Embed Size (px)
Citation preview
1 • _ •
,
FINITE ELEMENT ANALYSIS
OF TALL BUILDINGS
i .
FINITE ELEMENT ANALYSIS
OF TALL BUILDINGS
A Thesis
by
J.C. Mamet, M.Eng.
Submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for the degree of Doctor of Philosophy.
McGill University March 1972
@ J .C. Hamet 1972
.. ..
Finite Element Ana1ysis of Ta11 Buildings
J.C. Mamet
Department of Civil Engineering and App1ied Mechanics
ABSTRACT
Ph.D. March 1972
A new global approach to the ana1ysis of ta11 building struc
tures by means of finite e1ements and a substructure ana1ysis of the
f100r slabs is presented.
It may be used for a large variety of building configurations
comprising slabs, wa11s, co1umns, beams, diagonal braces, etc.
The program incorporates a set of new disp1acement functions
particu1ar1y suited for civil engineering prob1ems and an original use
of fictitious beams for the transmission of externa1 moments acting in
the plane of the e1ements. This formulation is used at the abutment
of 1inte1 beams into shear wa11s, where a semi-rigid connection may
be specified.
The program is used in a study of building structures consist
ing of end shear wa11s and intermediate frames under 1atera1 10ad, with
emphasis on the influence of the bending of the f100r slabs in their
own plane. It is confirmed that the usua1 assumption of the rigid
f100rs resu1ts in the under-design of the co1umns at the 10wer f100rs.
Analyse des Bâtiments-Tours par la Méthode des E1éments Finis
J.C. Mamet
Département de Génie Civil et de Mécanique Appliquée
RESUME
Ph.D. Mars 1972
Une nouvelle méthode d'ana1yse des bâtiments-tours par la
méthode des éléments finis est présentée. Les dalles des planchers
y sont considérées comme sous-structures.
Elle peut @tre utilisée pour une grande variété de structures
comprenant notamment des dalles, murs de refend, colonnes, poutres,
diagonales de contreventement, etc.
Le programme d'ordinateur fait usage de nouvelles fonctions
définissant les déformations des éléments et convenant particulière
ment aux applications du génie civil, ainsi que de poutres fictives
pour la transmission de couples externes agissant dans le plan des
éléments.
Celles-ci sont utilisées notamment pour étudier 1 1 encastrement
des poutres-linteaux dans les murs de refend. Un encastrement élastique
peut être considéré.
Le programme est utilisé pour 11 étude de structures de batiments
comprenant des murs d'extrémité et des cadres rigides intermédiaires et
soumis à des charges horizontales.
L'effet de la déformation des dalles dans leur plan est étudié,
et lion confirme que 1 1 hypothèse usuelle des planchers rigides conduit
à la sous-estimation des efforts des colonnes, aux étages inférieurs du
bâtiment.
- iv -
ACKNOWLEDGEMENTS
The work presented in this thesis was carried out under the
direction of Dean L.G. Jaeger to whom the author wishes to express his
deepest gratitude. His guidance and cheerful encouragements contributed
a large part to the development of this study.
The author would also like to sincerely thank the following
persons and organization for their contributions and for the help they
have provided:
Prof. P.J. Harris who acted as Research Director after the
departure of Dr. Jaeger;
Prof. A.A. Mufti, who initiated sorne parts of the work, and
devoted a lot of his time in fruitful discussions and guidance;
Prof. S.M. Mirza for his suggestions and encouragements;
Prof. A. Coull, of the University of Strathclyde, who communi
cated sorne of his results;
Dr. S.Z.H. Burney, for his encouragements and constructive
criticisrn of the manuscript;
the author's fellow graduate students, and in particular
Mr. A.Q. Khan, who was a patient and most pleasant room mate, and
whose skills in debugging computer programs were often called in
action;
Mrs. J. Vasseur who typed the manuscript in the most expert
and efficient way;
the staff of the Cornputing Centre and of the Structures
Laboratory at McGill University;
- v -
the National Research Council of Canada, which sponsored the
present work under Grant No.A2l67, and whose support is gratefully
acknow1edged.
ABSTRACT
RESUME
ACKNOWLEDGEMENTS
CONTENTS
LIST OF FIGURES
LIST OF TABLES
LIST OF SYMBOLS
1. Introduction
1.1 Preamb1e
.. vi -
CONTENTS
1.2 Methods of ana1ysis of ta11 buildings
1.3 Scope of the present work
2. Choice of finite e1ement formulations
2.1 Introduction
2.2 Element formulation
2.2.1 Membrane action
2.2.2 Transverse action
2.3 Prob1ems solved
2.3.1 Square plates under concentrated loads
2.3.2 Cy1indrica1 shells
2.3.3 Fo1ded plate structure
2.3.4 Coup1ed shear wa11s
3. Particu1ar Prob1ems of idea1ization
3.1 The inclusion of beam e1ements
3.2 The in-plane stiffness of the plate e1ements
Page
ii
iii
iv
vi
ix
xii
xiii
1
2
5
9
11
11
13
20
21
26
26
31
37
39
- vii -
Page
3.3 The ana1ysis of plane shear wa11s by finite e1ements 40
3.4 Examp1es of shear wall structures 46
3.5 Stress concentrations at the abutments of the 1inte1 beams 56
4. The analysis of three-dimensiona1 structures
4.1 The computer program
4.1.1 Description of the program
4.1.2 Capabi1ities of the program
4.1.3 Input-output
4.1.4 Particu1ar aspects of the program
4.1.4.1 The combination of beam and rectangular e1ements
67
67
71
74
80
80
4.1.4.2 Substructure ana1ysis 82
4.1.4.3 Solution of the system of equations of the vertical system 86
4.2 Examp1es of three-dimensional structures
4.2.1 Single storey structures
4.2.2 Multi-storey structures
5. The importance of the in-plane deformation of the f100r slabs
86
91
5.1 Description of the structures under investigation 110
5.2 Results
5.2.1 Deflections
5.2.2 Axial loads in the columns
5.2.3 Bending moments
5.2.4 Shears
5.2.5 Conments
115
115
119
122
122
131
- viii -
6. Conclusions and recommendations
6.1 Conclusions
6.2 Recommendations
BIBLIOGRAPHY
APPENDIX 1
APPENDIX 2
APPENDIX 3
134
136
139
145
148
149
.. ix ..
LIST OF FIGURES
1. Rectangular element. Dispiacements and force vectors. 12
2. Simply supported square plate under central concentrated load. Convergence curves for the central deflection. 22
3. Clamped square plate under central concentrated load. Convergence curves for the central deflection. 24
4. Isotropie cylindrical shell under ring load. Longitudinal moments. 30
5. Folded plate structure under distributed load.
6.
7.
8.
Geometry and finite element idealization. 32
Shear wall structure. Geometry and finite element idealization.
Shear wall structure. Lateral deflection.
Shear wall structure. Deflected shapes of slabs.
34
35
36
9. Combination of beams to a rectangular finite element. 38
10. Cantilevered beam subject to an end couple. 42
11 . Li nte 1 beam anchored in a wall. 45
12. Coupled shear walls of constant floor height 48
13. Lateral def1ections. Coupled shear walls of Fig. 12. 49
14. Moments in 1inte1 beams. 50
15. Vertical stresses in 1eft hand piers. 51
16. Convergence graphe 52
17. Coupled shear wa1ls of unequal height. Finite element and equivalent frame idealizations. 54
18. Lateral def1ections and moments in lintel beams. Example of Fig. 17. 55
19. Shear wall-frame structure. Finite element and equivalent frame idea1izations. 57
No.
20.
21.
22.
- x -
Shear wall-frame structure. Lateral deflection.
Zone of stress concentration at the abutment of a beam in a wall. Various finite element idea1izations.
Zone of stress concentration - horizontal disp1acement u.
23. Zone of stress concentration - vertical displacement v.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
Zone of stress concentration - horizontal stress
General f10w chart.
The combination of beam and plate elements.
Table structure.
Table structure of Fig. 27. Convergence graphs for the displacements at node 5.
Rectangular tab le under vertical load. Convergence curves.
Seven storey structure. Elevation and plan.
Seven storey structure. Photographe
Idealization of one quarter of the seven storey mode1.
Typical quarter f100r idea1ization. North-south loading.
Typica1 quarter f100r idea1ization. East-west loading.
East-west loading. Def1ections.
North-south loading. Def1ections.
Seven storey structure in torsion. Angle of twist.
Fifteen storey structure under 1atera1 wind load. F100r idea1ization.
Vertical system idea1ization for fifteen storey framed-tube building.
58
62-63
64
65
66
68
81
87
88
90
92
93
95
96
97
99
100
101
105
106
- xi -
40. Lateral def1ection of the frame-tube structure under horizontal load. 107
41. Framed-tube structure. Axial stresses in co1umns at third f100r 1eve1. 108
42. Eight storey, eight bay structure for wind ana1ysis. 111
43. Idea1ization of structures A6, 86. 113
44. F100r idea1ization for structures A6, 86. 114
45. Lateral def1ections of structures type A. 116
46. Lateral def1ections of structures type 8. 117
47. Axial loads in outside co1umns. 8 storey buildings. 120
48. Outside loads in outside co1umns. 4 storey buildings. 121
49. Moments in outside co1umns at the top of each f100r slab. Co1umns at 24 and 48 ft. from end shear wa11s. 123
50. Moments in outside co1umns at the top of each f100r slab. Co1umns at 72 ft. from end shear wa11s. 124
51. Moments in outside co1umns at the top of each f100r slab. Co1umns at 96 ft. from end wa11s. 125
52. Moments in outside co1umns. Four storey structures. 126-127
53. Structure type A6 .. Actua1 f100r def1ections and approximation. 133
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
- xii -
LIST OF TABLES
Moments along a diagonal of the square, simply supported plate under eoneentrated load.
Moments along a diagonal of the square, elamped plate under concentrated .load.
Radial displacements of the fixed ended eylindrieal shell under internal pressure.
Radial deflections of an isotropie cyl indri cal shell subjected to a ring load.
Longitudinal moments in an isotropie cylindrieal shell subjected to a ring load.
Folded plate structure. Defleetions at nodes 2 and 3.
Stresses in cantilever of Fig. 10a.
Loads in eolumns.
Seven storey model. Distribution of the total shear among the various structural elements.
Pereentages of total shear taken over by the various frames.
23
25
27
28
29
33
44
59-60
102-103
128-130
- xiii -
LIST OF SYMBOLS
Unless otherwise defined, the following symbols are used:
A
b
tC]
[ D]
d
dl
F, F*
[G]
[1 ]
[ K]
[KBB J
[KBI]
[ KIl]
* [KBB ]
L
m
Area of the element or cross sectional area of a beam
Suffix used to denote bending action
Matrix connecting the generalized displacements to the nodal displacements of the element
Elasticity matrix in bending. Also, diagonal matrix (chap. 4)
Depth of a beam
Distance from the centroid of a beam to the plane of an element
Modulus of elasticity in the x and y directions for orthotropic materials
Displacement functions inside the element
Elasticity matrix for shear
Shear modulus in the xy plane
Moment of inertia of a beam
Unit matrix
Stiffness matrix
Matrix of boundary forces due to unit boundary displacements
Matrix of boundary forces due to unit internal displacements
Matrix of internal forces due to unit internal displacements
Condensed floor stiffness matrix
Length of a beam
Lower triangular matrix
Suffix used to denote membrane action
[Q]
[S]
t
u, v, w
U, v, w
{ UB}
{ UI}
{ UB*}
Wij
aw w'x or ax x, y, z
{ (X }
(Xijk
8
Ôij
{ ô }
{ôB}
{ôI }
{-,b }
'Vxy ' 'Vyx
- xiv -
Moment vectors parallel to the coordinate axes x, y, z
Twisting moment
Moment in the k direction, at node i, due to a unit transverse displacement at node j
Matrix connecting the strains and generalized displacements
Matrix connecting the moments and generalized displacements of the element
Thickness of the element
Linear displacements parallel to the coordinate axes
Forces parallel to the coordinate axes
Boundary node vector
Internal node vector
Condensed floor load vector
Force W at node i, due to a unit transverse displacement at node j
Partial differential of w with respect to x
Coordinate axes
Generalized coordinates of the element
Generalized coordinates for the subregion of the element bounded by nodes i, j, k
Non-dimensional parameter
Kronecker delta
Nodal displacement vector
Boundary displacement vector
Internal displacement vector
Strain matrix
Poisson's ratios for orthotropic materials
{ ) [ ] ( ]T
[ ]-1
- xv -
Stresses in the x and y directions, and shear stress
Rotation vectors paral1e1 to the coordinate axes
Denotes a co1umn vector
Denotes a matrix or row vector
Denotes the transpose of a matrix
Denotes the inverse of a matrix
Chapter 1. Introduction
1.1 Preamble
- l -
The availability of digital computers and the growing demand
for high rise structures have resulted in considerable progress in the
analysis of tall buildings in the last twenty years during which new,
taller and more efficient types of structures evolved.
Modern buildings of the first half of this century used to
consist mainly of steel or con crete rigid frames connected by floor
slabs, but the contribution of the walls and partitions to the lateral
rigidity of the structure was ignored. As the need for more lateral
stiffness increased with the height of the buildings, vertical walls,
or vertical wall assemblies became the major components in the resis
tance to horizontal loads.
The term II shear wall ll is usually reserved to plane vertical
walls resisting the horizontal loads applied in that plane. Wall
assemblies normally found around elevator and utilities' shafts and
resisting horizontal loads from all directions are referred to as
IIcoresll.
The popularity of core structures in which the external columns
play little or no part in resisting the horizontal loads is closely
related to the advent of the modern slip form technology. Most recently,
facade panels consisting of closely spaced columns and floor beams have
been made to contribute significantly to the lateral stiffness of very
tall structures, hence the idea of the building being enclosed within
a structura l Il tube ll .
- 2 -
1.2 Methods of Analysis of Tall Buildings
This work is concerned with the linear, static analysis of
* tall buildings. Conferences (1) and excellent review pa pers (2)
have contributed to assess our knowledge in this field.
There is little need to recall the old methods of analysis
of plane frames based on Hardy Cross' relaxation technique and suited
for hand calculations (3,4). The I canti1ever" and "portal" methods
are still in use for preliminary design but computer programs have
taken over for more accurate work (5). Such is the case also for
the analysis of three-dimensional rigid or braced frame structures.
The essential prob1em of the coupled plane shear walls subjected
to horizontal loads has been the object of considerable attention during
the last ten years. Three main trends may be distinguished in the work
published so far. In the continuum approach, the lintel beams connect
ing the walls are replaced by an equivalent lamina and a continuous
shear function is substituted for the redundant shear forces in the
beams. This approach is best il1ustrated by the works of Beck (6),
Rosman (7), Coull and Chaudhury (8,9), and others. It is applicable
to shear walls having one or severa1 rows of openings (7,10) and to
regu1ar shear wall-frame structures of constant floor height (11).
More adaptable to other configurations is the equivalent frame
method where the shear walls are considered as wide columns (12); the
whole problem is then reduced to the analysis of a plane frame which
* The numbers in brackets refer to the bibliography.
- 3 -
may be performed by applying the general stiffness or displacement
methods calling for the solution of a system of linear equations
(5,13) or an iterative procedure (14). In this case irregular com
binations of shear walls and frames may be considered.
Less publicized are the capabilities of the finite element
method which is well suited for the analysis of shear walls of any
shape, at the cost of only minor refinements.
Symmetrical three-dimensional structures consisting of parallel
frames and shear walls may be reduced to an equivalent two-dimensional
wall-frame system if it is assumed that the floor slabs behave as rigid
diaphragms in their own plane and that no vertical shears are trans
mitted from one frame to another (15,16,17).
For the analysis of true three-dimensional structures, that
is structures that may not be reduced to two-dimensional configurations
by virtue of their symmetry, Jaeger et al (lB), Rosman (19) and Gluck
(20) have extendèd the continuum method of analysis of plane shear
walls and have obtained the deflections and internal actions by solving
a system of linear differential equations of equilibrium or compatibility
(19).
With the same approach, Michael (2l) has studied the torsional
behaviour of core walls coupled by lintel beams.
Winokur and Gluck (22) and Stamato and Stafford Smith (23), on
the contrary, call for the solution of a system of linear equations of
equilibrium obtained from a stiffness analysis. Coull and Irwin (24)
combine the advantages of bath continuum and Qquilibrium approaches.
- 4 -
In all these cases, coupling between the individual shear
walls or frames located in different planes is performed by the floor
slabs, which are assumed again to be infinitely rigid in their own
plane. This assumption is extremely helpful in reducing the number
of redundancies considered in the analysis, since only three rigid
body degrees of freedom, plus a number of vertical displacements at
convenient points are then needed to describe completely the behaviour
of the structure under lateral as well as vertical loads. Also, when
evaluating the interaction between these vertical bents,the warping
rigidity of the floor slabs is neglected. This assumption is not
compatible with the usual presente of heavy floor beams perpendicular
to the shear walls.
Dickson and Nilson (25) have considered the case of cellular
buildings made of continuous shear walls and slabs subjected to mem
brane actions only.
Goldberg (26) has performed the analysis of structures consisting
of parallel shear walls and frames. The warping stiffness of the floor
slabs was again neglected; however, the effect of their in-plane deform
ations was included. This work was extended by Majid and Croxton (27)
in order to take into account the axial strains in the load bearing
parts of the structure, as well as the effect of accentric vertical
loads. The analysis of a ten-storey building having a length/width
ratio of only 3.2:1 and a slab thickness of 3 in. was carried out
according to this approach. The wind was resisted by plane shear
walls situated at the extremities of the building and by interior
three bay portal frames. The results were compared to those obtained
- 5 -
by assuming rigid floors. A discrepancy of up to 100% was observed
in the evaluation of the moments et the base of the central columns (27).
This result means that the usual assumption of the floors being
rigid in their own plane may be uBsafe. The authors do not indicate,
however, the value of the length/width ratio at which the discrepancy
begins to be significant. The method they use is, in addition, restricted
to the study of one type of structures consisting of floor slabs and of
parallel shear walls and ,frames, in which torsional effects are resisted
by forces in these planes only.
1.3 Scope of the Present Work
These remarks emphasize the need for a more general method of
analysis of building structures such that all aspects of their behaviour
may be considered at once. The preferred method will be the one in which
several loading conditions may be taken into account at little extra cost.
Stafford Smith (2), Majid and Croxton (27) and others point out
to the finite element method as the most powerful available technique.
This approach was demonstrated by Majid and Williamson (28).
However, the very large number of equations to be solved did not allow
them to analyse more than very small and elementary structures.
It is proposed in this work that this number of equations may
be significantly reduced by recognizing that most tall buildings conslst
of a restricted number of columns and shear walls connected by a large
number of floor slabs of a few different types, and by treating the
latter as substructures. A computer program was written along these
lines.
- 6 -
The procedure calls for the following steps:
1. Each typical floor is idealized by means of discrete
elements, six degrees of freedom being considered at
each node; its stiffness matrix and applied load vector
are generated.
2. Those nodes common to the floor and to the vertical system
consisting of columns and shear walls are regognized as
boundary nodes; the floor stiffness matrix and 10ad vector
are condensed so that only those degrees of freedom at the
boundary nodes are retained.
3. Steps land 2 are repeated for all typical floors; their
reduced stiffness matrices and load vectors are stored on
disco
4. The stiffness matrix and the load vector of the vertical
system are generated. At the intersection with each floor,
the stiffness coefficients and load vector of the appropriate
typical floor are retrieved from storage and added.
5. The system of equations thus formed for the whole structure
is solved, and the displacements at all the nodes of the
vertical system are obtained.
6. If necessary, the displacements, stresses and strains inside
the floors may also be calculated.
In order to make the program as compact as possible, the same
formulation for all the plate elements, whether they belong to the floor
substructures or to the shear walls has been adopted. Rectangular elements
were chosen for their good convergence properties and simplicity. This
- 7 -
imp1ies that structures made of rectangu1ar para11e1epipeds, a10ne,
may be analysed by the program described herein. Shou1d the need arise,
however, this limitation cou1d be removed by the introduction of sub
routines for the derivation of triangu1ar (38), or even quadrilatera1
e1ement properties. This wou1d th en bring buildings of triangu1ar or
curved plan within the capabi1ities of the program.
Within this ' para11e1epiped context l, a11 possible configurations
may be accepted: frames, coup1ed shear wa11s, enc10sed shafts or coup1ed
core wa11s, slabs, whether mono1ithic or with ho1es, etc. The slabs and
wa11s themse1ves may be reinforced with edge beams or ribs and diagonal
braces may be inc1uded.
In order that a complete building structure may be represented
successfu11y by means of finite e1ements, a number of idea1ization
prob1ems have had to be solved. These are:
1. The selection of adequate disp1acement functions such that
a rapid convergence towards the true behaviour is attained
with a minimum number of e1ements and such that compatibi1ity
of disp1acements between adjacent orthogonal e1ements is
achieved. This point is dea1t with in chapter 2 and led
to sorne work on new disp1acement fields for the bending
of rectangu1ar plate elements.
2. The inclusion of the abi1ity to treat beam e1ements in such
a way that a variety of configurations such as rigid frames,
diagonal braces, 1inte1 beams, etc. may be considered within
a single idea1ization pattern.
- 8 -
The use of such beams for the evaluation of the in-plane
rotational stiffness of plate elements led to a new finite element
formulation of the plane coupled shear wall problem, in which semi
rigid connections between the lintel beams and the walls may be con
sidered. These points are treated in chapter 3.
The computer program itself, including the substructure analysis
is described in chapter 4, in which it is also used for the solution of
a number of three-dimensional structures.
In particular, in chapter 5, the analysis of various buildings,
four and eight storeys high, and resisting the lateral wind loads by
means of end shear walls and intermediate frames, is performed. The
number of bays in elevation and hence the length/width ratio of the
floors is treated as the variable.
This study provides additional information on the effect of
the deformation of the floor slabs in their own plane and demonstrates
the significant transfer of vertical loads between the shear walls and
the adjacent frames through the floor slab and beams.
Conclusions are drawn and recommendations made in chapter 6.
As may be seen, chapters 2 and 3 are concerned with very particular
idealization problems. They are treated as separate entities and may be
considered as the first part of the work.
- 9 -
Chapter 2. Choice of Finite Element Formulations
2.1 Introduction
A considerable number of finite element formulations have been
developed, both for membrane and plate bending problems (29).
In the analysis of building structures, however, the choice is
limited for the following reasons:
a) it is advantageous that the degrees of freedom considered
at each node be only the physical translations and rotations
in the direction of the coordinate axes;
b) mid-side nodes should be avoided in order to keep the band
width of the stiffness matrices to a minimum;
c) the edge displacements of adjacent elements situated in
orthogonal planes should be compatible. This requirement
is particularly important for the junction between floor
slabs and shear wa11s if a proper value is to be obtained
for the stiffness of their assemb1ed structure.
Membrane disp1acement functions satisfying these requirements
consist of 1inear po1ynomia1s for the disp1acements u and v a10ng the
x and y axes, for triangula ... e1ements, and of bi1inear polynomia1s
in the case of rectangu1ar e1ements (29,30,31).
Due to the higher order of the latter po1ynomials, and hence
to their better convergence properties, the rectangular e1ements have
been selected in this study.
In bending, Irons (32) showed that it was impossible to formu1ate
a conforming displacement function if the transverse displacement w and
- 10 -
the slopes ~, aw at the nodes, alone, were considered as degrees ax ay of freedom, and if Kirchoff's assumptions were to be satisfied at the
same time.
Other element formulations for the bending of triangles thus
evolved, in which the lateral deflection w and the rotations ex' ey were expres5ed separately; shear deformations were thus taken into
account. This allows compatibility conditions to be satisfied separately
for w and ex' ey .
Such formulations were obtained by Melosh (33), Utku (34),
and Dhatt (35,36) and were used in a variety of problems such as
plate bending, shell analysis, analysis of tubular connections, etc.
(37,38,39).
The development of rectangular bending elements has been more
rewarding than the triangular ones. The non-conforming displacement
function containing twelve terms is well known (31). Bogner, Fox and
Schmit (40) restored the compatibility of slopes in the direction normal
to the interface of adjacent elements by adding a fourth degree of free
dom (a2w/axay) at each node.
In order to obtain a conforming formulation with only three
degrees of freedom per node, shear deformations must again be included.
This was done by Wempner, Oden and Kross (41) who introduced
Kirchoff's constra;nts at a number of discrete points within the struc-
ture.
In this chapter, a new formulation for the bending of rectangular
elements with three degrees of freedom per node ;s discussed. Shear
deformations are included and the stiffness matr;x is derived by means
- 11 -
of an equi1ibrium a1gorithm.
Various resu1ts are given in section 2.3 and these compare
very favourab1y with those obtained by means of other formulations
having the same degrees of freedom.
The work described in this chapter was initiated by Mufti and
carried out by the author.
2.2 Element Formulation
A typical element, and a set of right-handed rectangular
Cartesian axes attached to it have been shown in Fig. 1. The positive
directions of the 1inear disp1acements u, v, w and of the rotations
ex' ey, ez are also indicated.
2.2.1 Membrane Action
The e1ement properties are developed within the scope of the
small def1ection plate theory, so that the membrane and transverse
disp1acements effects may be uncoup1ed. The displacements u and v
are given by:
{~} - [: ;]. {~m} (1)
where
F = [1 X Y xy ]
{OCml= (0(1 ~ ....... 0(8) T
for which a stiffness matrix D<rrJ may be derived easi1y by means of wel1
documented procedures (29,30,31). It will be noted that no provision
is made at this stage for the inclusion of the in-plane rotations ez'
and of the corresponding moments Mz'
- 12 -
z
FIGURE 1. Rectangular Element. Oisplacements and Force Vectors
- 13 -
2.2.2 Transverse Action
In the absence of shear deformations, i.e. when Kirchoff's
hypotheses hold, the assumption of a linear distribution for sx' Sy
results in a constant bending moment which would imply a state of zero
transverse shear. Conversely, the assumption of a constant traverse
shear would require a quadratic variation of the rotations or a cubic
distribution of transverse displacements.
When shear deformations are considered, however, independent
displacement functions which do not comply with these observations may
be chosen for w and for ex' Sy.
Two different formulations were investigated.
The displacement functions considered initially are as follows:
w
where
and again
F = [) X Y xy]
Also, let
{~} - [Cs J.{b} { ex b} :: [ Co l { S }
(2)
(3)
(4)
where
{ , } =
t 6i } --
- 14 -
T ('. ~2 S, !4)
( Uj Vi 'IIi 9xi 9y i Bzj ) T
Functions 2 and 3 may be determined unique1y in terms of the nodal disp1acements, hence compatibi1ity of transverse disp1acement and rotations is ensured a10ng the four edges of the e1ement. However, Kirchoff's assumption
(5)
is no longer va1id within the e1ement, as mentioned above. In this formulation, the contributions of the transverse disp1acements and the rotations to the strain energy of the e1ement are obtained separate1y; the correspondi ng sti ffnesses are defi ned as [KJ and [Kb]. The el ement stiffness matrix [K] may thus be expressed as
(6)
2.2.2.1 Bending Stiffness Matrix
The strain matrix)tb is defined by
(7)
and may be expressed in terms of {ab} by differentiation of 3 .
- 15 -
Its terms correspond to the curvature in the x and y directions
respective1y, and the twist. Let
For the orthotropic materia1, the e1asticity matrix [D] has the fonn
Dx D, 0
[1>] = D, Dy 0
where 0 0 D~
DJ( =a e)l / ( J - "")CoY Yyx )
Dy 18 Ey / (1 .. ~)(y "1yx)
01 c: ~yx E ~ / (1 ... ".Y "'Iyx ) o.)' = Gxy
(9)
The stresses at the top surface of the e1ement of thickness tare
given by
(10)
They give rise to bending moments per unit 1ength
(11 )
so that the bending strain energy may be obtained in c10sed fonn as
Ub '" t ~ {Mbt {tb} dA = t { t fib}~"tsbl~{Xb} dA
= t~{~ fT. [Cb] T [{[QJ ![J)]'[~b] dA].{Cb1-{ A} (12) 2.4
- 32 -
20' z
1~ 1..1
60'
Ixe d d e lae "-
-0 C\J
simple supporj/
fixed edae / V 60' r
FIGURE 5. Folded plate structure under distributed lQad.Geometry and finite element idealization (E = 3 x 106 psi,v= 0.2, thickness 5 in.).
1
lb .C\I
ct·
1
1
1 1
2 1
3 •
J l
- 33 -
TABLE 6
Fo1ded plate structure-def1ections at nodes 2 and 3
(in 10-3 in. and 10-3 rad)
Ana1ytica1 TRICUB TRILIN QUARTREC NEW
v2(in.) 0.5320 0.4752 0.4717 0.5027 0.5039
w2(in.) -0.3273 -0.2971 -0.2930 -0.3077 -0.3081
92(rad) -0.0439 -0.0341 -0.0297 -0.0357 -0.0296
v3(in.) 0.5246 0.4757 0.4728 0.4951 0.4963
w3(in.) 0.3127 0.2782 0.2761 0.2934 0.2939
93(rad} 0.01165 0.0091 0.0090 0.0090 0.0090
- 34 -
36"
el. 30" ~~~~,~ \
.. .f·V 1 1 1 .~
~---'----I ~ ~
1 ! : u 1 i ' ~-- -r- - --: 1 1 1 1 l ,
500 lb
el. 15" ~ , 10001 1 1 1
L. __ ........ ___ J 1 1 1 1 1 1 1 l , l , 1
1 1 1 1 1 r-- - --.- - --, 1 1 1
1 1 1 1 1 1
z
thickness: 0,75 in
FIGURE 6. Shear wall structure. G~ome~ry a~d finite e1ement "idea1ization (E = 3 x 10 pSl, ~ - 0.2).
elevations
30 in
- 35 -
equivalent frame left ier right pier
15
'---___ ~ ___ ---'-___ __L.,;;u~(in ) o 0.01 0.02 0.03
FIGURE 7. Shear wall structure. Lateral deflection.
w(in)
.005
wOn) ~0,01
-0.01
6
6
- 36 -
elevation 30 in
y=o y= 1.5 Y=3.75 Y=6.
15
...... _""
y=o Y=1.5 Y= 3.75 Y=·6.
elevation 15 in
FIGURE 8. Shear wall structure. Deflected shapes of slabs.
30 x(jn)
x(in) 30
- 37 -
Chapter 3. Particu1ar Prob1ems of Idea1ization
3.1 The Inclusion of Bearn Elements
In addition to the rectangu1ar plate e1ements described above,
linear, or beam e1ements are also needed in the ana1ysis of building
structures. The six displacements and rotations of Fig. 1 are consi~
dered at each end of the beam, and these 1ead to a 12 x 12 stiffness
matrix which is readily available in the 1iterature (5). The organi
zation of the program is as follows:
Each beam element is assigned to a rectangu1ar e1ement, of which
it connects two nodes. Conversely, one rectangu1ar plate element may
"support" up to six beam elements along its sides and diagonals. These
six elements are markeci "a" to "f" in Fig. 9. Isolated lintel beams,
columns or diagonal braces are likewise assigned to a plate e1ement of
zero thickness.
viz:-
The calculations are organized to proceed element by element
i) the stress, strain and stiffness matrices are generated
fûr the plate itself;
ii) the stiffness matrix of the beams is calculated, and both
are added together.
Should the thickness of the e1ement be zero, or should no beam
be assigned to it, then the corresponding part of the calculations is
omitted automatically. This approach has two major advantages; firstly,
the required computation time is very small; second1y, since in most
cases one of the principal planes of the beams is parallel to the plane
- 38 -
4 c x 3
b x
y
1 a x 2
FIGURE 9. Combination of beams to a rectangular finite element.
- 39 -
e1ement, no further definition of the orientation of the members is
required. A f10w chart describing this particu1ar aspect of the program
is given in chapter 4. Provision for offset beams is inc1uded, and the
complete stiffness matrix of a beam situated at a distance d above the
e1ement is given in Appendix 2.
3.2 The In-plane Stiffness of the Plate Elements
In the formulation of the e1ement, on1y expressions for the u
and v in-plane disp1acements have been considered.
Since the contribution of the moment Mz acting in the plane of
each e1ement is ignored in the eva1uation of its stiffness matrix, the
latter contains four rows and co1umns of zero terms. When a11 the
e1ements situated around one node are cop1anar, this causes a singu1ar
ity, for after transformation into the global system of axes attached
to the structure, on1y five out of the six equations of equi1ibrium
at that point are 1inear1y independent. When the plane of the e1ements
is para11e1 to a coordinate plane, the rows and co1umnsof zero terms
remain unchanged, and the prob1em may be overcome by adding large
numbers to the corresponding diagonal terms after formation of the
global stiffness matrix (31,37). For other orientations of the plane
common to the adjacent e1ements, arbitrary sma1l terms may be inserted
at the intersections of the zero rows and co1umnsof each e1ement ' s
stiffness matrix.
An expression was given for triangu1ar elements (48). It was
extended by Mufti for rectangular elements, and reads as follows (64):
- 40 -
l -0.333 -0.333 -0.333
-0.333 l -0.333 -0.333 = 8txcYCGxy • (23) MZ3 -0.333 -0.333 l -0.333 6Z3
MZ4 -0.333 -0.333 -0.333 l 6Z4
in which 8 is a small constant. The introduction of these terms removes the singularity problem connected with the 6Z displacement, irrespective of the orientation of the elements plane. It does not provide, however, for the transmission of a moment applied in that plane.
This is a major obstacle in the analysis of building structures. Such moments are transmitted by lintel beams to shear walls and also by twisted columns to floor slabs and are of the greatest importance.
In the remaining part of this chapter, the problem of the plane shear wall will be considered in detail.
3.3 The Analysis of Plane Shear Walls by Finite Elements In order to circumvent the problem outlined in section 3.2
regarding the transmission of moments applied in the plane of the elements, a number of special formulations have been developed.
McLeod (49) developed a new element by adding av/ax and au/ay as a degree of freedom at alternate nodes, thus introducing some asymmetry in the idealization (50). Spira and Lokal proposed additional displacement fields corresponding to unit rotations of each of the four nodes (51) whenever necessary; however, no results were given. Pole (52) used four degrees of freedom at each node thereby increasing the storage requirements considerably. Girijavallabhan, on the other
- 41 -
hand, replaced the lintel beams by a double row of slender rectangular
elements (53,54). Besides the enormous number of additional nodes
required by this method the stiffness of the lintel beams was exaggerated
so that the overall behaviour of the wall was slightly modified. Oakberg
and Weaver (55) used specialized elements along the edges of the wall.
Majid and Williamson (28) proposed to include flexural members along
the diagona1s of those plate elements whose in-plane stiffness was to
be considered.
In the present formulation, fictitious beams, having a zero
axial rigidity, and a non-zero flexura1 rigidity are added along the
sides of the elements as needed. A1though less systematic, this
procedure is preferred for its accuracy and its versati1ity. It
enables the ana1yst to reproduce the actual displacements and stress
distributions more c10se1y in a variety of circumstances, as is demons
trated in the two fo110wing examp1es.
In Fig. 10 a simple cantilever subjected to an end moment is
shown. Three idealizations were used. Idea1ization A was obtained
by adding 2 vertical beams joining the end nodes of the cantilever.
In idea1ization B the beams were p1aced along the diagonals of the
two last elements, as suggested in (28). In these two cases the
beams were of the same material as the cantilever, and their moment
of inertia was given a nominal value of 0.1 in4. The cross-sectional
area of these beams was set to zero, in order not to modify the in-plane
behaviour of the plate e1ements.
Idealization C was obtained by adding beams along all the
element boundaries, and was included to see if a uniform beam pattern
~ 6.5 in 18 elements l 1
/ A /
/ / /
B / / /
C ; nlDDDDDDDDODDDDDDD DDDDDDDDDDDDDDDD /
(a)
~
~
r-
1=
c ~. Q
C\J
M)
t = 0.1
E = 107 psi
v = 0.3 M = 1000 in.lb.
~
1.0 2.0 3.0 4.0 5.0 6.0 x
2)( 10-3
3xl0-3
deflections ( in)
A+C
( b)
FIGURE 10. Cantilevered beam subjected to an end couple.
- 43 -
cou1d be adopted for the solution of these prob1ems. In this case
the beams form a complete frame superimposed on the cantilever and
cou1d stiffen it considerab1y. A sma11 investigation showed that
a f1exura1 rigidity of 10-1 lb in2 was the most suitable. Fig. lOb
gives the deflections of the cantilever in these three idealizations.
Table 7 giving the stresses crx' cry and crxy emphasizes the ability of
the first idealization to depict the stresses more accurately near
the point of application of the moment. Idealizations Band C induce
direct stresses in the y direction and also shear stresses to the
detriment of axial stresses. This explains the deflected patterns
of Fig. lOb.
In fig. 11, a typical case of a beam, embedded in a plane
continuum is shown. Again, the formulation of the simple rectangular
element does not provide the rotational stiffness required at node A
to ensure the fixity of the beam. Additional beams are thus provided
to elements land 2. This may be done according to Figs. lla, llb or
11c. For a detailed analysis of the stress concentration, an appropriate
mesh size wou1d be such that the height h be made equal to one third
of the depth of the embedded beam. Scheme llc is then most adequate.
When a coarser mesh is adopted schemel1a is preferred, for the moment
at A is then resisted by vertical reactions only. No horizontal stresses
are induced in the wall so that its overall stress distribution is only
little affected. This is shown in detail in Section 3.4. Fig. lla may
also be used to demonstrate how semi-rigid connections between the beam
and the wall may be idealized.
- 28 -
TABLE 4
Radial defleetions of an isotropie eylindrieal shel1 subjeeted
to a ring load (in.)
Node Analytieal TRILIN QlIARTREC NEW
1 -0.01286 -0.010816 -0.01278 -0.01221
2 -0.00949 -0.00840 -0.00941 -0.00947
3 -0.00431 -0.00416 -0.00442 -0.00457
4 -0.001 09 0.00133 -0.00120 -0.00128
5 0.00028 0.00014 0.000048 0.000017
6 0.00047 0.00006 0.00015 0.00012
7 0.00022 0.0 0.0 0.0
Applied load: 200 lb/in.
- 29 -
TABLE 5
Longitudinal moments in an isotropie ey1indriea1 she11
subjeeted to a ring load (in. -lb)
Element TRILIN QUARTREC NEW
1 -17.077 -18.955 -19.431
2 3.298 3.166 3.249
3 6.884 7.976 8.312
4 4.971 5.897 6.164
5 2.306 2.493 2.545
6 0.373 0.616 0.845
10
7
-10
-30 M
6 5 4
FIGURE 4. Isotropie eylindrieal shell under ring load. Longidudinal moments, analytieal curve and values obtained with element NEW (in. -lb).
w o
- 31 -
the four types of elements listed above. Very good convergence is
achieved.
is 10-8•
A value Of)' in formula 23 suitable for this problem
2.3.4 Coupled Shear Walls
Finally, the analysis of the shear wall structure shown in
Fig. 6 is given as a further demonstration of the properties of the
finite element formulation proposed. The structure consists of two
shear walls rigidly fixed at their base and connected by two thin
slabs. Concentrated lateral loads are applied as shown. The lateral
deflections are plotted in Fig. 7, along with those obtained by the
equivalent frame method of analysis of shear walls (12). In the latter
case, the effective width of the slab was taken as the clear opening
between the walls.
Excellent correlation is achieved. In Fig. 8, the deflected
shapes of various lines drawn on the slabs are shown.
- 32 -
20' z
2
60'
Ixe d d e Ige
-'"
{ simple sUPQory
fixed edge / v 60'
FIGURE 5. Fo1ded plate structure under distributed 1Qad.Geometry and finite e1ement idea1ization (E = 3 x 106 psi,v= 0.2, thickness 5 in.).
1
lb .N
. • 1
ct·
1
1
1 1
2 1
3 •
j 1
- 33 -
TABLE 6
Fo1ded plate structure-def1ections at nodes 2 and 3
(in 10-3 in. and 10-3 rad)
Ana1ytica1 TRICUB TRILIN QUARTREC NEW
v2(in.) 0.5320 0.4752 0.4717 0.5027 0.5039
w2(in.) -0.3273 -0.2971 -0.2930 -0.3077 -0.3081
92(rad) -0.0439 -0.0341 -0.0297 -0.0357 -0.0296
v3(in.) 0.5246 0.4757 0.4728 0.4951 0.4963
w3( in.) 0.3127 0.2782 0.2761 0.2934 0.2939
93(rad) 0.01165 0.0091 0.0090 0.0090 0.0090
- 34 -
36"
el. 30" 1 i 1
~-- -r- - --: 1 l , l , 1 1 1 1
• • l , 1 1----,-- --, , . , : 1 1 l , 1
500 lb
, , 1 1 1 L __ .. _L.. ___ J • 1 1 • 1 1 1 1 1 1
1 • , 1 1 r----.----, l , 1
, l ' 1 1 1
z
thickness·. 0.75 in
FIGURE 6. Shear wall structure. GSome~ry a~d finite element "j dea l i za t ion (E = 3 x 10 pS 1 ,~ - 0.2).
elevations
30 in
- 35 -
equivalent frame left ier right pier
15
~ ______ ~ ________ ~ ________ ~u~(in)
o 0.01 0.02 0.03
FIGURE 7. Shear wall structure. Lateral deflection.
w(in)
.005
w(in)
~O.Ol
-0.01
6
6
- 36 -
elevation 30 in
y=o y= 1.5 Y=3.75 Y=6.
15
y=o Y=1.5 Y= 3.75 Y=·6.
elevation 15 in
FIGURE 8. Shear wall structure. Deflected shapes of slabs.
30 x (in)
x(in) 30
- 37 -
Chapter 3. Particu1ar Prob1ems of Idea1ization
3.1 The Inclusion of Bearn Elements
In addition to the rectangu1ar plate e1ements described above,
linear, or beam e1ements are a1so needed in the ana1ysis of building
structures. The six disp1acements and rotations of Fig. 1 are consi
dered at each end of the beam, and these 1ead to a 12 x 12 stiffness
matrix which is readi1y availab1e in the 1iterature (5). The organi
zation of the program is as follows:
Each beam element is assigned to a rectangu1ar element, of which
it connects two nodes. Conversely, one rectangu1ar plate e1ement may
"support" up to six beam elements a10ng its sides and diagona1s. These
six elements are markeci "a" to "f" in Fig. 9. Iso1ated linte1 beams,
columns or diagonal braces are likewise assigned to a plate e1ement of
zero thickness.
viz:-
The ca1culations are organized to proceed element by element
i) the stress, strain and stiffness matrices are generated
fûï the plôte itself;
ii) the stiffness matrix of the beams is calculated, and both
are added together.
Should the thickness of the e1ement be zero, or should no beam
be assigned to it, then the corresponding part of the calculations is
omitted automatically. This approach has two major advantages; firstly,
the required computation time is very small; secondly, since in most
cases one of the principal planes of the beams is parallel to the plane
- 38 -
4 c x 3 r-------------~~------~
b x
y
1 a x 2
FIGURE 9. Combination of beams to a rectangular finite element.
- 39 -
e1ement, no further definition of the orientation of the members is
required. A f10w chart describing this particu1ar aspect of the program
is given in chapter 4. Provision for offset beams is inc1uded, and the
complete stiffness matrix of a beam situated at a distance d above the
e1ement is given in Appendix 2.
3.2 The In-plane Stiffness of the Plate Elements
In the formulation of the e1ement, on1y expressions for the u
and v in-plane disp1acements have been considered.
Since the contribution of the moment Mz acting in the plane of
each e1ement is ignored in the eva1uation of its stiffness matrix, the
latter contains four rows and co1umns of zero terms. When a11 the
e1ements situated around one node are cop1anar, this causes a singu1ar
ity, for after transformation into the global system of axes attached
to the structure, on1y five out of the six equations of equi1ibrium
at that point are 1inear1y independent. When the plane of the e1ements
is paral1e1 to a coordinate plane, the rows and co1umnsof zero terms
remain unchanged, and the prob1em may be overcome by adding large
numbers to the corresponding diagonal terms after formation of the
global stiffness matrix (31,37). For other orientations of the plane
common to the adjacent e1ements, arbitrary sma11 terms may be inserted
at the intersections of the zero rows and co1umnsof each e1ement ' s
stiffness matrix.
An expression was given for triangu1ar e1ements (48). It was
extended by Mufti for rectangular elements, and reads as fo110ws {64}:
- 40 -
l -0.333 -0.333
-0.333 l -0.333 = BtXcYCGXY • (23) MZ3 -0.333 -0.333 l
MZ4 -0.333 -0.333 -0.333 l
in which B is a sma1l constant. The introduction of these terms removes the singularity problem connected with the eZ displacement, irrespective of the orientation of the elements plane. It does not provide, however, for the transmission of a moment applied in that plane.
This is a major obstacle in the analysis of building structures. Such moments are transmitted by lintel beams to shear walls and also by twisted columns to floor slabs and are of the greatest importance.
In the remaining part of this chapter, the problem of the plane shear wall will be considered in detail.
3.3 The Analysis of Plane Shear Wa11s by Finite Elements In order to circumvent the problem outlined in section 3.2
regarding the transmission of moments applied in the plane of the elements, a number of special formulations have been developed.
McLeod (49) deve10ped a new element by adding av/ax and au/ay as a degree of freedom at alternate nodes, thus introducing sorne asymmetry in the idealization (50). Spira and Lokal proposed additional displacement fields corresponding to unit rotations of each of the four nodes (51) whenever necessary; however, no results were given. Pole (52) used four degrees of freedom at each node thereby increasing the storage requirements considerably. Girijavallabhan, on the other
- 41 -
hand, replaced the lintel beams by a double row of slender rectangular
elements (53,54). Besides the enormous number of additional nodes
required by this method the stiffness of the lintel beams was exaggerated
so that the overall behaviour of the wall was slightly modified. Oakberg
and Weaver (55) used specialized elements along the edges of the wall.
Majid and Williamson (28) proposed to include flexural members along
the diagonals of those plate elements whose in-plane stiffness was to
be considered.
In the present formulation, fictitious beams, having a zero
axial rigidity, and a non-zero flexural rigidity are added along the
sides of the elements as needed. Although less systematic, this
procedure ;s preferred for its accuracy and its versatility. It
enables the ana1yst to reproduce the actual displacements and stress
distributions more c10se1y in a variety of circumstances, as is demons
trated in the two following examples.
In Fig. 10 a simple cantilever subjected to an end moment is
shown. Three idea1izations were used. Idealization A was obtained
by adding 2 vertical beams joining the end nodes of the cantilever.
In idealization B the beams were p1aced a10ng the diagonals of the
two last e1ements, as suggested in (28). In these two cases the
beams were of the same materia1 as the cantilever, and the;r moment
of inertia was given a nominal value of 0.1 in4. The cross-sectional
area of these beams was set to zero, in order not to modify the in-plane
behaviour of the plate e1ements.
Idea1ization C was obtained by adding beams along all the
element boundaries, and was included to see if a uniform beam pattern
65 in 18 elements ,. 1
/ A
/ / /
B / / /
~I
c ~. Q
C\I
~M) ~
t = 0.1 E = 107 psi
v = 0.3 M = 1000 in.lb.
1-
DDDDDDDDDDD~ r-
/ F := M) / DDDDDDDDDDOrnm - .....
c /
(a)
10 2.0 3.0 4.0 5.0 6.0 x
2)( 10-3
3<10-3
deflections ( in)
A+C
( b)
FIGURE 10. Canti1evered beam subjected to an end couple.
- 43 -
cou1d be adopted for the solution of these prob1ems. In this case
the beams form a complete frame superimposed on the cantilever and
cou1d stiffen it considerab1y. A sma11 investigation showed that
a f1exura1 rigidity cf ïO-1 lb in2 was the most suitab1e. Fig. lOb
gives the def1ections of the cantilever in these three idea1izations.
Table 7 giving the stresses crx' cry and crxy emphasizes the abi1ity of
the first idea1ization to depict the stresses more accurate1y near
the point of application of the moment. Idealizations Band C induce
direct stresses in the y direction and a1so shear stresses to the
detriment of axial stresses. This explains the deflected patterns
of Fig. lOb.
In Fig. 11, a typica1 case of a beam, embedded in a plane
continuum is shown. Again, the formulation of the simple rectangu1ar
e1ement does not provide the rotationa1 stiffness required at node A
to ensure the fixity of the beam. Additional beams are thus provided
to e1ements 1 and 2. This may be done according to Figs. lla, 11b or
11c. For a detai1ed ana1ysis of the stress concentration, an appropriate
mesh size wou1d be such that the height h be made equa1 to one third
of the depth of the embedded beam. Scheme l1c is then most adequate.
When a coarser mesh is adopted schemella is preferred, for the moment
at A is then resisted by vertical reactions on1y. No horizontal stresses
are induced in the wall so that its overa11 stress distribution is on1y
1itt1e affected. This is shown in detail in Section 3.4. Fig. lla may
also be used to demonstrate how semi-rigid connections between the beam
and the wall may be idealized.
x/L
A
B
C
A
B
C
A
B
C
- 44 -
TABLE 7
Stresses in cantilever of Fig.10a
(in lb /in. 2)
0.0845 0.254 0.423 0.592
Stresses (]x theory 7.5
7.230 7.226 7.227 7.228
5.78 5.777 5.778 5.780
7.230 7.225 7.226 7.227
Stresses (]y theory 0.0
0.0747 0.000007 0.00007 -0.000106
0.0596 -0.000402 0.00258 0.0902
0.0747 0.0037 0.000152 0.001037
Stresses (]xy theory 0.0
0.000888 0.000872 0.000824 0.000427
0.001048 0.000629 0.000137 0.0491
0.000938 0.000892 0.000839 0.000564
0.761 0.923
7.229 7.244
5.781 5.735
7.233 6.802
-0.000259 -0.00012
0.8180 -1. 741
0.06930 -8.446
0.000183 -0.000335
0.7748 3.433
-0.009368 2.2606
- 45 -
L
h 1
------- A d Eb1b ~
h 2
t----+-IA d .A-------I
( b)
1 wall thickness: t L
h 1
lA 1
d
h 2
1 1
(c)
FIGURE 11. Linte1 beam anchored in a wall.
- 46 -
Let suffi ces a and b refer to the two spans of the beam of
Fig. lla. Then the rotation of that beam at A is
(24)
Michael (56) has suggested that the rotation SA due to the elastic
stress concentrations in the wall be expressed as
(25)
in which d is the depth of the lintel beam and t the thickness of the
wall. Equating the right hand sides of equations 24 and 25 gives
the following expression
(26)
Equation 26 gives a realistic value of the inertia of the fictitious
beams added to the shear \O/all. When total fixity is assumèd at A a
large value is substituted for lb.
Ea and Eb correspond to the materials used for the beam and
the shear wall respectively, and are the same when equation 26 is used.
This is not a requisite of the program, however, and different values
of Ea and Eb may be specified. This would be the case of a stell beam
embedded on the side of a concrete wall. In this case, equation 25
would be replaced by another expression giving the true stiffness of
the connectors.
3.4 Examples of Shear Wall Structures
Several problems have been analysed using this simple procedure.
Three of them are included here; another example may be found in (57).
A special computer program was used for this investigation. It was first
- 47 -
given in (31), and was modified by Mehrotra and Redwood (58). It
was then deve10ped to accommodate rectangu1ar as well as beam elements.
The first problem is the analysis of coupled shear walls connec
ted by regularly spaced lintel beams. The dimensions and material pro
perties are taken from (54), but small modifications were made so as
to meet the requirements of the continuum method of analysis. The
cross sectional area and the inertia of the top beam were thus halved;
the app1ied load initially distributed along the left hand side of the
structure was divided so as to make the problem skew symmetric.
Fig. 12 shows a typical idealization of the wall, and Fig. 13
shows the deflected profiles obtained by means of the procedure described
above, as well 'as by the continuum approach (6) and the equivalent frame
method (12). Fig. 14 shows the moments in the lintel beams as obtained
by these three methods. Fig. 15 shows the vertical stresses in the walls
at various elevations. Fig. 16 gives the convergence for the lateral
deflection at elevations 30.5,62.5 and 94.5 when l, 2 or 3 elements
are used along the width and the floor height of the wall. In all these
cases, the shear deformations in the lintel beams were neglected. The
connection between the latter and the walls was assumed rigide This
was achieved by assigning to the moment of inertia of the fictitious
beams an arbitrary value equal to one thousand times that of the lintel
beams. This multiplying factor was found to be adequate for most pro
blems. In fact a value of only one hundred produces results within
0.5% of the former ones.
In addition, the effect of the elastic deformation at the
abutments of the lintel beams was also studied. In conformity with
- 48 -
4000 d c
1 1 0 --------
00 1 ~ a 80
1 1 0 -------
l 1 80 00 b a
1 1 0 --------
1 1 80 00 b a
1 J 0 ---------
1 1 80 00 b a
1 1 0 --------
1 1 80 00 b a
l 1 0 --------
1 1
" , '" 16" "- '" "16' "
beamtype a b
area (ft2) 3.0 0.0
d
1 1
b 1
1 1
1 1 b
1 1
1 1 b
1 1
1 1 b
1 1
1 1 b
1 1
1 1
" "16'\ '"
c
1.5
4000 lb
800 o
800 o
809 o
800 o
0 0: <D .Ç ~ ........
8~ 00
ln ~ -,,"
. +> oC cr. .,.... QJ
oC
~ o o r-
"-+> t: ltS +> III t: o U
"o III rrltS :3 ~ ltS QJ
oC III
"0 QJ r-0.. ::s o u
. N r-
thlckness of d elements:
0.0 type 1 0
l (ft4 ) 2.25 2250.0 1.125 1125.0 t(ft) 10 0
~ = L1x 1()7 nc;f a.-. •• ~ r- '-' .. v-=0.25
elevations (ft) 94.5
78.5
62.5
46.5
005
- 49 -
--- continuum methcx:l ------ equivalent frame metrod
finite elements - -- rigid connections _ .. - finite elements
semi rigid connections
lateral deflections 0.1 0.15 ft
FIGURE 13. Lateral deflections. (Coupled shear wa11s of Fig. 12).
- 50 -
--E$t-- continuum method
- - fi- -equivalent fra me
beam no.
6Ctop)
5
3
2
1
. -+-. finite elements rig id connections
,'-6-.. finite elements semi rigid connections
beam moments ~--2-5..LO-OO--5-0....JOOOL...---7-5..J,.OOO---10-0---00L--O- ( ft.lb)
FIGURE 14. Moments in 1intel beams. (Coup1ed shear wa11s of Fig. 12).
e1.82.5
el.66.5
el.50.5
el. 34 .5
psf
t2000
-2000
~OO -2000
hoo 8000
4000
-4000
20000
10000
el. 18.5~-
-10000
- 51 -
v+"'xy/0 TOVV
ov+~ o~v
[ + ~v v\j:L0t
o continuum method
+ finite elements
v equivalent frame
FIGURE 15. Vertical stresses in 1eft hand piero (Coup1ed shear wa11s of Fig. 12).
- 52 -
lateral deflection (ft)
Ol5 elevation 94.5
0.10 ~ elevation 62.5
0.05 '-1-_-1----.1 elevation 30.5
no. of elements 1..------I.1---.L.2---'-3-per side
FIGURE 16. Convergence graphe (Coup1ed shear wa11s of Fig. 12).
- 53 -
equation 26, a moment of inertia of 4.0 ft4 was assigned to the
fictitious beams; the 1atera1 def1ections and the moments were
computed and are shown in Fig. 13 and 14. The increase in def1ection
is of sorne 21% at the top of the wall, whi1st a simi1ar reduction in
1inte1 beam moments is obtained.
The second prob1em was solved in {53}. It consists of coupled
shear wa11s, six storeys high, with unequa1 f100r height, as shown in
Fig. 17. Two idea1izations were used, one by means of 70 rectangu1ar
e1ements {Fig. 17a}, the other by means of an equivalent frame {Fig.
17b}. Shear deformations were neglected and the connections between
the beams and the wa11s were assumed to be rigid. The 1atera1 deflec
tion and moments in the 1inte1 beams are p10tted in Fig. 18; good
correlation is achieved between both analyses. For the sake of compa
rison the results of Girijaval1abhan {53} have a1so been plotted. This
idealization comprised not 1ess than 264 e1ements and 334 nodes; yet,
the deflections obtained are sma11er and so are the moments in the
lintel beams. This contradiction is explained in the next section;
it is due to the approximation made by representing the 1intel beam
as an assembly of eight finite elements on1y. From these two examples
it appears that the idea1ization of the wa11s by means of a relatively
small number of elements gives results which compare very well with
those obtained with other methods of analysis. Fig. 16 shows that a
more refined mesh would not improve the resu1ts by more tha~ 5% in the
typical example of Fig. 12.
The third problem consists of a single shear wall-frame system
idealized again by means of rectangular elements and beam elements, in
20000 lb
40 000
40 000 •
40 000
40 )00
800 00
b a b 20000 c a c 1 1 0 1 1 --- --1 1 b a 1 b 1 L
1 1 0 1 1 --------1 1 b a 1 b 1 4
1 1 0 1 1 -------
1 1 b a 1 b 1 4
0000
1 C ,a C 1
r:--- a c 1'---
---~~~ ~lc==~==~a====~=c~1 0000
0000 fiS
1 1 0 1 1 -- -----
1 1 b a 1 b 1 4'
, ,
000 ~II cac Iid 1 1 0 1 1 ------1 1 b a 1 b 1 8 1 cac 1 0000 1 1 0 1 1
1--
------1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
element 0 1 type Il g thickness 08 10
(ft) ..
1 1 1 1 -7 1///1 717771 20' 20' 20' .. . .. . .
(a)
( b) 77 /-7/ / beam types a b c A (ft2 ) 2.0 0.0 200.0 1 (ft4 ) 0.667 66.67 466.67
FIGURE 17. Coup1ed shear wa11s of unequa1 height. Finite e1ement and equiva1ent frame idealizations.
<.n .J::o
d 20.0
666.67
beam no. - 55 -
6 (top) ~ \ ,
5 ~ \
4 " 1
3 1
Cf t ,
2 P 1
1
1 6
20000 60000
o. eJevations(ft)
60.
40.
1. /,
1
/ /
1
1 Q
1 1
1
1 1
/ 1
(/ /
0.01
,/
/ 1
1 ,,0 , " ,/
/ , ,
C 'r-
moments in cn of..) C • QJEr--. Or-(beams ft. lb.)
l
1 1
1
1
,P '" ,
E . "0 01 C'rra 1.1..
cnl+CO o
'r- Q) of..) rua. Q)E r- ra I+-x Q)LLI "0-
r-ra • S-cn Q)E
of..) ra raQ)
....J .Cl
. CX) r-
o Girijavalla bhan
v finite elements
+ equivalent frame
002 ft lateral deflections
- 56 -
which case the thickness of all the elements situated in the second
and third bays was made equal to zero, and also by means of the
equivalent frame method (12). Some problems were encountered due
to the large difference in stiffness between the wall and the frame.
They were overcome by specifying double precision computations.
Shear deformations were neglected in the beams and columns. Full
fixity was assumed between the beams and the wall. Fig. 19 shows
the idealizations used, and Fig. 20 shows the deflected shapes of
the structure. Also of interest is the comparison of Table 8 between
the axial loads in the columns and shear wall obtained in both analyses.
3.5 Stress Concentrations at the Abutments of the Lintel Beams
In the previous section it has been shown that good results
of the deflected shapes, and of the moments, stresses and axial loads
can be obtained from the analysis of shear wall structures by means of
the proposed idealization. Due to the simple overall stress pattern
encountered in shear walls, simple elements in minimum number have
been used without any noticeable loss of accuracy. This, however,
excludes the stress concentrations existing at the abutment of lintel
beams, or near any irregularity in the geometry of the structure.
It is proposed that these may be best tackled by a separate
detailed study of the zone concerned. Such an analysis may be achieved
along the lines followed by Michael (56) by application of the classical
theories or by means of a finite element formulation. The following
example is meant to show the suitability of the procedure proposed here
for such an analysis. Consider the abutment of a half lintel beam in a
.::
- 57 -
b a a 900 d a a 1 1 0 [TI 1 b
c a
1 1 D 1 1
kg
1 ~ i 0 cl[1] e 1800 a
1 ! ID - -~
-........ - If 1 Il .JL..--
::::[ - -: Il
D ~
1
1 1 0 0 1 1 b a a
! ID i ID i ID ,
ID 1
1 1800 1
[~ i 0 1[1] 2100 a 1
1 1
~ 0 c 0 c 1 1
1 e 01 0 C 0 C
1
: 4CX) 500 4CX) 2001 500 4CX)
,
beam types A~1000 in2) l (l< 106 in4 )
a 1.5 0.312
b 4.5
312.5
c 4.5 375
d 120.0 156.25
e 6.0
80.0
element 1 0 types
FIGURE 19. Shear wall - frame structure. Finite element and equivalent frame idealizations.
thickness 15.0 0.0 (cm)
- 58 -
4600 cm elevations
4000
3400
2800 finite elements
2200 equivalent frame
1600
1000
400
123 tateral deflection (cm)
FIGURE 20. Shear wall - frame structure. Lateral deflection .
Elevations
cm
0
400
700
1000
1300
1600
1900
2200
2500
2800
3100
3400
TABLE 8
Loads in columns (in 1000 kg)
Axial load, F.E. idealization
Central column R.H.S. column
-968 -34700
-1170 -33000
-1330 -32000
-1460 -27400
-1560 -24300
-1630 -21120
-1645 -17900
-1615 -14900
-1555 -12100
-1445 -9550
-1300 -7050
Axial load, equivalent frame
Wall Central column R.H.S. column
36500 -1470 -35500
34850 -1545 -33400
32370 -1617 -30800
29424 -1689 -27730
26224 -1734 -24500
22948 -1755 -21200
19720 -1737 -17980
16612 -1680 -14925
13708 -1590 -12100
11040 -1461 -9570
8628 -1296 -7350
1
1
,
1
U1 \0
Elevations
cm
3400
3700
4000
4300
4600
TABLE 8 (Contd .. )
Loads in co1umns (in 1000 kg)
Axial load, F.E. idea1ization
Central column R.H.S. column
-1122 -5400
-910 -3690
-682 -2220
-381 -977
Axial load, equivalent frame
Wall Central column R.H.S. co1umn
6476 -1105 -5370
4572 -888 -3687
2888 -655 -2235
1372 -378 -981
en o
- 61 -
shear wall and a vertical load acting at the tip of the beam. This
assemb1y can be idea1ized in severa1 ways by means of finite e1ements
{Fig. 21}. The corresponding disp1acements in the x and y directions
and the stresses in the x direction are shown in Figs. 22 to 24. A1so
shown in Fig. 22 is the tip def1ection given by Michael {56}.
It may be seen that those idea1izations where a beam e1ement
has been used for the cantilever give the best resu1ts, as far as the
tip deflection is concerned. Idealization d used in {53} makes the
cantilever itse1f too stiff, thus exp1aining the resu1ts obtained
for the second examp1e of section 3.4, where both the overal1 1ateral
def1ection of the structure, and the moments in the 1inte1 beams were
too sma11. As far as the e1astic stresses in the wall are concerned
idea1izations a and c give very close resu1ts. A1together idea1ization
b provides a good picture of the stress concentration in the wall,
a1though better resu1ts cou1d still be achieved with a 1arger number
of e1ements.
Such an idealization is shown in Fig. 21e; it has been derived
from idealization b by rep1acing the four central rows of square e1ements
by six rows of rectangular e1ements, 0.667 ft high.
In this manner, the total 1ength of the fictitious beams is
equa1 to two thirds of the depth of the 1inte1 beam. The disp1acements
in the wall in the x and y directions have been superimposed in dotted
1ines in Figs. 22 and 23. The stresses in the x direction have been
added to the lower part of Fig. 24 on1y.
Very good agreement is obtained with the slope 6M/Eh2 and the
tip def1ection given by Michael {56}.
5ft (a)
(c)
- 62 -
104 lb 1041b
1= 66.67ft4 1= 66.67ft4
/ ~2ft2 V /J"2 tt2 1
~ 1
---fOft---j ---'Ott---j
5ft (b)
1
(d)
FIGURE 21. E = 576 x 106 lb /ft2; V = 0.4 Zone of stress concentration at the abutment of a beam in a wall. Various finite e1ement idea1izations.
- 63 -
104 lb
1= 66. 67ft4
~ j/,"2ft2 l' 1
---'Oft---j
5ft
(e)
FIGURE 21 (cont'd). Zone of stress concentration.
" A 4il1 KI-;
y
x
FIGURE 22. Zone of stress concentration -horizontal displacements u.
,
" x a
A b
+ c o d
--- e --- --Michael
-4 o ,4clÇJ u (ft) ,
en +=-
- 65 -
y
x
- -.-- e x a  b
+ c 0 d
0 Michael 1 2~1O-3 v (ft) 0
FIGURE 23. Zone of stress concentration - vertical displacement v.
s::: o .... ..., ~ ~ ..., s::: cu U s::: o CIl UCII
cu CII~ CIl"" cu CIl ~ ...,~
CII~ ..., 't-s::: 00
N cu· ... S:::~ 00
N.s::::.
- 66 -
U1J<l.> + 0
o
"--. -1---
- 67 -
Chapter 4. The Analysis of Three-dimensional Structures
4.1 The Computer Program
4.1.1 Description of the Program
The computer work described in this chapter is a development
of· a finite element program for the analysis of plate structures and
cylindrical shells given in (43), in which six degrees of freedom are
considered at each node, and in which the system of equations is solved
in a tridiagonal manner. The matrices are inverted by Choleski's method
(59).
The same organization has been retained in the present work.
All the stiffness matrices and load vectors are thus also generated
partition by partition. A prime advantage of this scheme is that most
transfers of data between the in-core memory of the computer and the
storage devices may he performed very efficiently on large matrices.
The new program, whose general flow chart is given in Fig. 25,
consists of one major loop. For a structure containing n different
typical floors, this loop is executed n + l times. During the first
n times, the stiffness matrices and load vectors of the floors are
generated and condensed, in such a way that only those degrees of
freedom at the nodes common to the floors and the vertical system
of shear walls and columns are retained. The resulting matrices are
stored on disco During the (n + l)st execution, the vertical struc
ture consisting of the separate shear walls and frames is treated.
Its stiffness matrix is generated, again partition by partition.
Each partition corresponds to a full horizontal II s1ice ll of the
- 68 -
( READ NFLOOR 1 1
l NPROB = NFLOOR + 11 1
LA = l, NPROB 1
( READ INPUT 1 1
...... EACH PARTITION 1
FORM STIFFNESS MATRIX ++ AND LOAD VECTOR OF PARTITION
1 / LA "- NFLOOR :
t L FLOOR LEVEL '"
ADD FLOOR EQUIVALENT
STIFFNESSES AND LOADS T
/ LA 'NFLOOR : 1
FORM AND STORE KBI
FORM KII,UI
FORM D.LT
FORM AND STORE [D.LTrl 1 , " LA ~ NFLOOR 1
FORM R = KBI.[D.LT]-l SOLVE SYSTEM OF EQUATIONS
FORM KBB* = KBB-R.D.RT PRINT DISPLACEMENTS
FORM FI* = [L]-l.FI AND STRESSES
FORM FB* = FB-R.FI* 1 STOP 1
STORE lA,KBB*,FB* END 1 1
FIGURE 25. General flow chart. (++) indicates those parts of the program which are derived from the program of reference (43).
++
- 69 -
building; when the IIslice ll corresponds to a floor level, the equi-
valent stiffness and load vectors obtained above are retrieved from
storage and added. The stiffness matrix and load vectors thus obtained
are those of the entire structure, after condensation of the floors.
It is imperative that the order in which the nodes are numbered in
the floors and the vertical system be consistent.
The solution of the system of equations then proceeds by
forward elimination and backward substitution, as indicated above.
However, Choleski's method has been modified along the lines suggested
by Melosh (60) in order to remove square rooting operations.
Thus, the stiffness matrix of each partition is inverted after
decomposition into
(27)
where,
[L] is a lower triangular matrix of unit diagonal terms
[0] is a diagonal matrix
[L]T is the transpose of [L]
After inversion of [K], the displacements and stresses in the
vertical system are computed. This information is sufficient to cal
culate the stress resultants in the floor beams, due to the overall
displacements of the structure. The displacements, moments and shears
within the floor.slabs could also be computed without difficulty, at
a number of levels specified by the designer. However, the need of
such information seems questionable in view of its cost, since the
meshsize adopted for the finite element analysis will not necessarily
- 70 -
comprise those few locations which are critical for the design of
the floor slabs.
In substructure analysis, the equations are usually subdivided
according to
where,
[1<11
K1H KIB] . {bI} = {UI} KBS 68 UB (28)
KIl is the matrix of internal forces due to unit internal
displacements,
KBI is the matrix of boundary forces due to unit internal
displacements,
KBB is the matrix of boundary forces due to unit boundary
displacements,
UI includes all the loads applied at interior nodes,
UB includes the loads applied at the boundary nodes,
ôI is the internal displacement vector,
ôB is the boundary displacement vector.
* The computation of the condensed stiffness matrix [KBB ] and
load vector{uBl requires the inversion of [KIl] and the subsequent
multiplication [KBIJ . [KII-l] . [KIBJ. These matrix operations are
performed by decomposition following a method proposed by Rosen and
Rubinstein (61). Sorne of the calculations are outlined in the flow
chart, using the notations of equations 27, 28 and reference (61).
They are given in detail in section 4.1.4 below.
In Fig. 25, those parts of the program derived from (43) have
been indicated by a double plus sign (++). Modifications of these
- 71 -
parts are noted in section 4.1.4.
4.1.2 Capabi1ities of the Program
The computer program 1isted in Appendix 3 was coded in FORTRAN
IV language. It has been run at the McGil1 Computing Centre on an IBM
360/75 computer, with an available core memory of 400 kbytes.
For a structure containing n different typical floors, the program
is executed (n + 1) times. Hence, (n + 1) complete sets of data must be
supplied, of whom the n first belong to the typical floors and the (n + l)st
pertains to the vertical systems.
The number n is limited to 20 in the listing of Appendix 3.
For each of the separate data sets, the following may be spe-
cified: (single precision calculations)
maximum number of nodes 300
maximum number of elements 300
maximum number of boundary nodes per floor 16
maximum number of boundary nodes in the vertical systems 100
maximum number of loading conditions 4
maximum number of beam types 50
maximum number of element e1astic properties 10
maximum number of partitions 20
maximum number of nodes/partition 16
The partitions of the vertical system are taken as complete hori-
zontal Islices" through it. The maximum number of nodes/partition is
thus a1so the maximum number of nodes on the co1umns and shear wa11s that
may be considered in a structure.
- 72 -
This number is 16 for the above computer. Machines with in
core memories of 800 kbytes or even 1000 kbytes are readi1y avai1ab1e.
For 800 kbytes, the maximum number of nodes/partition rises to 30.
It will be seen in section 4.1.4 that this number, which commands
the size of the most core-consuming matrices, is indeed the critica1
figure of the who1e program.
The number of partitions of the vertical system is proportiona1
to the number of storeys, but increasing this number does not increase
the core space required.
Boundary conditions for the vertical system may be specified at
each of the 100 nodes in terms of a110wed or restrained disp1acements,
or in termsof prescribed disp1acement components a10ng each of the six
degrees of freedom considered.
Such boundary nodes may be situated on the actua1 boundaries of
the structure or in its planes of symmetry.
App1ied 10ads may be specified at each of the 300 nodes of the
typica1 f100\~s and vertical system, in terms of their six usual compo
nents, as shown in Fig. 1.
Vertical as we11 as horizontal loads or moments may thus be
considered, for up ta four loading cases.
For example: dead, live, wind loads or combinations thereof
as recommended by the codes of practice.
A large variety of structural configurations may be analysed
by the program, provided they may be inscribed in one or several adjacent
rectangular parallelepipeds.
- 73 -
For the f1oors: plain slabs; slabs supported by ribs and
beams; slabs with rectangu1ar drop panels and openings.
For the vertical system: single or coup1ed shear wa11s with
one or severa1 rows of openings; cores; three-dimensiona1 rigid frames
or trusses; diagonal braces, and the combinations thereof inc1uding
semi-rigid connections between wa11s and 1inte1 beams.
The configuration of the vertical system may vary from storey
to storey.
Co1umns and wal1s are norma11y considered to be fixed at the
base of the building, but pinned co1umns may be specified and the effect
of elastic foundations may be inc1uded by considering a fictitious base
ment storey whose e1astic properties are equiva1ent to those of the actua1
foundation (14).
Orthotropic properties may be specified for a11 the plate e1ements.
Since six degrees of freedom are considered at each node, due account is
taken of torsiona1 effects, of the axial deformations of the co1umns and
wa11s, and of the warping rigidity and in-plane deformations of the f100r
slabs, which are neg1ected in most other works.
The ca1cu1ations are performed on the initial geometry of the . system, and hence the secondary overturning moments due to the axial
loads in the def1ected geometry are neg1ected (27).
Typica1 computing times, inc1uding compilation, are as fo11ows:
ExamE1e 1 ExamE1e 2 Exame1e 3
No. of typica1 floors 1 1 1
No. of nodes/typica1 floor 63 85 69
No. of nodes in the vertical system: 64 154 128
No. of storeys
CPU time (min.)
1/0 time (min.)
Double precision calculations
- 74 -
7
5.16
3.63
yes
13
7.68
3.14
no
15
7.60
2.80
yes
It may be noted that the time required for input/output is only
a fraction of the CPU time.
4.1.3 Input-output
The data required for each structure are as follows:
No. of Cards
1 NFLOOR : number of typical floors in the structure
system:
NPOIN
Then for each of the NFLOOR typical floors, and for the vertical
NPART
NPOIN
NELEM
NBOUN
NCOLN
NFREE
NCONC
NYM
NBEAM
X(I ,J)
number of partitions
number of nodes
number of elements
number of boundary nodes
number of loading cases
number of degrees of freedom per node (6)
number of nodes at which loads are specified
number of different sets of element elastic
properties
number of different types of beams
the coordinates of node l, where (J = 1,2,3),
(I = 1,NPOIN)
NBEAM
NELEM
NBOUN
NPART
EB{I}
AB{I}
IXB{ I)
IYB{I)
IZB{I }
GB{I}
HCB{I}
NOD{I ,J}
THICK(I)
R{I}
BEAM(I,J}
NF{I}
NB{I ,J}
BV{I,J}
NSTART(I}
NEND{I}
NFIRST (I)
NLAST(I}
- 75 -
modulus of elasticity of beam 1
(1 = 1,NBEAM)
cross section of beam 1
torsional constant of beam 1
moment of inertia of beam 1
moment of inertia of beam 1
shear modulus of beam 1
distance of centroid of beam 1 above the
middle plane of its supporting element
nodes at the corners of element 1
(J = 1.4), (I = 1,NELEM)
thickness of element 1
index defining the positive normal of
the element
indices defining the beams supported
by element 1 (J = 1,6)
number of boundary node 1 (1 = 1,NBOUN)
indices for boundary conditions at node 1
(J = 1,6)
prescribed displacements at node 1
(J = 1.6)
number of first element of partition
1 (I = l, NPART)
number of last element of partition 1
number of first node of partition 1
number of last node of partition 1
NYM El(l)
E2(1)
PO(I)
P2(1)
GE(I)
- 76 -
modulus of elasticity of element 1
in x-direction (1 = 1, NYM)
modu1us of e1asticity of element 1
in y-direction
Poisson's ratio of element 1 in
x-direction
Poisson'~ ratio of element 1 in
y-direction
shear modulus of element 1 in
x-y plane.
For each loading case J (J = 1,NCOLN):
NCONC K
U(6K-5,J) to U(6K,J)
number of node at which load is applied
six components of the load applied at
node K for loading case J
A structure is referred to a global right-handed set of rectan
gu1ar cartesian axes Oxyr, such that the Oxz and the Oyl coordinate
planes are para1le1 to two of its vertical facades.
The coordinates of all the nodes must be given in that system,
however, the Z coordinate of the nodes located in the typical floors
needs not be specified.
The floors are partitioned so as to have a consecutive nodal
numbering system; only one partition line, which needs not be straight,
should pass through an element.
The boundary nodes of the typical floors are the points of
intersection between the floor slabs and the vertical system. No
boundary conditions are considered at these nodes. At any given
- 77 -
elevation, the number of nodes/partition in the vertical system must
be equal to the number of boundary nodes of the floor intersected,
and these nodes must be numbered in the same order.
The numbering of the nodes of an element in array NOD must be
sequential, as shawn in Fig, l, and m~st be such that the direction
1-2 corresponds to the positive direction of one of the global axes
Ox or Oy (Ox only for the typical floors).
When the nodes are numbered anti-clockwise, as seen from the
positive global axis perpendicular to the plane of the element, index
R is set equal to 1. In the clockwise direction, R is equal to o. Element properties are referred to a local right-handed set
of rectangular cartesian axes Ox'y'z', such that axis Ox' is along
the side 1-2 of the element and axis Oy' a10ng 1-4.
It may thus be seen that R defines the orientation of the
local set of axes with respect to the global axes.
The array BEAM(I,J), (J = 1,6), corresponds to the six beam
locations a to f connected with the element l, and shown in Fig. 9.
A non-zero index BEAM(I,J) = N indicates the presence of a beam of
type N at location J of element I(N ~EAM).
The properties of the beam are given in the local system of
coordinates attached to the element. lYB is the moment of inertia
for bending out of the plane of the element; IZB is the moment of
inertia for bending in that plane. HCB is positive when the z' coor
dinate of the centroid of the beam is positive.
The boundary conditions are expressed by two arrays NB(I,J)
and BV(I,J), where (J = 1,6) corresponds respectiv~ly to the six
- 78 -
degrees of freedom u, v, w, ex' ey , eZ in the global axes. NB(I,J)
is equal to l when the displacement of node NF(I) is allowed alon~
the Jth degree of freedom and is equal to 0 if restrained. BV(I,J)
gives the magnitude of the prescribed displacements, if any, in the
case where NB(I,J) = 1.
The applied loads are specified by their six components.
It is imperative that the number of loading cases be the same for
all the typical floors and the vertical system, but the number of
points of application of the loads may vary from one typical floor
to another.
The output of the program consists of the following:
a) a print-out of all the input data;
b) the residuals of the solution of the tridiagonal system
of equations, defined as the difference between the applied
load and the product of the stiffness matrix by the dis-
placements;
c) the components of the displacements of all the nodes of
the vertical system with respect to the global axes.
This list is repeated for all loading cases;
d) for all the elements located in the vertical system:
one line comprising the number of element, and the plane
and bending stresses at its centroid. Then for each beam
supported by the element, one line comprising the type of
the beam, the number of the loading case, the axial load,
twisting moment, bendinq moments and shears.
Each individual line is repeated for all loading cases.
When the thickness of the element is zero, zeros are printed
- 79 -
for the stresses.
e) for all the elements located in the vertical system:
one line comprising the number of element, the number of
its nodes, the local coordinates of its centroid, the
cartesian and principal stresses at the centroid, computed
at the top face of the element; a second line comprising
the same information, the stresses being computed at the
bottom face. Each line is repeated for all loading cases.
f) an identical list giving the cartesian strains at the
centroids of all the elements of the vertical system,
for the top and bottom faces of the elements. Each line
is repeated for all loading cases. Zero strains are printed
for the elements of zero thickness.
The elements stresses and stra~ns are referred to the local
system of axes of the element.
The sign conventions of the beam actions are referred to local
axes attached to each beam and shown in Fig. 9.
A positive axial load means tension in the member.
A positive bending moment at either end means that the action
of the beam on the end node is in the positive direction of the local
beam axes.
The shears have the sign of the algebraic sum of the moments.
A positive twisting moment acts on the origin of the local
system of axes in the positive direction.
- 80 -
4.1.4 Particu1ar Aspects of the Program
4.1.4.1 The Combination of Bearn and Rectangu1ar Elements
An out1ine of the program has been given in section 4.1.1.
Details of the stage "form stiffness matrix and load vector of par
tition" indicated in Fig. 25 are given in Fig. 26 where the emphasis
is laid on the particular combination of the beams and rectangular
elements. As the computation proceeds, element by element, the
strain and stresses matrices are formed for the rectangular element,
as well as its stiffness matrix. The stiffness matrices of all the
beams supported by the element are also generated and combined to
that of the element. When the thickness of the latter is zero, all
the calculations pertaining to it are skipped. Conversely, a plain
continuum is analysed by skipping all the calcu1ations related to the
beams. The test "sum of beam indices is zero" refers to the expression
6
1: BEAM( l ,J) j = l
in which BEAM(I,J) has been defined in section 4.1.3.
Although less convenient for automatic data generation, this
approach to the combination of beams and rectangular elements has two
major advantages. Firstly, by not considering the beams as separate
elements, the computational effort is reduced considerably. Secondly,
in the case of three-dimensional prismatic structures, the principal
planes of the beams are generally parallel and perpendicular to the
plane of the rectangular e1ement so that no additional definition of
their orientation is required.
- 81 -
1
EACH PARTITION r
EACH ELEMENT 1
NO BEAM IN PROBLEM) 1
-< SUM OF BEAM INDEXES IS ZERO l 1
CALCULATE BEAM
STIFFNESS MATRICES 1 1
THICKNESS OF ELEMENT IS ZERO
1 CALCULATE STIFFNESS MATRIX OF
RECTANGULAR ELEMENT 1 1
ADD BEAM AND ELEMENT
STIFFNESS MATRICES 1
FORM GLOBAL STI FFNESS MATRI X 1
!
FIGURE 26. The combination of beams and plate elements.
- 82 -
The stiffness matrix of each partition is formed by assembling
those of the individual elements, after transformation from the local
to the global system of axes.
This transformation of coordinates is not performed for the
typical floors, however, since both systems are then superimposed.
These partition stiffness matrices are kept in core for use
in the substructure analysis, or for the tridiagonal solution of the
system of equations, in the case of the vertical system of the structure.
4.1.4.2 Substructure Analysis
The starting point for the floor substructure analysis is given
by equation 28 which is repeated here for convenience:
[KIl KSI
KIB] . {b 1 } ={UI} Kas a» V8
in which the stiffness matrix may be decomposed into the triple product
(61) :
[ KU Kra J_[ L 0 }[~ o }[LT RT
] (29) KSI KBS - R l K- 0 1
where:
[ L} is a lower triangular matrix of unit diagonal terms
[LT] is the transpose of L
[ Dl is a diagonal matrix
(1) is the unit matrix
From 29 , the following equations are derived:
[KIl] = [L][D]{LT] (30)
[KBI] :: [R}[D]{LT] (31)
- 83 -
l K8S] = [K-] + [R][O ] [RT] (32)
Equation 30 is a straight decomposition of KIl • Subst'j tuti ng 29
into 28 gives
[~ o J {6·} { ur *} K- · h; = us'" (33)
in which
{!i}=[~T ~T]. {!: } (34)
{~:} == [~ 0 l {~:'.} l
(35)
Equations 34 and 35 define a coordinate transformation between the
"starred" and '~nstarred" systems of axes. Hence, the second equation
of 33 ,
(36)
represents an independent relationship between the boundary forces and
displacements in the "starred" system.
From equations 34 and 35 :
f b;} = {ha} (37)
{ U I} = [ L J.{ U l ~ } (38)
{UB} =[R] {UI"}+{Ur.*} (39)
- 84 -
From which
(40)
ln addition, 33 gives
(41)
whereas {b I} may be obtained from 34 :
(42)
Equations 41 and 42 are on1y used if the interna1 disp1acements
are needed. The various stages of the program are as fo1lows:
a) re-organise the partition stiffness matrix into its
components [Kil], [KBI], [KBB] of equation 28;
b) decompose [Kil] according to 30. Since [Kil] is a
banded matrix, [L] is also banded; [D]and [OLT]= [O}[LT]
are obtained;
c) i nvert [ OLT] .
These three operations are performed partition by partition,
as shown on Fig. 25. Terms belonging to [KBB1 are kept in core in a
separate array; [KBI]which is sparse is stored on disc in compact
forme [Kil] is kept in core.
For a partition comprising N degrees of freedom, the maximum
storage space required is N x 2N. Such is also the space required to
store [Kil]. Adding up another N x N space for [KBB] gives a total
memory requi rement of N x 3N words for stage a) which i s performed
in subroutine SHUFFL.
- 85 -
Oue to the banded nature of [KIl] and CL], the core requirement
for stage b), performed in subroutine OLTMAT, is not increased.
An algorithm was developed in order to perform stage c) within
a core of N x 3N words. One direct access file is used for the storage
of [OLTJ-l which is a complete upper triangular matrix, and which is
therefore subdivided in NPART x (NPART + 1) records of N x N terms each, 2 where NPART is again the number of partitions in the floor.
d) From equations 31 and 36 , form
[ R ] = [K BI] . [DI..T1-t
{OIO} = [ ri] . {ur} (43)
(44)
These matrices are obtained in subroutine MULTPL in which
{KBI]is called from storage and then multiplied by the con
secutive records of IOLT]-l . [UI*]T is obtained from
e) The product [R].[O].[RT] is performed in subroutine REFOST
and the condensed stiffness matrices [K*] and {UB*} are
obtained from equations 32 and 39.
These matrices are stored and will be combined to those
of the vertical system when the latter will be analysed.
- 86 -
4.1.4.3 Solution of the System of Equations of the Vertical System
This section refers to the box "solve system of equations,
print displacements and stresses" of Fig. 25.
This part of the work follows very closely the program given
in (43), but the following modifications are introduced:
a) in the programming of the tridiagonalization algorithm
(subroutine SOLVE), an improvement suggested by Khan led
to saving up to 25% of the core requirements;
b) Choleski's square root decomposition has been replaced by
the improved "LDLT .. scheme;
c) the STRESS subroutine for the calculation and printing
of the displacements, stresses and strains has been
simplified and algorithms for the calculation of the
moments, shears and axial loads in the beams have been
added (subroutine ACTION).
4.2 Examples of Three-dimensional Structures
4.2.1 Single Storey Structures
A number of three-dimensional structures have been analysed in
order to check the program described above and demonstrate its use, and
also to provide further tests for the element formulation of chapter 2.
Among them, simple table structures were considered first since they
constitute the basic ingredient of most multi-storey constructions.
Such a table consisting of a square slab, edge beams and corner
columns is shown in Fig. 27. It is subjected to two equal horizontal
loads of l lb each. Fig. 28 shows a comparison between the displacements
- 87 -
18"
+' :::: '+ lb . (OLO 1
':::::'d CV')
• 1 --f'... • ~
•
0.5" 0.5" z 1
4 3 1 '\'l'
1\\ X " ~"2
P E=450000 psi 7 8 3 - - :..... - _::1:_ - - ...:. =-t==""" -= - - t- -..4---.-
1 ~ 1 1
: LO -L' 1 l '.
H- -1-' --i,I. :: i: 00 ,1 i ' :1 ~
- -1-- [--i' ~ 1 : 1
5 '--c-.=l9---:L_--1' 6 1z 2
FIGURE 27. Table structure
\):0.38
P=11b
-- new element
ln. -·-quartrec element
0.08 new+ substructure \
08
o
,
" """-.-
007 wx103 0.7
0.06 0.6
CP
-ex ,ey ~
U/V 0.05 ;=:; --;-. , 0.5
0.04
, , " elements 2 4 6 Iside
, , '" elements 2 4 6 Iside
FIGURE 28. Table structure of Fig. 27. Convergence graphs for displacements at node 5.
- 89 -
u, w and e at node 5 obtained with the element formulations QUARTREC x and NEW defined in section 2.3. Convergence curves are given for an
increasing number of elements per side of the slab. In addition, the
results obtained by substructure analysis are given for the element
NEW.
In this case, the idealization of the vertical system is reduced
to eight nodes numbered l to 8 and two vertical elements 1-2-6-5 and
3-4-8-7 of zero thickness, whose function is to define the corner
columns. The slab including the edge beams is idealized separately
as one typical floor, and its stiffness matrix is thus condensed so
as to contain only terms connecting the 24 corner nodes' degrees of
freedom, as explained is section 4.1. All calculations for this example
are performed in double precision.
Another table consisting of a plain slab and four columns has
been studied. It was analysed by Davies (62) and has the following
dimensions:
slab: 20 x 10 x l ft3
columns cross section: l x l ft 2
total height: 10 ft
E = 3 x 106 psi
v = 0.15
loading: 10,000 lbs acting downward at centre of slab
deadweight of slab (150 lbs/cu.ft.)
Fig. 29 shows the displacements ex and ey at one corner as well
as the vertical deflection w under the concentrated load. These values
are obtained again for increasing numbers of elements per side of the
- 90 -
central deflection (ft)
/ 1
2
-3 e (10 rad) y
1.1
10
0.9
2
/
-3 ex (10 rad)
0.3
/
"
","
" "
4
.-----.--- ... -- .....
6
.... .e-_ " --" 0""_ -
4 6
w
8 10
A
~II::::'<-:_:::--__ -,,:+ --
8 10 12
----~~~~--~~~~+ ,.~---V -0_--
0.2
fi 0.1
2
" ,/ /
/
4 6 8
o
6 +
À new with condens3tion new without condensation quartrec Davies triangular elements Davies rectangular elements
10 12 no of elements per side
FIGURE 29. Rectangular table under vertical load. Convergence curves.
- 91 -
slab, and by the various formulations indicated in the legend of the
figure. In particular, for 8 elements per side, the following corner
displacements were obtained:
(10-2ft ) ( -3 ex 10 rad.) -3
ey(lO rad.)
NEW element; substructure anal. 0.26195 0.22799 0.93651
NEW element; no substructure 0.26388 0.22947 0.94171
QUARTREC element; no substructure 0.26388 0.23006 0.93567
In this process, a stiffness matrix, 486 x 486 in size, was reduced
to a 24 x 24 one, and the relative error on the rotations was not more than
5 x 10-3•
In Fig. 29, the values for the corner rotations obtained by Davies
(62) have also been indicated. They are obtained by hand calculations,
by expressing the moment equilibrium at that point; the stiffness charac
teristics of the slab resulted from a separate finite element study.
Davies gives these characteristics from an investigation using
49 nodes and triangular elements; significantly different values were
obta;ned for the same number of nodes by means of rectangular elements.
In the end, good agreement ;s obtained with the complete finite element
formulation.
4.2.2 Multistorey Structures
A seven storey model described in (18) and shown in Figs. 30
and 31 has been ;nvestigated. It consists of identical slabs made of
thin steel plate intersected by columns made of steel bars, tube and
strips and assembled by brazing.
~ïœ
~
20-14.1'2"
:::
o <D
. -
"'" r
LO L!1
1---
~
10e
- 92 -
1 -1
12"
·
1
~
· <0 ~
· Il Il Il 1
· , elevation
12" 1 .....
T / 1J~udia . 6.0"
"o.d. .j~ - ---
1 Y2- Ya fe ~
plan
FIGURE 30. Seven storey structure. Elevation and plan.
· ., 1
- 93 -
FIGURE 31. Seven storey Structure. Photograph.
- 93 -
i L ..
FIGURE 31. Seve~ storey Structure. Photographe
- 94 -
Jaeger (18) has given an analytical solution by the continuum
approach and has compared it with experimental values of the displace
ments obtained for two loading conditions (concentrated lateral load
at the top and at one intermediate floor) in each of the principal
planes of the structure.
A comparison is made here between the finite element solution
and experimental values obtained for the displacements under six loading
conditions:
- concentrated lateral load at the top and load equally
distributed between all storeys, acting in each of the
two principal planes;
- twisting moment acting at the top and twisting moment
distributed equally between all storeys.
The finite element idealization used for the vertical system i5
shown in Fig. 32. The idealizations of the typical floor, for bending
in the N-S and E-W directions are shown in Figs. 33 and 34. Only one
quarter of the structure has been idealized. All the columns are treated
as beam elements. In Fig. 33, a distinction is made between real and
fictitious boundary nodes. The latter are introduced for the sole
purpose of specifying the boundary conditions along the axes of symmetry.
Beams are also introduced in order to enforce boundary or compa
tibility conditions. For instance, in the floor idealization of Fig. 33
(N-S loading), beams are introduced:
- between nodes 4 to 7, in order to ensure that their displace
ments are compatible with those of a cross section of the
corner column;
49
17
9 Z
59.//
/ /
/
- 95 -
FIGURE 32. Idealization of one quarter of the seven storey model.
- 96 -
f~9 30 31 32 33 34 35 ... 1jr" -- - -- -- . ..... ,..-1 1 1
: 19 20 21 22 23 2A 1 , 1
)~22 23 24 25 26 27 28
be am '13 14 15 16 17 18 .. ...
0 0
~1 ... 35 16 17 18 19 20 <0 , " 1 fi ttit ous bounc arv bde 1
n 1 , 1 1
7 8 9 10 11 12 1 1
~ 8 9 10 11 12 13 14 , 1 1 1 ,
1 2 3 4 5 6 1
real: Ibou hda rv r ode
~~1 2 3 4 ~ :2 6 7
~5" .45" .~" .75"" .,
;>.?~" ,50'
- ...... x
3.00"
FIGURE 33. Typical quarter floor idealization. North-south loading.
- 97 -
y 57 58 59 60 61 62 63 ~-- -- .... -- .. :43 44 45 46 47 ~ Ln 1 h'1 l'51.) 51 ~2 53 54 55 56 1 41 ~~ Ln 1 1 en 1
1 48 ~9
1 1 1 1
1 1 1
~ II' 1 1
:::
1 0 1 L() Il ~ 1 1 1 1 1:15 16 17 18 1
:7 8 9 :8 9 10 11 12 13 14 [1 ~ 1 1
2 3 4 5 ~ 11 2 3 6 -- ... ;2 Q _7 x
.55" :45" 50' .75" 2.75" 3.00"
FIGURE 34. Typical quarter floor idealization. East-west loading.
- 98 -
- between nodes 1 and 29 in order to force w = 0 displacements
along that edge, and
between nodes 29 and 35 in order to force ex = 0 displace
ments along that edge.
In Figs. 35 and 36, deflected profiles of the building are given
for its bending in the two principal planes. They are obtained:
a) experimentally;
b) by Jaegerls method (18) (for tip load only);
c) by finite elements using the actual dimensions of the steel
parts (thickness of floor slabs: 0.117 11, inertia of central
tube: 0.037 in4);
d) by finite elements using the nominal dimensions of the
steel parts (thickness of floor slabs: 0.125 11).
The last two cases are given in order to emphasize the importance
of the floor slab stiffness, expressed here in terms of its thickness, on
the overall behaviour of the structure;given the additional stiffness of
the experimental model due to the brazing beads, the agreement is found
to be very good. The effect of the in-plane floor deformation is negli
gible for this structure whose length/width ratio is only 2.0:1.
In table 9, however, the distribution of the total shear between
the parallel resisting bents is given for the E-W and N-S loadings. Con
trary to the assumption often made in simplified methods of analysis,
this distribution is not constant over the height of building.
In Fig. 37, the angle of twist is given for the torsional case
of loading.
"0 2 ::::s
.Cl
-
\. \.
'\ '\
\
L (1) CJ) (1) CU
--:> .......... .......
- 99 -
C .-
"0 .Cl Cfl CU 0 0 0 -- - )(
(/)
$ .0 (/) C (1)
~ C\J E -'\ W , , ,
'\ ~ \
.0 0 -C (1)
E U
, ~ \ ... ... ... , u ... .... \ CU
\ ,
C\J
<.0 LO C\J
FIGURE 35. East-west loading. Deflections.
- 100 -
7 dis ributed . load.
expe imenJ
6 /
5
4
3
2 4
/
/ /
/
Jae er
N-S
total la d 10 lb
6 8 )(10 in
FIGURE 36. North-south loading. Deflections.
- 101 -
7 dist ibuted m ment
6
5
4
3
experl ents
finite lemen actua dimen
10-3 rad
FIGURE 37. Seven storey structure in torsion. Angle of twist.
TABLE 9
Seven store~ model
Distribution of the total shear among the various structural elements
actual dimensions actual dimensions nominal dimensions nominal dimensions tip load distributed load tip load distributed load
exterior inter- central exterior inter- central exterior inter- central exterior inter- central storey bent med. bent bent med. bent bent med. bent . bent med. bent
bent bent bent bent
7 44.2 21.4 34.4 6.4 49.1 44.5 48.3 16.3 35.4 18.0 35.2 46.8 -'
6 50.3 18.6 31.1 40.5 25.7 33.8 54.9 15.5 29.6 46.6 19.8 33.6 0 N
5 49.1 19.4 31.5 48.2 20.1 31.7 52.9 15.5 31.6 52.0 16.1 31.9
4 50.8 18.1 31.1 52.6 16.7 30.7 54.0 14.8 31.2 55.4 13.9 30.7
3 54.0 15.7 30.2 57.2 13.4 29.4 56.7 13.1 30.2 59.2 11.5 29.3
2 60.3 11.7 28.0 63.8 9.4 26.7 62.5 10.0 27.5 65.5 8.4 26.1
69.9 5.0 25.1 71.2 3.8 25.0 72.0 43.3 23.7 73.0 3.4 23.6
North-south loading
TABLE 9 (cont'd)
actual dimensions actual dimensions nominal dimensions nominal dimensions tip load distributed load tip load distributed load
storey exterior central exterior central exterior central exterior central bent bent bent bent bent bent bent bent
7 48.8 51.2 88.25 11.75 48.3 51.7 85.3 14.7
6 45.2 54.8 52.9 47.1 45.5 54.5 52.6 47.4
5 46.2 53.8 47.2 52.8 46.5 53.5 47.3 52.7 --'
4 45.1 54.9 43.5 56.5 45.5 54.5 44.1 55.9 a w
3 42.0 58.0 39.3 60.7 42.6 57.4 40.2 59.8
2 35.0 65.0 31.8 68.2 35.9 64.1 32.8 67.2
17.6 82.4 15.5 84.5 18.4 81.6 16.2 83.8
East-west loading
- 104 -
Finally, a finite element analysis was made for a fifteen
storey perspex model described in (62). The structure is of the framed
tube type; the idealization used for the typical floors is shown in
Fig. 38, along with the actual floor and column dimensions. The idea
lization of the vertical system consists of 128 nodes and 90 elements
of zero thickness, as shown in Fig. 39. All the columns are treated
as beam elements.
In Fig. 40, deflected shapes of the building are shown. They
were obtained:
- experimentally by Coull and Subedi (66);
- by the present finite element analysis, using substructures;
- by Coull and Subedi's approximate method (66).
In this method, the total structure is replaced by an equivalent
plane assembly consisting of the rigid frames parallel to the plane of
bending and frames made of the leeward and windward beams and columns.
The latter are supposed to resist axial load only, and the connection
between the two components is by means of fictitious horizontal members
of large flexural rigidity that ensure the compatibility of vertical
displacements. The effective stiffness of the floor slab was determined
experimentally. That approximate method is expected to behave in a much
more flexible manner than the actual structure.
In the finite element model, shear deformations in the columns
were neglected. The element dimensions were also calculated between the
centre lines of the floors and colum~s. This representation of the struc
ture is thus too rigid. The second cause is particularly important in
this case due to the sizeable thickness of the columns and floor slabs.
63 64 65 66 67 68 69 , ;=- - -- --= - ~.- ::::LI,F"'"'"-- - --== -._-= - ---- -- --~-------
ln """ beam IYB r 1. 59324 in . 1
8 48 49 50 51 52 53 l ~ ---------------1 C\J thickness:O. 1 a : ---- ----------1 ~ 36 37 38 39 40 41 ' o 41 42 43 44 45 46 47 48'
~ 29 30 31 32 33 34 35 1
Ln :.1 (\J Ln 1 (() C\J Ln d 1 ~ :1 C\J : ~I
K} beams IXB=OB3906 in4 o IY 8 = 1. 95324 in4 .1
~ Y p ~I o ln ~ 11 2 3456 7 ~: 1 r () ,1 x 2 3 14 r
0.65625" 3cQSO"
3.65625 "
E = 4.2)( 105
psi '\1 = 0.35 thickness: 0.125"
15 6
1o.2~'
r 17 1 __ 6. 1 J
0.75" 1 0.50"
direction of wind action
FIGURE 38. Fifteen storey structure under lateral wind load. Floor idealization.
--' o (J'l
- 106 -
FIGURE 39. Vertical system idealization for fifteen storey framed-tube building.
- 107 -
elevation storey 31.87515
(in)
_ theoretical (Coull & Subedi)
--+- finite elements
o experiments
load: 1 Ib/ in
~----~------~----~------~ 300 "10 ..... in 100 200
FIGURE 40. Lateral deflection of the frame-tube structure under horizontal load.
+ o~<~--~~~~
+ +
r----------------------, 1 1 1 1 1 1 1 1 1 1 1 1 1
~--- -
1 1 l , 1 1 - - -- --
1 1 +1 1+
L ____ - - - __ - - - - - - - - - -- - - - -. ~
+ t
200 psi
100
o
o
+
1-
theoretical (Coull & Subedi)
experiments finite elements
FIGURE 41. Framed-tube structure. Axial stre~ses in co1umns at third f100r 1eve1.
..... o 00
- 109 -
The fact that the f100r stiffness was overestimated is confirmed
by the increased vertical loads in the lateral frames obtained in Fig. 41.
In such an ana1ysis, the proper slab characteristics should have been
determined by a pre1iminary study of a typica1 column-s1ab connection
in which bath would have been idea1ized by para11elepipedic e1ements.
- 110 -
Chapter 5. The Importance of the In-plane Deformation of the Floor Slabs
5.1 Description of the Structures under Investigation
Fig. 42 shows the elevation, plan and cross section of a typical
eight storey building whose members were dimensioned for the following
loads:
1 ive load
snow load
wind load
50 psf
43 psf
14.5 psf, uniformly distributed over the height
of the building
The elastic modulus is 3000 ksi and Poisson's ratio is taken
to be 0.15.
The building consists of two plain end shear walls and seven
intermediate two bay frames of varying column sizes; its length/width
ratio is 4.8:1. The seven frames are located between eight identical
spans; this building will therefore be referred to as IIA8 11 . By removing
two, four or six intermediate spans, shortened structures may be derived,
that have length/width ratios of 3.6, 2.4 and 1.2:1, respectively. These
structures will be called IIA6 11 , "A4 11 and IIA2" with regard to their own
number of longitudinal spans.
Another group of structures may be derived from this series by
considering only the top halves 'of these buildings. These four storey
buildings are referred to as "B8, B6, B4 and B211, respectively. All
structures are assumed fixed at ground level and they have been analysed
under lateral wind load by means of the program described in chapter 4.
The aim of this work is to gather data on the influence of the
deformations of the floor slabs on the axial loads, moments and shears
lA , --" co
o (\j
.0 (\j
8"shear wall
- 111 -
SECTION AA plan
l24~24" ,30~ 3d',
'24~ 24 1/ 130/~ 30"1
B {
r--
1 ( l,
Il 1'-rA
1
" Il 1
1 Il
1 li ELEVATION
1 1.
12~21"
SECTION 8B cross section
FIGURE 42. Eight storey, eiqht bay structure for wind analysis (type AB),
- 112 -
in the columns. In particular, the horizontal forces Fij resisted by
each transverse frame i at all the floors j were determined.
For the sake of comparison, data were also obtained for the
same structures when the floors were assumed to be rigid in their own
planes. Since the program used does not accept relations between nodal
displacements as boundary conditions, an alternate scheme was adopted
by modifying the load distribution applied to the building.
Each vertical frame was thus subjected to horizontal loads
equal to -Fij' whereas the end shear walls received the differences n
!(WJo -oE FiJ·)' where WJo is the actual wind load resultant at level ,=1
j and n the number of frames.
The finite element idealization used for structures A6 and
B6 is shown in Fig. 43; Fig. 44 shows their floor idealization. For
structures containing 6 and 8 bays, only one quarter of the building
was idealized and the longitudinal axis of the floor slabs was consi
dered as their neutral axis for bending around a vertical axis.
All the computations were performed in single precision.
The scheme for eliminating the bending in the plane of the
floors is only approximate since the new loading pattern induces a
different distribution of moments and shears in the frames. It will
be seen in the next section that this scheme was indeed effective
since the amplitudes of the lateral deflections of the floor slabs
due to bending in their own plane was brought within a fraction of
their actual values.
Results obtained with the modified load pattern will be
referred to as "A2R, A4R, ... , B2R, ... , B8R", where R stands for
- 113 -
24' 24'
24'
FIGURE 43. lon of Idealizat" structures A6 , 86
- 114 -
1 61 62 63
64 ~ ~ t·- --= --=-=-.:-~~==-= ~
145 46 47 48 1
l, , 5 6, 57 58 59
1 1 , 1 1 1 1
1 , 1 1 F "- -1 1 1 1 1 1 1 1
1 1 1
1 , 1 ,.. r-1 1 1
1
y 1
1 1
6 1 7 8 1 1 1 2 3
1 ,
2 3
20'
FIGURE 44. Floor idealization for structures A6, 86.
,-
9
4
601 1 1
" :1 l' , 1
., 1 1 , 1 1
ii li
1 1
1
1 1 1
" , 1 1 l'
101
1
4 l'
51
-
~ N
... "f' N
...
~
X
- 115 -
"Rigid floors". Due to the residual bending mentioned above, it is
reasonable to postulate that the discrepancy between the "R" and
"noRu results may usually be interpreted as a lower bound of the
error made when assuming that the floors are rigide This statement
does not hold, however, if the structures display markedly different
types of behaviour. This point is particularly valid for the low
four storey structures of series 8, and will be discussed in the next
section.
A short summary of the notation used to denote the structures
under investigation is thus as follows: the notation consists of three
digits, of which:
- the first, an alphabetic character, refers to the number
of storeys (A for 8 storeys, B for 4 storeys);
- the second is the number of longitudinal spans (2, 4, 6
or 8 with corresponding lenqth/width ratios of 1.2, 2.4,
3.6 and 4.8:1);
- the third, the letter R, if any, indicates that the results
refer to the loading scheme meant to keep the floor slabs
as rigid as possible. Results corresponding to this scheme
will normally be plotted in dotted lines.
5.2 Results
5.2.1 Deflections
Figs. 45 and 46 give the lateral deflections obtained for the
various structures.
Under the uniformly distributed wind loads these are markedly
- 116 -
......., '+-
0 ~
Il ... ... .-
. c:(
<LI c.. >, -I-l
III <LI ~ :::1 -I-l U :::1 ~
-I-l III
~ 0
III t: 0 .,.... -I-l U <LI r-~ QJ '0
r-tIj ~ QJ
-I-l tIj
....J
. Ln <:::t
l.LJ 0:: ::::> (.!l ..... I.i..
11 11 ->' 1 fi ~ 1.'/1" /' ~ 1 r,' '/~ ~ JI 1 j l _,;~;;~ J ~ .1 ~ :;;»
B2 84 86 B4R B6R
88 B8R
1"= O. 5 x 10--1 ft
FIGURE 46. Lateral deflections of structures type B.
......
- 118 -
larger towards the middle of the buildings than at the end shear
walls. At the first storey, the difference is of sorne 50% for pro
totype A4 and of 100% for A6 and A8; it is reduced to less than 10% at
the fourth storey of A4, does not reduce below 10% for A6, nor below
20% for A8.
These observations confirm those of Majid and Croxton (27)
who noticed that the lower floors were the most affected by the in
plane floor deformations. However this effect is already well marked
for a length/width ratio of the slabs of 2.4:1, as opposed to the ratio
of 3.2:1 mentioned in (27). It is neqligible for structures A2 and B2,
and therefore these wi 11 not be anal ysed under the If Rif l oad i ng scheme.
On Fig. 45, it may be seen that the deflection patterns indi
cated by the If Rif curves are essentially similar to those obtained under
uniformly distributed loads; their spread is indeed much reduced. A
certain amount of back bending occurs in the top storeys of structures
A6R and A8R, but its amplitude is very small and creates only a very
limited increase of the bending moments. Tt serves as an indication,
however, that a second adjustment of the load for a further reduction
of the bending of the slabs would not necessarily produce the desired
effect.
On Fig. 46 the effect of the bending of the slabs in their own
plane is very much more pronounced. The larger amplitude of the floor
deflection is due to the smaller cross section of the vertical columns
and the smaller resistance they offer to the lateral loads.
In the "R" loading scheme, the deformed shapes of structures
R6R and R8R differ considerably from the curves of B6 and B8. This
- 119 -
difference is also pronounced in the figures showing the axial loads
and moments in the columns. Very clearly, the "R" loading scheme is
not applicable here, and no conclusions will be drawn from these curves
which are given for completeness only.
5.2.2 Axial Loads in the Columns
Figs. 47 and 48 show the magnitudes of the loads in the exterior
columns of the building. The load distribution along the length of the
structures is plotted for each floor and each prototype: A8-A8R, A6-A6R,
etc.
Fig. 47 shows the results for the 8 storey structures. The
graphs display a maximum value of the load for the frames situated at
24 1 0" from the shear walls, and another at mid-length, in the plane
of symmetry of the structures. The first peak is due to the transfer
of vertical load from the shear wall to the frame, the second one seems
due to the additional bending in the frames, due to the larger amplitude
of the deflections at the centre. This second peak disappears when the
floors remain more rigide
The discrepancy between the results under uniformly distributed
load and the "R" loading scheme is otherwise quite small, except for the
longest structure (A8-A8R) where it reaches 20% at the first and second
storeys.
Fig. 48 shows similar results for the 4 storey structures but
the discrepancies are much larger due to the differences in behaviour
observed above.
N(kips) ct 18.0 \. /1
)
A8,A8R
6.0
4.0
----4 5 2.0
1 - "" '" /
'\: " ~--5
o
~6
'-.... ~ __ =7 ~ ___ ---~8
241 481 72' 96'Y
FIGURE 47. Axial loads in outside columns. 8 storey buildings.
'---_____ <t.
storey ~\A6,A6R1
2
7 ----8
24' 48' 72'Y
A4JA4R <t -" N o
Cl A2
1 +
2+ 3 ....
4+
5+ 6+ 7+ 8-4--y 241
N( kips) t
1.2 ./ 1
B8,B8R
0.8 , , 2
0.4 , ", 1 ,
3
" / 2
':~~ ~--~ " , -----4Y ~==-= --- - -----
24' 481 72 1 961
FIGURE 48. Axial loads in outside columns. 4 storey buildings.
·storey
1 \ ~
1 ~ \7 B4,B4R •
~~ (
L 1 \\ / , , \ ,
3 2S-3 '\
4 ~ " , "-4 -- ,,' ......... -.... "-~y --, 24' 48'
IB2
N --'
~
1 ... 2 ... 3 4 4 .. 24' Y
- 122 -
5.2.3 Bending Moments
Figs. 49 to 52 show the bending moments obtained in the outside
columns, above the floor slabs. For convenience, each drawing shows
the moments at a given distance from the end walls, for the various
prototypes.
On Figs. 49 to 51, the agreement between the moments in the
actual structures and those with rigid floors is quite good, from the
8th storey down to the 3rd. Below the 3rd floor, the moments increase
much faster for the actual structures, and the discrepancy with the
moments of the "R" schemes reaches sorne 25% at a distance of 24 and
48 ft from the shear wall, and up to 40% at 72 and 96 ft in the case
of the longer structures. This is slightly less than the value of
50% observed by Majid and Croxton, but their test structure consisted
of columns of constant cross section.
By superimposing the various graphs, one might observe that
the moments in the various frames of a single structure have comparable
magnitudes, at least from the 8th ta the 3rd floor.
On Fig. 52, similar results are given for the four storey
structures.
5.2.4 Shears
On table 10, a comparison is made between the percentages of
the total shear taken by each frame at each level, for the loading
cases considered. Again, it rrray be seen that these figures vary with
the height of the building, but that they do not vary appreciably from
one frame ta another.
storey 8
5
4
3
2
1
, 1 1 1 1
columns at y= 24.0'
A4,A4R
A6A6R
M
8 'if '
71h~ ~
5
~
3
2
1
A4 A4R j
A6 columns at y= 48.0'
A6R 1
A8R ,
4.0 8.0
N W
M( kip. ft)
FIGURE 49. Moments in outside columns at the top of each floor slab.
8
7
6
5
4
3
2
- 124 -
storey
A6
1 1 1 1 1
1 1 1 , ( 1 \ \
\ \ ~A6R \ ) ) 1 1
1 A8R! 1 ( , ,
columns at y= 72.0'
, , , l , " .... " ........ -" ........ -" -.... " -~--1~ __ ~~ __ "~~~~~ ____ ~ ____ .... ~ ____ ~ ____ ~ __ ~~
4.0 8.0 12.0 16.0 M(kip ft)
FIGURE 50. Moments in outside co1umns at the top of each f100r slab.
storey
7 , ,
5
4
3
2
n
, , , '\ , , , , \ \ \ \ \
1 1 , 1 , \ ,
A8R, , ,
4.0
.... ---
columns at y=96.0'
-- '""'-------8.0 12.0
(kipjft) 16.0 M
FIGURE 51. Moments in outside co1umns at the top of each f100r slab.
N (J1
4
1
3,
2
3
2
1
4
3
2
, , , ,
- 126 -
,
2.0
2.0
B8R B8
4.0
86
4.0
columns at y= 96.0'
6.0
y =72.0'
6.0
y=48.0'
1L-~~~~~ __ ~~~~ ____ ~ __ ~~_ 6.0 M
(kip ft) 2.0 4.0
FIGURE 52. Moments in outside co1umns at the top of each f100r slab.
storey
4
3
2
columns at y=24.0'
82
B4,B4R
B6,B6R B8.B8R
11 \ " .~, ""'b~' ~ lO 2.0 3.0 M(kipft )
FIGURE 52 (Cont'd)
~
N '-1
storey
8
7
6
5
4
3
2
1
4
3
2
1
- 128 -
TABLE 10A
Percentages of total shear taken
over by the various frames
Frame at 24 1 -0" Frame at 48 1 -0" Frame at 72 1 -0" Frame at 96 1 -0"
actua1 R actual R actua1 R actual R loading loading loading loading loading loading loading loading
Structures A8, A8R
7.55 8.29 7.42 8.65 6.78 8.15 6.22 7.92
4.06 3.90 4.70 4.45 4.92 4.61 4.48 4.64
6.08 6.42 5.75 6.15 5.33 5.45 4.62 5.28
4.45 4.27 4.96 4.42 4.80 4.35 4.50 4.28
4.66 4.40 4.80 4.15 4.80 3.80 4.44 3.60
3.82 3.14 4.51 3.17 4.92 3.11 4.84 3.02
3.89 3.06 4.97 3.33 5.72 3.40 5.80 3.44
4.62 3.50 6.40 4.40 7.55 5.01 8.20 5.24
Structures B8, B8R
2.50 1.66 2.89 1.36 2.72 1.05 2.98 3.14
1.84 0.66 3.14 0.25 2.60 0.25 3.30 1.84
3.71 2.00 5.76 2.10 4.80 2.14 5.86 2.08
4.28 2.55 8.35 2.66 6.74 4.32 8.92 4.62
Storeys
8
7
6
5
4
3
2
1
4
3
2
1
- 129 -
TABLE lOB
Percentages of total shear
taken over by the various frames
Frame at 24'-0" Frame at 48'-0"
actua1 R actual R loading loading loading loading
Structures A6, A6R
8.75 9.35 9.10 10.3
4.52 4.16 5.75 4.68
6.80 7.25 6.66 7.35
4.92 4.85 5.26 5.23
4.92 4.82 4.92 4.67
3.80 3.20 4.15 3.18
3.98 2.98 4.72 2.86
4.84 3.85 5.60 4.95
Structures B6, B6R
2.39 2.06 2.42 2.09
l.40 0.71 1.65 0.55
2.96 1.86 3.70 1.57
4.29 2.33 6.27 3.04
Frame at 72'-0"
actua1 R loading loading
8.40 9.40
5.20 4.70
5.90 7.04
4.96 5.22
3.80 4.38
4.02 3.08
4.78 2.78
3.46 2.56
2.65 2.02
0.93 1.60
l.85 0.65
7.00 3.32
- 130 -
TABLE 10C
PercentaQes of total
shear ta ken over by various frames
Frame at 24 1 -0" Frame at 48 1 -0" Frame at 24 1 -0"
actual R actual R actual loading loading loading loading loading
Structures A4, A4R Structure A2
8 9.78 10.8 10.4 9.6 4.90
7 4.56 4.17 4.94 4.58 4.32
6 7.80 8.15 7.80 8.64 8.64
5 5.37 5.12 5.54 5.42 5.48
4 5.42 5.46 5.28 5.64 5.94
3 3.80 3.58 3.74 3.64 4.04
2 3.60 4.90 3.62 2.94 2.44
1 4.70 3.36 5.36 3.56 3.98
Structures B4, B4R Structure 82
4 2.34 2.32 2.36 2.44 2.40
3 1.09 0.95 1.08 0.96 0.90
2 2.30 1.75 2.17 1.69 2.27
l 3.35 1.78 4.36 1.92 2.25
- 131 -
As expected, the percentages of the shear resisted by the
frames are larger near the base of the structures for the actual load
than for the "R" loading scheme.
5.2.5 Conments
These computer results indicate that the usual assumption of
the floors behaving rigidly in their own planes (18,19,20,22,23,24)
is not accurate, and may lead to the bottom parts of structures being
seriously underdesigned.
In this study, very stiff end walls have been used. Such walls
would not be used for 4 or 8 storey buildings, but they \'/ould be conmonly
found in higher buildings for which these conclusions are expected to
hold.
If a structure were to consist of a central, rigid core and
weaker frames in both s!des, opposite conclusions might be anticipated,
the discrepancy between the axial loads, moments and shears becoming
greater near the extremities.
From the observations made in the previous sections, an appro
ximate method of analysis might be envisaged as follows:
1. Assume that each floor takes a given percentage of the total
shear. This figure might be a constant, (5% for example) or
might vary linearly along the height of the building.
2. Determine the lateral deflection of the shear walls under
the remainder of the load.
3. From the beam theory, determine the lateral deflections of
the floor slabs in their own planes.
- 132 -
4. Combine the 1atera1 def1ections of the wa11s and slabs,
and get the def1ected shapes of the frames.
5. Compute the frames as separate units, for the values of
the sways obtained above. This may be performed by finite
e1ements as was do ne in section 2.3.4 or by means of an
equiva1ent plane frame system, using the values given by
Qadeer and Stafford Smith (63) for the appropriate stiffness
of the f100r slabs.
6. Compute the percentage of the total shear taken over by
each frame at a11 1eve1s, and check their average value
against the figure assumed in step 1. In case of discre
pancy, repeat steps 1 to 6 with an adjusted percentage.
7. Knowing the vertical displacements of the frame joints,
and those of points situated in the end wa11s, a correction
for the axial load in the frame nearer to the wall may be
eva1uated.
A significant saving in the cost of computation is expected,
and sma11er computers may a1so come into use.
In Fig. 53, the def1ected shapes of a11 the f100r slabs of
prototype A6 are given, for uniformly distributed wind load. Chain
1ines indicate the corresponding profiles obtained by the method out
lined above.
An average of 4% of the total shear has been assumed to be
carried by each frame. Clearly, the percentage used was dictated by
the results obtained above. More research ;s needed ;n th;s regard.
- 133 -
starey 0.60
--=-~·~-~-~-------t-·-----_·==-----l8 ----- . - -
.- ~ _______ ----+-------,7
:::::::~:::::.:-==~~==--:.=-~·,6 --=.:: . ---. --- -
.-- 5 --
-----+-----13 __ -------1
L-____ ----~-------~----Îl o 24 48 72ft
distance fram wall
FIGURE 53. Structure type A6. Actual f100r deflections and approximation.
- 134 -
Chapter 6. Conclusions and Recommendations
6.1 Conclusions
In the present work, a new approach to the analysis of tall
building structures, making use of finite elements and a matrix
substructure analysis of the floor slabs, has been present~d.
The computer program, which is given in Appendix 3, has been
tested in the solution of several numerical problems and the results
compare very well with those of other investigations, whether experi
mental or anlytical.
It may be used for a large variety of usual building configu
rations comprising slabs, shear walls, columns, beams, diagonal braces,
etc. Openings in the walls and floors are taken into account, and so
are the effects of torsion, axial load and shear deformations.
The cost of computation is very moderate.
The only limitations are that the structure must conform to a
rectangular parallelepiped grid, however irregular it may be, and that
the size of the structure, in particular the number of nodes in the
vertical columns and the shear walls, is linked to the size of the
in-core memory of the computer used.
The input data for the program is easy to set up; the output
consists of the deflections of all the joints situated on the columns
and walls, of the axial loads, moments and shears in the columns and of
the stresses and strains in the shear walls.
In the course of this work, two original formulations were also
developed.
- 135 -
The first is a set of new disp1acement functions for the
bending of rectangu1ar e1ements, with the fo110wing advantages:
- simple disp1acement functions for both membrane action
and bending;
- six physica1 degrees of freedom that can be transformed
according to the vector 1aw of transformation, at each
node;
- no additiona1 nodes a10ng the sides of the e1ements;
- conformabi1ity between adjacent e1ements, and compatibi1ity
of disp1acements for orthogona11y situated e1ements;
~ shear deformations inc1uded.
The convergence characteristics compare very we11 with other e1ement
formulations. So far, on1y the stiffness matrix has been generated;
applications are thus 1imited, for the moment, to 1inear static prob1ems.
A1so, singu1arity may occur for very thin plates subjected to
large def1ections but such occurrences are not expected in building
structures. More work is required in this area.
The second original deve10pment is a new finite e1ement treatment
of the ana1ysis of plane shear wall structures and their combination to
plane frames.
Externa1 moments are transmitted in the plane of the elements
by means of fictitious beams having a large bending rigidity; semi-rigid
connections between beams and wa11s may be considered, and the effect of
local distortions at the abutment zones is inc1uded. This procedure com
pares favourably with other avai1ab1e methods of shear wall ana1ysis.
- 136 -
A study of building structures consisting of end shear wa11s
and intermediate frames subjected to 1ateral wind loads was undertaken.
The number of these frames was varied, a10ng with the 1ength/width ratio
of the f100r slabs. This study has shown the necessity of taking the
bending of the floor slabs in their own plane into consideration.
A comparison with results obtained when keeping the floors rigid
has demonstrated that this often used assumption leads to a considerable
underdesign of the columns in the lower storeys of the structure. This
effect has been found sizeable for length/width ratios of only 2.4, but
affected on1y the first three storeys of the buildinysconcerned. Axial
loads, moments and shears were all increased by the deformation of the
floors.
The results have also emphasized the significant transfer of
vertical loads from the end walls to the columns of the adjacent frames.
6.2 Recommendations
Recommendations for future work include further refinements to
the computer program described herein, as well as the extension of its
use to research problems which are neither static nor linear.
First to be imp1emented shou1d be the introduction of subroutines
for the eva1uation of the stiffness matrices of triangular or quadri
lateral elements, so that buildings of arbitrary plan may be considered:
curved, circular or triangular structures, etc.
Then, a close look at the representation of shear walls by means
of finite elements should be taken. The present representation implies
a fairly large number of elements and nodes, which could be reduced by
- 137 -
either of the following schemes:
a) by a substructure analysis of the portions of shear
walls comprised between consecutive storeys that would
eliminate all the intermediate nodes, or
b) by adopting the stiffness matrices derived by Heidebrecht
and Swift (64) for various shapes of walls, and which
connect actions and disp1acements at end nodes only.
A considerable saving in computer core requirement might ensue,
especially for structures with multiple walls.
Dynamic ana1ysis is likely to become a field of application of
the present work. Such studies are indeed hampered by the large size
of the matrices whose eigenvalues and eigenvectors must be found. The
present substructure analysis provides a considerable reduction of that
size, and is thus a step in the right direction.
The response of structures to earthquake motions consists mainly
of horizontal displacements. Further reduction in the size of the problem
might consist in e1iminating the other degrees of freedom at each node by
static condensation. The additional work to be performed in this line
wou1d comprise:
i) the determination of the consistent mass matrices of the
typical f1oors;
ii) their condensation along the lines described above;
iii) the computation of the consistent mass matrix of the column
and shear wall system, to which the equivalent masses of
the floors must be added.
- 138 -
In the field of materia1 non-1inearities, step by step
procedures might evo1ve from the present work; howe1er, the present
substructure ana1ysis is especia11y advantageous if severa1 identica1
f100r slabs may be found in the structure.
This situation might not subsist after non-1inear deformations
have ta ken place.
- 139 -
BIBLIOGRAPHY
1. COULL, A. and STAFFORD SMITH, B., "Tall Buildings", Proc. Symposium held in the Dept. of Civ. Engrg., Univ. of Southampton, April 1966, pergamon Press, Oxford, 1967.
2. STAFFORD SMITH, B., "Recent Developments in the Methods of Analysis for Ta11 Building Structures", Civ. Engrg. and Public Works Review, 1970, 65(Dec), 1417-1422.
3. KANI, G., "Analysis of Multistorey Frames", Frederik Ungar, New York, 1956.
4. TAKABEYA, F., "Multistorey Frames", W. Ernst & Sohn, Berlin, 1965.
5. GERE, J. and ~JEAVER, W., "Ana1ysis of Framed Structures Il ,
Van Nostrand, Princeton, 1965.
6. BECK, H., "Contribution to the Analysis of Coupled Shear Wall Sil ,
J. Am. Concr. Inst., 1962, 39(Aug), 1055-1069.
7. ROSMAN, R., "Approximate Analysis of Shear Wal1s subject to Lateral Loads", J. Am. Concr. Inst., 1964, 41(June), 717-732.
8. COULL, A. and CHOUDHURY, J.R., "Stresses and Deflections in Coupled Shear Wa11s", J. Am. Concr. Inst., 1967, 64(Feb), 65-72.
9. COUlL, A. and CHOUDHURY, J.R., "Analysis of Coupled Shear Wall Sil ,
J. Am. Concr •. Inst., 1967, 64(Sep), 587-593.
10. SCHWAIGHOFER, J., "Door Openings in Shear \~alls", J. Am. Concr. Inst., 1967, 64(Nov), 730-734.
11. CARDAN, B., "Concrete Shear Walls combined with Rigid Frames in Mu1tistorey Buildings subject to Lateral loads", J. Am. Concr. Inst., 1961, 58(Sep), 299-315.
12. SCHWAIGHOFER, J. and MICROYS, H., "Analysis of Shear Walls using Standard Computer Programs", J. Am. Concr. Inst., 1969, 66(Dec), 1005-1007.
13. ClOUGH, R., KING, I. and WILSON, E., "Structural Analysis of Multistorey Buildings", J. Str. Div. Am. Soc. Civ. Engrs., 1964, 90(Jun), 19-34.
~ 140 -
14. KHAN, F. and SBAROUNIS, J., "Interaction of ShearWalls and Frames", J. Str. Div. Am. Soc. Civ. Engrs., 1964, 90(Jun), 285-335.
15. THADANI, B., "Analysis of Shear Wall Structures", Indian Concr. J., 1966, 40(Mar), 97-102.
16. TAMHANKAR, M., JAIN, J. and RAMASWAMY, G., "The Concept of TwinCantilevers in the Analysis of Shear Wal1ed Multistorey Buildings", Indian Concr. J., 1966, 40(Sep), 488-498 and 516.
17. CHAN DRA , R. and JAIN, J., "Analysis of Shear Walled Buildings", Indian Concr. J., 42(Dec), 506-515.
18. JAEGER, L.G., FENTON, V.C. and ALLEN, C.M., "The Structural Analysis of Buildings having irregu1ar1y positioned Shear Walls ll
, paper presented at the October 1970 Meeting of the Canadian Capital Chapter of the Am. Concr. Inst.
19. ROS MAN , R., "Analysis of Spatial Concrete Shear Wall Systems", Proc. Inst. Civ. Engrs., 1970, Supplement vi, paper 7266S.
20. GLUCK, J., "Lateral Load Analysis ofAxisymmetric Mu1tistorey Structures ll
, J. Str. Div. Am. Soc. Civ. Engrs., 1970, 96(Feb), 317-333.
21. MICHAEL, D., IITorsional Coupling of Core Walls in Ta" Buildings ll,
The Structural Engr., 1969, 47(Feb), 67-71.
22. WINOKUR, A. and GLUCK, J., "Lateral Loads in Asymmetric Multistorey Structures", J. Str. Div. Am. Soc. Civ. Engrs., 1968, 94(Mar), 645-656.
23. STAMATO, M.C. and STAFFORD SMITH, B., IIAn Approximate Method for the Three-dimensional Ana1ysis of Tall Buildings··, Proc. Inst. Civ. Engrs., 1969, 43(Jul), 361-379.
24. COULL, A. and IR~lIN, A.W., "Analysis of Load Distribution in Multistorey Shear Wall Structures", The Structural Engr., 1970, 48(Aug), 301-306.
25. OICKSON, M.G. and NILSON, A.H., IIAna1ys1s of Cellular Buildings for Lateral Loads", J. Am. Concr. Inst., 1970, 67(Dec), 963-966.
26. GOLDBERG, J.E., IIAnalysis of Multistorey Buildings Considering Shear Wall and Floor Deformations ll
, Tall Buildings, Pergamon Press, Oxford, 1967.
.. ,/
~ 141 ~
27. MAJID, K.I. and CROXTON, P.C.L., "Wind Analysis of Complete Building Structures by Influence Coefficients", Proc. Instn. Civ. Engrs., 1970, 47(Oct), 169-184.
28. MAJID, K.I. and WILLIAMSON, M., "Linear Ana1ysis of Complete Structures by Computers ", Proc. Instn. Ci v. Engrs., 1967, 38(Oct), 247-266.
29. HOLAND, I. and BELL, K., "Finite Element Methods in Stress Analysis", Tapir, Trondheim, 1969.
30. AR6YRIS, J.H. and KELSEY, S., "Energy Theorems and Structural Analysis", Butterworths & Co. Ltd., London, 1960.
31. ZIENKIEWICZ, O.C. and CHEUN6, Y.K., "The Finite Element Method in Structural and Continuum Mechanics", lst edition, Mc6rawHill, London, 1967.
32. IRONS, B.M. and DRAPER, K.J., "Inadequacy of Nodal Connections in the Stiffness Solution of Plate Bending", J. Am. Inst. Aeronautics and Astronautics, 1965, 3(May), 961.
33. MELOSH, R.J., liA Flat Triangular Shell Element Stiffness Matrix", Matrix Methods in Structural Mechanics, Proc. Conf. held at the Wright-Patterson Air Force Base, Ohio, Oct. 1965, 503-514.
34. UTKU, S., "Stiffness Matrices for Thin Triangular Elements of Non-Zero 6aussian Curvature", J. Am. Inst. Aeronautics and Astronautics, 1967, 5(Sep), 1659-1667.
35. DHATT, 6.S., "Numerical Analysis of Thin Shells of Arbitrary Shape", Proc. 2nd Cano Congress of Appl. Mech., Univ. of Waterloo, Ontario, May 1969, 965.
36. DHATT, 6.S., "Numerical Analysis of Thin Shells by Curved Triangular Elements based on Discrete Kirchoff Hypothesis", Proc. Symp. on Application of Finite Element Methods in Civ. Engrg., Am. Soc. Civ. Engrs., Nashville, Tenn., Nov. 1969, 255-278.
37. MEHROTRA, B.L., "Matrix Analysis of Welded Tubular Joints", Ph.D. Thesis, Dept. Civ. Engrg. and Applied Mechanics, Mc6ill Univ., Montreal, 1969.
38. MEHROTRA, B.L., MUFTI, A.A. and REDWOOD, R.6., "Analysis of Three-dimensional Thin-Walled Structures", J. Str. Div., Am. Soc. Civ. Engrs., 1969, 95(Dec), 2863-2872.
- 142 -
39. MUFTI, A.A. and HARRIS, P.J., "Matrix Analysis of Shells using Finite Elements", Trans. Engrg. Inst. of Canada, 1969, 12{Nov), A-9.
40. BOGNER, F.K., FOX, R.L. and SCHMIT, L.A., "The Generation of Interelement Compatible Stiffness and Mass Matrices by the Use of Interpolation Formu1ae", Matrix Methods in Structural Mechanics, Proc. Conf. held at the Wright-Patterson Air Force Base, Ohio, Oct. 1965, 397-443.
41. WEMPNER, G.A., ODEN, J.T. and KROSS, D.A., "Finite Element Analysis of Thin She11s", J. Engrg. Mech. Div. Am. Soc. Civ. Engrs., 1968, 94{Dec), 1273-1294.
42. MUFTI, A.A., "Matrix Analysis of Thin Shells using Finite Elements", Ph.D. Thesis, Dept. Civ. Engrg. and Applied Mechanics, McGil1 Univ., Montreal, 1969.
43. MUFTI, A.A., MAMET, J.C., JAEGER, L.G. and HARRIS, P.J., "A Program for the Analysis of Three-dimensional Structures using Rectangu1ar Finite Elements", Report No. 70-1, Structural Mechanics Series, McGill Univ., Montreal, April 1970.
44. UTKU, A. and MELOSH, R.J., "Behaviour of Triangular Shell Element Stiffness Matrices Associated with Polynedral Deflection Distributions", 5th Aerospace Science Meeting, Am. Inst. Aeronautics and Astronautics, New York, Jan. 1967, paper 67-114.
45. TIMOSHENKO, S. and WOINOWSKY-KRIEGER, S., IITheory of Plates and She1ls", 2nd Edition, McGraw-Hill, New York, 1959.
46. KRAUS, H., "Thin E1astic Shells", Wiley and Sons, New York, 1967.
47. GOLDBERG, J.E. and LEVE, H.L., "Theory of Prismatic Folded Plate Structures", Proc. Internat. Assoc. for Bridge and Structural Engrg., 1957, 17.
48. ZIENKIEWICZ, O.C., PAREKH, C. and KING, I.P., "Arch Dams Analysed by a Linear Finite Element Shell Solution Program", Paper No. 3, Symposium on Arch Dams, Instn. Civ. Engrs., Southampton, 1968.
49. MACLEOD, I.A., "New Rectangular Finite Element for Shear \~all Analysis", J. Str. Div. Am. Soc. Civ. Engrs., 1969, 95(Mar), 399-409.
- 143 -
50. HRENIKOFF, A., Discussion on Reference 49, J. Str. Div. Am. Soc. Civ. Engrs., 1969, 95(Oct), 2323-2325.
51. SPlRA, E. and SOKAL, V., Discussion of Reference 49, J. Str. Div. Am. Soc. Civ. Engrs, 1970, 96(Aug), 1799-1802.
52. POLE, G.M., Discussion on Reference 49, J. Str. Div. Am. Soc., Civ. Engrs., 1970, 96(Jan), 140-143.
53. GIRIJAVALLABHAN, C.V., "Analysis of Shear Walls with Openings", J. Str. Div. Am. Soc. Civ. Engrs., 1969, 95(Oct), 2093-2103.
54. GIRIJAVALLABHAN, C.V., "Analysis of Shear Walls by Finite Element Method", Proc. Symposium on Application of Finite Elements in Civ. Engrg., Am. Soc, Civ. Engrs., Nashville, Tenn., Nov. 1969, 631-641.
55. OAKBERG, R.G. and WEAVER, W., "Analysis of Frames with Shear Walls by Finite Elements ll
, Proc. Symposium on Application of Finite Elements in Civ. Engrg., Am. Soc. Civ. Engrs., Nashville, Tenn., Nov. 1969,567-607.
56. MICHAEL, D., "The Effect of Cross Wall Deformations on the Elastic Interaction of Cross Walls Coupled by Beams", Tall Buildings, pergamon Press, Oxford, 1967.
57. MAMET, J.C., MUFTI, A.A. and JAEGER, L.G., "New Developments in the Analysis of Shear Wall Buildings", Proc. lst Canadian Conf. on Earthquake Engrg. Research, Univ. of British Columbia, Vancouver, May 1971.
58. MEHROTRA, B.L. and REDWOOD, R.G., liA Program for Plane Stress and Plane Strain Analysis", Report No. 4, Structural Mechanics Series, Dept. of Civ. Engrg. and Applied Mechanics, McGill Univ., Montreal, 1968.
59. WEAVER, W., "Computer Programs for Structural Analysis", D. Van Nostrand, Princeton, 1967.
60. MELOSH, R.J., IIManipulation Errors in Finite Element Analyses", Lecture Notes, Cornell Univ., 1969.
61. ROSEN, R. and RUBINSTEIN, M.F., IISubstructure Analysis by Matrix Decomposition", J. Str. Div. Am. Soc. Civ. Engrs., 1970, 96(Mar), 663-670.
62. DAVIES, J.D., "Analysis of Corner Supported Rectangular Slabs", The Structural Engr., 1970, 48(Feb), 75-82.
- 144 -
63. QADEER, A. and STAFFORD SMITH, B., "The Bending Stiffness of Slabs Connecting Shear Wall Sil , J. Am. Concr. Inst., 1969, 66(June), 464-473.
64. HEIDEBRECHT, A.C. and SWIFT, R.D., "Ana1ysis of Asymmetrica1 Coup1ed Shear Wa11s", J. Str. Div. Am. Soc. Civ. Engrs., 1971, 97(May), 1407-1422.
65. MUFTI, A.A., Private Communications.
66. COULL, A., Private Correspondence.
[C,J
- 145 -
Appendix 1. Matrices for Bending Action
Strain matrix
Stress matrix
[Sb] = t'
12
o
o
o
0
0
0
o
o
0
0
Dxy
(Cb![f[QLJT[D] [~dA] [Cb] ::
le
0 0 0 0
0 Dxy 0 :Jo.l>,cy 2.
0 0 D,)' ~ 2.
0 Yc D,y "'c.D,y x: l)y +ystPlsy 2- 2. ~
0 0 0 0
0 0 -DI )CC: P, ... -2.
0 -D. 0 )'c.lh,~ ~ 2-
0 _ )cc: Dxy Ye.!)1 _ XcYe(P' + Dy) --2- Z. 4
with l)x DI 0
[D J = D, Dy 0
0 0 Dx.1
o o
-1 -x
o
• 'Dl -xDI
-D'y -xDy 0 JD~J
0 0
0 0
0 DI
0 _ )CC~' '2.
0 0
0 D)C
0 0
Yc DI( '2
o o .y
o o o o
o o -1 -'X
0 Dx 0 yD"
0 DI 0 yD,
0 0 -~ -xDxy
0 0
- D"1 -~ 2
0 .':k. DI 2-
Yc~~ _ "!:Itç.(DI 1-DxJ [C,,] - 2 ~ ~
0 0
0 ~ 2.
D)(,Y ~ z. 2.
~ Xc Dx.>'+y:~ 2 '3
W" "'''''33 -~(2.W'I+W2~) ~(Wzl"t2W51) W11 0 ~)J%I 2W~1 -)'c ~, XC W,\ ~1. - :t~ W"$2 0 I~ 2. -il :3
Y. ~ ~ -y 2 -:t.2. W
CD
0 0 0 Yc W31 'k'lc W )!2W
en
-i 1V}I .• Wt, i ~W3tW31..) Xc{C W;51 -t }\ ~Wn -~ 52- 0 en
~ JI 2 4 3 QI cT
- ~(W21+Z\J31) -tW1\ ~W )(2 'II.
~
lC~ (,\,+1.~I) ~W31 0 ~Vl3\ Y-èic.W 0 0 0 ~.
- 'le J\ x
-- 11 T 3\ 'l. i -il
0 ~
Wtl
0 Xc.W Wu -~ Wn -i 'Ntl W~2. -~ W31 0 0 0 0 1 I~ -- tJ 2 2
~ ~.
1. 0 .!s:.. YJ32.
y,2\aJ 1:3
0 0 0 Ycw YcW 0 0 0 0 -- 11 -- 11 - -S:. 32-
2. 4 2 4 I~ --' ~
cT 0'1
XC W21 0 x2 xc:.W 0 x2 a 0 0 0 0 0 I~· - -f.. W11 -- 21 -~WZf
t . 'l 1 2- 4
lKs1=-JcW~1 -xc VlJI ~2. \'111+ W'3 1(Z~I+Wu) - i('NZI+2WjI) W21 7.~yçl 2..W'31 ";/cWJ1 0 0 -~~,
2-
-Ye. W'I _ YJW31 Xt)'e W :J,.w. y?w 0 .Y; (Z~l+ \VI.)) -YI (2W.nt'llsl) XcYsW 0 0 0
2. T JI -t ~t -~ J1- 1 11
XC W31 xe:>'c W 2 -~(~1 .. 2W3\) 'I.c'.fc W )C2. oz.
- 31 -~W31 0 0 0 T 31 --1(Wfl+ 2 'rJll) r-Wfl 0 _~c ~I 2 '2.
W31 ':iC.W31
0 0 0 0 W11 0 ~ \\1 W2.'Z. Yc'N Xc W 2 2- 2" JZ. T 11
)Jc W"j2 y2 .J 0 0 0 0 0 0 0 Ye. ~ Y,}W 0
_.J::. t 11. - 2. -- 32-
1- 4 'l. 4
0 0 0 0 0 0 Xcw 0 x2W XcW 0 x2
-2 tl _-S.. 2.1 - 21 -~WZI 4 2 4
in which:
~,
W '1
W 2t
- 147 -
o
o
EAII EAICd.' _liA. 0 0 0 _ EAxa' 0 ):a
0 0 0 0 "0 -r- L 1.. L "0 ID
12 rIe '(lt ~
_/ZE/iJ.' 6fli _12'[~ 12lIrd' 0 Q..
0 0 0 0 0 -x L' L3 LZ L Ll L'l N
12.'1:1 _6~1~ _n'Iy _'~I;t .
0 0 0 0 0 0 0 0 LJ L2- L' LZ aJ ID QI
_ I~'/i. J' G:J + /2 E;tcl/2. _ 6G1i d' /2 &IiJ.' _Gj_/2;Iir 0 _ 6EIl,.d' 3 0 0 0 0 0 (1) LJ L L [.2. L3 L L [.2 C"'t' ....
-fi -fi
EAlId.' _ 6EI.y 4Eli+EA J' 2. _ bAUd' 'Ely 2'~€Ad'''' II~ 0 0 0 0 0 0 L l7. [. L. L L1. L. L 3: QI 1
0 6Glr 0 _'~IÈdl 0 ""'li; 0 6&1~ 0 '~l"d' 0 ?GI~ /13. ~ ~
L'2.. L2 L - --;-r- - x Ct) L lZ L
EAw 0 0 0 _ f"AKd, 0 .fAx 0 0 0 G'AIC cl' 0 -y-
L L L
0 /2GI~ 0 12 ~Iiï JI 0 6~/:a 0 12fI~ 0 _12~Iè cl' 0 6CJ1; - [.J --L3 L'2. LJ L3 L2-
0 0 _ Il'Iy 0 6&/.1 0 0 0 12bIy 0 'b"Iy 0 L'
[.2- L3 l"
0 12fI~ JI 0 <;j ItlZ d ,
7-0 6~I-i:d' 0 _ J1.fIj d' 0 G:r 12 'I. ri''' 0 GE~d' _~_ i -rLF Ll L L~ LZ. LJ L&.
_ éA)I ([' 0 _ 6fI:!
0 2 Efy_fA 1ld''L- 0 EA"d' 0 6~[~ 0 4~ EA.cll. 0 'L L .... L L L L1- L ... L
0 6f"I -l
0 _6EI~d' 0 2f:T~ 0 'G:Ia 0 '~ltd' 0 4~Ilr -- --- -LI,. L1- L L1.- L" L
- 149 -
Appendix 3. Computer Program Listing
G LEVEL 20 MAIN OAlF • no78
c
I~TtGER BEA'·' RF AL K1,IXS,IVB,IZB;KijB,KBIl OI~ENSlaN X~300,31/KE!4/3I,NFt100I,"B!10016I,BVI10C,611
1 T~iC~!300I,~STA~T!,Ol/~E"D~20l,NFIRSTIZOI,~LASTI20l/HI3,~I, 1 ~EI!3.31/·124.24) ,RI3001 ,XEOI1,3I,5BEI24,2 41; 1ALFAI12,12);NTYDFCI70),~~!96,4) C~H~J~ STIQ6,192);KRI119b,96),KBBI9b,961
1 ,Kl(24,24),C(i2;12),AI12,121,CI3,121/CCNI3/1ll/EI~/31 ! 1 clle 1 3/12 1 l '1 112,12). Enet! 1 1>,24 " tDOC lb, 24 1 ,1)( 1800,4) ,M~JNOP6 1,OZTI3,4),OZBI3,41,nSZT(3,4)IOSZBI3,41,h316,~I,A2Ib,61
l,~111OI,E2110l,poll~I,~211DI,CE(10),N[P(3DO),LK l,9El4(30D,b),IXG(5a),IYÜ(5D),IZBI50),EBI5~),A~150I,GS1501~HCB(51)1 l",::;, 1 ~0f\,4) E:UIVALr~CE (IIBC l;ll,'J( Isn0,,+I) OfFI~E ~ILE 151127S;1764,J,III) CALL ERRSET 1207,75~,~111,1,209)
RE,q 'l') 17 NPTY" • 0 """1\');1 • lb ~""':~PI'I • l'tl::Op*~ M~~,P7 c 12*~~NCP M'l';:);) 1 • M""iOp~1
R~A~ 15,101 NFL~OR ~pR~B • NFLnOR + ï
10 FCiUAT 110I4,2F1b",81 O~ 20 LA • l,"PROB RF'lil'ln 1 "-E o' 1!10 2 tl.Eh FIO 3 II,f .. 1 :10 4 IF (LA,LE,NFLOOP) r,~ TO 4 RE"l"," 17
4 RF AD 15.101 apART;NPOIN,NELEM,NBOUN,NCOLN,NFREE,NeO'IC l,~Y'''Ne~AH
'IV'" a 1 .... R ITE 1 ",101 NP ART l "PO 1 Il, NEL Ei'I, ~;BC:UII, ~ICnLN,';FR EE, NCONC
1"y ... ,·IB~AM D~ 30 1 a l~NpOIN RE:'~ 15,351 XlI>lI,XI!,2I,X"!I.3)
30 "PITE 1!l,35) XII,lI,XII,2),X(!,31 3' F~~:1AT 15F14.6) 4' F~R~AT 14t4.2Flb,B,141
REA' 1~,lO) IlCA~O IF I~CA~O~NPU!Nl iln,i11,110
110 STOP 111 Cn~:T l'lU!'
IF 1:IFIEAI1.EO.O) GO TO l09 1C4 Or, 20h tal,~aEA~
R~A~ 15.205! Fall',ABlI),!X~1!I,!VBIII,IZBI!),G~(II,~C8IJ) l::5 F~~o'Al IbEl2",Fb:l, 2el> ~RITE Ib,207) !,EPI!I;Aall),!XB!II,IYBII),IZ811I,GBIII,~tB(rl 207 Fr,IWAT 11H ,I4,I,E12,5,Ft-,Z)
REAl) 15.101 NCAPD
208 209
210
2lB
211 213
120 121
5"
46 217
60
64 99
6
68
69
65 34 33
HAIN
IF lNCARO-N~EAHI ZOB/209,108 S"OI" DO 218 !.l,~E'EH
DAT! • '720'711
RFAD t5,2101 lNnolI;JI,J.l,41,THIC~11I,~IJI~NEP1JII llrEAHII,JI,J-l,bl F~R~AT 1414,2F16,8,14;16,,141 NEPClI _ 1 W~ITE 16,2111 I,INUnlr,JI,J-l,41,THICKIII,RIJI,~EP(II,
lIPEAMII,JI,J-1,61 FC'RI'AT (p-l ,I4,t6;3r4;2F16,II,14,16"I41 READ 15,10) ~CAPO
IF INthRD-~ELEH) 120illl,110 STiJl' C"NT 1~IUF. 00 sa 1.1I:l'lOUI~ READ 15,46) ~FII);NR(I,\I,N8tl,2),N8IJ,3),NR(J,4"NBII,";NPel,6', IBVII,II,BVI!,ZI,BVI!,~),BVI!,4',BVI!",,8VI!,61
WPITE Ib,461NFI!I;NRI!,ll~NBIJ,ll,N81!,31,NPIJ/41,N811/"i"8Il,~I, lBVI!,11.BVI!,ZI,BVI!,3),BVII,4I,BVII""BVlr,61 FrR~AT 1714,b~7,4) (j~ t.0 h1,tJI'ART REA" 15,10) ~~ThRT(!)~NEND(!I,NFIRSTII),NLASTIII,NTVPFLIJI WRITE 1~,101 HSTART~II,NEND(I),NFIRSTIII,"LtSTI!I,NTyoFLII) 00 64 1.1,NYM RFA~ 15,991EIII)~E211),POIII,PZIJI,GEIII WPITE 1I'I,99IEIIlliE2111,PO(!I,Plll),GEII) F(lR~'AT 15F14,3) NPOIIlI> • NP~I~*b OC 65 J-l,NCQLN 00 66 1 • 1,NPO!N6 UII.J) • 0, IF I"CQNC,EO.~I GO TO 65 00 69 l-lINCONC RFA!) 15,341 K,IJ e 6*K-', J) lU 16*K-4' J l ,U 16*K-:h 01 !lue 6*K-Z, JI' lU(6*K·l,JI,UI~*K,J) wRrTe Ib,33) K,1J16*K-!,JI,UI6*K-4,J),UI6*K-"JI,U(6*K-Z,J;, lUlb~K·l,J',UI6.K,JI CN"TPIUE FOR~AT 114,6Fl1.4, FnR~AT Ir4,~E16,BI H;TF.R • 0 HISUP • 1 r~!ot.c - 0 IK • 0 DO ?4 r_l,HNNOP6 DO l4 J-l,HNNOP6
-10
U1 o
24 KP811~J) • 0,0 00 70 rI - 1,~PART WRne Ib,251 JI
l5 FORHAT 11H 111 1 PA~TITIONI,14111'
" DO 75 J-l,~~N~P7 O~ 75 1.1,~~~~Pb STII,J) - 0, NST • NSTA~Tlrll
G LEVEL 20
NE)/ - 'lENDC!I; K2 a ~"IPSTC!II L - N~A5TCIlI
MAIN
IF CII-~pARTI 17,l8;15 17 LP1 - NLA5TCII+11
Ge Ta 19 18 LPl .. L 19 .. l':US a K2-1
Dr BO L~ - ~ST,NEN H"I a LK-I:-<TFOI. Oc 85 ! - 1.4 J J a ':0:)( LI(; 11 XEII.lI a XIJJI1I XEC!,21 - XIJJ,21
85 XECI,31 a XIJj,31 TH a TI"'ICI( hl( 1 Jc"'E;lCL~1 YV laEllJI Y)'Z-EZCJ) PPl.pOCJI PPZ-PZCJ) Gc:;~ C Jl IF CLA.LE.HFLOOR) G~ TO 231
DATE a 7Z075
Al- CXEIZ,21-XEl1;Zj).CXEl3,31-XEC1,311.cxel3,ZI.XEC1,21,*IXElz,31 1 -XEIl,311 8la-IXEI2,ll-~EC1;1Il~(XEI3,31-XEI1.311
1 +IXEI3,ll-XEll;111*IXEIZ,31-XEI1,311 Cl - IXfI2,ll-XEC1,ill*IXEI3.21-XEll,211
1 -IXEI3,l)-X~Cl,lll*lxEC2,21;XEI1,211 I~ IAlI 11,-:,11
9 IF lal1 11112111 11 QI- SCRT IIA1**21 + 191.*211
pla SORT 11~1*.2) + IR1**21 + IC1.~2)1 IF IRI~~II e3,64,S3
83 nll,11 _ -51/Cl 1111,21 - Al/Ql HCZ,ll a.Al*Cl/lpi*nl1 HlZ,Z) a .S1*Cl/lpl*Qil HIZ,ll - CA1**Z+81**21/Ipl*011 Gr Ta 8~
84 Hl1,11 -81/01 HOC l,z) --A1/Ql HIZ,11-Al*Cl/IPl*C1I HCZIZI-~1*Cl/(P1*~11 HCZ,3Ia-CA1**2+~1**21/IP1·Q1I
86 HC 1,31 _ o. HI3,lI - Al/Pl HI3,21 _ 91/1'1 HOl3,31 _ CIIPl Ge- T:l ~30
lZ 0(' 14 1.1,3 DA 14 J-1,3
14 HII,JI a o. IFI~ILKIlla4,183Ii84
184 Hil,llal, H12,Zl al, HC),3I al, GCI TO 130
183 HOIl,U_l. HI2,21--l. HI),31"-1,
130 XfOCl,llaXEll;ll XEOll/21aXE~1/21 XFOll,31 aXEI1,)I
HAIN
13 Dr: 1 lal,3 XEII,llcXEII,11-XEOil 11 Xr CI,ZlcXEII,21-XEOil ZI XEII,31aXEII,31-XEOll 31 DO 2 r*1,3 DO Z j a l,3 XEIcl,JlaO, OC 2 Ka l,3
2 XEIIIiJlaXEICI'JI.HIJ;KI*XEII;KI D~ 3 ra', 3 00 3 ;'-1,3
) XEII,JlaXEliIiJI 231 DO 218 '-1,24
DAT! • 12078
DO 21B J al,24 ~ 218 SREII,JI a 0,0 ~
XCaXEIZ,1'-XEil,li YC-XEI3,ZI-xEI1,2i IF IN~EAH,EO.O' GO TO 246
220 IPEAM a 0 DO 221 J-1,~
221 IREAH - IBE~H • 8eAHICK,JI IF IrBEAM,E~,OI Gn TD 246
222 CALL REA~S ISBE,A(F~,XEI 246 Cr.IlTINUF
IF eTHI 223,223,219 223 DO 224 r-1,24
DI) Z24 Jal,24 224 KlII,JI - 0.0
OIlX - O.5*XC OPV _ O.5*VC IF(LA,L~.NF(OORI GO TO 227 DO 225 1-116 DO 225 Jal,24 EOC~I!I,.11 - 0°,0
225 EDQCII,JI _ O~O IF IMM) 227,227,226
226 WRITE (lI IlEQCNIIIJI;I-1;61;J-1,241,'IEDQCIJ~JI,I.1,61;J.1,241, lORX,~RY
Ge: TO 227 219 CALL FE~pIXe,VH1,VH2,PR1,PR2,G,TH,HHI
CALL FEMBIXE,VM1,VM2,PR1,PR2,G,TH,MH,LA,NPLnQRI ALPllA - 0,1}'.\ °
BETAaALPHA*C*TH*XC*VC KU 6,061 a8ETA
G L~V:L 20
1(1I1Zi121-!l~TtI KlCBdRI-BQA KlC24,Z41_!lëTA
MAIN
Kil 6il?I--n.~333~3.B~TA 1(11 ~,1~1--r.~333!3~B~TA Kil b,Z41--C.33!33~ABETA ~IIIZ,b )=-~.33~33:'n6ETA
KlllZ,l~I_-~.333333~B~TA
K1112'2·1.-~.3!l3~?~9ETA
Klll~,b 1--~,3333?JuBETA
~11Ie,lZI·-r,33~333var:TA
Klll~,2412-C.331333.S:TA
1(112 4 ,6 1--~,3333~3aB~TA
1(112411Z)--~.333333ftBr,TA
KI124,l~)a-n.3333~3.B~TA
227 D~ 30~ 1-1,24 DO 3CC' J=l,24
3~0 KIII,J) 2 KIII,JI • S~EII,JI
IF (LA.LE,~FLOO~1 Gn TO 234 DO 232 1-1,24 OC Z32 J-I,?4
232 TII,JI - o,~ Or 301 K-l,Z2,3 K3-V.-l 0'" 301 1-1,3 00 301 ,,-l,') Ir - 1+0:3 Jr - J+r<3
301 T'IIT,J1 1 - HI,JI IFI~~1214,214/212
212 ~RITEI11 CITII,JI;J-lïZ41,1-1,241 214 Cr.I:T1':UE
00 302 hl,24 C~ 302 .)-l,?4
302 SOcCI,Jl - ",0 00 3'13 '-1,24 D'J 3!)3 J-l,24 DO 3;)3 1(-1,?4
303 S~~II,JI - saEII,jl+Klll,KI*TIK,JI OC 304 l-l,?4 0(' 304 J-I,24 Klll,JI·O. IiC 3:l4 Kal,7.4 /
3'14 K1II,JI-KlC!,JI.TiK,Ij.SB~IK,JI 234 OC eo LL-l,4
OC 80 1(,1\-1,4 l~ CNCCILK,KKI-K21 ~Oï131;131
131 IF (~CDILK,I(KI - Cl 132,132,80 132 M_~FREE~IYO~ILK,KKI_K~1
N-~FRFE.I~~DILK,LLl~K21
I.NFREE*tKI( - Il J-~:rREE*(LL - 11 IFI~1 en.~cr,~on
9DO 00 5 NJ-l,HFREE
DATE • 721)78 HAIN DATE _ 72075
OC 5 '~I.liNFREE H'II-"'+Ml N'!J-N+NJ l'''hI+:~I J~;J .J+NJ
, STtMMI,NNJ) - STCMMI,»NJI + KlI1Hl,JNJI 80 CN.TH!UE
235 H:TER - r.E!--l MI-HFRE~.MI~US + i NJ-'IFREF*L M_~IJ-tll+l
IFIII-NPARTI 115,116,il' 115 NA-~FREE.tNLAST(11+11 - MINUS)
G~ TO 117 116 fIoA-t1+1 117 N./ill-PI
M~'-"I+l IF tLA.CT,NFLOOP.l Gn TO 236
'.~ '.' ',.
CALL SHUFFL~K2,L,INSUP,NFREE,NCOLN,LP1,NF;NBDUN,IINSUQ,1INSUS,
111101SUTI IFIII"SuQ,e~,~1 GO TD 70 1 Il • 1 K+l C'LL DLTMAT tll,NPART;IINSUQ,IINSUS,IINSUT,IKI CILL OLTINV tll,NPA~T;llNSUQ,INDAC,IK)
GC' Tn 7'1 236 IF tNTV~FLtlll.FQ:Ol GO TO 245
NTYP _ NTVPFL~Itl IF INTYP,NE,NPTvPI GD TO 244 8AC'<sDAr.e 17'
244 RFAO (17) N5IlE,I~K88il,JI,J-l,NSIZE),1.1.NSllE), IIIUStl,JI,J-l,NCOLNI,I-l,NSIZEI
NpTyp - NTYII DD 239 hl,r:SIZE Dli 238 J-l,~SlZE
238 STtl,JI • STtl,Jl • KB811,JI 11''''1 _ " 00 239 I-MI;l~j lI'M1 • 1 MH1+1 00 Z39 J.l,~cnLN
239 UII,JI • Utl,Jl • U8CIHHliJ) 24' OC 290 l-l,N30UN
MX • NFIIl-K2 "'''x - NFt 11 -1 IF I~XI 2QO.242,242
242 IF tr'\lI-MN'hJPll 243,:11.3'290 243 OQ 230 J-l.HFREE
IFt N8tl,JIl 230,34';230 34' N~I • NFREE.~X+J
STI"HI,~MII - STINMI,NMI) •• le+12 00 233 JJ-l~~eOLN JNJ - NFP.EE*~~X+J
213 UlJ~J,JJ) - ST(~MI.~Ml).BVCl,J)
230 Cr.:n l 'IVF. 290 CONTlNUE
...a U1 1\)
G LEVEL 20 MAIN
or. 54 1.1,M IF (ST(I,I)) 54,53/~4
53 STll,!1 a 5Tll/11 +O,lE+l6 3'0 Cê'\TINU.
DATE. 7Z078
237 ~gITE 141 M,~;(tSTCT,~I,t~l~~I,J.l/H)/11UlI;J)/!aHI/NJ),J.l;NCO(N) ;,g l'!'E (41 (5T( t,JI. t-l.MI,J-HM,NAI
7C C:-'''I'IU~ H :!.A.LE,"lFLODR) r,O TO 240 t:~~ pm 1 i\E~I'lD 2 QE"!'lD 3 Rr~I~!D 4 C~LL SOLVEINPART,NCDLNI RF"I~:J : CtL!. ST~Essl~PAPT;NFIRST,NLAST,NCOLN,NELEM,~FREE/NPOIN) G:- T: 2':\
2'oC "~L~C • IK CHL ~'UL TPL n:PART,'~B')U!~/NBLOC,IlFREE, tx,NRNI,NCOLNI CILL REFOST INaLcc,NCOLN,IXI
2:: cr'TP:U. 15 Fr~~AT(qE14.51
e~b F=~~AT (lH ,9E13,5) STO;> E~·o
/ /
FF.MP OAT! • 'Z078
5UBROUTINE FEMP (XE, YM1.l YMZ, PU, PRZ,C,TH.MH) REAL Kl,KBII,KBR DIMENSION X~14,3) CCM"ON ST(9~,192);KKli(96;96)iKeeC96,961
1 ,~lI24,24I,CC'Z;12);A(12,lZI,Q(3,lZI,QCNC3,lZ"e(3,3' l,nOCI3,12I,~112,12).E~CNI6,241,EOOCI6/Z'o1,U!leo~,4),~~~OP6 l,OZT(3,4),DZBl3,41,OSZT(3,41,'SZB(3.41,A3(6,6I,AZ(6,61 l/El110I,E2110i,poilnl,P2(10I,GEI10),NfPI3001
13 EX~V/11 EV-VH2 VXY"PRl VVXaPRZ GXY .. G WI!ITE Ib,ZOl TH on 1 la1.6 OC) 1 J a l,24 EOCI'I(!,JlaO~ EOUCII,JlaO'. on 2 1"1.124 on l. Jal,l4
Z K1!t,JlcO. OD 3 ral,e oc 3 Jal,8 CII.Jla",
3 An.JlaO. XC aXEI2,ll-Xeil,li Yt-XEI3.21-xell,zi CIl,llal. C(2.lI a -l./XC CIZ.3IhC(Zill C(3,lI a-l./vC CI3,7I a-CI3.1I CC4,llal,/IXC·VC, tI4,31a-CI4.11 C"14,51 aC 14, 11 C 14.71--C (1;.11 CI5,Z)al. C(6,ZlaC IZ, l' tH"'tl a -CI6.21 CI7.ZI- C13.1I C"( 7 ~ 81 a -C (7. ZI tl8,ZIa Cl4,ll CIB,41--Clô,ZI t18,61- C(8.21 C"(8,a,--CI8,ZI DO 4 r-l,3 on It Jal,e
4 QU,JI .. n, Q"ll,ZI-l. Q(l,41.".5.YC Q"IZ,7I al, QjZ,8lan,OS.XC" QI3,)I"l, Qt3.41 a t:l,5.xC
~
U1 W
G L~VH Zo FEMP DATE - 72079 PEHP DAT! w 120711
QC3,b)-1. 00 10 J -1,8 Q·13.81-~.'*VC on 10 J -118 O~ 5 J-l,e An,Jl-O. D~ 'i 1*1,3 Dr. 10 hl~8 QCl! l, JI- O. 10 AIl,J)-Alr'J)+CIKill*nll(,J) Do: 5 K-1,8 oC' 11 1-1,3
5 QC~~I'JI.:C~II'J)~QII;KI"'CIK,J) H-O 00 b 1-1,3 NaD 0':1 !> J-t,3 oC' 11 K.l,24,6
b EII,JI.~. K3-K-l O~~.1./11.-VXV*VYXI HorN+l e'l l,! I-JE~"'!:X Na"·l EC2,ll·VYX*~Cl,1) KZ-1 EIl,Zla~(Z,l) Otl 11 J-H,N EI2,Z,,,":"*EY JL aa.:2+K3 E"c3,3Ia';XY EOCHIJ,JL)-OCNI!,l) Oc· 7 .Jal,A 11 1{2 aKl+l 0'1 7 1-1,3 DO Il !a1l3 OCCII,JI-O. H.~ Ct' ., Kal,3 N_O
7 OCCII,J,-OQCII,J)+eiliKI*OCNÎK,J) OCI Il K-1I24,6 OPx.o.s*lIe K~al{ .. l CAvao.s*ve M·~~+l AA-xc*yC :-h~+1 CX-O.S*AAIIIXC K2-1 -.. CVa').S*/lA*VC: DO lZ J-M,N 01 STX*O.333333*AA+CxC",*Z) JL-KZ.K3 ~ Slv-0,333~3).AA*IVe",,,,Z) EOQCll,JLI-DQCCI,JI SIXVaO.2S*(dA*"'Z) lZ K1-KZ+l A(2,Zla~(l,ll*AA ,..wO A(3,3,a;(~,?,*AA N.O A(4,Z,aEI1,ll·CV DO 15 K-l.19,6 AI4.31-FC3,3'*CX N.M+1 AI4,4)-eI3,~,·SIX+E~1;l)*SIY M-N+1 A(b,),*el?,?I*Ah K2 al( .. 1 A~b,4'-eI3,?'·CX K"-l Alb,b'aFI),),,,,AA 00 15 hN,,.. A~7,2,aEI1.2,·hA 1I.:-1(6.KZ AI7,4)-EI1,z,·CY NI-O AI7/7)-~IZ,2,.AA HI-O AIS,Z)-EI1,7.,*Cx DO 16 K3-1,i9j6 Alg,))_E(3,?,~CV K4&K3-1 AIS,4'·IEI3,3,+elï,z)i*SIXY ~H-Ml+l Ala,6'·~(3,~'·Cv Hl·N1·l AIS,71-EIZ,Z'.CX K5-1 AIS,6I aelZ,7,*SIX.EI3;31*SIY OC 16 J-N1,M1 on 8 1-1,8 J(-K5.K4 DO R J-\,A KlI1L,J(I-Air;Ji*TH
8 A(1,J! .... IJ,I' 16 K5-K'.l DO 9 1-1,8 15 Kil sKb.l 00 9 Jal,8 ZO F~~~AT Il~El1~2) DI I.J 'a/), 21 F[1RIl4T 18E12.Z) OC 9 K.l,8 RFTUitN
9 OII,J'-DII'J).AI!;KI*eIK,J) END
G LEVE\. 20 FEMB o~TE _ 72078
s ~paUTINE ~:~BIXE,V~i,VM2,PR1,PR2,G,TH,M~,lA,NFLOORI R AL KI/KBI~,KBB
~ ~ENSI"~ XE(4,~);L(12)/HI12) ( ~HJ~ ST(9~,192);K~ll(9b~9~),KeBI9b,9b)
,Kl(24,241,C(12;12).~112,12),Q(3,12),~CN(3,12),E(~/3)
,~:CI~,12),~112,12).EaCHlb/24),eD~C(b/24)/UI1800,4),MHNOP6
.rZT(3,4),~!g(3.4),~SZT(3/41,~SZB(3,4),A3(6,b),~Zlb,b)
,EI(10),E2(lOI,?0~ln);P2110),GE(lO),NEP(300)
EX-VM1. Eysv~'Z VXV=oRl VVX.PR2 Gxv_C ~E~.1./(1.-VXV*VVX)
0:- 2 1-1112 O~ 2 J-l,lZ
2 CII,J)-O. XC-XEI2.1)-X:ÎI'li vr·xE(),2)-YEll,2) CI1,lI-l. CI2,5)-1. CI3,9)-1. (1411)-1. C'(4" )-XC C·15,5)-1. CI5,b)·xe (lb,91-1. (lb,lOI-XC ("17,11-1. C17,Z)-XC (('7,3)·~·C
C 17,41 .xc*ye (IS,51-l. c·(e,b)·XC C(ô,7)·vC C(S,ôl·xe*vc (19,0)-1. CI9,1~I~XC (1If,lll.VC Cco,IZI.XC*VC (·Il~,l"l. (llO,3)-VC , CI11,5).1. / CCll,7).VC C't12,9)_1. C 112, 11 ) - vC CALL MI~V (C,12;00,(,M) OPV.0.51C<YC ORX.O.5*XC il:' 3 hl,3 0[1 3 J a l,12
3 OII,J)-O. Oil,10).1. Oil,l2)-ORY
Oi2,7)--1. 0(2,Bla-ORX 0(3,61-1. Q(3,BI·rp.y 0(3,11)--1. 013,12).-ORX DO 4 J-l,12 DO 4 1.1112 OCNll,JI-(l. DO 4 K-l,12
FEHB
4 OC~II;JI.O(Nlt'J).O~I;K)*CIK.JI Dr 41 1 a 1/3 DO 41 J-l,12
41 OCNiI/JI-0.5*,H*OCNil;JI DO 1 1-\,3 on 1 J a l.3 EII.J)an. EI1.1)-FX*OEN E~l,ZI-VYX*E(l'll EI2.1I aEIl.2) E IZ. Z 1 -EV*OFN EI3.31-r,XY Do 5 J"1.12 0[1 5 1-\.3 DC(II.JlaO. DO 5 Kal,3
5 oQCII,JlaOOCII,JI+EiliKI*aCNiK,Jj AbaXC*VC CX-~.5*(XC**21*YC CvaO.~*YC*IVC**?1 SIX-O.~3333~*IXC*.3j*yC SIY.O.333333*XC*(YC**31 SIXV.O.25*(XC**21*(VC**ZI DO b lal.12 DO 6 Ja1,12
6 AII,J)aO. A(b.6I aE(3"I*AA AI7,71.EI2.ZI*AA AIB.61-E(3,31*CV AIB,71-E(2.2)*CK A(B.9IaE(Z,ZI*SIX+E~3;31*5JV
AI1n.7)a-EI1,21*Ab AI10,el·-E(1~21$CX AI1n.101-EI1,11*AA Al1l.bl.-EI3.~).AA AI1l.81--E(3,11*CV Alll,11)·EI',31*AA
·AIIZ.61 - -F(3,~I*C~ AI12,7)a-EI1.21*CV AI12.81.-(E~1;21+E(3,311.SIXV AI12.10)aEI1,1)*CV A(lZ;111~E(~.~I·CX A~12,12IaE(1,11*SIY+El3,31*SlX DO 61 1-1,12
DATE. ?Zon
~
Ut U1
:; LEVEL 20
DO 61 J-l,12 61 AI!,J)-AIJ,I)
O'J ~Z !.1I12 D~ ClZ J.1,12
FeMB
62 Ail,JlaAII,J)*ITH**3)/lZ. 0,= q 1-1'12 O~ 9 J-l,12 OIl,J).o. C: ~ 1(-1,12
9 D~I,JI-~II,JI.AII;Kj*CI~,J) ;::~ 10 1-1,12 cc- 10 J-l,12 AIl,J)co. C':' 10 Kc1, U
10 AlI,J)-AII,J).CIK;II*~IK,J) C~ - TH/12.0*AAI GVZ-G GxZ-O~5*V~1/11 •• ppri Oi 200 1-1,12 :;(' 2nn J-1,ll
ZOO CiI,J)-O. CIl,l) • CH~IGXZ*lVc**z).nvZ*'XC**21) C11,41 • -CH.Gxz*lVc*.ZI (14,4) • C~*IGXZ*IVr.**21.GYZ.IXC**2» C14,1) • -CH*GVZ*IX~**2) CI7,7) • CIl,lI (11,2) • -0.5*VC*CI4,1) CI1,3) _ 0.~*X(*Cl1,41 Ci1,b) • O.·~xC*Cl1;4i (11,10) - CI4,7) Cll,ll) - -C.5*vC*Ci4j7) CI2,z) • -Q.2~*CI4,7).IVC**21 (IZ,10) - J.s*Ve*cI4,7) CIZ,11) - -~.?5*lvC**~)*ci4,7) CI3,3) • -0.2,*rll")*lxC**Z) C13,4) _ -0.S*XC*CI1,4) Cl3,b) • -0.Z5*IXC**Z)~CII,4) CI4,5) • -O.5~YC*CI4,7) C14,bl • -o.s~xr*CIl,4) CI4,9) • -O.5*ye*cl,,71 CI5,S) • -0.Z5*IVC**ZI*C(4/1) CI5,7) a O.~*VCDCl4~71 C15,81 • -O.ZS*IVr.**ZI*CI4/7) tlb,b) • -O.Z5*IXC**Z)*tll,4) CI7,d) • O.~*VC*Cl',71 CI7,9) • -0.5*XC*cll,4) C11,101 - Cl1,41 (1 7,121 • -n.~*XC*Cll,41 CI5,8) _ -0.Z5*CI4,7)*I~C**?) CI9,9) • -0.ZS*Cll")*IXC.*2) Ci9'lQ) - O.S.XC*Cll") CI9,lZ) - -r.l5'IXC**2)*Cl1,4) CIln,101 • CI4,4) C110,111 - r..S*VC*CI4i71
DATE - '72078 FI!HB
C,10,12) - O.'.XC*Cl1~4) Ci11/11) - -0~25.iYc.*Z).CI4;1, CilZ,lZI - ~0:Z5.iXc**ZI.C(1~4) 0[1 2n2 1-1112 DO 20Z J-lI12 C·,J,I)-CII,JI
ZOZ Ail,JI-AII,J)+Cll;JI OC 14 1-113 "'1-0 N.O DO 14 K-3,Z4,6 1<3-K-1 Ml· tl+l ~hI11+~ K2-l 0[1 14 J-HlIN JL-V.Z+K3 ECC~II+3,JLj.QC~I!,~)
14 K2-I<Z+1 Dn II 1-1,3 Ml·!) Il_O DO t.z K-3,Z4,6 K:,:I-K~1 li1·ll+l N-M1+Z 1(2-1 [11'1 12 J-Ml,t-: JL-K2+k3 ECQCII+3,JLI. OCCll;Ji
1Z K?-K2+1 IFILA.LE.NF(onR) GO TO 13 IF U'HI 13,13;114
DAT! Il 720TS
114 WRITE III l'E~CNlf,J);I-l,61,J-l,241,IIEDQCil;JI,I-1,61;J.l.Z41 1,ClRx,ORV
13 Cl:lnTWUE 112 •. ,
IIZ .. 0 00 7 K • 3,2116 Ni' -:1Z+1 M2 - llZ+Z kZ "K-1 KE- -1 OC; 7 1-~12, M2 IL - 1o:6+KZ N1 .0 1'1)-0 un B k3. 3,21;6 K40 • 1<3-1 Ni • "1+1 Ml-Nl+Z K5 -1 OCl e J .. ~llI Ml JL _ KS+K4
rn (»
G Ln~L 20
klltL,JL' • Ail,Ji e 0(5 • K5"1 7 ~t- • 1</)+1
R~TU~~ E~C
,/
HMB DATE • 72078 BEAMS
SUBRQUTINE ~EAMS(SBF,ALFA,XEj REA( IXB,IVB,JZB,CE~G;KBB,KBll,Kl I~TEGER BEAI-4
DATE • '7Z078
DI~ENSlnN ALFAC1Z;121;SRBI12,lZI,SBE(Z4,Z.'~S~111Z/1ZI,SBllI1Z'lZI l,XEII,,31 CO~"OH STI9t-,19?I;K~liI9b,9bl,KBBI96,961
1 ,VIIZ4,241,CI1Z;lZI,AI12,lZl,O(3,lZI,OCN(3,lZl,!(3," l,OQCI3,lZ)/nllZ,121,eJCMlb'24),EOQCI6,241IU(lBOO,4',~NNOP6 l'CZTI3,41,D1BI3,4I,nSZTI3,41.DSZB(3,4l,~lC6,61,AZ(6,61 l,F.II10I,E2C10l,POllOl,P2Il01,GEC10),NEPc3nO),LK l,UEAHI300,bl,tXnI501,TYaI501,IZR(501,EBI5~I,AB(50),~B(5n';HeB~5~, xc • XEi2,11-xEllill VC • XEI3,21-XEI1;ZI
627 00 ~Ol r-1,12 DI') "01 J-l,lZ
6~1 ALFAII,j, a J~O D(I (,2" ra1l24 Co fsZb J a l,24
626 S&EII,J) • n.o IF 18EA~ILK,1)-01 6nZ;603i60Z
60Z LAMDA a 1 MU • IIEAKILK,iI LfNG a S~RT(XC**ZI ALF4Il,\! _ 1.0 AlFAIl,21 a 0.0 G[I TO 607
603 Ir IOEA~ILK;21-01 604;605,604 601t LAI'IDA • Z
HU • ~EAM (LKiZI LENG a SQRTIYC •• ZI ALFAll,11 - 0'.0 ALFAll,21 • 1.0 GtI TiJ bt\7
6~5 IÇI~F.AHILK,31~O) 60b,631,606 606 LAMA - 3
Mil a REAM IlK,31 LENC - 5Q~Tixc •• ZI ALFAI1,11 - -1.0 AI.FAIl,ZI - O~O GO TO f!(17
631 IF IUF.AM(LK,41-0, 63Z;633,63Z 63Z LAnnA - 4
MU • I\EAMILK,.) LF~G • SQRTIVC.*ZI ALFAC1,1I a 0.0 ALFAI1,21 - -1.~ GO TO 607
633 Ir (BEA I1ILK;51-n, 634;636;634 634 LAHIlA - 5
MU - BEAMILI(,~, Lf\& • SQ~T~(XC*.2'+(YC •• 2', ALFA Il,11 • xC/LENG ALFA Il,21 - YC/LENr. Gf'I TO 6'17
l ' -01 ~
G LrVEL Zo BEAMS
636 IF IBEAHILK;61-01 637;698;637 631 LM'IOA • 6
Ml! • DEAMILK,6) LE~G • SQRT}IXC~.2'+IVC~*Z» AlFAI1,l) • -xC/LE~R ALFA Cl,ZI • VC/LENG
6~7 c~rlT l'lUE ALFf.IZ,2) • ALFAI1,tl AtFAIZ,ll • -hLFAl1,ZI ALFAI3,3) - 1.0 D" 60R LB-l,::! Dt' b08 1.1,3 DQ 608 J-l," Kp • 1 + 3~L!l Je • J + 3*LB
6CB ALFAIKB,JSI • ALFAII,jl Dr 609 1-1,12 or. 609 J-l,lZ
6Q9 saOll,JI - ~.~ xe1 • Ef\I!iL')UBIHIJ)iLENG x~z • Gql~U)*IX9IMU)/CE~G Z~3 • lZ.0*F!lIMU)*IZB1HUI/ILENG**3) ZFZ - 0.5*LE~G*lB~ ZOl • Z.O*EQI~Ul*IZP.I~U)/LENG 'fG3 • lz.~*E!l(l~iJ)*IVB·IMU)"LENG.*31 vez - 0.5*LE~G*vB" VFl • Z.0*E~14UI*IVRI~U)/LEMG Sf\!l11,1l • )tôt S~~11,5) • X9l*HC!llrU) SRAI1,7) .. -x!ll 5BSI1,11) - -XB1*~C~IHUl Se:!IZ,Z) • lR3 S~PIZ,4) • -Z~3*HCB/~U) SPB/Z,6) • lBZ Sr.S(z,8) • -ZR::! 5"il:2,l~) • ZIl3 0 HrB1MIJ) S~O(Z,12) - ZflZ SPB/3,)) • VB3 SB9C),51 • -V!lZ SPQ13,91 • -V1\3 S~B13,lll • -vBZ SB!!14,41 • XBZ + ZB3*}HCB~MUI)*'Z S?BI4,6) • -ZAZ*HCBcMUI S~S14,81 • Z3".~C~(~uj 5R~14,101 • -xB2 - ~B"./HCBjMUll**2 SP~14,121 _ -ZB?*HC!lIMUI S~B15,51 • ~.~*VB1 + XB1~jHCBIMU)'**Z S~3/5,11 • -XB1*HC9cMUI 5~5CS,9) • VBZ SP9CS,111 • val -x91-iHCe1MU»)--Z S?eC~,6) • z.~*zBi SRè(6,81 • -zrz S22/6,1 0 ) • zez*HCBiMU) S~P/61l21 • Zf\l
DATE - 7Z01R BEAMS
S8B17,71 - ICBl 5RBI1,11) • XR1*HCBiHU) SeB'18~BI - ZB3 . SABIBil0) - -ZB3*HCRIMU) S~S(8,12) • -lB2 511519.9) - VR3 51\&/9,111 - VRZ Spall~,lOI - ICB2 + ZB3*/HCBIHU»)**Z 5P~II0,12' - lBZ*HCRIMYI
DAr! • '72078
SPBlll,111 - 2.0*VRl + XB\*IHCBIHU)I.*Z SRBI1Z,lzl - 2.0.zai on 656 t.l,12 If.41 -1-1 on 656 J-l.I:-11
656 SP.BII,JI - SSBIJ,I) Dr! 610 1-1, \2 DO 610 J-blZ SP.IIIIJI - 0.0 Dn 610 K-l,IZ SSIII,JI • ~BI(I,j, _ SBBèl,KI*ALFACKIJI
610 tONTIt.lUE DO 611 l-l,1Z 01:1 611 J-l.12 SRIlII,J) • 0.0 0[1 b11 K-l,lZ SPI111,J) - 5RllIJIJ) + A~FAiK,I)*SBIIKIJ,
611 CO~iTI tlUE IF ILAMnA-11 61,,613,615
613 DO 614 1-1,lZ Dn "'14 J-blZ
614 5AECI.JI • SaJllIïJi Gt'I TO 61i3
615 IF ILAMnA-ZI 619,616,619 616 O~ 617 t-l,12
on 617 J-l,lZ _ 617 5PEII+6,J+61 - SBE(J+~,J+6' + S8I1CI,J)
GO TO 60S 619 IF ILAMnA-31 6S5,620,655 6Z0 DO 621 l.l,lZ
to ~21 J-1,12 621 S&EII+12,J+IZI • SBEII+1ZIJ+121 + SBI111,J)
GO TO 631 6~S IF ILAHr,A-41 6'7,659,657 659 Dr. 639 1.1,~
DO 63A J-bt. 63B SHEII+18,J+lBI • SBE(J+1B,J+1 8 )+SBI1Cr,JI
OC 639 j.7,12 639 S~EiI+18,J-61 ·SBECJ+1S,J~61 +SBlllr,jl .
DO 642 1-7,12 OC 6101 J-l,6
641 S~EII-6,J+1P) • SBEil~6,J+IBI + SBI1IIIJI on 642 J.7,12·
64Z S~Elr-6,J-bl - SBECJ-"',J-6) +SBI1II,J) G:o TO 633 .
-U1 Cl>
G LEVEL
657 ~~O
643
64lt
!l45
6lt6
!!58 t;f.l
6lt7
649
649
~50 ~51 65, 653 654 698
70 BEAMS
IFILAMOA-51 658,660,658 oc 644 1-1,6 DO 643 J a 1,6 SFEII,J) • SBEII,jl + SBI1IIiJ) r.: 644 J-7,lZ S~E(I,J+6) • SBECl'J+6) + SSllCl,JI or 646 1-7,12 DD 645 J-,,6 SeE(I+6.J) • SBEli+6,j) + SaIlcI,J) 00 646 J-7,12
DATE - 120'78
SBECI+6'J+6) - SBEII+6'J+61 + SBI1II,J) GD TO 636 IF (LA~~A-6) 69~,661.69B C:l !'4B hl.6 0" 647 J-l,6 SPéCI+6,J+6) a SBEII+6,J+61 + sBI111,J) DO 648 J-7,12 SRE(1+6.J+12) - SSECI+6.J+12)+SBllll,JI cn /';0 1_7.12 OC 649 J-l.b S~ECI.12,J+bl - SAECI+l2,J+6) + sBllll,J) ilO 650 J.7.12 S~:Cl+12,J+1Z) - SS~CJ+12,J+l2) + SBI111,J) F:,~,"AT ·C lH ,3Fl0.t-) FC'~AT Cl~ .12 F8.5) F~R~AT Cl~ ,12EIO:3) F~R4AT C1H ,BE15.6) RETURN Et.[j
/
sHUFFi,; DATE • n078
sUBROUTINE SHUFFLlK~,(,INSUP,NFREE,NCOLM,LP\,NF,NROtiN,IINSUQ, \ IINSUS, IINSUT;
REAL K1,KBBÎKAliKÏl;KBI1 Ot~ENSlnN KtI19b,192IiKBI16,Z881,UlllBOO,41,UBI9b,iol,NFC100l,
ITFHP(96) t~MHON STI9~,i921;KAllI9b~9bl,KABI9b,961
1 ,KIIZ4,241,C,iZi1ZI;AI12,lZI,QI3,lZI,nCN(3,121,E(1"i 1,DQCI3,lZI,nC12,12)~EotNI6,24),F.DQCC6,24),UC1800,41,MNNOP6 1,~ZTI3,4I,DZBr3,4),r~ZTr3,41,OSZBI3,4),A3r6,b),A216,b1 1,E1i1nl.E2CIOI,POc1nl.P2110I,GEI10I,NEPr3no),LK eCUIVAL~NCE (Kll(i,il~S·~1,1Il,CUII1,1),UC1;111 DO 319 lal,HNNOPb Dn 319 J-l,'~N!IOP6
319 K~llCl,J) a O~O HI"'''D a 0 lKBn - 0 I~PART _ K2-1NsUP H SIIRE _; lti'5lJP Il,PPAR a 1"<1'1 ART Ir 1 K2-1) 233,33~,~32
33, R~AD(2) 11USUQ, IKaI2,lNPPAR 333 DC ~40 t-K2,L
Il a Cl-ll.'!FREF. 1t1!-.~i) a 1'1:1"0 + 1 H:SIJPl - C ItISUP-l j*NFREE INND01 _ Il~NnO-ll.NFREE IINSUP a (I-INsIJP~I~IPARTI.NFRee, IlhSUW - CI-l"SUPI.~FPEe IFC~FIl~SU~)-11 305Î3n6,305
305 on 313 Kal,~FREE on 307 J-1,"cnL"l
31)'7 \JICJI"lSU~+K,JI - "It1+K,JI 11<811 a 0 JN~IJP a INSUP JJNSUO a 0 JNNC'D - Immo on ~49 INDXal;HNNOP6
349 Te"PCINQXI - 0.0 on 312 J-I,(P1 JJNSUP • IJ-JNSUP;INPARTI.NFREE J!H~,)Dl - 1 J"NOO-11*~IFREe IFCNFIJNSUP;-JI 31)9;~~B,309
30B JNSUP1 • IJ!,'SUP-lI.'!FREE IF CJ-LI 3~6,33~,117
31'7 ~o 31B N.l,~FRE~ 31B kDIIIIINSUP+K; IkBJ\+N) • sTiINNOD1.K,J~N001.NI
11<811 a lK811 + NFRee 3~6 Jf,SUP a JNs'IP + 1
GO TO 311 309 IF iJ~GT.LI GO TO 347
on 310 Nal,NFREE 310 KIIIIINSUP+V.,JJIISUP+NI a STIJNNOD1+k,jNNOD1+NI
Ge TO ,311 • 347 D~ 348 Nal,~FREE
-~
G L~'iH
348
3n 312
35:> 313
3~~
32: ;37 323
32;'
321 322
329
324 325
32~ 327 334
:135 34;)
344 342
339
343
20 SHUFFl
T~X~IJJHSUQ+~I a STIINNOC1+K,JNNOOl+NI JJ~5UQ • JJ~SUQ+NFREe J':,,')\) • JNNr.D+l C::\TPIUE O~ 350 INOX.l,MNNnP6 ~IIIIINSUP+K,~N~cr6+1NDXl • TEMPIINOXI Cl1T!'I'JUe: ü~ TO 340 1~IV2-l) 32?,323,32R IF IIK612) ;23/323,337
DATF. • 'l'207ft
~EAD (2) IIKBIIK,~),J.l,IINSUQ),K.l,NFREE) :~ 32'1' ~.I,~F~Ee D~ no J.I,"COLtl U~:I~SUPI+K,Jl .• Ultl~K,J) J'.S'Jp • 1~~Sl'RE
J~""'D • 1 J:' 321> Jal(Z,LPI J' S.JDI • IJ~!SUP-l)*r~FREE J'-Yl:ll • IJ'!'J"O~l ,*~:FR.EE IF l'JFIJNSUPl-J) ~2q,321,3Z9 :l~ 322 '!.l,,"~p~E ~~S( P~S'.IP1+K,J1>I5UP1 ... N) • STI PINOD1+K,Jf-<1l0Dl.Nl J'SUI' • JNS~P + 1 û" TO 325 JJ',SUp • IJ-J'!StlP';'I':PPARI*NFREE ~r 324 N.t,~FREE (~IIK,JJ~SUP+~1 • STIINNUOl+K,JNNOD1+N) J'.~~~ • J~'J~O + 1 JJ\SUR • ILPl + 1 - JtiSUP - INPPARl*NFRH C::',-rPIU" CC'HI'IUE ~~ITe 1~,3511 INPPAR,JJNSUR ~R 1 TE 1 31 l~ippAR, jJ';SUR, 1 IKR Il K, J), Jal, JJNSI'R) ,hh 61 1"5,11' • lfolSI;P ... 1 Cr\Tl'~UE Il",SUO • IL+l-ltlSUP.INPART)*NFRre 11\SUS • II~SUQ ~ 1 1l',5,IT • ILP1.l·J~;S"P-ltIPART).NFREE II~SJV • IltSUT-IIN~UO "R.ITE 16,353) 1!t,S\JQ IF IIINSUQ,EQ,OI GO TO 344 l~ II-LPII 342,34\,14î .... ~ 1 TE 1",3551 1 H;SIJ,~,l1nSl!S,IINSUT ~:IT~ It,35~) II~SU~, IK~Il,INPART "f ~ t'II) ? ~RITE 121 It~SUQ,IK~li,INPART IF (1INSUQ,EQ~OI GO TO 339 l~ IIIN~UT.lE,lIhSU51 GO TO 338 IF IIKBIl.E~,~) GO TO 339 ~CIV • IKOIll"FItEl: O~ 343 K.l,~OIV ~~IVI • CK-ll*NFREE + 1 ~:IVb D 1~-ll*~FRfE + 6 ~~ITE 121 IIK8IlII,Jl,l.l,II~SUQ),J.NOIV1,~OIV6)
339 REWINO Z GO TO 398
3~l NSIZE • NFREE*NBOUN DO ~45 l.l,~StZE DO 345 J.l,NStZE
345 Ke3IJ.Il • KBRII,j) WP.ITE 1~.356) NSIzE WPITEI6,3S71 HSIZE RE~I'IO 2
SHUFFL DAT! • '720711
WPITE 121 ~SIZEll(KB8(IIJ).J.1INSIZEIII.1INSIZEI, l(IUAII,JI,J D1,NCOLNI,I·l,NSIZE)
351 F"R~AT IlH ;ZI4,IKBllj 352 F~~~AT IlH ,9~13,51 353 F~R~AT IlH ,14,'U!') 354 FOR~AT IlH .2I4,lrlt'l 355 F~~~4T 11~ ;3t4,IKlllj 356 Fr~~AT 11H ,14,'KeR'1 3~7 Fr~"AT IlH ,I4,'Upl) 358 F['lRI1AT I1H .314,IK811') 398 RETtl~N
END
~
0> o
G LE\'ëL 20 OLTMAT DATE _ 7Z078
S~9P.DUTI~E ~LTMAT IIIïNPART,N,lINSUS,lINSUT,lKI ~:~L KSI1,KPo,K1 ~l~E~SIQ~ A~96,96j/~196,96),Ol~6),BEI96,96)
C~~":J~ STI9~,192)ïKlIliI9~/%l/KRBI9~,961
l ,KlI24,?4l,ylt2;12I,VI12,12),CI3,lZI,~CNI3,lZ),el~,3)
l'::CI3,12I,~llZ,lZI,EOC~16,24),EDCCI6,Z41,UllPOO,4),MHNnp6
E:: J 1 VAL E\C E (t. 11111, ST ( 1111 " 'c BE ( l, ll, ST ( l, ~7l', (e ( lIU, K~ 1\ li., \lI
l' (::(ll,!:C;:lCllïlll !- (IK.EQ,ll GO TO 355 ~~A~ III M,NX,II~II,JI,J.l,~Xl,IE1,Ml
~E,;l'::l 1 S!CI(SPAC: 4 ~~An l4l ~,ICII);I.l;MI
.';'.lTë 1~,3771 /':,t,
.:;, .J 35; 355 ::- 356 Ial,~~~OP6
;) 1: l • (). 0 ~~ 356 J.l,~N~0~6
356 ~Cl,JI • C.~ ,1-1
3~7 C:O'Tl'lUE ,.~ :re (1',37/>1 N ~:' 3b5 I-l,~l
Il - 1-1 S!,'1 • O.C OC '3511 l.l,'"
358 S~u • SUM .. IBIL,II*.ZI*O~LI
~ll,II - All,tl - SU~ IF II.EQ.ll GO TO 3~O :::' 359 L-l,1l
359 ~(J,Jl _ AII,II ~ IAIL,II**ZI*AIL,L) 360 ::-',rpIUE
1~ (.\II,Jl.L='.O~O) r.O TO 368 3e.l :111 - 1+1
II' CI .E!:l.tll Ga TO 36~
;l:) 3b4 J-IPl,N Jl'l - J+1 St,,· • c.o :i, 362 l-I,"
36Z s~,,, • S'JH + SH,Ii*[lIL,Jl*OIL! AC l,JI _ AI!,JI - SUM I~ (I.E~.ll GO TO 3b7 Po ~63 L-l,1l '
36' All,JI _ AIJ,JI ~ AlL;ll.AIL,Jl~AIL,L)
367 AC:,JI - AII,JI/Ail,Il J64 ~(J,Il - AII,J) 365 tt"'T1tlU~
OC' ;IbO 1-1,,.. 0') "69 J-l,~'
~~q ~il,JI _ ~II,JI*Dili IF CI!.E~.NPA~Tl GO T~ ~66
IF,(IINSUT.~e.l!NSU~1 GC TO 373 14 1 • ~t:j'!Cjl6
='r 374 lat,t:
DCT"'AT DATE. non
DO 374 J-l,f"l 374 ~EII,J) • D~O
GO TO 385 373 Ml - !IMSUT-IINSU5+1
OP 312 lalltl 11 • 1-1 DO ~72 J-lI"l IF II.Eo.t) GD TO 3'4 DO 353 C-l'Il
'-.,
353 BEI l,JI a BEII,J) - AiL,I)*SEIL,J)*AIL,L) 354 BEII,JI • BEII,J),Ail;II 37Z CO'iTItlUE 38~ ~RITE Il) ~,Hl,(1Bell'J);J-1~tll),I.l;N)
Rr;~~HIO 1 NRITE Ib,377) N,KI
36~ -RITE 141 N,iAII;II,lal,N) "'PITE 16,37bl N IF 1~.EQ.ll GO TU 352 Nl • N-l DO 371 lal,~a 1111 • 1+1 i)O 371 J-IPl,N
371 A(I,JI • AII,JI*Ailil, 3'2 C'JtlTltlUE
Gr) T:J 37a
\\.\" .' .. \;'
368 ~q1TE 16,3801 1,11,AII,I, ,
3~O F~R'AT IlH ,I4,'TH nlAGCNAL TER~ OF PARTITION 1,14,115 LE;ZERD AND lSUSROUTI~E FAlLS AIT,II a',F6.Z)
STOP 376 FC'!R'IAT 377 F[)R:'IAT 375 Fn..HIAT 370 RF.T"R~1
E~:O
Ilfl1l4,· AlhJI OLT~AT., IIH ,ZI4,'BCl,jl OLTMAT 1)
«1H ,9E13.,)
;; LEVEL 20 DL TINV
SUB~OUTI~E DLTtNviII,NPART,N,INDAC,IK) ~ë~l KBll,K~8;Kl
DATE • '7Z078
')II'~~IS Ir~ A (9~,lled 1 ~ 1'16,96), X (96) IC (96,</(\' cr~4~~ STIÇ~,192,iKAIlI96,9!'1I,KBB(9b,961
1 ,KIIZ4,24),V(tZ~IZ',V(12,IZ),~(3,IZ"~CNI3,lZ,,EI3,3' l,~QCI3,12',~112/1Z',EQC~16,24"EOQCI6,24"Urleoo,4',HNNOP6 eCUIVALE~CE IA I1,11,STll,l'),1 Cll,I,,5Tll,97",1811,I',K8IlI1,I')
l,IX 1 1 1, E ~ C :l"1 1, 11 , P-lSLlCE • 6
4' 0 C":',T! NU!' Ir: IIK.f~.l) GO TO t.5Q I~l - 11<-1 Dr. 459 1<7 - I,IKI I~ClC - INDIC + 1 I~DACI - I~DAC + 1 - IK t~LL DIRACA 1(,M,INDACl,NSlICE,5,21 ~otTE 1~,475' L,M 00 lo53 1-I,L or. 451 J"lI~l XIJ) - (1.0 0':1 451 ,..-b."
451 XI~) - XIJ'-CiI,K'.PIK,J) (J~ 4'2 .1_l,l!
452 CII,J' - XIJ) 453 cC":rr 'lUE
oc 454 l.l,L 454 CII,l) _ Cllill/Ali,il
IF 1~.EQ.ll GO TO 4t15 :l" 457 J-z,'j CC 456 l-l,L JI - J-l ~: 45'5 I(sl,Jl
455 CII,JI - CII,JI-cil,KI.AIK,J) 456 CII,JI • CII,JI/AiJ,JI 457 C!:'lT!'IUI' 465 CALL OIRAC4 IC,N,INoAC ,NSLICE,6,21
~~ITE 16/47~1 l,N "!le CC\JTINUE 4~9 I~~4C • I~OlC.l
IF 1~.EJ.ll GO TO 46b Ni - N-l Di: 462 r.l,~jl AII,I) _ l./Afl,ll In • 1+1 H'2". 1+2 AII,IPII • -AII,IP11.AII,I)/AIIPI,IPl) AII;>l,II - n.n IF Ilr2.GT.~1 Gn TO 462 0(.1 Iobl J.IP2,~1 JI • J-l AII,JI • -AlriJI*AII,II oC' lobO K.IP1,jl
460 AIl.JI • AIl,J).Air,Ki.AIK,JI lIIJ,1) .O.CI
DLTINV
461 Atl,J) • AII,J)/Ai~;JI 462 CrNTINUE 466 AiN,N) • 1./AlN,NI 4QO ftRITE 16,47PI N,~ 491 C~LL DIRACA IN,N,yNOAC ,NSLICE,6,1) 475 FeRMAT Il~ ;2I4,IC(I,J,I) 476 FCR~AT ilH ,9E13.5) 478 FORMAT 11H ,214,'A(y,J)') 479 F~~~AT I1H ;Z14,I.Cir;JI.')
RET\lil.~j Erli)
DAT! • non
-0> 1\)
,1
Il !.~v~!. 20 MU!.!!'!. UAI!: a 'ri!n711
S~2RJUTINE vUL·~L 1!;rART/~S~U~/NBLJC,NFRE~,IX,NSN1,NCOLN) REAL ~SII/K~3iKl,KBI OIMENSION KBIl4~,2B~);CI96,96),RI96~96);I9E~INelO),IENOC10), lI~C~XK(10I,UJl9h,4) ccr~~~ ST(9~,192);K'Ill~b,96),KB8(96,96)
l ,Kl(24,Z41,Vli2;12)iV(12,12),QI3,12),~CNI3,12),Ee3,3) 1,~~CI3,lZI,~IIZ,12);E0CNI6,24),EOOC(6i24),U(lPOO,4),~~Nnp6 E~0IVALE~CE 1~~t(l,t);STI1,l))/IC(1/1),k8Rel,1)),
!IOll,l),KDI111,l)) R~,t~ll' 1 I\f'" t,'ll' 3 ;:t~,;I'JO 16 N~~l • HNNap~/12 IF C:·lil'Il.LE •. lBO'.IN) r.0 TO 526 "P. 'Il • 'JS1U!1
'26 Cr::';TPI'J~ \St.ICE • 6 Ix • 1
520 ~l~"!> • 6.~J81;1
o~ 521 KKa1,NBNl KK1~1 a IKK-1)*6 • i KK1P6 a IKK-l)*h • h RrA' 131 INP?AI\,JJN~UI\,(IKBI(t,J)/Jal,JJNSUR),IaKK1Pl,KK1P6) wc l TE 1 b, 550) IIIPPA;:t, JJt:SUR IF-E::iPIIKI<.) • INPPAR6~jFREE
521 IE~DIKKI a t;eGtNlKK) • JJNSUR w~ tTE 16,!'~I'\) IFEClt:, iE~1D ~~~~~b • ~il:U~.h l' J~C a 0 l'::EXN • 0 or 535 l~al,N~LOC o~ 522 ~Ka1,~BNl
522 It')EX~II(IO a (I l'~')EXIJ • 0 D" 53'1 J.l,11crL~1 D: 53'1 l.l,H~NO?6
539 ~J(I,J) a 0.0 Oc 538 JJe1,IM I~OAC a I:-IOAC+l CALL DIRACB IM,H/I~nAC,NSLICE) kRIT~ 16,550) M,N IF IJJ.'!E,ll GC TO 525
523 I~~5X~ • INrExN + N If.C=X'1 a 0 . C!: 524 1-1,':iHI6 CC 524 J.l,~1
524 R1I,Jl _ o.n 525 l~JFX~ a tN~EXM • M
"D IrE 1 6,55(,) It:OEXH,INOEXN D~ 532 K~-l,~BNl ~~lPl a IKK-l)*~ • 1 ~~lP6 a IKK-l)*6 • h I~ CI~E~I~(I(~).GE:I~OEXM) GD TC 531 IF II~~EXM.GT.IENOeKK)1 GO TO 531
DO '29 I*KK1Pi,KKiP6 DO 529 J*lI~ DO '528 Kal,M Ke a INDEXKiKKI • K
MULTPL
'?oB Rlt,J) - RII,J) + K~liI,K81*C~K/J) 529 CC"THIUe
I~JEXKIKKI a INOEjK1KKI • H 531 Ct:i\TI~jUE 532 CeNT! "lue
IF IIX.GT.l) GO Tn 540 IF INBOIJN.N~.NBfj1) GO TO 543
540 on ~42 t a l,"COLN Dr. 542 Jal," D~ 541 Ka1,Po
DAn a 72078
541 UJeJ,II a UJe~,11 + UIINOEXU+K,I)*CIK,J) 54Z CO<,jTI~jUE
I~OEXU a INCEXU + M 543 C~'lTPjUE 538 C:l~TINUE
I~ IIX.GT.l) GO T~ 533 l'/P.lTE e 11 rIBN6,N;nR·lt,J)'Jal,~"1*1,N8N.1 IF CN~OUN,LE.NB~l) GO TO 544 GO TO 535
'33 WRtTE (l~) ~aNb,~;eiR1I,J)'J.l,~),lalIN8~6) '44 ~~tTE (lô) N,liUJeI,J),Ja1,NCOLN),I.1,N) 535 CCltlTINUE
I~ IIX.Ge,Z) GO TO '37 r.r "Il a !lB!'!'J~;-tIBNl IF(~a~l.LE.~1 GO TO 537 IX a IX + 1 GO Ta 520
550 F:lRtlAT Clfl ,H1ULTPLI,4I4' 551 Fr.t\IIAT HH ;~E1:.1,~) 537 RHUR'j
E~·D
-0> w
G LEVE!. 20 Rl'FOST
Sl3~QUTt~E ~EFOST (HB(Oc,NcaL~,txl
~rAL K8tl,K~BIK1,wBt
DUE a non
D!HE~SI"N R!96,q61,"19b1,U1~96,41,U8196,4I,~l(96,96'
Cr.'·"JI~ 5TI9/>,192IiK:\Ii 19b,9bl,K~B 196,961 ~ ,KlI24,~41,C(12;12),AI12,lZ1,OI3'12,,~C~13,\21,EI3,3
1
!,C:CI3,lZ',~ 112IlZ)IEDCNlb,2411~DOCI6,Z"lu!\eOOI4),~NN~P6
ECUIVALE~CE IRll,ll;K~tl(111)),!Rl(l,l),ST(l,l)),IDll),U(Z50,Z)',
~ III 1 1,1 ',UIl5"']) " I;'.I~I l,l',UIZ50/41) R~~I'l~ 1 RE~I"[) 2 ~E'"I~~!'I • RE"I'm 16 D~ '03 lal,~~~OP6 ~:: '7)1 Jal,'
?Cl J!II,J) a 0.0 Dt 703 Jal,~"lIIOP6
7~3 ~l!l,J) a 0.0 '~2 :~ '1~ IMal,N~LJC
7(4 RU,~ III H,N,'IIRli'JliJaliNI,lal,H) ~~ITE 16,72n l M,N tFI:X,E~.ll GQ TO 714 R~~~ (16) ~l,";liK8Bll~JI,Jal,N),lal~Ml'
~~lTE 16,7Z~) "11" 0:' '06 lal'~'l C::- '06 Jal,'.
'7C~ Rlt+'1,Jl a KBBlI,jl !'I2 a toi + Hl W~IT~ 1~,7Z0) HZ," Go: T~ 715
714 ~2 • '1 '15 itE~:"' 141 ~, l!ll n;la1;N) ,
~~l~ (16) N'IIUII,J),Jal,NCOlN),I-l,~)
,~:!ITE I!"nt:') N 7~5 ~: 709 lal'~2
~" 709 .!-l," ~: 707 Lal,~CJL"
'0 7 U11!,L) a U1II,L) + R1I,JI.UtJ,L).OIJ) OC 708 Lal,:~2
70~ R1II,L) a R1II,L) • RII,J).RlL,J).OIJI 7:'9 C :\!Tt 'lUE 71:l Ct",T l'lUE
,"P ITE Ib,nO) H2,H2 ~PITE 16,720) HZ,NCnLN R~~~ (2) ~SIZE,11KPB~1'JI'Jal,~SIZI'I/Ial;N~IZEll
1 IIUBII;J),~al,NCOLN1,lal,NSIZEI ~RITE 16,7201 NSIZE,N51ZE ~alTe 16,72~) N51ZE,NCOLN or '13 lal,~StZ~ 0:' 711 Jal,"CCLI:
71\ WH I,Jl a \IP 1 t,Jl -!Jl'! l,JI 0:- '12 Jal,':SIZE
712 K~9II/JI a KB~II,j)-R1Il,j) 713 C:::-.Tl'lUE
~~lTE 16,720) N51ZE,N51ZE
REFOST DATE. n018
WRITE 16"201 NSIIE,NCOL~ WRITE(l7) NSIZE/(iK~BiI/J"J.l/NSIZE"1-1;NSIIE1, lIIU~(!/j)iJ-lï~COCN;/t·l,NSlZEI
720 FC~MAT IlH "REFO~T'/4141 721 F~~~AT IlH ,9El3,51
RrTURN E~D
-"
~
G LEVEL 20 SOLVE DATE _ 72(1'78
c
c
S~9ROUTtNE SOLVE lNrART,NCOLNI R~~~ ~a!l,KBS;KX Ot~:~SI:~ A~196,96);a~19b.96),TFI96,4),RSI9h,4),FI96'.'/OISI9~,4) cr~~a~ STIC~,lq2IiKPIlI9b,9~),KPAI9b.961
! ,KXI24,24I,CI12;1~),AI1~,12I,CI3,12),~CNI3,12),EI3,3) ~,r~CI3,12),D(12,12);E~Ct;lb,24),EOQC(b,24IiU(leOO/4),M~NrPb ECUIVALE~CE(,~ll,l),STI1,1)I,IBHI1,1),STIl,C7)),ITFI1111,UIl,l)),
t 1 l' 1 5 ( l, li • U '( 1.2 1 ) ~ 1 R S ( 1. 1 ) ,li 1 1,31 ) ,IF 1 111) ,U (114) ) 0::- 140 l.l.~"I~orb O~ 141 Je1,~C~LN
141 TFII,J)-O. O~ :4n J-l.~N~OPb
140 B;o( l,JI - \)',0 O~ 144 (L-l.~DART RF~)14l ~,~,IIAMlt,J);I-l,H),J-l,H),IIFII,J),I-l,M),J-l,NCOLN)
150 O~ 424 !-l," OC' 425 J-l,r'COLN
42~ F(I.JI - FII,JI • TFII'J) 0::' "Z4 J-I,"
424 Ar(I,JI-AHII,JI-BMII,J)
C6LL (DUINVI~,LL)
REA~(4) 118~II'J);I-l;H),J-1,N) Da 100 l-l,H OC 10~ K-l,~COLN CISII,KI-(l. 0: 100 J-l,M
100 OISll,KI-(lI~II,K)+AMII,J).FIJ,K) 411 n=l ITE(2) H,~I,I (h,.,'II;JI, l-l,Mj,J-lIH), 1 IB"I'I I,J), 1.\lHIIJ.1iNlI
1 IIFII,J),I-l,M);Jel;NCOLN) 1~I~PlRT-LL) 437,437,~32
432 DO 1'1 I_l,u O~ 1\)1 K-l,NCOLN DIS 1 1, K 1-0. Dr 101 Jel/~ .
101 0ISII,KI-(lI51I'K)+A~II/JI.FIJ,KI D=' 102 t-l,'~ Dr 102 K-l,~COLN T~II,K)-O.
D~ 102 J-l,M 102 TFlt,~I.TFli,l()+SHlj,i).DISIJ;K)
DO 5 1-1.'" Ol~ R J-l,'J RSIJ,ll • 0,0 D:l 4 K-l,H
4 RSIJ,ll • RSIJ,I) • AHII,K).SMIK,J) B Cr·'TPl:.JE
C:l 5 J-l,': 5 A~II.JI • ~Slj/l)
DO 12 Kl.l,N O~- ~ r2-1I~1
9 ~SIIZ.11 - 0.0 DI) 10 K-l,N
Snl.VE
oc 11 L-1,M "_ 11 R~I~,l) - RSIK,l) • BMIL,K).AHIL,K1) 10 CONTINUE
cr 12 K-lIN 12 A~CK.Kl) - RS~K,lj
DO 13 hl,N D[I 13 J-lIN
13 BMIJ,J)_ A'~liJ, 144 Crt.TINUE 437 R[liINQ 4
W~ITEI31 I~OrS~liJ),t-l,H).J.l,NeO(N' Irl~PART-l) 600,bOO;6~1
601 NA-"PART-l Dr 441 LL-l.'l~ BACKSPACE Z BACl"SPACe Z
DAT! • '2078
., .",
REAOIZ) H,~,liAHlr,j)il.liH),J.l,"',C~BHCI,JI,I.l,M',J.l,NI, 1 IIFII,J),J-ï,MI;J-l;NCOLN)
0(1 103 J-lI~ DO 103 ~-l,"COLN TFII,KI_n,
0(1 103 J-1.N 103 TFII,K)-TFII,K).BHII,j).DrSIJ;l()
Dr 444 J-l,MCOLN DrI 444 l-llH
444 FlI.J)-FII,J) • TFlr,J) DO 104 1_1,14 0[1 104 K-l,IICOLN 01511,1<1-0, DO 104 Jel,"
104 OISII,KI-OISII,KI~A~II,J).P~J,kl 441 WRITE (3) IIDrSllïJi,l-l,H),J-l,NeOLNI
WPITE 16,515) 515 FO~'IATI 10H RESIDUAL.S)
DO 14 1-1,M~~~P6 DO 14 J-l,ilCOLN
14 RSII,j) • 0,0 DO 500 LL-liNPART ~EAn141 M,N,11A~lr'J);I-l,HI,J-I,M),C~FII,JI,r-1,H',J.1.NCOLN' RFA~(4) IIB"II,J);I.l;H).J-l,N) BACKSPACE 3 "-_ READ 131 ICOISI!,j);I-l.MI,J.l,NCDLN) 8ACKSPACE 3 BACk.SPACE 3 REAO ~3) IITFil~J),r.i,N);J.l;NCOLN) DO 510 J.l,IICOLN DI) 5111 t.l.··· FII,JI-FII,J) - RSC"JI DO 512 K.l,H
512 FII,JI-FII,JI • AHCI,KI.DISIK,JI DO 510 L-lIN
510 FII,JI- FII,J) • BHI!;L).TFILiJ) ~O 105 r.lI'-' OQ 1~5 K-l,~COLN
-rn
..
G LEV!:L 2:)
~S<I,KI-O, C: 1~5 J-l,Y,
saLVE DATE - 12018
1~5 ~S'I,~'-~S(I,K'+BM(j,t'*DIS(J,K' 'co ,.·tTE C'H311 (CFCt,J,it-lIMI,J-lINCOLN)
31 c:'~"ATC l~ IlZE9.zi 6~C C'=~~TI\4t.!~
l\oT~,~·.
E":
,/
LDUINV DATE • 12018
SUBROUTtNE LDUINV1N;L[1 CCMMON AN196,9~1 . 1-1 IFIANII,III 8;8,2
2 on 3 J-Z,N 3 A~(I/J, - A~II/J)/A~II/I)
DO 15 h2/N Il - 1-1 Dr, 5 L-lIIl
, A~cr,l, - ANct,t) -CANCL,II**ZI*ANIL,C) I~CA~lt,II) 8~8,4
4 11'1 - 1+1 DO 7 J-tPl,~! Dr, b L-l, Il
b A~(I,J' - AHII,J) • ANIL,II*AN(L,J)*ANIl,L) A~t!,J' - AN(I,J)/ANlt,r)
7 C[l~l'l'H~UE l!i Cr:NTINUE
0[1 l~ l_l,N IPl - r+l Ip2 • 1+2 AM(IP1,rl - -ANIJiIP11 Dr. lb J-rp2,tl JI - J-l A~CJ,rl _.AH(l/J) DC lb L-rPllJl A~IJ,tl - A~IJ/I) • ANll,JI*ANIl,r)
1~ COI'TltlUF. Dr, 17 I-liN JPl - r+1 At·II/II - l,/ANII;rl 0[1 18 K-IP1#~~
18 AN(I,II • AN(I,II .~AN(K,II**ZI/ANIK,KI Dr 19 J.rPliN Jl'l - J+l A~(I,JI - AH(J,l)/A~(~,JI DO 19 L-JP1,N
19 A~(r,J) - AM(J,J).ANIC,rl*ANC~,JI/ANIL,C) 17 CrNTlNUE
DO 20 r-l,N IPl - r+l oc 20 J.IP1,N
20 Ah(J,II - ANII,J) Gr) TO 25
8 "RITE 1I'},121 IILL
.,1.'. • \:0.
-m
12 FDR~AT 11H ,l4,ITH DIAGONAL TERH L.E.lERD rN PARTITrD~',IJ""ND S lUPR~UTrNE FAreS'I
WDlTE (6,2~I A~CI;II ~PJTE Ib,261 CIAN~I;Jj,J.l,NI,I.l,NI STOP
25 CONTINUE 26 FI'R!!AT 'c HI ,6[12.51
RrTlIR:~ END
~ L"VEL 2:: STRESS DATE • 72078
S~~RaUTINE STREsslN~ART,NFlRST,NLAST,~COLN,~ELE~,NFREE,NPolNI I\':'EGER BEho.: ~E~L KBll,K~B,Kl,iX~,lYB,IZ~ OJME~SION NFJRSTI201,NLASTIZOI,TcZ4,24',CIIZ4,.1
~'~~llb,4' ,~Sib,4',~SAI3,41,ZI24141 C':'''~J'l ST 19b 19 21 ;K~ Il (Ob,96' ,K!;B 19b,9b)
l ,K1IZ4,241,cI12;lZ),AI12,12),OI3,lZ"~CNI3,12,,EI3,3) 1,~~CI3,12',~112,121.EOC~Cb,24),EOCClb,24',UI1800,4',MHNnPb 1,~Z-I~,4"JZ913.4),~SZTI3,4),OSZB(3,4),A3Ib,b"A2Ib,b) 1,E1C10I,E2110"poilC,;PZ(lO),GEC101,NEPI3001,LK 1,?~ft~13nO,b',tX~(50',!YQI50',JZaI50',EBISO),A~ISO',GBISOI;HCBl5?1 1,~~"130:l,4) E:JIVALE\CE IC1Cl;I),STll,I)1 :~ bJO Il.l;~PART JJ.',PAi1.T+l-il ',.' ~~,FE .. (';FtRSTIJJI - 11 + 1 ...:. ';FI(E:*'~Lt,S'!'( JJ 1
~()C R~A') (3, 1 (1'(J,J);I-lo\;N),J-l,tlCOLNI wCITElb,b90)
~qo F:-P"t.TI'l'l ~~!T= (t>,bl~)
biS F~~~AT C'J.,INOnE X-OJS~LACEHENTS V-OISPLACFH~NTS Z-DISPLACEHEhTS lC~-~ISPLACEPENTS nV-OtSPLACEMENTS OZ-DISPLACE~ENTSI, ~~:TE (~,32) I(J,~I~*J-5,J)/Ulb*I-'/JI,IJI6.:-3,JI,Ulb.I-2;J),
: ~ (~-I-l,J),Jlb·J;J',I-l,NrOlNI,J·l,NCDLN' 32 F='~AT(lH ,J4;bElb,AI
RF"t~" 2 Rf .. tWI 4 ,.PITElb,bI)O) '~qTE lto,71~)
715 F~puATI'JI,IE(EM X;PLA~ESTI(ESS V.PLANESTRESS XY-PLANESTRESS lX-' '.: "l!) l':GSTI\E sS V-BF~n ItIGsnESS XV-r.ENDHlGSTRESS' 1 "~!TE 1~,,7Z5)
"'2:; Fr-""AT 1 lH ,'BF.AM AXIAL LOAD lNI' t'O"E"Tl INPMOHEtlT2 ~T'5T "O~EHT oTr HOME~Tl OTP MOMENTZ INPLN SH~AR QUTPL 2 SHEA'l' 1 ~~ 20 LL-l,HE(EH R!4~ It) (lEQCNIY,Jlil-l~61;J-l,Z4',IIEOQCCl,JI,I-l,ft)iJ.l,Z.',
li:,X,iJRY "~:'l' III (ITll,J),.I-i,241,I_l.241 OC bZO J-l,~'C"LN i>t 020 1-1,. J';-',JD(Ll,l) Zi~.I-5,J)-U(~*JJ-S,J; Z 1"-1-4, J I-U( f,*,IJ-4, J 1 Z C ~. 1 - 3, J , _1 Il b. ,1 J -:; ; J ) Z 16-1-2, JI-l' 1 ~.JJ .. Z, J i l (e.*I-1, J hll( ~.JJ-l, J)
b20 ZC~.I'JI-Ulto*JJ,J) 0:' b21 t-l,24 DO b21 J.l,NC~LN Cl Il,J)-O, o~ 021 K-l,24
621 C1II,JI.ClI1,J).Tll;KI*ZIK,J)
STRES!
00 630 J-1,NCCLN DO 630 Y-l,6 oBII,JI-O. OBlII,J,-O. DO b30 K-l,24 O~ll,JI-DBII,jl+EoQell,KI.ClIK,JI
oAT! - no'78
630 oel(I,J,-OB1IY,JI.EQCNII,KI.C1(~,JI
1
2
3
WRITE (~.3Z' lLL,l09(I,JI~I-l~61,J-l,NCOLNI OC \ 1-113 on 1 J-lIIICOLN OZTII,JI_O. Dzell,JI-O. DSZTlI'JI-O'. D5ZBll,JI-O. DO 2 J-lI"1CCLN DO Z 1-113 OZTII.J)-OBIliJ'+oBjl.3,JI DZBll,J)-091I,Jl e o8il+:;.JI CALl PRINIOZT;A1,NC~LNI CALl PRl~I018,Az,HC"LNI WPITEIZ'lL,C~~O(l[,J';J-l,4"ORX,ORV,ffOZTII,JI,I_1,",IA'I!,JI,
1 hl, 3', J-l,I'COll!); 1 LL,INOOILL,j';J-1;4"ORX,ORY,CIOlBCI,JI;Z_1,31,CA2CI,Jl, 1 le l, 31, J-l, r'C~Lrll
.JO 3 J-l,UC~L'I DO 3 Y-1I3 05ZTII,JI-091l1,Ji+OB111+3,JI DSZSII,J)aOBlfl,J'-OB111+3,Jl WPITEf4'LL,INnOIL(,JI;J-l,41,ORX,ORV,lIOSZTI!,JI,l a1,'l,J-1.NCOLNl l' LL,IN~DILL'JI;J-li4,.ORX,ORV,IIOSZBII'JI,!-1,31;J.1,NCO(NI
IF INBEAH.EO.OI GO TO 19 IIlEAH - 0 Dr:! 18 hl,b
18 IBEM' a IBEAH + BEA~'IÜ,ll IF IIBEAH,Eo.ol GO TO 19 CALL ACTIONILL,NCoLN,nRX,ORVI
19 CONTINUE 20 C[I"TItlUE
RE'WINn 2 REWIND 4 HP !TE 1 tu b90, WRlTE 16,6251 HRlTE 16,b3'1
b25 FORMATIIOI,IECEHENT NODES CCCRD! OF CENTROIOI. PRlr~CIPAL STRESS!S l' CARTESIAN STRESSES
2'PRINCIPAL' , 635 FORMATII ','NUHRER
11 X-STRESS V-STRESS Z' ANGLE"
XV-STRESS
Dr 2Z LL.l,~ELEH
)( .. COOR" STPESS-l
V.CO[1Ro'i STRESS.2
l,
" REAOl21 LL,I~nDIL(,JI;J-1;41;ORX,DRV,IIDZT't,Jlil.l,31,IA311,~J; lI-l,3I'J-l,uCOL~I; 1 , LL,~NOOIL(,JI;J-I,4IJORX,ORV,(IDZ~lr,JI,I-l,31,fAZII,J" 11-1,31,J-l,NCOLM, ,
G LEVEL 20 STRESS DAT! .. 7Z0711
Io,'l:TEICI,311 l ILL,i~~OILC,KI,K_1,41,ORX,ORY, I~ZTII,JI,I-1,3\,IA311,JI, !I·1,31,J-1,McnL~1
22 i\ O ITElb,31l 1 CLL,I~OOILC,KI;K-1~4\,ORX,ORY, IOZBlr,JI,I-l,lIIIA2CI,JI, 11-1,31,J-l,~COL~)
ftc:'!'E CblbBI .=ITE C~,b2"1
~Zb :·~~ATC 'O','ECEHE~T NaDES caa~o. OF ~.G CARTESIA~STRAIN 15 ' 1
~27 F:~uATC'O"INUM~ER x COORO, V COORO ! 1 ST~AIN V STRAI~ XY STRAINI) ;: 23 LL-l,~ELEM R~A~I~1 LL,IN~DIL('J);J-1,4),ORX,ORY,IIOSZTCI,JI,I.1,31;J-I,NCD(NI
11 LL, C'H')DILltJ liJ-1I41,Oil.l\,ORY, IIDSZSC l,JI, 1-1,3),J-l,NeOLNI .;ITE C~,3'"
iLL,iN~DILL,K);K.l,4I,ORX,Oil.V, InSZTcI,J\,lal,31,Jal,NCOLNI 23 .,cITE 16,34)
ILL,i~nDIL(,K);K.1i4\,aRX,ORV, IDSZBII,J\,1.1,3),Ja1,NCOLNI 31 c:'R"ATI' ',14,415 ,Fl2.4,F10,4,5E12."F7.?) 33 ~:~~ATCIH~,~E12,5)
34 F-~uATC' ',14;415 ,~12.',F10.4,3E1Z,5) RETUR'I E~.~
PRIN
SUBROUTINE PRINIDB,A,~COLNI DIMENSION OBC3,b)iAib;CI\ 00 100 J-l,NC~LN T1 aCOBI!,J) • ORC2,j)1/2. T2a~IDBI1,J\·OBC2;Jj)/2,1.·Z T3 a IDBI3,J)\.*Z T4-2.*DBC3,J)/I~Bil;JI-OBI2,JII Ai1,J)DTl • S~RTCTZ+T3) AI2,JI - Tl - SeRT(T2.T3) T~A.O~'.ATA~(T41 T~8.THA • 0.5*3,14159
DATI! .. 720711
SIG • Tl.SQ~TIT2)*Cr5i2,.THAi + SQRTCT31.SltICZ,.THAI IFIAC1,JI.SIGI 1,2';
2 AI3'JI-THA.18~./3:14159 Gn Ta 100
1 AI3,J).THB*lSO,/3:14159 100 CQ~TINUE
RETURN END
0> ())
G LEVEL 20 DiRAC A DATE - 72078
s~le:taIlTtNE ['l'tAtA (~,IJ,tlREC,NSI.ICE,IRW/tAC)
DIME~SI~N Ai9b,Qb)/CI9b,96) CrM~JN ST(9b,l9~)
EQUIVALENCE IAI1,l),STI1,l»,ICI1/1),STI1,97» NPECl • I~REC-l).IN~LICE+l)
I~DEX • ~~ECl + 1 le ( IR~.CT.SI CO To 'tE~~ 115'I~rEX) H;N G:: T::J 2
1 Ile; ITE 115'1'·[)EX) MIN
2 ""5 • "/~5L1CE O~ 4 1.!,~!SL ICE I~~~x - INC~X + 1 Ml • ~~5.II-l) + l "2 • I$V'I;S IF (I~~.GT.51 GO TO 3 !tE!.!: C15 ' J),rEX) nC'CK;J)/J-lIN"K-"Il,M21 Gr r:l 4
~ I~ (lAC.GT.l) Gn TO 5 ~~lTEC15'INDEXI 11AiKiJI,J-l,HI,K-Ml,M2) c;- T" 4
5 .;IT~Cl~'IN[)EX) IlCCK;Jl,J.l~N),K-Ml,M21 " C"'ITl~~UE
RFT'_'R'j E")
/
oiRAca
SUB~OUTINE OIRACB (H/N/NREC,NSLICe) DIMENSION Ci96/96)
OAT! • no.,.
C~MHQN ST(96/192);KRIiI9bi96)~KBBC96,961 ECUIVALENCE ICI1,i);KBBI1,1» NPECl - INREC-l)*lNSLICE+l) INDEX - NRECl + 1 REAO C151INOEX) H;N
2 M~S • M/NSLICE DO 4 I.lI~'SLICE l~Dex - IHOEX + 1 Hl • MNS.II-l) • i HZ • t.Mt-\S REAQ C15'INOEX) 11CiKiJ),J.l,N),KaHl,H21
" CotITI'~UE RETURN END
-Cl) CD
G LeVEL 20 AeTION
SUBROUTTNE ACT InN (LL;NCOLNioRXiORYl l',TEGFR BEA"
DATE • 7Z07B
R~AL lX~,IYP,TZ~,Kl;KRB,KRll,LE~G
D:~E~SInN ALFh(3,~),X~12"),Zl2.,41
C~I'H:1'l ST (9b 192) iKIl Ii t 9t.,%) ,KSSt 96,9b)
1 ,KltZ.,24',Ctl2;12),A(12,lZ),Qt3,12I,OCN(3,lz,iE(3,31 1,OQC(3,12"r.(lZ,12),E~Ct:(~,2.),EDQC(b,24),U
(1800,4I,HNNOP6
l,DZTt3,.),Ozsi3,4"rSZT(3,."DSZBt3,4"h3Ib,bl,AZlb,61 l,Elll0',EZ(10),PO~ln);P2(lO"GEtlO),NEPI300I,LK
l'PEA~13~O,bl,IX~(501,IV~1501,IZB(50),EBI50"ABt50),GB(50);HeBf50)
l' ~,Ol) t 300,,) ECUIVALF.NCE (ALFAil,II,KBI1(1,l'),(XI1,11,K~BI1,1)I,(ltl,
ll,STll,l
11) Or. fiOCl Y-l,3 D~ 800 J-l,?>
800 ALFAI1,J' • O~O IF IBEAM(LL,l).EQ~O) GO TO BOL L~M,)A • 1 MU • BEAMILL,iI LENG • 2,0* soRTIORx**Z) ALFA( 1, 11 - l,a ALFA(l,Z) - 0',0
Ni - 0 N2 • 6 GO TO 806
801 IF IBEAM(LLizi.EQ:O; GD TO BOZ LAMI)A • 2 MU D BEAM(LL,2) LENG - 2,0* SQRT(ORv**ZI ALFt,(l,l) • 0.0 ALFAI1'ZI - 1,0 IH - b ti2 • n GC Ta BOb
80Z IF IBEA~(LL,31.EQ:0) GO TO B03 LAMDA • 3 MI) • SEAMILL,3) LENG _ 2,0* SQRT(ORx**Z) ALFA( 1, 11 --l,a ALF,\! 1,2) • O~O NI " 12 N2 .. 18 GO 1'0 806
e03 IF (BEAMILL,4i.EQ~Oi GO TO B04 LA~:DA • 4
MU - BEAHILl," LFNC • 2,0* SORT(ORv •• Z) ALFAl1,l1 D 0.0 ALFA(lIz) --1.0 Ni • 18 NZ • ° GO TO BOb
804 IF IBEAM(LL,SI.EQ:O, GO TD Ba' LAHOA - ,
AeTIDN
MU • BEAHILC,5) LfNG - z.o* SORTIDRx*.Z + ORY**ZI ALFAI1,l) - 2,0*oRX/LENG ALFAll,21 - 2.0*oRV/LENG Nl • 0 NZ - 12 GO TO 80b
eo, IF fBEAM(LLi6j.EQ:Ol GO TO Bl' LAtlDA • 6 HU • AEhMILl~6)
LFNG • 2,0. SORTlnRx**z + ORy.*Z) ALFA(1,11 • -z.n*nRx/CENG ALFA(l,zl. Z.O.ORv/CENG Ni • li N2 • 18
80b ALFA IZ,21 • ALFAll;li ALFA 'CZ,ll - .'ALFAI1,U ALFh 13,31 • i.o
'00 807 J.l,~COLN 00 807 1.1112
807 X!I~Jl • o.n Do 912 J-l,NC~LN 00 811 Kal,4 Kx • CK-11*) IFIK.LE,ZI GD. TO eOIl :::j KZ • ~IZ + KX • 6 0 Gn TO 809
BOB K2 a NI + KX BC!9 C[lNTINUE
DO 810 1-1,3 OC! R10 L-l,)
Bl0 XIKX+l'Jl a XIKX+i,jl + A~FAC1,LJ.ZCKZ+L,JI BU CONTINUE 812 CO~TINUE
DO 813 Jal,NCDLN AL - ~EBIMU)*AB~MUI/LENGJ*IXi',JI.X'l,JI • HceCHU)*'XC11,jl.xl',JI
111 B~PLI • IZ.O*EBIMU'.IIBIMUI/ILE~G*.ZII*13~O.CX(8,JI.XI2'J"
1·3.0*HCAIHU1*lXll0'JI.X(~,J»·LENG*IZ.0.XI6,JI+XI12,JJII
B~PL2 • BMPll + 2~0.(EBCHUI*IZBIMUI/LENGJ.(XI6,JI-XllZ,Jlj TM • IGBIHU1*IXBIHUi/[ENGI*'XI10,JI-XI4,JII B~OPl • 12.0$EBIMU!*IYB(MUI/ILENG**ZIJ.13.0.CXI3,JI_XI9'JII
1 • LE~G*(z.n.XI"jl+Xll1,J)Jl
BMOPZ • BHUPl + Z:0*IEBIHUI*IVBIHUI/LENGI*IXC',JI-XC11,JI; SHRl • tBMPL1+SMPCZ1/(ENG SHR2 - IBHUP1+BHOPZ)/LE~G WR ITE 16,8110) HIl, j, AL.BHPO,BHPLZ,TH,BMOP 11 AMOPZ,SHRlI S"'U
eu C['tITINUE IF'ILAHnA.EQ.l1 GO TO 801 IF,CLAHOA.EO.Z' Gn TD BOZ IF (LAHDA.EQ.31 GO TO 803 IF fLAHOA.E~.41 GO TD B04 IF ILAMnA,EO.5) Gn TO 805
814 FnRl'AT ClH ,14lIZ;BEl!5," B16 FrRMAT 11H ,lIEla.51 B15 RET1.lRN
E~;O
