolution
Subjected to Symmetrical and Non=Symmetrical Loads",
Journal of Applied Mechanics, Sept o 1964 0
17 0 .. Bud1ansky l) B 0 and Radkowski, PoP 0 9 "Numerical Analysis
of Unsymmetrical Bending of Shells of Revolution",
AoloSoS~l) Volo 1 Noo 8, August 1963 0
18 0 Krausl) Ho 9 liA Review and Evaluation of Computer
Programs for the Analysis of Stresses in Pressure
Vesselsn 9 Welding Research Council No o 108, pp oll=28,
1965 0
19 0 Soare, Mo s " Application of Finite Difference Equations
to Shell Analysis"9 Pergamon Press 19670
20 0 Turner, MoJol) Cloughl) Ro~o$! Martin, HoC., Topp, LaJo,
"Stiffness and Deflection Analysi~ of Complex Structures",
Jo Aero. Science, Sept. 19560
210 Argyris 9 J.Ho and KelseY$l SOl) "Energy Theorems and
Structural Analysis"l) Butterworths Publishers, London
1961 0
87
22. Wilson, Ea: "Finite Element Analysis of Two Dimensional Structures"g Ph.D o Thesis~ University of California p Berkeley~ 1963 0
23. Gallagherg R.Hai nA Correlation Study of Methods of Matrix Structural Analysis"~ Pergamon Press p Oxford 1964.
24. Grafton g PoE.~ Strome, DoR.p "Analysis ofAxi=symmetrical Shells by the Direct Stiffness Methodng AoIoAoA. Journal Vol. 19 2347~ 19630
25. Clough9 RoWap and Johnson p CoPov "A Finite Element Approximation for the Analysis of Thin Shells"p Into Jo Solids Structures~ Vol. 4 g Pergamon Press p 1968.
26. Cloughp RoWo p and Tocherp JoLo g "Analysis of Thin Arch Dams by the Finite Element Method"v Proc. Sympa on Theory of Arch Dams 9 Southampton~ 1964 g Pergamon Press g
1965.
27. Utku p Senol p "Stiffness Matrices for Thin Triangular Elements of Non~zero Gaussian Curvature"p ADIDADAo Journal p Vol D 5~ Noo 9 9 Sept. 1967.
28. Zienkiewiczg O.C og and Cheung g YoK0 9 nThe Finite Element Method in Structural and Continuum Mechanicsnp McGraw=Hill g 19670
29. Mehrotra 9 BDL. g Mufti g A.AD9 and Redwood g RoG.~ "A Finite Element Analysis for Three Dimensional Flat Plate Structures and Shells"9 Approved for PU91ication in AoSoCoEo Journal~ Structural Divisiono
300 Mehrotra g BoLog Mufti 9 AoA0 9 and Redwood 9 ROG Dp "A Program for the Analysis of Three Dimensional Plate Structures"g Structural Mechanics Series Noo 5 p
88
Dept. of Civil EngineeriDg and Applied Mechanics~
McGill University, Sept. 1968·.
31. Mehrotra, B.L., and MUtti,A.A., "A Modified Tocherls
Function for the Analysis of Tbree Dimensional
Plate and Sbell Structures", Structural Mechanics
Series No. 8, Dept. of Civil Engineering and Applied
Mechanics, McGill University, Nov. 19680.
32. Mufti, AoA., and Harris, P.J.~ "Matrix Analysis of
Shells by Finite Eléments", Paper prese~ at Second
Canadian Congress of Applied Mechanics at Waterloo
University~May 1969 0 (Also to be published in Trans.
E.I.Co)
33 0 Melosh, R.J o, "A Flat Triangular Shell Element Stiffness
Matrix", Airforce Conference on Matrix Methode in
Structural Mechanics at Wrigbt-Patt.rson Airforcè Base,
Dayton, Ohio, Oct. 26m 28, 1965 0
34 0 Utku, So~ and Melosh, R.Jo, "Behaviaùrof Triangular
Shell Element Stiffness Matrices Associated with
Polyhedral Deflection Distributions",. A.I.AoAo 5th
Aerospace Sciences Meeting, New York, Jan 0 1967 0
350 Mufti, AoAo, and Mehrotra, B.L. ~ "Development and
Evaluation of Matrices for Triangular Element .
Characteristics Using Linear- Ful'lcti-ons"$) Dept. of
Civil Engineering and Applied' 'Mechanies, Structural
Mechanics Series No o 9, McGill Univers·itY9 Deco 19680
36. Tocher$) J.L., "Analysis of Plate Bending Using
Triangular Elements"~' PhoD. Tbesis, University of
California, Berkeley, 1963.
·e·
89
37. Przemieniecki~ J oSo» "Theory of Matrix Structural Analysis"~ McGraw-Hill, 1968.
380 ·Lekhnitskii, S.G.~ "Theory of Elasticity of an Anisotropic Elastic Body"~ Holden~DaY9 Inc. 9 San Fransiscos 1963 0
39. Timoshenko~ S.P. SI and Goodier p JoNo~ "Theory of Elasticity"p 2nd. Ed. McGraw-Hill~ 19510
400 Timoshenko p SoP. ~ and Woinowsk.y-Krieger~ So p "Theory of Plates and Shells"~ 2nd o Ed. McGra~Hill~ 1959.
41. Marguerre p Ko SI "Zur Theorie der Gekrummten Platte Grosser Formanderung"~ Fifth Int o Congress for Applied Mechanics~ edited by J.P.Do Hartog~ John Wiley. and Sons Inc.~ Vol. 5, New York~ 1939.
42.0 .. J.&e38r g LoGo ~ "Cartesian T~nsors in EllJineering Science" ~ Pergamon Press, 1966.
43. Archer p Gog "The Analysis ofAxially Symmetric Spherical Shells with a Central Circular Opening"g Mo Engo thesis, McGill UniversitY9 Oct. 19650
440 Melosh 9 RoJ op "Basis for the Derivation of Matrices for the Direct StiffnelJs Method"p AoIoAoAo Journal
1
Vol o 79 July 1963 0
45 0 Dunne 9 PoCo, "Complete Polynomial Displacement Fields for Finite Element Method ", The Aeronautica1 Journal of the Royal SocietY9 Vol. 72, No. 687, March 1968.
46 0 Mufti 9 AoAo and MehrotJ;'a~ BoLo Sl Discussion of "Analysis of Plate Bending byTriangular Elements", byWoYoJo Shieh, Seng""lip Lee and RoAo Parmellee, Journal of the EH Division Proco AoSoCoEo, Volo 95 p April 1.969'0
90
l~7 0 Harris p Po J 0 p "~rical Analysis ofAxially Symmetric Spherical Shells", Trans. EoloCo, Vol. 10, No. A~l, Febo 19670
480 Harris\) P .. Jo p "Stresses in a Spherical Shell under Narrow Ring Loads", Trans o EoIoCo Volo·ll p Noo A-4 p Novo 19680
490 Zienkiewicz p OoCo p and Cheung p YoKo p "Finite Element Method of Analysis for Arch Dam Shells and Comparison with Finite Difference Procedures", Proco of Symposium on Theory of Arch Dams, Southampton Il 1964 11 Pergamon Press\) 19650
500 Private communication between Professor PoJo Harris and Professor OoCo Zienkiewiczo
510 Gallagher, RoHo p et al, "Stress Analysis of Heated Complex Shapes"p ARS Journal ll MaYII 1962.
52 0 Mendelson p Ao\) and Hirschberg, Mo p "Analysis of Elastic Thermal Stresses in Thin Plate with Span=wise and Chord=wise Variations of Temperature and Thickness"S) NâCA TN 3778\) Nov 0 19560
530 Gallagher p RoHoS) "The Development and Evaluation of Matrix Methods for Thin Shell Structural Analysis lt p PhoDo thesisS) State University of New York at Buffalo, 196.6 0
54 0 Clough,l RoW0 9 and Tocher ,l JoLo lI "Finite Element Stiffness Matrices for Analysis in Plate Bending"l1 Matrix Methods in Structural Mecbanics ,l Proc o Conf 0 held at Wright=Patterson Air Force BaseS) Ohio ,l 19650
91
55. Wempner, G.A.~ Oden, J.T. and Kross, D.A., "Finite Element Analysis of Thin Shells", J.
EH Division Proc. AoS.C.E.~ Vol. 94~ Dec. 1968. 56. Mebrotra~ BoLo and Mufti, A.A., Discussion of
"Finite Element Analysis of Thin Shells", J.
EH Division, Proc. A.SoCoEo, Vol. 95, Augo 1969.
92
APPENDIX 1
Element Characteristics and Equivalent Nodal Forces
z y
x
Figure 34. Forces and Deformations
Let F ~[F1 1:2 F3]
where Fi = [Ui Vi· Wi '\i ~i MZi] (i-1,2,3)
Positive directions of Xi and ~ are identical. In order to find the staticaîîy equivalent nodal
forces corresponding to the boundary stresses, the principle of virtual work may be applied.
For a virtual displacement 0* at the nodes, the external work done by the nodal forces is [o*J fF] and the internai work done by stresses is f [e*Jr~]d(VOl.):
vol. L Equating these two work quantitiesr
[ô*J J [e*] ! <1] d(vo1.). volo
=
93
Since [e*] = [BJ f ô* 1 and [e*J = [ ô* 1 T [BJT = [6*J [ 8JT
[ô*J f F J = [ô*J J [BJT [<1 l d(vol.)
vol.
This is valid for all values of [ô*J and therefore
= J d(vol.) vol.
Therefore
Hence
K = ~
J !T E A d(vol.) vol o
FO = - J ~T,B ~o d(vol.) vol.
Where ,!S is the stiffness matrix and 1:0 the thermal load
vector
94
APPENDIX 2
-Sb and ~ Matrices
Sb = 0 0 0 -2 0 0 -6x -2Py 0
0 0 0 0 0 -2 0 -2Qx -6y
0 0 0 0 2 0 0 4 (Px+Qy) 0
~= hll T Sb Eb Sb dA
Let
°x = E t 3
D = E t 3 x , ~
12(1- Vxy Vyx) y
12(1- Vxy Vyx>
,
K64 :::1 2°1 x2 Y3
-K74 = 20 x x2 Y3 (x2+ x3)
- (x2+ x3)] K84 = l x2 Y3 [p °x Y3+ Q Dl 3
- y~ K94 = 2°1 x2
KSS = 2Dxy x2 Y3
95
KS5 =
-K76 = 2Dl Y3 x2 (x2+ x3)
-K86 = ~ x2 Y3 [p Dl Y3+ Q Dy (x2+ x3)]
2 = ï Dl x2 Y3 (x2+ 2x3)
= X~Y3[r2 Dx Y~+ P Q D1Y3(x2+2x3}+ Q2Dy(X2+X2X3+X~} + 4 DXy[p2(x~+ x2x3+ xj}+ Q2yj+ P Q Y3(x2+ 2X3}J]
2 = x2Y3 [2P Dl Y3+ Q Dy (x2+ 2x3) ] -y-
AlI the elernentsof Eb not given above are equal to
zero 9 also Kijb = Kjib'
APPENDIX 3
Eguilibrium Algorithm
~~--------------~~
Q t ~L/2
Figure 35. Eguilibrium Algorithm for Beam
96
Figure 35 shows the distribution of the unbalanced
bending moment caused by a unit transverse nodal
deflection in a clamped beam. Note that the unbalanced
moment caused by the end shears is being equally
resisted by the nodes and the nodal resistive bending
moment vectors are perpendicular to the beam axis.
x ..
Figure 36. Eguilibrium Algorithm for Triangular Element
97
Figure 36 shows the extension of this concept to triangles. The unbalanced moment, after a sign change is equally distributed between the node that is moved and the other two.
A typical example is shown in Figure 36.
Hence for node 1 the moments are
This can be written concisely as [27J
IIlxij = ... ~(l-Ôij){fijYij)+ Ôij(fi+l,jYi+l,j+ f i +2 ,jYi+2,j)]
(6.k)
where subscripts i,j May take integer values of module e 9
ego. i + 1 is 1 when i = 3 and Ôij is the Kronecker delta. x ij= XiœXj eg. x32 = x3- x2' f ij here is defined on page 29 as an element of !lls'
(6.1.)
98
once [~lJs and [K31Js are obtained via (6.k), applying the same equilibrium algorithm to [K2lJ; and [K3lJ; the total shearing stiffness matrix is obtained.
w
= t 4A
1 1 ___ L __ --1 __ _ {6.m} 1 1
---~--~--1 1
The stiffness matrix thus obtained may be rearranged to suit the arrangement of the displacement,~.
APPENDIX 4
Computer Program
The computer program given in this thesis was
coded in the FORTRAN IV Level G language. It has been
run at the McGill Computing Center on the IBM 360/75.
99
The program is restricted to the analysis of thin plate
and shell structures but thermal loads, non-homogeneous
and orthotropic material properties and arbitrary
boundaries are possible. It requires 3l8K bytes p and 6
disks for a max~ number of 500 elements, 300 nodes,
20 partitions of a structure, 4 loading conditions, ~O
prescribed boundaries and 10 different elastic ând thêrmal
properties.
The flow chart for the program is shown in Figure
37 and the main sequences are described below.
* Read and print input datai
A structure is idealized by means of triangular
elements meeting at the nodes. These elements and nodes
are numbered as illustrated in Figure 38. A structure
should be partitioned so as to have a consecutive nodal
numbering system. The tot~l number of nodes per par~ition
should not be more than 18. Only.oneparti~ion line,
which need not be a straight line~ should pass through an
element.
* Input data refers to global axes.
100
Start
! 1 Number of problemS!
l Read and printj input data
~ 1 Number of partition~
~ r-1 First element,last element
l Elemènt characteristics Transformation matrix Thermal load vector Stiffness matrix Stress and strain matrices
J Transform thermal load vector and stiffnessmatrix of an element to global axes to form overall thermal load vector overall stiffness matrix
1 J.
lnsert boundary conditions 1 •
Solve equilibrium equations to obtain Global displacements
FG_ ~G ôG Print residuals R =
t Transform global displacements to Local displacements Compute stresses and strains Print output
1
'" 1 End 1
Figure 37, Flow Chart
101
z ,/" Partition 2
--- -
y
CD
Parallel Circle . Element Number'Î'
i= 2 \V x
j= 3 k= 8
Figure 38. ldealization of Shell
102
The following sequence of punched cards numerically
defines a structure.
Cols Notation
A Problem 1 ... 4 NPROB
B S t!:".lcture Properties 1-4 NPART 5-8 NPOIN 9 .. 12 NELEM 13-16 NBOUN
17 ... 20 NCOLN 21-24 NFREE 25-28 NCONC 29-32 NYM
C Nodal Point Array 1-14 X(I,l) 15-28 X(I,2) 29-42 X(I,3)
D Element Array
1 ... 4 NOD(I,l) 5-8 NOD(I,2) 9 ... 12 NOD(I,3) 13 ... 28 THlCK{I) 29 ... 44 R(l)
45..,52 TEMPAV(I) 53 ... 60 TEMPDF(I)
61-64 NEP(I)
E Boundary Array
1 ... 4 NF(I)
5 ... 8 NB(I~l)
9-12 NB(I,2)
Description
(14) One card Number of problems to be solved
(914,2F16.S) One card Total number of partitions Total number of nodal points Total number of elements Total number of points with prescribed displacements Total number of different loadings Degrees of freedom Total number of concentrated loads Total number of different elastic properties
(5F14.6) One card per nodal point X-ordinate Y-ordinate Z-ordinate
(314,2F16oS,2FSo3,14) One card per e1ement
Nodal point number i of an element Nodal point number j of an element Nodal point number k of an element Thickness of element o for clockwise counting of nodes of an element 1 for counter clockwise counting of nodes of an element Average temperature of an element Temperature difference between top and bottom surfaces of an element Number of elastic properties relative to an element
(714,6F7.4) One card per boundary _ point
Number of a nodal point with prescribed displacement o for translation not allowed in x-direction 1 for translation allowed in xdirection o for translation not allowed in y-direction 1 for translation a1lowed in y ... direction
Cols Notation
13-16 NB(I,3)
17-20 NB(I,4)
2l~24 NB(I,5)
25.28 NB(I,6)
29 .. 35 BV(I,I)
36 .. 42 BV(I,2)
43-49 BV(I,3)
50-56 BV(1,4)
57 .. 63 BV(1,5)
64 ... 70 BV(I,6)
F Partition Array 1 ... 4 NSTART(l)
5=8 NEND(I)
9 ... 12 NFlRST(I)
13 ... 16 NLAST(l)
G Elastic and Thermal
1-14 El(l) 15",,28 E2(1) 29=42 GE(l) 43=46 PO(I) 47 ... 50 P2(l) 51=60 ALPHX(l)
61=70 ALPHY(l)
103
Description
o for translation not allowed in z-direction l for translation allowed in zdirection o for rotation not allowed in x-direction 1 for rotation allowed in x-direction o for rotation not allowed in y~direction 1 for rotation allowed in y-direction o for rotation not allowed in z-direction 1 for rotation allowed in z-direction Magnitude of prescribed translation in x-direction Magnitude of prescribed translation in y-direction Magnitude of prescribed translation in z-direction Magnitude of prescribed rotation in x-direction Magnitude of prescribed rotation in y-direction Magnitude of prescribed rotation in z-direction
(914,2F16.8) One card per partition Number pertain~ng to first element in a particular partition Number pertaining to last element in a particular partition Number pertaining to first nodal point in a particular partition Number pertaining to last nodal point in a particular partition
Properties One card per elastic . property
(3F14.3,2F4.3,2flOo8) Modulus of elasticity in x-direction Modulus of elasticity in y-direction Shear modulus Gxy Poisson's ratio in x-direction Poisson's ratio in y-direction Coefficient of thermal expansion in x-direction Coefficient of thermal expansion in y-direction
Cols Notation
H Load Array
1 ... 4 K 5-15 (6*K-5,1) 16-26 (6*K-4,1) 27-37 (6*K-3,1) 38-48 (6*K-2,1) 49",,59 (6*K-l~1) 60-70 (6*K,1)
104
Description
(14,6Fll.4) One card per loaded point
Loaded nodal point Load in x-direction* Load in y-direction Load in z-direction Moment in x-direction Moment in y-direction Moment in z-direction
*Vectors in positive directions of axes are considered as positive.
1
105
Transformation matrices
The transformation matrix for each element is
calculated from the global coordinates of the nodes
of an element. lt is used to transform the global
coordinates, stiffness matrix and thermal load vector
to the local axes o Transformation matrices are stored
on a disk for subsequent use. lt may be noted that the
line joining the first two nodes i and j of an element
(see Figure 38) forms the local x-axis which ~st be ~ G G the plane parallel to the global xy plane i.e o Zi- Zj= O.
Stiffness. strain and stress matrices
For each element, stiffness, strain and stress
matrices are calculated in the local coordinates of which
the strain and stress matrices are stored on a disko
Thermal loads
Thermal loads are calculated for each element with
respect to local axes o
Formation of overall (global) thermal load vector
and stiffness matrix and introduction of boundary
conditions
The thermal load vector and ov~~~ll ~~if~n~s~ .. ~tri~
are formed by adding inc;1i,,~4u~l _~h~~.l. J~oB:d ~nd stiffness
matrices of the elements after transformationo The
formation of the overall stiffness matrix, because of
106
its large size, is done in the tridiagona1ization manner
by means of appropriate partitioning of a structure. The
boundary conditions are substituted as exp1ained previous1y.
The modified load vector (FG_ F~) and the stiffness matrix
are stored on disk in the order of the partition number.
Solution of egui1ibrium eguations
The equi1ibrium equations are solved by
tridiagona1ization togive disp1acements with respect
to global coordinates.
- To check the errors introduced in the solution of
the equations due to round-off and truncation in the
arithmetic operations the residua1s are ca1cu1ated by
the subtraction from the global load vector of the
product of the overal1 stiffness matrix and the global
disp1acement vector. These residua1s are compared with
the overa11 load vector, and may be used again as
applied loads if better accuracy is desired.
Ca1cu1ation of strains and stresses
The global disp1acements are transformed back to
local axes and the strains and stresses are ca1cu1ated - -
at the centroid of each e1ement.
107
Output information
The following output is printed with appropriate titles.
A. Node number and its displacements with respect to glo1;>al axes (lH. ,14,6E16.8)
B. Element number and corresponding plane stresses and bending- stresses- computed at the centroid of- the elerp,nt_ (lH ,14,6E16.8)
c. Element number, node number and the displacements with.respect to local axes (lH ,214,6E16.8)
D. Element number, node number, coordinatès of the centroid of the element, top and bottom stresses, principal stre~se~ and principal angle ._ ('0',14,217,16,F12.4,FIO.4,5E12.5,F7.2)
E. Element number, node number, coordinates of the centroid of the element, top and bottom strains (Format same as D)
Timing
Typical running t imes
10 18 nodes 20 elements l partition 2.26 20 55 nodes 80 elements 9 partitions 4.20
minutes
minutes 3. 86 nodes 134 elements 13 partitions 6~11 minutes
•
C C
C C C
10
30 3S
110 III
40
45
120 121
50
MA 1 N PROGRAM REAL KI DIMENSION X(300,3',XE(3,3),NF(60),NBC60,6),BVC60,61,NOO(500,3',
1 THICK(5001,NSTARTC20J,NENOC20J,NFIRSTC20J,NlASTC20),H(3,3), 1XEI(3,3),TC18,18J,P(18,18t,R(500),XEOC1,3),EC2,2)
COMMON ST(108,216), 1 K1CI8,18',C(9,9J,OBA(6,6J,DBC6,6J,A(9,9J,B(3,6J, 1 QIC(3,9J,OQIC(3,9),BA(3,6'fOQ2C(6,18I,DQ3C(6,18),U(1800,4), 1 oZTC3,4J,oZB(3,4J,OSZT(3,4"OSZB(3,4),A3C6,6),A2C6,6) 1,El(10',E2(10"POClOJ,P2C10),GEClOI,NEP(SOOJ,AlPHX(10),AlPHYC101 1,TEMPAV(500),TEMPOFCSOO),TFORCEC18,2"TlOAOCI08,1)
CAll PGMCHK
REAC CS,10) NPROB FORMAT (914,2FI6.8) DO 20 LA = 1,NPROB REWIND 12 REWIND 1 REWINO 4
READING AND PRINTING CF DATA
, .
108
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
'RHS RHS RHS RHS
REAC (5,10) NPART,NPOIN,NElEM,NBOUN,NCOlN,NFREE,NCONC RHS 1,NYM RHS
WRITE (6,10' NPART,NP(IN,NElEM,~BOUN,~COlN,NFREE,NCONC RHS I,NYM RHS
DO 30 1 = 1,NPCIN RHS READ (5,35' XCI,1),X'I,2"XCI,3' RHS WRITE (6,35' X'I,1I,XII,2),X(I,3' RHS FORMAT '5F14.6) RHS READ (5,10) NCARo RHS IF 'NCARo-NPOIN' 110,111,110 RHS STOP RHS CONTINUE RHS DO 40 l=l,NELEM RHS READ (5,45' (NOD' I,J)'J=1,3t, THICKU hRCI) RHS
I,TEMPAV'I),TEMPDFCI),NEP'I) RHS WRITE '6,45' (NOoU,J),J=I,3),THICKU),R(1) RHS
1,TEMPAV(I),TEMPDF(I',NEP'I. RHS FORMAT (314,2F16.8,2F8.3,14' RHS READ (5,10' NCARo RHS
IF CNCARD-NElEM) 120,121,120 RHS STOP RHS CONTINUE RHS DO 50 I=l,NBOUN RHS READ 'S,46' NF' IJ,NB(I,1),NBCI,2),NB'I,3J,NB(I,4),NB(I,5),NBCI,6),RHS
1BVCI,1),BV(I,2),BV(I,3J,BVCI,4),BV'I,SI,BV(I,6) RHS WRITE (6,46)NFCI),NB(I,l"NBCI,2),NBCI,3),NB'I,4J,NBCI,5),NBII,6"RHS
1 B V CI, 1) ,BV ( l ,2) , B V ( l , 3) , B VII, 4) ,BV 1 1., S J ,B VI 1,6) RHS
1 2 3 4 S 6 1 8 9
10 Il 12 13 14 15 16 11 18 19 20 21 22 23 24 2S 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50
109
"6 FORMAT (1I4,6F7.4J RHS 51 00 60 l=l,NPART RHS 52 REAO (5,10' NSTART(I),NENO(I"NfIRSTCI.,NLASTCI. RHS 53
60 WRITE (6,10. NSTARTII.,NENOCI"NFIRSTIII.NLAST(I' RHS 54 DO 64 Is~,NYM RHS 55 REAO C5,99IEl(IJ,E2(I.,GECII,POIIJ,P2CI"ALPHXCIJ,ALPHYIIJ RHS 56
64 WRI TE (6,99 lEU IJ.E 21 J) ,GE 1 Il, POC 1), P2C 1 J .ALPHXCI hAlPHYC 1) RHS 51 99 FORMAT C3F14.3,2F".3,2FlO.8. ' RHS 58 6 NPOIN6 = NPOIN*6 RHS 59
DO 65 J:l,NCOLN RHS 60 00 68 1 :: 1,NPOIN6 RHS 61
68 ue l ,J) = o. RHS 62 IF (NCONe, 66,65,66 RHS 63
t6 DO 69 l=l,NCONC RHS 64 REAO' (5,34' K,UC6*K-5,U ,UC6*K-4,l"UC6*K-3,1"UC6*K-2,U, RHS 65
1U(6*K-l,1),UC6*K,1. RHS 66 69 ~R 1 TE (6',33' K ,UI 6*K-5, U ,UC6*K-It,lI.UC 6*K-3, 1) ,Uc'6*K-2,1' , RHS 61
1U(6*K-1,11,U(6*K,I' RHS 68 65 CONTINUE' RHS 69 34 FORMAT CI4,6Fll.". RHS 1U 33 FORMAT CI4,6E16.8J RHS 71
C RHS 72 C FORMATION OF MATRICES RHS 13
INTER = 0 RHS 14 DO 10 11= 1,NPART RHS 15 DO 15 J = 1,216' RHS 16 00 15 1 :'1,108 RHS 17
15 STCI,J. = o. RHS 18 00 2000 I~I, 108 RHS 79
2000 TlOAOCI,ll=O. RHS 80 NST':: NSTARTCII' RHS 81 N,EN :: NENoU 1 ) RHS 82 K2= NFIRSTCIl' RHS 83 L = NLASTCl I) RHS 84
MINUS = K2-1 RHS 85 00 80 lK ='NST,NEN RHS 86 MM = LK-INTER RHS 81 00 85 1 :: 1,3 RHS 88 JJ :: NOoCLK,II RHS 89 XECI,l) :: XeJJ,l) RHS 90 XECI,2J = XCJJ,21 RHS 91
85 XEe!,]) = X(JJ,3. RHS 92 TH = THICK eLK' RHS 93 TEMP1=TEMPAVILKJ RHS 94 TÈMP2=TEMPOFCLKJ RHS 95 J=NEPCLK' RHS 96 YMl=E1eJ' RHS 91 YM2=E2eJ) RHS 98 PR1=PO(J' RHS 99 PR2=P2(J) RHS 100
SO-59.tt
c c
c C C
G=GEeJ' ALPHA X=ALPHX (J' AlPHAY=ALPHY (J'
DEVELOPMENT OF COORDINATE TRANSFORMATION MATRIX
RHS RHS RHS RHS RHS
Al= (XE(2,2'-XE(1,2".(XE(3,3'-XEC1,3)'-(XE(3,2)-XE(1,2))*(XE(2,3'RHS 1 -XEU,3))
Bl=-CXEC2,1'-XEll,1".(XE(],3)-XE(1,3" 1 +(XE(3,1'-XE(1,1,,*(XEI2,3'-XE(1,3') Cl = (XEI2,1'-XEIl,1')*(XE(3,2'-XE(1,2"
1 -(XE(3,l'-XE(1,1"*(XE(2,2'-XE(l,2" IF eAU Il,9,11
9 IF (BI' Il,12.11 Il Q = SQRT"(eA1.*Z' + (81**Z.'
Pl= SQRT ((A1**Z' + (B1*.2' + (C1**2') IF (R(LK') 83,84,83
83 H(l,l. = -BI/Q Hel,2) = Al/Q H(Z,l) =-Al*Cl/(P1*Q, H(Z,Z' = -B1*CI/CPl*Q, H(2,3) = (Al.*Z+Bl.*2'/CPl*Q, GO TO 86
84 H(l,l' =BI/Q H(l,Z' =-A1/Q H(2,1'=Al*C1/(Pl*Q, H(Z,Z)=Bl*Cl/(Pl*Q) H(Z,3'=-(Al·*Z+Bl·*2,/(P1*Q)
86 H(1,3' = O. H(3,!) = Al/Pl H(3,Z) = BI/Pl He3,3' = Cl/Pl
FORMATION OF ELEMENT COCRDTNATES W.R.T. LOCAL AXES
GO Ta 130 12 DO 14 1=1,3
DO 14 J=I,3 14 HU,J. = O.
IFCRCLK))184,183,184 184 H(1,1.=1.
H(Z,Z'=l. He3,3'=1. GO" Ta 130
183 H(l,l'=l. H(2,Z'=-1. He3,3)=-1.
130 XEO(l,l'=XE(l,l' XEO(1,2'=XEel,Z' XEOCl,3)=XE(1,31
13 DO 1 1=1,3
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS ~HS RHS RHS RHS RHS
110
101 102 103 104 105 lOé 101 lOS 109 llCl 111 112 113 114 115 116 117 Ils 119 120 121 1Z2 1Z3 124 125 126 lZ7 12S 12~ 130 131 132 133 134 13~ 13t 131 13e 13<3 14(] 141 142 143 144 145 l4E 147 l4S l4c.i l5(J
l.\ , j' # .... ~;) -~
111
XE( 1,1)=XE(I,1)-XEOI1,U RHS XECI,Z'=XECI,Z)-XEOC1,Z) RHS
1 XECI,3'=XE(I,3)-XEO(I,3J RHS DO Z 1= 1,3 RHS DO Z J=1,3 RHS XElfl,J'=O. RHS DO Z K=l,3 RHS
Z XEI(I,J'=XEICI,J.+HCJ,K'*XEfl,K) RHS DO 3 1=1,3 RHS DO 3 J=I,3 RHS
3 XE(I,J'=XEICI,JI RHS XZ=XECZ,l) RHS X3=XE(3,1J RHS Y3=XEC3,Z' RHS
C CALCULATION OF ELEMENT STIFFNESS MATRIX RHS C *****************************************************************RHS C ******************************************.***********************RHS
CALL FEMBCXE,YM1,YMZ,PR1,PRZ,G,TH,MMJ . RHS CALL FEMPCXE,YM1,YMZ,PRl,PRZ,G,TH,~M. RHS
C***********************************************************************RHS C FORMATION OF ROTATIO TRANSFORMATION MATRIX RHS C***********************************************************************RHS
DO 300 1*1,18 . RHS DO 300 J=1,18 RHS
300 T( l ,J )=0. RHS DO 301 K=1,16,3 RHS K3=K-l RHS DO 301 1=1,3 RHS DO 301 J=1,3 RHS If = I+K3 RHS JT = J+K3 RHS
301 TCIT,JT' = HCI,J' RHS IFCMM'214,214,21Z RHS
212 WRITECI2.C(TCI,JJ,J=1,18J,I=1,18. RHS Z14 CONTINUE RHS
C***********************************************************************RHS C FORMATION OF STIFFNESS MATRIX OF ELEMENT IN GLOBAL COORDINATES RHS C****************************~*********.********************************RHS
DO 302' 1=1,18 RHS DO 30Z J=1,18 RHS
302 P ( l ,J )=0. RHS DO 303 1=1,18 RHS DO 303 J=I,18 RHS DO 303 K=1,18 RHS
303 P(I,J'=PCI,JJ+KICI,K'*TCK,J' RHS DO 304 1-=1,18 RHS
A DO 304 J=1,18 RHS _ KlCI,J'=O. RHS
DO 304 K=1,18 RHS 304 K1(I,J)=K1CI,J'+TCK,I'*PCK,J' RHS
151 15Z 153 154 155 156 1.57 158 159 160 161 162 163 164 165 166 161 168 169 170 171 172 113 114 175 116 171 178 179 180 181 182 183 184 185 186 181 188 189 190 191 19Z 193 194 195 196 191 198 199 ZOO
C C C
, FORMATION OF OVERALL STRUCTURE STIFFNESS MATRIX
DO 100 1=1,18 DO 100 J=1,2
100 TFORCECI,JJ=O. DO 2001 1=1,2 DO 2001 J=1,2
2001 E(I,J'=O. TElA=!./( 1. -PR1*PR2) EC1,1'=YM1*TElA EC1,2J=PR1*YM2*TElA E(2,2,=YM2*TElA FACTl=0.5*TH*TEMP1*CEC1,1'*AlPHAX+E(1,2'*AlPHAY) FACT2=0.5*TH*TEMP1*CEC1,2'*AlPHAX+E(2,2)*ALPHAY' TFORCEC1,!J=FACTl*Y3 TfORCECZ,1,=-FACTZ*CX3-X2' TFORCEC1,lJ=-FACT1*Y3 TFORCEC8,1)=FACT2*X3 TFORCEC14,1.=-FACT2*X2 01=EC1,2.*CTH**3'/12. OZ=EC1,1'*CTH**3'/12. 03=E(Z,2'*CTH**3'/12. FACT3=0.5*TEMP2*C01*AlPHAX+03*AlPHAYJ/TH FACT4=0.5*TEMP2*C02*AlPHAX+Ol*AlPHAY'/TH TFORCEC4,lJ=-FACT3*CX2-X3' tFORCEC5,1)=FACT4*Y3 TfORC~CIO,1'=-FACT3*X3 TFORCEC11,1'=-FACT4*Y3 TFORCEC16i1J=FACT3*X2
15 FORMAT (1H ,9E9.3J DO 305 1 =1, 18 TFORC E( l ' ,2'=0. DO 305 K=I,18
305 TFORCECI,2.=TFORCECI,2'+TCK,I'*TFORCECK,11 C FORMATION OF OVERAll STRUCTURE STIFFNESS MATRIX
DO' 1000,1=1,3 IFCNoOClK~I'-K2' lOOO~1001,1001
1001 IF(NOOClK,I'~L) 1002,1002,1000 1002 Il=NFREE*CNOOClK,ll-K2'
12=NFREE* CI -1) ob 1003 13~1,NFREE 14=11+13 15=12+13
1003 TLOAOCI4,1'=TLOAOCI4,1'+TFORCECI5,2. 1000 CONTINUE
A Ofl 80 L"'=1,3 ,., DO 80 KK=1,3
IF CNODCLK,KK)-K2' 80,131,131 131 IF (NOOCLK,KK' - l' 132,132,80
112
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
t.,{-) '--" RHS .... .1-)' .RHS
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
201 202 203 204 205 206 201 208 209 210 211 212 213 214 215 216 211 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 231 238 239 240 241 242 243 244 245 246 247 248 249 250
132 M=NFREE*CNOO(LK,KK'-K2J N=NFREE*(NOOCLK,LL.-K2' I=NFREE*'KK - 1) J=NFREE*(LL - l' IF (N' 80,900,900
900 DO 5 NJ=l,NFREE 00 5 MI=I,NFREE MMI=M+MÎ NNJ=N+NJ IMI=I+MI JNJ=J+NJ
5 STCMMI,NNJ) = STCMMI,NNJJ + K1CIMI,JNJ) 80 CONTINUE
I5=NFREE*L I4=NFREE*MINUS + 1 13=0 DO 53 1=14,15 i3=13+1
53 U'I,I'=U(I,1'-TLOAoCI3,1' 16 FORMAT(6E14.5.
C * * * * * * * * * * * * * * * * * * * * * , C INTRODUCTION OF PRESCRIBED DISPLACEMENTS ·c
242 243
345
233 230 290
115
116 111
52
DO 290 l=l,NBOUN M=NFC 1) -K2 MM=NF(I'-l IF(M) 290,242,242 IF(M-17' 243,243,290 DO 230J=1,NFREE IFCNB(I,J') 230,345,230 NMI=NFREE*M+J .. STCNMI,NMI' :;: ST(NMI,NMI' •• lE+12 DO 233 JJ=I,NCOLN JNJ=NFREE*MM+J U'JNJ.JJ'= STCNMI,NMI,*SV(I,J' CONTINUE CONTINUE INTER :;:'NEN MI=NFREE*MINUS + 1 NJ=NFREE*l M=NJ-MI+l IF(II-NPART' 115,116,115 NA=NFREE*(NLAST(II+l' - MINUS' GO TO 111 NA=M+l N=NA-M MM=M+ 1 DO 51 1=I,M IFCST(I,I"51,52,51 ST(I,IJ=STCI,I'+0.lE+18
l '; "'. , ., . ',1
• *
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
113
251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 216 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
114
51 CONT 1 NUE -' ,RH-S· 301 C
RHS 302 10 WRITE(4J M,N,(ST(I,J),I=l,M),J=l,M),C(ST(I,JI,I=l,M),J=MM,NA), RHS 303 1 «(U(I ,J), I=MI ,NJ. ,J:z1 ,NCOLNI RHS 304 REWINO 1 RHS 305 REW INO 2 RHS 306 REWINO 3 RHS 301 REWINO 4 RHS 308 REW INO 12 RHS 309 C • • • • • • • • • • • • • • • • • • • • • • RHS 310 C SOLUTION OF TRIOIAGONAL MATRICES AND CALCULATION OF RESIOUALS RHS 311 C RHS 312 CALL SOlVE(NPART,NCOlNJ RHS 313 REWINO 3 RHS 314 C CAlCUlATION 'OF STRESSES RHS 315 CAlL STRESS(NPART,NFIRST,~LAST,NCOLN,NElEM,NOO,NFREE,NPOIN) RHS 316 20 CONTINUE RHS 311 STOP RHS 318 END RHS 319 SUBROUTINE FEMP(XE,YM1,YM2,PR1,PR2,G,TH,~M. RHS 320 C SUBROUTINE FOR FORMATION OF ELEMENT STIFFNESS AND STRESS MATRICES RHS 321 REAL K1 RHS' 322 DIMENSION 0(3,3),BTOBA(6,6),XE(3,3J,IXC3"IY(3. RHS 323 COMMON'ST(108,216', RHS 324 1 Kl(18,18"C(9,9"OBA(6,6"OB(6,6.,A(9,9.,BC3,6', RHS 325 lQ1C(3,9"OQIC(3,9"BA(3,6),OQ2CC6,18"OQ3C(6,18J,UCI800,41, RHS 326 1 0IT(3~4"0IB(3,4J,OSIT(3,4.,OSZBC3,4,,A3(6,6J,A2(6,6' RHS 321 00 20 J=I, 6 RHS 328 Da 21 !=1,3 RHS 329 B( l ,J '=0.
' ,RHS 330 OB( I,J)=O. "RHS 331 21 OBACI,J'=O. RHS 332 DA 20 1=1,6 RHS 333 A(I,J'=O. RHS 334 BTOBA (l, J '=0. RHS 335 20 C ( 1! J )=0. R HS 336 DO 22 J=I,3 RHS 331 00 22 1=1,3 RHS 33B 22 O(I,J'=O. RHS 339 ORX=(XEC1,1' ~ XEC2,l. ~ XEC],l)' •• 333333 RHS 340 ORY=(XEC1,2' ~ XE(2,2. ~ XE(3,2)J*.333333 RHS 341 00 5 1=1,3 RHS 342 XE'I,l) = XE(I,l'-ORX RHS 343 5 XE(I,2' ='XE(I,2'-ORY RHS 344 IXCl' = XE(2,2'-XEC3,2' RHS 345 IX(2.= XE(3,2)-XEC1,2' RHS 346 IX()': XE(1,2,-XE(2,2' RHS 341 IY(l) = XE(3,1)~XE(2,l' RHS 348 IY(2' = XEC1,1'-XE(3,1' RHS 349 ZY(3' = XE(2,1'-XE(!,1) RHS 350
ZK=xe(2,l.*XE(3,2. - XE(3,l.*XE(2,2. Z=3.*ZK A(l,l' = ZK/Z A(2,1'=ZX(I'/Z A(3,1'=IY(I'/Z A(4,2'=A(l,l. A(5,2J=AC2,1. A(6,2'=A(3,U A ( l , 3 • = ZK / Z A(2,3'=ZX(2'/Z A(3,3'=ZY(2./Z A(4,4'=A(1,3J A(5,41=AI2,3) A( 6,4'=AC 3, 3' AU,5'=ZK/Z A(2,5'=ZX(3)/Z A (3,5 )= ZV (3 J / Z A(4,6'=A(1,5J A(5,6)=A(2,5) A(6,6'=A(3,5' B(I,23=1. B(3,3'=1. S (3,5 '=1. B(2,6)=I.
75 OEN=Cl.-PRl*PR2' OCl,l)=YHl/OEN O€2il'=PRl*VM2/0EN OC1,2'=PR2*YM1/0EN 0(2,2'=YH2/0EN 0(3,3'=G
72 DO 30 J=1,6 DO 30 1 =1 ,3 DO 30 K=I,3
30 OSCI,J)=OBCI,J' + O(I,K'*SCK,J' 00 40 J=I,6 00 40 1=1,3 DO 40 K=1,6
40 OBA(I,J.=OBA(I,J. + OeCI,KJ*ACK,J) 00 44 1=1,3 M=O N=O 00 44 K-=1,18,6 K3=K-l M=N+l N=M+l K2=1 DO 44 J=M,N Jl=K2+K3 OQ2CII,Jl'=OSA(I,J.
44 K2=K2+1
;.-. RHS . , ~. <.) ... ; • •. _j . ./~tiS
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RH$ RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
115
351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
c c c
DO 41 1=1,3 DO 41 J=I,6
41 BACI,J'sO. 00 42 J=1,6 DO 42 1=1,3· DO 42 K=I,6
42 eA(I~J'=8ACI,J)+BCI,K'*ACK,J' DO 43 1=1,3 M=O N=O DO 43 K=1,18,6 K3=K-l M=N+l N=M+l K2=1 DO 43 J=M,N Jl=K2+K3 OQ3CCI,JlI=BACI,J'
43 K2=K2+1 * * * * * * * * * * * * * * * * * * * * * * STRESS MATRIX IS FORMED
IFCMM) 126,126~127 121 WRITEC1' (COQ2C(I,J),1=1,61,J=1,18),(COQ3C(I,J),1=1,6"J=I,18',
10RX,ORY
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
C COMPLETE STRESS AND STRAIN MATRICES WRITTEN ON THE OISC 1 SILE (3,18)
RHS RHS RHS RHS
C C C
126 CONTINUE VOL·= .5*TH*Z DO 50 J=I,6 00 50 1=1,6 DO 50 K=1,3
50 BToBACI,Jt=BToBAC I,J' + BCK,I'*OBACK,J'*VOL DO· 60J=1,6 00 60 1=1,6 00 60 K=1,6
60 C(I,JI= CCI,J' + A(K,I'*BTOBA(K,J'
STIFFNESS MATRIX C IS FORMEo
M=O N=O 00 2 K=1,13,6 N=M+l M=N+l K2=K-l i<6=1 DO 2 I=N,M IL=K6+K2 Nl=O Ml=O
RHS RHS RHS RHS RHS RHS
t,Il ~Œ\HS '- l.~ ... ..) ~ tts
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
116
401 402 403 404 405 406 407 408 409 410 411 412 4i3 414 415 416 411 418 419 420 421 422 423 424 425 426 421 428 429 430 431 432 433 434 435 436 431 438 439 440 441 442 443 444 445 446 441 448 449 450
C
DO 1 K3=l,13,6 K4=K3-1 Nl=Ml+l Ml=Nl+l K5=1 DO 1 J=Nl,M1 JL=K5+K4 KI CIL , J L , =C CI, J )
1 K5=K5+1 2 K6=K6+1
15 FORMAT C9El4.5' 3 FORMAT (6E16.5.
1
RETURN END SUBROUTINE FEMBeXE,YMl,YH2,PR1,PR2,G,TH,MM'
SUB ROUTINE FOR FORM~TION OF ELEMENT STIFFNESS REAL KI JIMENSION OC9,91,L(9),Me9"XE(3,3"Q1(3,9),EMC3,3' COMMON ST(108,216), l' Kl(18,l8J,C(9,~),OBA(6,6),OB(6,6),A(9,9"BC3,6', 1 QIC(3,9),OQICC3,9),BA(3,6),OQ2C(6,18"OQ3C(6,18',UC1800,4), 1 OlT(3,4),OlB(3,4J,OSlTC3,4),OSlBC'3,4J,A3C6,61,A2e6,6'
X2:XEC2,lJ X3=XE(3,1. Y3=XE (3,2) OEN=TH**3/(12.*(I.-PRl*PR2JJ 02=YM1*OEN 03=YM2*OEN 01=YM2*PR1*OEN 012=G*TH**3/12. DO 1 J=l,9 DO 1 1=1,9 OCI,JJ=O. AU,J' = O. CCI,JI=O. C ( 1,1 J= 1. C(2,4)=I. CC3,71=1. C(4,1)=1. C(4,2J=X2 C (5,4'= 1. C(5,S'=X2 CC6,7'=I. CC6,8'=X2 CC7,l,=1. C(7,2'=X3 C (7,3 I=Y3 CC8,4J=1. C(8,5'=X3 CC8,6)=Y3
RHS RHS RHS RHS RHS
,:'RMS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
117
451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 411
,472 473 474 475 476 417' 478 419 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 491r 498 499 500
C
C C C
C CT C
C C C
CC9,1)=1. CC9,8'=X3 C(9,9,=Y3 CALL MINV CC,9,O,L,M'
STRAIN $ STRESS MATRIX IN FEMB ORX=(XEC1,1,+XE(2,l'+XE(3,1»)*.333333 ORY=(XEC1,2'+XEC2,2'+XE(3,2)~.333333
DO 9 1=1,3 DO 9 J=1,9 OlCCI,J.=O.
9 0 li l , J , =0. 01(1,8)=1. 01(2,6)=-1. 01(3,5'=1. 01(3,9'=-1. DO 10 J=1,9 DO 10 11=1,3 DO 10 K=1,9
10 Q1CCl1,J)=Q1CCl1,J,+Q1CI1,K'*CCK,J. STRAIN MATRIX(Q1C. IS FORMED IN CURVATURE $ TWIST
STRAIN MATRIX IN USUAL FCRM DO 17 1=1,3 DO 17 J=1,9
18 OiC(I,J)=.5*TH*Q1C(I,J' 17 CONTINUE
STRAIN MATRIX IN USUAL FORM IS FORMED TO LOCATE STRAIN MATRIX FROM FEMB AT PROPER PLACE
DO 13 1=1,6 DO 13 J=1,18
13 D03CCI,J)=0. DO 14 1=1,3 M1=0 N=O DO 14 K=3,18,6 K3=K-1 Ml=N+1 N=M1+2 K2=1 DO 14·J=Ml,N JL=K2+K3 003CCI+3,JL'=0IC(I,J)
14 K2=K2+1 STRAIN MATRIX OF FEMB PLACEO AT PROPER PLACE IN COMPLETE STRAIN
STRESS MATRIX FOR BENCI~G DO 50 1 =1,3 DO 50 J=I,3
50 EM(I,J'=O.
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS R'HS
-' ~HS RHS RHS RHS RHS RHS
MATRRHS RHS RHS RHS RHS RHS
118
501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 521 528 529 530 531 532 533 534 535 536 531 538 539 540 541 542 543 544 545 546 541 548 549 550
C
16 C
EM( l, U=02 EMC1,2'=01 EM( 2, U=OI EMC2,2J=03 EM(3,3J=012 DO 15 J=I,9 qo 15 1=1,3 OQICe l, J)-O. 00"15 K=li3
15 OQICCI,J'=OQICCI,J.+EMCI,KI*QICCK,JI STRESS' MATRI)( C OQIC 1 IS FORMEO IN MT5.$ T\!fI ST ING MOMENT
DO 161-1,3 DO 16 J=I,9 OQICCI,J.=12.'CTH**31*OQICCI,J.
ST~ÉSSMATRIX IN USUAl FORM IS FORMEO
RHS RHS RHS RHS RHS RHS RHS RHS
~~'-,,'.r.)~~ RHS RHS RHS RHS RHS
C Ta ',lOCA TE STRESS MATR 1 X FROM FEMB 1 N COMPLETE STRESS MATR IX OF AN ElRHS C
c C
11
DO 11 1=1,6 DO Il J=1,18 002C CI, J '=0. 00'12 1=1,3 MI-0 N=O, ' DO 12 K=3,1896 K3=1(-1 Ml=N+l N==Ml+2 K2=1 DO 12 J=Ml,N Jl=K~.K3 OQ2CCI+3,JlJ=OQ1CCI,J'
12 K2=K2+1 25 FORMAT (9E14.3'
STRESS MATRIX OF FEMB, LOCATEO AT PROPER PLACE IN'COMPlETE ;
AR fA: X2*Y3/2. AC5;5.=AREA*012 AC6,6'=AREA*03 Ae8'6'=-AREA.Ol AC'8'8'=' AREA*02 AC9,5'=-AREA.OI2 A(9,9'= ÀREA*012 DO' 2 1=1,9;'·' DO '2' .1=1 .. 9 '
2 ACI',J'=ACJ,U DO '3 1=1,9 od :3 J=1,9 OU,J"a'O. DO 3 K=!,9
3 : 0 ( l" J J = 0 CI, J • + A CI, K) *c C K, J J
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
STRESSMARHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS, RHS RHS RHS
119
551 552 553 554 555 556 5'7 558 559 560 561 562 563 564 565 566 567 568 569 570 571
'572 573 574 575 576 577 578 579 580 581_ 582", 583 5~4 585 586 587 588 589 590 591 592 593 594 595 596 591 598 599 600
~ 1
1
00 4 1= 1,9 00 4 J=I,9 ACI,J' = O. 00 4 K=1,9
4' A CI , J , = AC l, J J +C C K ,1 , * 0 C K , J' CH=TtUC 4.*AREA' GYZ=G GXZ=0.5*YM1/Cl+PR1' DO 101 1=1,9 DO 101 J=I,9
101 CCI,J'=O. cCl,i'=CH*CGXZ*CY3**2'+GYZ*C'X3-X2'*.2" CC1,4.=-CH*CGXZ*(Y3**2'+GYZ*CCX3-X2,*X3" C(I,7'=CH*CGYZ*CX3-X2'*X2' CC4,4)=CH*CGXZ*CY3**2'+GYZ*(X3**2'J CC4,7'=CH*C-GYZ*X3*X2' C(7,7)=CH*CGYZ*CX2**2" C(1,2'=-0.5*(CCl,7'*Y3' C(1,3'= O.5*CCC1,4i*X2+CC1,7'*X3' CC1,6'=O.5*CCCl,4J*X2' C(1,S'=-0.5*CCCl,1'*Y3' CC1,9'=+O.5*CCCl,1'*X3' CC2,2,=-o.25*'CC1,1'*(Y3**2" CC2,3'=O.25*CC(1,7'*X3*Y3' C(2,1'=0.5*CCC1,1'*Y3' CC2,S'=~0.25*(CC1,1'*CY3**2" CC2,9'=0.2S*CC(I,7'*X3*Y3, CC3,3'=-0.2S*'C(I,4'*CX2**2'+CC1,1'.(X3**2" C(3,~)=-0.5*(CC1,4'*X2' CC3,6'=-0.25*tCC1,4'*'X2**2.' CC3,1'=-0.5*(CC1,1,*X3' C(3,S'=0.25.CCCl,1'*Y3*X3' t(3,9'=-O.25*(CC1,1'*(X3~*2)' C(4,5'=-0.5*CCC4,1'*Y3' CC4,6'=0.5*CCC4,1'*CX3-X2'-C(1,4'*X2' C(4,8'=-0.5*'CC4,1'*Y3' .. C C 4, 9 J = 0.·5* CCC 4, 7 • * , X r X 2 , • C(5,5'=-0.2S*CCC4,1'*CY3**2" CC5,6,=-o.25*CCC4,1'*(X2-X3'*Y3J ·C(S,7'=0.5*(C(4,1'*Y3' .C , 5, 8 , = -0 • 2 ~* ( C ( 4 ,1 , • ( Y3** 2 » » CCS,9'=0~2S*(Ct4,1'*CX3-X2'*Y3' C(6~6'=-0~2S*(C(1,4'*(X2**2'+CC4,1J*CCX3-X2'**2') CC6,7,=O.5*CCC4,1.*CX2-X3'. CC6,8'=-0.25*CCC4,7,*Y3*(X2-X3" CC 6,9'=-0.25*( C C 4, l' * cc X3-X2'**2' , C(1,81=0.5*'C(1,7'*Y3+CC4,1,*Y3' C(1,9'=-0.2S*CCCl,7.*X3+CC4,7'*CX3-X2"
C(S,8'=-0.2S*CCC1,7'.CY3**2'+CC4,7i*CY3*.2" ~C8,9'=0.25*CCCl,1'*X3*Y3+CC4,7'*CX3-X2'*(Y3"
i ..
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
. RHS . c- ;~HS
RHS RHS RHS RHS RHS RHS RHS RHS
120
601 602 603 604 605 606 601 608 609 610 611 612 613 614 615 616 611 618 619 620 621
'622 623 624 625 626 621 628 629 630 631 632 633 634 635 636 631 638 639 640 641 642 643 644 645 646 641 648 649 650
1 ?1 ... -...
1
e C(9,9'=-0.2S*'CC1,7'.'X3**2.+CC4,7'*C,X3-X21**21. RHS 651 DO 102 1=1,9 RHS 652 DO 102 J=1,9 RHS 653 CCJ,I'=CCI,J, RHS 654
102 ACI,J)=A(I,JJ+C(I,J' RHS 655 5 FORMAT (9E12.4) RHS 656
DO 6 1=1,18 RHS 657 DO 6 J:1,18 RHS 658
6 KU I,JI=O. RHS 659 M2 =0 DU~ 660 ""J
N2 =0 RHS 661 DO 7.K = 3,1:;,6 RHS 662 N2 =M2+1 ~HS 663 M2 = N2+2 ,ÙfS 664 K2 =K-I RHS 665 K6 =1 RHS 666 DO 7 I=N2,M2 RHS 667 Il = K6+K2 RHS 668 NI =0 RHS 669 M1=0 RHS 670 DO 8 K3=3,15,6 RHS 671 K4 = K3-1 RHS -672 N1 ~ M1.1 RHS 673 M1=Nl+2 RHS 674 K5 =1 RHS 675 DO 8 J=tU ,Ml RHS 676 Jl. = K5+K4 RHS 677 K1(ll,Jl' = ACI,J' RHS 678
8 K5 = KS+1 RHS 679 7 K6 = K6+1 RHS 680
RETURN RHS 681 END RHS 682 SU8ROUTINE SOLVE (NPART,NCOLN' RHS 683
C SU8ROUTINE FOR SOLUTION OF EQUATIONS. CAlCULATION AND PRINTING OF RHS 684 C 1 RESIOUALS RHS 685
REAL KI RHS 686 DIMENSION AMC108,108"B~(108,108"YMCIOe,108"TFC108,4),RS(108,4),RHS 687
lr(108,4J,0IS(108,4' RHS 68.8 .OMMON STU08,2161, RHS 689
1 -Kl(18,18"C(9,9J,OBAC6,6),OB(6,6"A(9,9"B(3,6', RHS 690 1 QIC13,9"OQIC(3,9"8A(3,6),OQ2C(6,lSJ,OQ3C(6,lS,,U'1800,4), RHS 691 1 OlT(3,4"OlB(3,4"oSITC3,4),OSl8C3,41;A3!6,6"A2C6,6' RHS 692
E QUI V Al EN CE ( AM' l, li ,S TC l, 1 • ), C 8M' l , 1 J , S TC l , 109 ) , , C TF ( l ,1) ,U ( l , 1) ) , RH S 693 1 (OIS(l,II,UCl,2,),(RSC1,1),U(I,3,),(FC1,l),U(l,4)' RHS 694
DO 140 1=1,108 RHS 695 DO 141 J~ 1, NCOLN RHS 696 TF(I,J,=O. RHS 697
141 R5CI,J'=0. RHS 698 e DO ·140 J=l,-l08 RHS 699 140 YM( I,-J'=O. RHS 700
C
C
00 144 ll=l,NPART REAO(4' M,N,(CAMCI,J),I=l,M',J=l,M),CCBMCI,J),I=l,M"J=l,N), 1 (CFCI,J),l=l,M),J=l,NCOlN'
150 00 424 1=1, 'M 00 42 5 J= l, NC Ol N FCI,J'=FCI,J)-TFCI,J)
425DISII,J)=FCI,J) DO 424 J=l,M
424 AM(I,J'=AMeI,J)-YMCI,J.
CALL OCOMPeM' CALL INVERTIM)
00 100 l=l,M ob 100 K=l, NCOLN D'Ise I,KI=O. DO 100 J=l,M
1000ISCI,K'=OISCI,K)+AMeI,J'*FCJ,K)
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
411 WRITE(21 M,N,ceAMel,JJ,I=l,M"J=l,M),eCBMeI,JJ,I=l,M',J=l,N', 1 ((FCI,J),I=l,M"J=l,NCOLN'
RH5 RHS RHS IFCNPART-LL) 437,431,432
432 DO 101 l=l,M pb 101 K=l,NCOLN OIS(I,K'=O. 00 101 J=l, M
1010IS(I,K.=0Isel,K)+AMeI,JJ*FeJ,K' DO 102 i=l,N DO 102 K=l, NCOlN
TF(I,Kt=O. DO 102 J=l,M
102 lFCI,K'=TFCI,K'+BMeJ,I'*OISCJ,KJ DO 110 J=l,N DO 110 l=l,M YM(I,J'=O. 00 110 K=l,M
110 YMel~J)=YMeI,J' + AMCI,K)*BMCK,J) 00 111 J=l,N DO 111 l=l,N AMCI,J'=O. 00 III K=l,M
111 AMeI,J.=AMCI,JJ + BMCK,I,*YMCK,J) DO 112 l=l,N DO 112 J=l,N
112 VMCI,J,=AMCI,J' 144 CONTINUE 431 Rf:WINO 4
WRITE(3) CCOISCI,J',I=l,MJ,J=l,NCOlN) IF(NPART-1) 600,600,601
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
:.-' . ~HS :., 'l. c:. )". • , 1 rMS
'. RHS
4It601 NA=NPART-1 DO 441 lL=l,NA
RHS RHS RHS
122
101 102 103 104 105 106 101 108 109 110 111 112 113 114 115 116 111 118 1i9 120 121
.122 123 124 12'5 126 12' 128 129 130 131 132 133 134 135 136 131 738 139 140 141 142 143 144 145 746 141 748 149 150
c
•
103
444
104 441
515
512
510
105 500
31 600
BACKSPACE 2 BACKSPACE 2 REA 0 l 2) M , N , C CAM e It J) , 1 = l ,M • , J= l , M, , C (B Mel, J) , 1 = l , M) ,J = l , N) ,
1 lCFCI,J),I=I,M"J=I,NCOLN' QO 103 1=I,M 00 103 K=I, NCOLN
TFCI,K)=O. ~O 103 J=I,N TFCI,K'=TF(I,K'+BM(I,J)*OISCJ,K) 00 444 J=1,NCOlN 00 444 1=I,M FCI,J.=FCI,J'- TFCI,J' 00 104 1=1,M 00 104 K=I, NCOLN OISCI,K.=O. 00 104 J=1,M OISCI,KJ=OISCI,K'+AMCI,J.*FCJ,K) WRITE'(3) C(0ISCI,JI,I=1,M),J=1,NCOLNJ WRITE (6,515) FORMATC10H RESIOUAlS) 00 500 lL=l,NPART
, ,. __ :O.J'"
RE~O (4. M,N,(C~MCI,J),I=l,M),J=l~MJ,C(eMCI,J),I=l,M),J=l,N), 1 (CFll,J',I=l,M),J=l,NCOLN. i
BACKSPACE 3 ReAC (3) '(OIS(I,J),I=l,M),J=I,NCOLN) BACKSPA(:E 3 BACKSPACE 3 REAO C3' CCTFCI,Jt,I=I,N),J=I,NCOLN' 00 510 J= l, NCOLN 00 510 1=I,M FI!,JJ=FCI,J) - RSCI,J. DO 512 K=1, M FCI,J)=FCI,J) - AMel,KJ*OISCK,J) DO S10 L=l, N FCI,J)='F(!,J) - BMCI,L'*TFCl,Ji 00 105 1=1, N 00 i05 K=l, NCOLN ~SCI,K.=O.
00 105 J=l, M RSCÎ,K'=RSCI,K)+BMCJ,I)*OISCJ,K. WRITE C6,31) CCFCI,J),I=l,M),J=l,NCOlN) FORMATC1H ,12E9.2. CqNTINUE Rf;TURN eND SUBROUTINE OCOMPCN)
THIS DECOMPOSES THE SYMMETRIC MATRIX ACN*N • COMMON ANCI08,108. DO 1 1=I,N
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
·RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RH~ RHS RHS RHS RHS RHS RHS RHS RHS RHS D"'~ l"~ ,....,
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
123
151 152 153 154 155 756 151 158 759 760 761 162 163 164 765 166 761 168 769 710 711
. 712 773 774 775 716 717 118 17? 180 181 182 183 784-785 780 7é7 788 789 790 791 192 793 194 195 196 197 198 199 BOO
C
C
C
2
DO 1 J=I, N . SUM:: AN(I,J) K1= 1-1 IFU-1' 3,3,5
5 DO 2 K=l,Kl SUM = SUM-ANCK,I'*AN(K,J.
3 1 F ( J- l' 4,6, 4 6 IF(SUM) 7,7,8 7 WRITE(6,501.
IF(SUM' 503,503,504 503
502 WRITE (6,502' l, J,KI ,K ,SUM,ANCK,I' ,AN( K, Jl
FORMAT C414,3F10.2' STOP
8 TEMP=1.0/SQRTCSUM) AN(I,J'=TEMP
4 1
501 504
2 1
4
3
GÇl TO 1 AN(I,J'=SUM*TEMP CONTINUE FORMATC'b','SUM IS lE ZERO AND SUBROUTINE FAllS" RETURN END SUBROUTINE INVERT(NJ
COMMON Al(108,1081 Il=N-l OQ 1 1=.1, i 1 Jl=I+l DO l J=J1, N SUM :=0.0 Kl=J-l OQ 2 K=I,Kl SUM = SUM-AlCK,I.*AllK,J' ~l(J,IJ=SUM*ALCJ,J'
DO 3 l=l,N DO 3 J=I,N SUM =0.0 DO 4 K=J,N SUM = SUM+AlCK,I'*Al(K,JJ Al CI, J) = S UM Al.CJ,I'=SUM ~ETURN END
RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RH5 RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
SUBROUTINE STRESSINPART,NFIRST,NlAST,NCOlN,NElEM,NOO,NFREE,NPOIN) RHS RHS RHS SOBROUTINE FOR CALCUlATION OF STRESSES
OtMENSION NOOC500,3"NFIRSTC201,NLASTC201,T(18,18J,C1CI8,4J, 1 .DBU6,4J,Z(19,4)
COMMON ST(108,216), 1 KIC18,181,C(9,9"OBA(6,6J,OB(6,6),A(9,9),B(3,6J, ; 1 Q1C(3,9J,OQICC3,9"BAC3,6),OQ2CC6,18),OQ3CC6,18',U(1800,4)"
RHS RHS RHS
\ .. ,RHS .'~ .) 'RK.S
124
801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821
·822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 84a 849 850
C C
1 PZT'3,4J,OZBC3,4J,OSZT(3,4',OSIBt3,4J,A3C6,6J,A2C6,61 1,~1(10),E2CI0J,POCI0),P2CI0),GEC10.,NEP(500"AlPHX(lOI,AlPHY(lO) I,TEMPAVC~),TEMPOF(500"TFORCEC18,2.,TlOAO(108,11
RHS RHS RHS
00 600 II=l,NPART JJ=NPART+I-II M=NFREE*CNFIRSTCJJI - 1) + 1 N= NFREE*NlASTCJJJ
600 REAO (3' CCUCI,JJ,I=M,NJ,J=l,NCOlNJ WRITE(6,690)
690 FÇJRMAT('l') WRITE(6,615)
RHS RHS RHS RHS RHS RHS RHS
615 FORMAT C' J' ,'NODE X-DI SPlAC EMENTS V~DI SPlAC EMENTS'-Z-DI SPlAC EMENTS 10X~OISPlACEMENTS OY-DISPlACEMENTS OZ-DISPlACEMENTS'.
RHS RHS RHS RHS RHS RHS RHS RHS RHS
WRITE C6~32. (CI,UC6*1-5,JJ,UC6*1-~~JJ,U(6*1-3,Jl,uC6*I-2,J), 1 tiC6*1-1,J',U(6*I,J),1=1,NPOINJ,J=1,NCOlN)
32 FORMAT(lH ,I4,6E16.8" 625 FORMATC'O','ElEMENT NOOES
l' CARTESIAN STRESSES 2'PRINCIPAl')
635 FORMATC' ','NUMBER FIRST SECOND T~IRD l' X-STRESS Y-STRESS XV-STRESS 2' ANGLE'.
COOROS OF CENTROIO', PRINCIPAL STRESSES .. , ..
, ,
X-CODRO STRESS-1
Y-COORO' , '. _ RHS STRESS-2 >,~~<~HS
.. ,'../ . RHS
PR 1 Ne IPAl REWINO 13 Rf=WINO ,2
ANGLE IS THE ANGLE BETWEE~-Y AXIS ANO'~TRESS-1 RHS RHS RHS
REWINO 4 WRITE(6,690) WRITE U6,7151
715 FORMATC'J',IElEM X-PlANESTRESS Y-PLANE STRESS lX-BENOINGSTRESS Y-BENDINGSTRESS XY-BENOINGSTRESS')
DO 20 ll=l,NElEM J=NEPCll' YM1=EICJ) YM2=E2CJ' PR1=PO'J) P~2=P2(J) AlPHAX=AlPHXCJ) AlPHAY=AlPHV( J J TMSTNX=AlPHAX*TEMPAV'lll TMSTNY=ALPHAY*TEMPAVell) TBSTNX=O.5*AlPHAX*TEMPOFCll) TBSTNY=0.5*AlPHAV*TEMPOFIllJ
RHS RH$ RHS ~HS
XY-PlANESTRESS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS RHS
TElA=1./( 1.-PR1*PR2) TMSTSX=(YMl*TElA*AlPHAX+PR2*YMl*TELA*ALPHAY)*TEMPAVCllJ T~STSx=tVM1*TelA*AlPHAX+PR2*VMl*TELA*ALPHAVJ*TEMPDFCll.*.5 TMSTSY=CYM2*TElA*ALPHAY+PR2*YM1*TElA*ALPHAX'*TEMPAVCll. TBSTSV=CYM2*TElA*AlPHAY+PR2*VM1*TElA$ALPHAXJ*TEMPOFCll'*.5 4It REAO Cl) CCOQ2CCI,J),I=1,61,J=1,18J,CCOQ3CCI,J),I=1,6J,J=1,18',
lORX,ORV
RHS RJ;S RHS RHS RHS RHS RHS
1 •
125
851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871
, 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900
REAO 112' CCTCI,J"J=l,18I,I=I,18) DO 620 J=l,NCOlN DO 620 1=1,3 JJ=NOOIll,11 ~(6*1-5,J'=UC6*JJ-S,J) Z(6*1-4.J'=UC6*JJ-4,JJ Z.C6*1-3,JJ=UC6*JJ-3,J' zC6*1-2,JI=UC6*JJ-2,J' ZI6*I-l,JJ=UI6*JJ-l,J'
620 Z(6*I,J'=UI6*JJ,JI DO 621 1=1,18 DO 621 j=l,NCOlN CICI,J'=O. DO 621 K=1,18
621 tlll,J'=CICI,JI+TII,K'*ZIK,J' WRITE 1131 CIIll,NOoCll,JI,CIC6*J-S,KI,CIC6*J-4,KI,
lC116*J-3,KJ,Cl(6*J-2,KI,ClC6*J-1,KI,CIC6*J,KI),K=1,NCOlN),J=1,3) 00 630 J=I,NCOlN DO 630 1=1,6 otH l, J.=O. OB1CI,J'=0. DO 630 K=l,18 OBII,J'=OBCI,J'+OQ2CCI,K'*C1CK,J'
630 DB1C~,JJ=DBICI,J.+oQ3CCI,K'.CICK,JJ OBl1,1'=OBC1,!)-TMSTSX 08C2,1'~OBC2,1)-T"STSY
,oBC4,l'=OB(4,1'-TBSTSX PBCS,1'=OBC5,1'-TBSTSY OPIC1,lJ=DBIC1,l'-TMSTNX OBIC2,11=OBIC2,I'-TMSTNY OB1C4,1'=OB1C4,I'-TBSTNX oBICS,IJ=OBIC5,l'-TBS1NY WRITEC6,32) Ill,CIOBCI,JJ,J=I,NCOlN),I=I,6JJ
C COMPLETE STRAIN'MATRIX(OB1' FORMEO DO 1 1=1,3·' DO 1 J=l,NCOlN OlTCI,J'=O. olBCi,J'=o. DSlT(I,J'=O.
1 OSIBII,J,=O. 00 2 J=l, NCOlN 00 2 1=1,3 OlTCI,J'=OBCI,J'+OBII+3,JJ
2 OlBCI,JJ=OBCI,JJ-OBCI+3,J' CALl'PRfNfOlT,A3,NCOlN' tALL PRINCOlB,A2,NCOlN' WRITEC2'll,CNOOCll,JJ,J=1,3J,ORX,ORY,C(OlTCI,J,,1=1,3',CA3(I,J',
lI=1,3',J=1,NCOlNI, _ .1 ll,CNOoCll,J),J=I,31,ORX,ORV,((C~e(I,J),I=I,3"CA2CItJ',
1I=1,3~,J=I,NCOlN)
126
RHS 901 RHS 902 RHS 903 RHS 904 RHS 90S RHS 906 RHS 907 RHS 908 RHS 909 RHS 910 RHS 911 RHS 912 RHS 913 RHS 914-RHS 915 RHS 916 ~HS 911 RHS 91a RHS 91~ RHS 920 kHS 921 RHS '922 RH5 923 RHS 924 RHS 9?5 RHS 926 RHS 927 RHS 928 RHS 929 RHS 930 RH5 931 RHS 932 RH~ 9~3 RHS 934 RHS 935 RHS 936 RHS 931 RHS 938 RHS 9~9 RHS 940 RHS 941 RHS 942 RHS 943 ~HS 944 RHS 94S RHS 946 RHS 947 RHS 948 RHS 949 RHS 9S0
DO 3 J=l,NtOLN RHS DO 3 1=1,3 RHS OSZTCI,JI=OB1CI,JI+oB1(I+3,JI RHS
3 OSZBCI,JJ=OB1CI,J'-OB1CI+3,J' RHS WRITEC4'LL,CNOOCLL,JJ,J=1,31,ORX,ORy,CCOSZT(I,JJ,I=1,3J,J=l,NCOLNJRHS
l,. LL,CNODCLL,J"J=1,3"ORX,ORY,C(OSZB(I,~"I=1,3),J=1,NCOLN'RHS 626 FORMATC'O','ELEMENT NODES COORO. OF C.G CARTESIANSTRAINRHS
lS " RHS 627 FORMA TC '0' , 'NUMBER F IR ST SECOND "fHl RD X COORD. y COORo RHS
1 X STRAIN' y STRAIN XV STRAIN" RHS .Fe NCOLN - 1 J 110,20,110 ~HS
110 DO 640 J=2,NCOLN RHS DO 640 1=1,3 RHS OBACljJI=O. RHS DO 640 K=l,J RHS
640 QBA(I,JI=OBAeI,JI + OBCI,K) RHS êALL PRINCoBA,A,NCOLN) RHS WRITE C6,31'C(DBACI,JI,I=1,3"CACI,JI,I=1,3),J=2,NCOLN' RH$ W~iTE (6,33' CACI,NCOLN),1=1,3' RHS
20 CO~TINUE RHS REWIND 13 RHS REWIND 2 R~S REWINO 4 RHS WfUTE (6,690) !:.; -, ,_.)\HS "'~ITE (6,8151 --,,-.).,-I:f,S
815 FOR MA TC ,'J' , RHS l 'ELEM NODE X-OISPLACEMENTS V-OISPLACEMENTS Z-DISPLACEMENTS RHS 10X-OISPLACEMENTS Oy-oISPLACEMENTS OZ-oISPLACEMENTS" RHS
42 ~ORMAT (IH ~214~6E16.8' RHS DO 21 LL~l,NELeM RHS
READ C13' (CCLL,NODC'LL,J"C1C6*J-5,KJ,C1C6*J-4,KJ, RHS lClc6*J-3,KJ,C1(6*J-2,K),CIC6*J-1,KJ,C1C6*J,K",K=1,NCOLNJ,J=1,3' RHS
21 WRITEC6,42' C(CLL,NOOCLL,Jt,Cl(6*J-5,K"C1C6*J-4,KJ, RHS 1Cl(6*J-3,K"CIC6*J-2,K',Cl(6*J-1,KJ~ClC6*J,K»,K=1,NCOLN),J=I,3J RHS W~ITE (6,6251' ~HS WRITE C6,635' RHS 00 22 LL=l,NELEM RHS REAOC21 LL~(NOO(LL,JJ,J=1,3"ORX,ORV,(CCZT(I,JJ,I=1,3J,(A3(I,JJ, RHS
1I=1,31,J=1,NCOLNJ, RHS 1 LL,CNOOCLL,J"J=1,3"ORX,ORy,C(CZE(I,JJ,I=1,3),CA2CI,JJ, RHS lI=I,3"J=I,NCOLNJ RHS
WRITEC6,31' , RHS l, . LL,(NOO'LLtJJ,J=1,3J,ORX,ORy,(CDIT(I,J"I=I,~J,CA3(I,JJ, RHS lI=1,~',J=I,NCOLN. RHS
22 WRITE(6,31' RHS 1 LL,CNOOCLL,JJ,J=1,3"ORX,ORy,C(OZBCI,J),I=1,3J,CA2(I,Jl, RHS 11=1,3',J=1,NCOLN' RHS ~RITE (6,626. ' RHS e WRITE (6,621' RHS DO 23 ll=l,NELEM RHS
127
95l 952 953 954 95S 956 951 95'8 959 960 961 962 963 964 965 966 967 968 969 970 911
·972 913 914 915 976 911 978 919 980 981 982 98~ 984 98' 986 981 988 989 990 991 992 993 994 995 996 991 998 ~99
1000
REAO(4, ll,CNOOCLL,J',J=1,3"ORX,ORy,(CCSZTCI,J"J=1,3),J=1,NCOLN,RHS 1001 1, LL,CNOOCll,JI,J=1,31,ORX,ORV,CCDSZBCI,J"I=1,3J,J=1,NCOLN'RHS 1002
WR ITE (6,311 - RHS 1003 l' LL,CNOOCLL,J),J=1,3),ORX,ORY,(COSZTCI,J),1=1,3),J=1,NCOLN'RHS 1004
23 W~ITEC6,31) 'RHS 1005 1, LL,CNOOCLl,JI,J=1,31,ORX,ORV,CCOSZBCI,J),1=1,3),J=1,NCClN'RHS 1006
31 F9RMATC' ',I4,217,I6,Fl~.4,F10.4,5E12.5,F7.2' RHS 1007 33 FORMATCIHP,5E12.5. RHS 1008
~ETURN RHS 1009 ENO RHS 1010 SUBROUTINE PRINCOB,A,I\CCUH RHS 1011
C THIS COMPUTES THE PRINCIPAL STRESSES AC3,NCOLN) GIVEN THE RHS 1012 C CARTESIAN STRESSES OBC3,NCOLN' FOR NCClN LOAOING SYSTEMS RHS 1013
DIMENSION OBC3,6J,AC6,6) RHS 1014 DO 100 J=l,NCOLN RHS 10~5 Tl=COBC1,J' + OBe2,J' '/2. RHS 1016 T2=(COB(1,J'-OBC2,J"/2 •• **2 RH$ 1017 T3=(OB(3,JJ.**2 RHS 1018 T4=2.*OB(3,J./COBC1,JI-OBC2,J)) RHS 1019 AC1,J.=Tl + SQRTCT2+T3. RHS 1020 A(2,JI = Tl - 'SQRTCT2+T3) RHS 1021 THA=0.5*ATANCT4. RHS 1022 THB=THA'+ O.5*3~14159 RHS 1023 SIG = Tl+SQRTCT2'*COSC2.*THA. + SQRTCT3.*SINC2.*THA' RHS 1024 IFCAC1,J.-SIG'1,2,1 RHS 1025
2 AC3,J'=THA*180./3.14159 RHS 1026 GO TO 100 ' RHS 1027
1 AC3,J)=THB*180./3.14159 RHS 1028 100 CONTINUE RHS 1029
RETURN RHS 1030 END RHS 1031
129
APPENDIX 5
1,
2,
3.
Programs to generate some of che input data for the
computer program of Ap~endix 4
The fo11owing three programs are'provided to finda
The nodes of the e1ements of,an idea1:J,.zed structure.
The coordinates for a spher~cal she11.
The coordinates for a plate.
1, A program to f1nd the nodes of the element
Input
A Structure Properties (414) One card
Cols Notation
1-4 Nl
5-8 NPOIN 9'·12 NPC 13-16 NELEM
Description
Number of elements per paral1el circ1e, Total number of nodes Total number of parallel circ1es Total number of elements
B Thickness card (10F7.3) One card per ten e1emeats
These cards read the thickness of e1ements which lie between
the first two para11e1 circ1es o
Printed output (418,2F14 0 1)
Element number, nodal number, thickness,and transformation number (0 for clockwise nodal numbering, and 1 for counter clockwise nodal numbering).
Punched output (314~2F1608,2F8o3) One card per e1ement
Nodal number, thickness of element, transformation number.
The following should be Observeda
10 The parallel circles (or straight lines in the case
of a plate), as shown in Figure 38, must have the same
130
number of nodes except at the crown.
2. The thickness of elements between the first two
parallel circles must be given as input data.
3. The structure must be open (i.e. full cap of sphere
is not valid but three-quarters, half, or one quarter of
a sphere is allowed).
This program was used to generate the element array
for all the problems except the spherical shell triangular
in plan.
2. A program to find the coordinates for a spherical shell
Input
A Structure Properties (514) One card
Cols Notation Description
1-4 5-8 9-12 13-16
17-20
NPûlN NPC N LL
LLI
Total number of nodal points Total number of parallel circles Number of node. per parallel circle Parallel circle after which further subdivision is required Parallel circle before which further subdivision is required
B Structure Properties (5F14.6) One card
Cols
1 ... 14
15-28 29-42 43 .. 56 57-70
Notation
v OX OY R RN
Printed output
Description
Numger of total subdivisions between two para1lel circ les Co-latit~d~ in radians Longitud~ in r~dians Radius 9f ctirvature Number of equal divisions in the co-latitude
. (I4,3F16.4) , , .
Node number, x-coordinate, y-coordinate, z-coordinate
Punched outpùt (5F14.6)
x-coordinate, y-coordinate, z-coordinate The restrictions" land 3 of previous program apply.
3, A program to find the coordinates of plates
Input
A Structure Properties (3I4,2FIO.6) One card Cols
1-4 5-8 9-12 13-22 23-32
Notation
Nl NPC NPOIN DX DY
Printed output
Description
Number of nodes per parallel circle Number of parallel circles Total number of nodal points Regular interval in x-direction Regular interval in x-direction
(I4,3F16.4)
Node number, x-coordinate, y-coordinate, z-coordinate Punched output (3F14.6) One card par node x-coordinate, y-coordinate, z-coordinate
The restrictions 1 and 3 of Program (1) Appendix 5 must be observed.
C PK:".Ir;)i~AM Tt) FI"-O Ncoes Of ELEMENT CI ~E t\'S ION NOD (300,3' ,.fU 300) • T"(3O.0·' WRITE (6,615) .
615 FORMAT Ct ELEMENT NOD1 ~Z • R ., REAO (5,95' Nl,NPQIN~NPt·,:t4.L": 95 fORMAT (414' .
C Nl.~UM8E~ OF ELEMENTS' IN ' .. TITION C NPCI N:aTOTAL HUMBER Of ··t(OOo.·: C NPC i S HUMBER OF PÂR.'LL~" '".1;''' C' NELEM = HUMBER OF TOTAL' EL EtltINT..,
NIt= (NPC-l '.Nl N2:al N3-1 INDEX-O' REAO C 5,101 (nu 1 " h.I,'Nl)
10 FORMAT(10F7.3J 00 101 [al,NElEM
101 R( 1'=1. 00 100 lal,NElEM [laI+Nl If (II-NELEN'" ..... ':.
99 TH(ll'aTHeIJ 100 CONTINUE
Nooe1,l'=1 NGOCl,2J-2 NOD(1,3)~Hl/2+3 00 1 LL-I,NELE" IF (LL-N4J 14,14,15
14 IF (N2-Nl) 2,',2 2 IF C INOEX-l' ,zl,'I~m'
21 INDEX-INDEX+1 . NOO(LL+l,l!-NODClL.3t NODIlL+l,2taNOOCLl+l,lI~1 NOOtLL.lp3)·NOD'LL~1' GO Ta 6
22 fNOE~=INOEX-l NOO(Ll+lwl'=NOOCLl-l,2J NOOClL+l.2J-NOOILL+l, 1'+1 NOOILl.l,3J·HODCLL~1,3'+1 GO Ta 6'
5 N2::1 Lll==Ll-Nl+2 NOOelL+l, i)~N06"('['L~~ 2'.' . NOOfLL+lp2'=NOO'lL+t.l,.t1 .. NOD(lL+lp3J=NOOCll+l,l'+H11Z.1 IHOEXaO GO Ta 13
15 NOO(ll+lf.lJ·~qltcLt·f.'I,: .. ", \' .)
NODtll+l p 2'=NODCll+l,1'+1 NOD(ll+1,3J=NPOIN GO TO 13
6 N2=N2+1 13 WRITE (1,25) NOOCll,1',NOOCll,2),NOO(lL,3J,TH'LlJ,RCll'
WRITE (6,4 J ll,NOOCll,1',NOOClL,21,NOOCll,3),TH(ll),RCll' 1 CONTINUE 4 FORMAT C4I8,2F14.1'
25 FORMAT C314,2F16.8,2FS.3J END
133
~ COOROiNATES OF SPHERICAl SHElL DIMENSION X(150,3) REAC i5,100) NPOIN,NPC,N,LL,LLI WRITE(6,100' NPOIN,NPC,N,LL,lLI
100 FORMAT(5I4) C NPOIN=TOTAl NUMBER OF NODAL POINTS C NPC=NUMBER OF P~RALLEl CIRClES C N=NUMBER OF NODES PER PARTISION C lL=P.ARALlEl CIRClE NUMBER AFTER WHICH SUB DIVISION C lll=PARAllEl CIRClE NUMBER BEFORE WHICH SUB-DIVISION READ (5,9' V,OX,Oy,R,RN
WRITE(6,9) V,OX,Oy,R,RN C V=NUMBER OF TOTAL SUB DIVISION 8ETEWEN TwO PARAllEl CIRClES C OX=CO-lATITUDE C OY=LONGITUDE C R=RAoIUS OF SPHERE C RN=NUMBER OF EQUAl DIVISION IN OX
WRITE (6,615' 615 FORMAT (' NODE X-COORD Y-CODRD Z-COORO') Xl=O.
Yl=O. Zl=R* Cl-COS (OX, ) OYO=O. l=NPOIN-(N*NPC' M=N MI=1 1=0 oOX=OX MII=M DO 1 K=l~NPC RM=M-l oDY=OYO DO 2 J=MI ,MIl 1=1+1 P=SIN(OOX. Q=SIN(Oovt S=COS(DOX) T=COS(OOY) X(I,l'=R*P*T X(I,2'=R*P*Q X(I,3)=Zl-R*(I.-S' DOy=oOY+OY/RM WRITE (1,9. X(I,lJ,X(I,2),XU,3)
2 WRITE (6,4) I,X(I,ll,X{I,2)yX(I,3) MI=M MII=2*M-l IF (K-lL' 1 .. 3,3
3 IF (K-Lll' 8,7,1 8 oOX=DOX-OX/(V*RN)
IF (K -lll) l 'i 7 , 7
13/
~ DOX=DOX-OX/RN 1 CONTINUE
IF (l. 5.6,5 5 WRITE (7.91 XL,Vl,Zl
WRITE (6,41 l.XL.Vl,ZL 6 CONTINUE 4 FORMAT (14,3F16.4.
9 FORMAT (5F14.6) END
• c
C C C C C
DIMENSION X(300,3) PROGRAM TO FINo COORo. OF PLATE
READ (5,15) NI,NPC,NPOIN,oX,OY NI= NUMBER OF NOD ES PER PARAlLEL CIClE
NPC =NUMBER OF PARAllEl CIRClES NPOIN = NUHBER OF TOTAL NODAL POINTS
DX=REGUlAR INTERVAl IN X-DIRECTION oY=REGUlAR INTERVAl IN Y-DIRECTION
15 FORMAT (314,2FIO.6) WR 1 TE (6, 615 J
615 FORMAT (1 NODE X-COORO J3=O DO 5 I=l,NPOIN XU,l'=O. X( l ,2 )=0.
5 X(I,3J=O. DO 1 I=lpNPC X(l,lJ=O. DO 2 J=l,NI Jl=J+l
3 XeJI,lJ=X(J,l)+OX XeJl,2J=XeJ,2) J3=J3+1 WRITE (7,4' X(J,1),X(J,2J,X(J,3) WRITE (6,lOO)J3,XeJ,lJ,X(J,2J,X(J,3)
2 CONTINUE 1 X(I,2)=X(I,2J+OY 4 FORMAT(3F14.6) 100 FORMAT(I4,3FI6.4t
END
Y-COORD' l-COOROI)
• 127
APPENDIX 6
Typica1 strain gauge and dial gauge readings
1. Spherical Shell with Hole at the Crown. Load (total) 200 lb. on Meridian passing through Center of the Lnaden Arc
Strains (Micro-inch/inch).
Gauge Location
lA
lB
lC
2A
2B
2C
3A
3B
3C
4A
4B
4C
SA
SB
5C
Top Surface
181 average 10 13
10
1~1 2~J 15
- 7) -19> -16
-21~
-140(
-1811 -141
-101)
Bottom Surface
-17) average
-30 t -24
-25
_31} -56 -38
-26,
57
1 53 57
61
113
1 152 124
106
11 Gauge . Location
(cont. ) 6A
6B
6C
7A
7B
7C
8A
SB
8C
9A
9B
9C
TOJ2 Surface
_40j -58 -48
-46
~j 1
138
Bottom Surface
42] 35 37
33
-6] -40
-17
-21
-35] -59 -44
-38
Radial deflection under center of loaded arc .0023 inch.
2. Spherical Shell with Hole at the Crown. Load (Total) 200 lb. on Meridian passins throuSh Center of the un10aded Arc
Gauge Location TOJ2 Surface Bottom Surface lA 1~ average 6] average lB 13 14 2 7
lC 15 13
2A l;j 15] 2B 14 12 13 2C 20 12
Gauge Location
(cont.)
3A
3B
3C
4A
4B
4C
5A
5B
5C
6A
6B
6C
7A
7B
7C
8A
BB
8C
9A
9B
9C
Top Surface
29} 23 29 35
23} 21 21
20
~~llB 19
15} 15 14
13
l~} 9 13
~ 4
139
Bottom Surface
301 2~ t 21
0)
2B} 24 27
19
34} 26 31
33
20} 15 18
18
l~} 12
15
- 31 -~J -3
Radial def1ection under center of un10aded arc .0002 inch.
140 ., 3. Spherical Shell triangular in Plan. Load 20 lb. at the Crown
Strains (Microœinch/inch).
Gauge Location Top Surface Bottom Surface lA 014] average 067] average lB -14 -14 -69 -67 lC -14 -64
2A
:] OS] 2B 7 -56 -57
2e -56
3A
-21 - 4] 3B -17-19 - 7 - 7 3C -14 -10
4A
02] ~] 4B o -12 2
4C ",16
5A
4:] 0
34] 5B 33 -29 -34
5C 47 ",40
6A 38] 04
°1 6B 37 35 -34 =34 6C 30 -27
•
. -'.',
141
Deflect10n in inches
Crown Side Between Crown and Support Sl S2 S3 Al A2 A3
~000108 -00011 -0.0088 -.0002 -.0002 "00003 ~o0075 -.0098 ... 0.00023
40 Spherical Shell triangular in Plan, Load 20 lb. (9) = 50 42')
Strains (micro-inch/inch).
Guage Location Top Surface Bottom Surface lA 5 -54 2A -16 -84 3A -40 -58 4A -22 6
SA 28 .46
6A 26 -58,
Deflection in inches
Crown Side Si Between Crown and Support
..".0032 C> .0059 .., o 001/~