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CONCRETE PONDING EFFECTSIN COMPOSITE FLOOR SYSTEMS
Item Type text; Thesis-Reproduction (electronic)
Authors Peña-Ramos, Carlos Enrique, 1962-
Publisher The University of Arizona.
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Download date 02/08/2021 03:23:00
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Concrete ponding effects in composite floor systems
Pena Ramos, Carlos Enrique, M.S.
The University of Arizona, 1987
U M I 300 N. Zeeb Rd. Ann Arbor, MI 48106
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International
CONCRETE PONDING EFFECTS IN
COMPOSITE FLOOR SYSTEMS
by
Carlos Enrique PeRa-Ramos
A Thesis Submitted to the Faculty of the
DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS
In Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 8 7
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of the source is made. Requests for permission for extended quotations from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
Signature
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
: /
Reidar Bjor&ovde, Professor Department of Civil Engineering and Engineering Mechanics.
</Di Date
ACKNOWLEDGMENTS
I would like to express my sincere appreciation to
Dr. R. Bjorhovde for his generous time, effort and
invaluable advice throughout the development of this
thesis. Special thanks are due to Dr. M. R. Ehsani and Dr.
P. D. Kiousis for their review of the manuscript, as well
as their helpful comments and suggestions.
I would like to thank my parents for their moral
and financial support throughout my graduate and
undergraduate work. Their love and support have been a
driving force behind my academic career.
I would also like to thank my lovely fiance, Vicky
for her love, patience and understanding, as well as her
help in preparing the final drafts of the manuscript.
Last but not least, I would like to thank everyone
who supported me throughout my graduate study, especially
the CIAD Research Institute and the Ministry of Education
of Mexico for their financial assistance.
This thesis is dedicated to my parents, brother,
sisters, fiance, and especially to my son, Carlos, who has
been my inspiration and reason for wanting to become a
better engineer, a better person, and a better father.
iii
TABLE OF CONTENTS
Page L IST OF ILLUSTRATIONS v
LIST OF TABLES x
ABSTRACT xi
CHAPTER
1. INTRODUCTION I
2. ANALYTICAL MODEL FOR THE FLOOR STRUCTURE 5
2.1 Methods of Analysis for Floor Structures 5 2.2 Analytical Model for The Supporting Floor
Structure 6
3. ANALYTICAL LOADING MODEL 12
4. INCORPORATION OF ANALYTICAL MODEL INTO COMPUTER PROGRAM 20
5. RESULTS AND DESIGN RECOMMENDATIONS 30
6. SUMMARY AND CONCLUSIONS 40
6.1 Summary 40 6.2Conclusions 41
APPENDIX A: DESIGN CURVES 43
APPENDIX B: PROGRAM GRIDS 105
B.l Introduction ' 105 B.2 User Manual _ 107 B.3 Program Listing 112
NOTATION 121
LIST OF REFERENCES 123
i v
LIST OF ILLUSTRATIONS
Figure Page
1 Degrees of Freedom of a Typical Grid Element 7
2 Structural System Considered in this Study 11
3 Loading Stages of The Floor Structure 14
4 Analytical Loading Model 19
5 Floor Structure Computer Model 29
6 Total Interior Floor Beam Deflection 34
A1 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 45
A2 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 46
A3 Maximum Momemt vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 47
A4 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 48
A5 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.0 in.; Sb=5.0 ft.) 49
A6 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.; Sb=5.0 ft.) 50
A7 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.0 in.; Sb=5.0 ft.) 51
A8 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.; Sb=5.0 ft.) 52
A9 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=5.0 ft.) 53
A10 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.; Sb=5.0 ft.) 54
v
vi
LIST OF ILLUSTRATIONS - Continued
Figure . Page
All Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=5.0 ft.) 55
A12 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.; Sb=5.0 ft.) 56
A13 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=5.0 ft.) 57
A14 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.0 in.; Sb=5.0 ft.) 58
A15 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=5.0 ft.) 59
A16 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.0 in.; Sb=5.0 ft.) 60
A17 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.5 in.; Sb=5.0 ft.) 61
A18 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5 in.; Sb=5.0 ft.) 62
A19 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.5 in.; Sb=5.0 ft.) 63
A20 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5 in.; Sb=5.0 ft.) 64
A21 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=10.0 ft.) 65
A22 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=10.0 ft.) 66
A23 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=10.0 ft.) 67
A24 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=10.0 ft.) 68
A25 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.0 in.; Sb=10.0 ft.) 69
vii
LIST OF ILLUSTRATIONS - Continued
Figure Page
A26 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.; Sb=10.0 ft.) 70
A27 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4? Ts=6.0 in.; Sb=10.0 ft.) 71
A28 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.? Sb=10.0 ft.) 72
A29 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=10.0 ft.) 73
A3 0 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.? Sb=10.0 ft.) 74
A31 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4? Ts=6.5 in.; Sb=10.0 ft.) 75
A32 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.; Sb=10.0 ft.) 76
A33 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=10.0 ft.) 77
A34 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.0 in.; Sb=10.0 ft.) 78
A35 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=10.0 ft.) 79
A36 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.0 in.; Sb=10.0 ft.) 80
A37 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.5 in.; Sb=10.0 ft.) 81
A38 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5 in.; Sb=10.0 ft.) 82
A39 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.5 in.; Sb=10.0 ft.) 83
A40 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5 in.; Sb=10.0 ft.) 84
vlii
LIST OF ILLUSTRATIONS - Continued
Figure Page
A41 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in. ; Ts=5.5 in.; Sb=15.0 ft.) 85
A42 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4? Ts=5.5 in.; Sb=15.0 ft.) 86
A43 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=15.0 ft.) 87
A44 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=15.0 ft.) 88
A45 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.0 in.; Sb=15.0 ft.) 89
A46 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.; Sb=15.0 ft.) 90
A47 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.0 in.; Sb=15.0 ft.) 91
A48 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.; Sb=15.0 ft.) 92
A49 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=15.0 ft.) 93
A50 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.; Sb=15.0 ft.) 94
A51 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=15.0 ft.) 95
A52 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.; Sb=15.0 ft.) 96
A53 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=15.0 ft.) 97
A54 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.0 in.; Sb=15.0 ft.) 98
A55 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.0 in.; Sb=15.0 ft.) 99
ix
LIST OF ILLUSTRATIONS - Continued
Figure Page
A56 Maximum Moment vs. Beam Stiffness (1—2000 to 20000 in.4; Ts=7.0 in.; Sb=15.0 ft.) 100
A57 Maximum Deflection vs. Beam Stiffness (1-200 to 2000 in.4; Ts=7.5 in.; Sb=15.0 ft.) 101
A58 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5 in.; Sb=15.0 ft.) 102
A59 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=7.5 in.; Sb=15.0 ft.) 103
A60 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5in.; Sb=15.0 ft.) 104
LIST OF TABLES
Page
Table
4.1 Beam Size Classification 23
4.2 Concrete Weight as a Function of Slab Thickness and Metal Deck Geometry (Normal Weight Concrete) 25
4.3 (a) w-Loads (lbs/ft) for Floor Structure (5 ft. Beam Spacing) 27
(b) w-Loads (lbs/ft) for Floor Structure (10 ft. Beam Spacing) 27
(c) w-Loads (lbs/ft) for Floor Structure (15 ft. Beam Spacing) 28
x
ABSTRACT
During the first placement of concrete, the
typical floor structure will deflect under the concrete
load. This will produce an uneven slab surface that must
be leveled before curing and finishing procedures can be
performed. In order to avoid the uneven slab surfaces, the
slab is releveled with additional concrete until an
equilibrium position is reached where additional
deformations of the structure are negligible.
The increase in load and deformation resulting
from the releveling of the slab surface has usually been
neglected in the design of floor structures. The purpose
of this study is to predict the increase in load and
deflection caused by the ponding of the concrete during
the releveling procedure, and to prepare design
recommendations that incorporate the effects of these
i ncreases.
xi
CHAPTER 1
INTRODUCTION
The composite floor system consisting of a
concrete slab with metal deck reinforcement and supported
by an arrangement of steel beams and girders is commonly
used for steel structures. During construction, the first
permanent loading the steel framing and metal deck receive
is the one caused by the placement of the concrete slab.
If temporary shoring is not used during the placement of
the concrete slab, the steel framing and metal deck will
deflect under the additional loading. If the concrete
floor slab was placed to a uniform thickness, the result
would be a nonuniform slab surface whose shape would be
defined by the deflected shape of the structural system.
In order to avoid the above condition and assure an
acceptable slab surface, the following procedures are
normally used:
1. The entire floor system is shored during the concrete
placement.
1
2
2. Camber is given to floor beams to counteract the
deflections produced by the concrete placement.
3. The slab surface is releveled, and the additional
concrete that is poured results in a nonuniform slab
thickness.
Of the procedures given above, releveling the slab
surface is probably the most common. This is an iterative
procedure where the additional concrete used to relevel
the slab surface will cause an additional deflection of
the supporting floor structure, which in turn will require
additional releveling. The procedure is continued until
an equilibrium position is reached, presumably before any
local or overall failure of the floor structure occurs.
The equilibrium position is reached when the floor
structure undergoes negligible deflections with the
placement of additional concrete.
A previous study on the ponding of two-way roof
systems caused by rainwater was done by Marino (1).
Subsequent work by Ruddy (2) found this approach to be
suitable for the analysis of floor systems when releveling
is used. Since the structural compositions of roof and
floor systems are very similar, the only possible problem
was the assumption that concrete, like water, will seek a
horizontal level when placed on the deflected structure.
3
This assumption is the basis of the analytical loading
that is model developed in Chapter 3. However, it is
precisely the fact that the concrete will not seek a
constant level that causes the uneven slab surfaces.
Despite this, Ruddy found that the results obtained for
the deflection values using the rainwater ponding analogy
were reasonably accurate when compared to actual field
measurements. Therefore, given the reliability of the
analogy for the analysis of floor systems, this approach
was used to develop the complete analytical model of the
floor system, given in Chapters 2 and 3.
Chapter 4 treats all the assumptions that have
been made, and the load values used in implementing the
analytical model into program GRIDS, a finite element
computer program developed in this study. The listing of
the program along with its User Manual are given in
Appendix 6.
The purpose of this study is to investigate the
deflection and strength characteristics of the floor beams
when subjected to the loading associated with the
releveling of the slab surface. The beam spacing, span,
and required slab thickness are considered variables, in
order to incorporate as many floor structure compositions
4
as possible, and the effect of this variability has been
investigated.
. Finally, the results of the investigation are
given in Chapter 5, along with design recommendations that
are needed to avoid some of the problems that may arise
when the load and deflection increments associated with
releveling of the slab surface are neglected. From the
results obtained from program GRIDS, a series of design
curves giving maximum deflection and moment values as a
function of the beam spacing, beam span, slab thickness,
and beam stiffness have been constructed for the whole
range of beam sections indicated in the AISC Steel
Construction Manual (3). The complete set of design curves
are given in Appendix A.
CHAPTER 2
ANALYTICAL MODEL FOR THE FLOOR STRUCTURE
2.1 Methods of Analysis for Floor Structures
Since the theoretical methods of analysis given by
classical theory of elasticity have proven to be
impractical when analyzing large structural systems such
as a typical floor structure, engineers have been forced
to devise alternative approximate methods of analysis.
Various approximate methods of analysis have been
proposed and found to be reliable and useful. Before the
use of the computer in structural design, approximate
methods based on mor.ant distribution (4) and plate analogy
(5) theory were commonly used in the design office. As the
computer began to gain widespread use in the structural
design office, previous approximate methods of analysis
that seemed impractical due to the large set of
simultaneous equations that had to be solved became more
attractive to the engineer. Methods based on slope defle
5
6
ction (6) and finite element theory (7) began to be used
for the analysis of such system.
Since the finite element method represents the
most advanced and powerful tool of analysis available
today, it was chosen in this study for the development of
the complete analytical model of the floor system.
The purpose of the finite element method is to
simplify an otherwise complex structural system into a
series of smaller structures called finite elements.
Mathematical models are developed for the simplified
structures in such a way that these can be easily
reassembled. As a result, a more complex mathematical
model is created for the actual structural system from the
models developed for the simplified structures.
2.2 Analytical Model of The Supporting Floor Structure
The support system of a typical floor structure,
due the loading conditions to which the system is
subjected, can be characterized as a grid type structure.
A grid type structure defines a structural system where
all the loading the system is subjected to is applied in a
plane perpendicular to the plane of the structure. Due to
this loading condition, the individual elements of the
7
grid structure are subjected to torsional, bending, and
shear forces, which in turn cause two types of rotations
and a linear displacement at the ends of the grid
elements. Figure 1 illustrates the end displacements or
degrees of freedom that a typical grid element is
subjected to.
Figure 1 Degrees of Freedom of a Typical Grid Element
8
In Figure 1, the rotations along the x-axis are
caused by the torsional forces, and the rotations along
the y-axis, as well as the linear displacement
perpendicular to the plane of the structure are caused by
the bending forces. The assumption is made that axial
deformations, as well as the shear deformations of the
grid elements are negligible.
By recognizing that the floor beams and girders of
a typical floor structure can be modeled as grid elements,
an analytical model for the support system of the floor
structure may be developed from the mathematical model of
a grid element. Since the mathematical model for the grid
element has already been developed through a finite
element solution (7), the basis for the support system
model has already been established, and the development of
the analytical model becomes a simpler task.
The analytical model of the support system is
based on the structural system given in Fig. 2. The system
represents an interior bay of a typical floor structure,
consisting of equally spaced floor beams, simply supported
at the ends by girders. The perimeter members of the bay
are considered to be simply supported on columns at the
ends, and identical structural arrangements are considered
to occur on all sides of the bay. The floor beams are
assumed to have the same moment of inertia, and the moment
9
of inertia of the girders is considered to be related to
the moment of inertia of the floor beams by the following
formula:
Ig o lb + 500 (2.1)
Where:
Ig = Moment of Inertia of the Girder (in4)
lb = Moment of Inertia of the Beam (in4)
Equation (2.1) is simply an approximation that has been
used in order to avoid having to obtain Ig from standard
design procedures each time a new lb, slab thickness, or
beam spacing is used. The effect of using Eq (2.1) in the
model is explained in Chapter 5.
The modulus of elasticity for all of the members
of the bay is assumed to be the same, and remains constant
regardless of the load intensities the support system is
subjected to. In other words, the system is assumed to be
elastic. The assumption has been found to be reliable
(1,2), as will be explained in more detail in Chapter 5.
Since all of the members of the bay and therefore
the entire support system are considered to be simply
supported, the system is treated as having no torsional
rigidity. As a consequence, the torsional displacements
and forces are neglected in the analysis.
10
Finally, the analytical model of the support
system was developed by subdividing the members of the bay
into a series of grid elements. The number of grid
elements per member is directly related to how the load is
applied to the support system, as will be explained in
Chapter 3. The development of the load model is described
in detail in this chapter.
11
Lg
equal equal equal
METAL DECK ^ Lb
ORIENTATION
Figure 2 Structural System Considered In This Study
CHAPTER 3
ANALYTICAL LOADING MODEL
In order to simulate the loading conditions the
floor structure is subjected to, a loading model has to be
created. Since the floor will deflect under the concrete
placement, the loading model would have to be a function
of the deflected shape of the structure, and since another
concrete placement would be needed to relevel the slab
surface, the loading model would have to be able to
accommodate the corresponding increase in load.
It is important to note that the concrete is in
its fluid state during the construction stage. As a
consequence, the effect of the concrete on the supporting
structure can be modeled by considering its loading role
only. In other words, the concrete weight is the only
contribution of the slab to the structural system and any
composite action has not come into being.
12
13
Figure 3 shows the loading sequence for the floor
structure during the releveling procedure. The loading
condition illustrated in Fig. 3(a) represents the first
stage of the sequence, that is, the type of loading the
floor structure is subjected to when the first placement
of concrete is made. The next loading stage is illustrated
in Fig 3(b). Here, the floor has already deflected under
the load defined in the first stage, and the distribution
is now nonlinear, with its shape defined by the deflected
shape of the structure.
The third stage illustrated in Fig. 3(c) defines
the loading conditions of the stmacture when an additional
placement of concrete is made. The load is defined by a
combination of the loads of the first and second stages.
The linear portion of the load reflects the increased
loading given by the additional concrete placement. The
nonlinear portion reflects the loading conditions just
prior to the placement of the additional concrete. The
assumption that the concrete seeks a constant level is
represented by the straight line that defines the boundary
between the linear and nonlinear loading. (If this
assumption is neglected and the concrete is allowed to
behave in a realistic fashion, the result would be a
nonlinear boundary between the two loading conditions
14
w-Load
' \ 1 ' \ I \ 1 . 1 . 1 ' ' f 1 1
/J&T ~
(a)
(b)
(c)
Figure 3 Loading Stages of The Floor Structure
15
which will greatly complicate the development of a loading
model for the floor structure. The effects of the
simplification are negligible.)
As was explained in Chapter 1, releveling of the
slab surface is an iterative procedure. In the loading
model, the iterative procedure is incorporated by defining
a loading cycle in which the second and third stage
loading conditions are alternated. An increase in
deflection with a corresponding increase in loading is
then defined at the second loading stage each time the
next loading cycle is started.
Theoretically, the loading cycle could be repeated
until collapse of the floor structure occurs or the
deflection increases become acceptably small. However, it
was found that the increase in deflection from one cycle
to the next became smaller with the number of loading
cycles. As a consequence, the deflection converged to a
constant value and an equilibrium position was reached.
This meant that a leveled surface of the concrete slab can
be achieved given that the resulting loading and
deflection increments will be small enough, and thus will
not violate established design values.
The analytical model for the loading is
illustrated in Fig. 4. A typical floor beam is subdivided
16
into smaller grid elements. To accommodate the first
loading stage, a uniform load w is applied thoughout the
beam length. This load is a function of the floor beam
spacing, the design slab thickness, the unit weight of
concrete, and the beam self weight. The numerical value of
w can be computed using the following formula:
w =(Wc x Ts) x Atw + DL X Atw + Wb (3.1)
Where:
Wc = Unit Weight of Concrete.
Ts = Slab Thickness.
Atw= Tributary Area Width.
DL = Additional Dead Loads. (Construction
Loads, Partitions, etc.)
Wb = Beam Self-Weight.
In Fig. 4, the second and third loading stages are
considered in one single step. First, the deflections
caused by w are computed. Then the deflection values
defined at the nodal points are converted into load values
by adding their respective numerical values to w without
regard to the units involved„ In general, for the i-th
loading cycle:
17
W^+1 Wj_ + Wc X Atw X (d^+1 - d^) (3.2)
Where:
w = Uniform Load.
d •= Deflection.
Convergence to an equilibrium position is assured if the
second term on the right hand side of Eq. (3.2) converges
to zero.
It is important to note in Fig. 4 that the
nonlinear loading curve defined in the second loading
stage is approximated by a series of linear segments,
where each segment is defined by the length of a grid
element. Therefore, care must be exercised when choosing
the number of grid elements per floor beam. If too few
grid are used, a rough approximation of the loading curve
will result, and unconservative values for the forces and
deflections will be obtained. On the other hand, if too
many grid elements are used for each floor beam, the
loading model itself will be reliable, since a closer
approximation to the nonlinear loading curve will be
achieved. However, the complete analytical model of the
floor system will become impractical for programming
purposes, due to the large number of nodal points that
result. This is particularly true in the microcomputer
environment, but may not represent a problem if a computer
18
with greater computing power is used to perform the
analysis. In general, the reliability and efficiency of
the analytical loading model will depend on good
engineering judgement when choosing the number of grid
elements that will define a floor beam.
Finally, the complete analytical model for the
floor system is created if the model for the loading and
the support system are considered as a single unit. In
Chapter 4, this model is modified to account for the
different compositions of floor systems and programmed for
a computer using the program GRIDS.
19
w
1 3 ft Z
Figure 4 Analytical Loading Model
CHAPTER 4
INCORPORATION OF ANALYTICAL MODEL
INTO COMPUTER PROGRAM.
Once the complete analytical model for the floor
system was assembled, the next step consisted in
subdividing the model into grid elements and numbering the
resulting nodes and elements so that a computer model
could be created.
Figure 5 illustrates the computer model used in
this study, including node and element numbers. Although
the nodes can be numbered in an arbitrary fashion, it has
been found (8) that minimizing the node number separation
greatly enhances the efficiency of the computer model. The
node number separation is the mathematical difference
between the numbers of adjacent nodes. In Fig. 5, the
maximum node number separation is five. On the other hand,
the way in which the elements are numbered has not been
found to affect the performance of the computer model, but
20
21
may simplify the task of writing the data file represen
ting the model.
Boundary conditions were defined for the computer
model in the following fashion (refer to Fig. 5):
1. Nodes 1, 5, 18, and 22 are considered to be res
tricted against linear displacements, to simulate
the column supports.
2. All other nodes were free to displace on any al
lowable mode.
It is important to note that the interior floor
beams will be supported on free nodes instead of simply
supported nodes due to the assumed boundary conditions.
This was necessary to allow the floor structure to deflect
in a realistic fashion. However, once the results of a
computer analysis are obtained, the settlement of the beam
supports can be substracted from the maximum deflection at
the midspan, therefore, creating a simple support effect.
This effect is discused in more detail in Chapter 5.
In order to implement the analytical model into
program GRIDS, the following assumptions were made:
1. All members of the floor system have constant mo
ment of inertia and have no initial deflections
prior to the first placement of concrete.
22
2. The stiffness of the floor system is comprised of
the stiffnesses of the beams and girders only. In
other words, the metal deck and the concrete slab
have no contribution to the stiffness of the sys
tem.
3. The uniform load w is carried by the beams only.
The girders carry the beam load at the nodal
points commom to both, but carry no uniform load
4. All other assumptions made during the development
of the complete analytical model apply here too.
Once the computer model was created for a general floor
structure, specific cases representing a wide variety of
floor structure compositions could be easily created by
varying some specific characteristics of the floor
structure. For this study, close to 1800 variations of the
floor structure of Fig. 2 were created by varying the beam
size, the beam span, the beam spacing, and the slab
thickness.
The beam size was determined from the the
W-sections listed in the AISC Steel Construction Manual.
Since using all of the individual W-sections would be
impractical, the sections were divided into a total of
seven groups using the moment of inertia as the governing
criterion. The average weight of all the sections belon-
23
ing to each group was computed and that value was taken as
the self-weight of all the beams belonging to the group.
Table 4.1 gives the seven groups, the number of
sections belonging to each group, and the average weight
of the beams.
Table 4.1 Beam Size Classification
Moment of Inertia No. of W-sections Average Weight
(in4) (Lbs/ft)
200 32 21.0316
600 26 49.6923
1000 16 68.6872
2000 21 95.2852
4000 21 140.7148
10000 21 152.5720
20000 13 226.5388
The value of the moment of inertia defines the
maximum value that a section can have in order to belong
to the group. The minimum value of the moment of inertia
for the members of a group is defined by the value
categorizing the preceeding group. For instance, there are
24
21 W-sections whose moments of inertia fall between 2000
and 4000 in^.
In choosing the group ranges, it was attempted to
keep the number of shapes in each category close to the
same as that of adjoining groups. By doing this, sparsely
populated groups next to highly populated ones could be
avoided. The advantage of this procedure becomes clear as
the design curves are developed, as shown in Appendix A.
The design curves are more or less evenly spaced which
allows the user to perform a better interpolation with
specific "l"-value in between the regions defined by the
various curves.
The beam span was varied in five foot increments,
starting with a beam span of five feet, and continuing
until the resulting deflections or bending moment values
became too large for practical purposes. Since the
deflection is a function of the moment of inertia of the
beams, the lighter sections, such as the ones belonging to
the first two groups, required less increments in the span
length to reach large deflection values than the heavier
sections. As a result, the design curves for the lighter
sections were defined using fewer data points than curves
for the heavier sections. Since the curves were computer
drawn, the distances between data points were approximated
25
by straight lines. This resulted in better curve approxi
mations for the heavier sections than for the lighter
sections. The inconvenience can easily be avoided by
drawing the best curve by other means before actually
using the curves.
For the beam spacing, values of five, ten, and
fifteen feet were chosen. The reason for using these
particular values was that the most commomly used spacings
will either fall on or in between these values.
Interpolation can therefore be performed from the values
obtained from the design curves defining the beam spacing
range of the user.
The final variable considered in this study was
the slab thickness. Slab thicknesses of 5.5 in, 6.0 in,
6.5 in, 7.0 in, and 7.5 in were considered. The concrete
was assumed to be of normal weight (145 pcf). The load
that the concrete exerts on the floor has been computed as
a function of the slab thickness taking into account the
metal deck geometry (9), and is given in Table 4.2.
Table 4.2 Concrete Weight as a Function
of Slab Thickness and Metal Deck Geometry
(Normal Weight Concrete)
Slab Thickness (in.) 5.5 6.0 6.5 7.0 7.5
Concrete Wt. (psf) 40 46 52 58 64
26
Values similar to those of Table 4.2 can be
obtained from the term in the paranthesis in Eq.(3.1).
However, this term predicts the weight of concrete of a
solid slab, and these values are therefore somewhat
higher. Despite this, Eq.(3.1) is still valid for
computing the w-load if the term in parenthesis is
subtituted by one of the values given in Table 4.2,
depending on which slab thictaiess is being considered.
To make the total loading more realistic, some the
of the most common dead loads a floor structure can be
subjected to during construction and its service life were
also included in the model. Typically assumed values for
these loads were chosen as given below:
Construction Load: 5 psf
Carpets : 5 psf
Ceiling Load : 10 psf
Partitions : 20 psf
Total Load : 40 psf
This extra dead load is incorporated in Eq.(3.1) by the
"DL" term.
Given all of the information needed to compute the
w loads form Eq.(3.1), a set of loads were prepared for
the different floor structures of the study. These are
given in Tables 4.3(a), 4.3(b), and 4.3(c).
Table 4.3(a) w Loads (lbs/ft) for Floor Structure
(5 ft. Beam Spacing)
"I" Value Slab Thickness (in. )
(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5
200 421. 0316 451. 0316 481. 0316 511. 0316 541. 0316
600 449. 6923 479. 6923 509. 6923 539. 6923 569. 6923
1000 468. 6872 498. 6872 528. 6872 558. 6872 588. 6872
2000 495. 2852 525. 2852 555. 2852 585. 2852 615. 2852
4000 540. 7148 570. 7148 600. 7148 630. 7148 660. 7148
10000 552. 5720 582. 5720 612. 5720 642. 5720 672. 5720
20000 626. 5388 656. 5388 686. 5388 716. 5388 746. 5388
Table 4.3(b) w Loads (lbs/ft) for Floor Structure
(10 ft. Beam spacing)
"I" Value Slab Thickness (in.)
(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5
200 821. 0316 881. 0316 941. 0316 1001. 0316 1061. 0316
600 849. 6923 909. 6923 969. 6923 1029. 6923 1089. 6923
1000 868. 6872 928. 6872 988. 6872 1048. 6872 1108. 6872
2000 895. 2852 955. 2852 1015. 2852 1075. 2852 1135. 2852
4000 940. 7148 1000. 7148 1060. 7148 1120. 7148 1180. 7148
10000 952. 5720 1012. 5720 1072. 5720 1132. 5720 1192. 5720
20000 1026. 5388 1086. 5388 1146. 5388 1206. 5388 1266. 5388
28
Table 4.3(c) w Loads (lbs/ft) for Floor Structure
(15 ft. Beam Spacing)
"I" Value Slab Thickness (in.)
(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5
200 1221. 0316 1311. 0316 1401. 0316 1491. 0316 1581. 0316
600 1249. 6923 1339. 6923 1429. 6923 1519. 6923 1609. 6923
1000 1268. 6872 1358. 6872 1448. 6872 1538. 6872 1628. 6872
2000 1295. 2852 1385. 2852 1475. 2852 1565. 2852 1655. 2852
4000 1340. 7148 1430. 7148 1520. 7148 1610. 7148 1700. 7148
10000 1352. 5720 1442. 5720 1532. 5720 1622. 5720 1712. 5720
20000 1426. 5388 1516. 5388 1606. 5388 1696. 5388 1786. 5388
Given the different possible combinations of
loading and structural arrangements, exactly 1785
different floor compositions were created and analyzed by
program GRIDS. The end result is the set of design curves
that is presented in Appendix A.
29
Figure 5 Floor Structure Computer Model
CHAPTER 5
RESULTS AND DESIGN RECOMMENDATIONS
From the computer analysis performed for the range
of floor structures considered, the first conclusion that
can be drawn is that all of the structures converged to an
equilibrium position. As a result, a leveled slab surface
could be assured in all cases.
The rate of convergence was found to be high, and
only three to five iterations were required to achieve
converged deflection values for up to four significant
figures. In general, the heavier the initial load and the
longer the span length of the floor beams, the more
iterations were needed to arrive at an equilibrium
position. However, only a two iteration difference was
found between the lightest and heaviest load as well as
for the shortest and longest span considered in the
analysis. Therefore, under typical dead loads and span
lengths, only three to five relevelings of the slab surfa
30
31
ce will be needed in order -to reach an equilibrium
position.
Probably the most important finding of this study
is that only a maximum increase of about 3% was found to
occur for the maximum moment and deflection values from
the initial to the equilibrium state of the structure. The
3% increase was found to apply for maximum dead load
deflections of about 3 in. and below, as well as for
maximum dead load moments of about 4000 kip-in. and below,
for groups having moments of inertia less than 4000 in4,
and 9000 kip-in for groups having moments of inertia above
4000 in4. Converged deflection and moment values falling
above these limits were found to have greater than 3%
increases when compared to the initial values. However,
these increases rarely exceeded 10%. In the design curves,
this can be observed by the larger discrepancy that exists
between initial and converged values after the approximate
limits have been reached. Before reaching these limits,
the initial and converged data points almost coincide at a
same point, since due to graph accuracy limitations, the
maximum 3% difference is almost undetectable.
The 3 in. limit on maximum dead load deflection
seems to be of no concern for typical span lengths, since
deflections approaching this limit will have violated
32
design values during the first placement of concrete and
the slab releveling procedure therefore would be
meaningless. On the other hand, it may be possible that
the dead load moment capacity of a beam exceeds the limits
for the maximum dead load moment given above. If this is
the case, the design curves may provide the projected
increase above 3% that the beam will have after
termination of the slab releveling procedure.
Since the increases in moment and deflection
values due to slab releveling were found to be rather
small, the assumption that the system is perfectly elastic
(made during the development of the floor analytical model
in Chapter 2) will not be violated if the system was
within the elastic range before the slab releveling is
performed. Also, due to the small increases involved, slab
releveling should be considered a feasible alternative to
achieve a leveled slab surface, if the system is not
overstressed to begin with. This is also correct if the
system just barely complies with maximum design values for
which the associated small increases become significantly
large to cause the system to violate maximum design values
after the slab releveling is terminated.
During slab releveling, the maximum moments were
found to occur at the girder midspans for the short floor
33
beam spans (40 ft. and below), but as the spans were
increased, a critical span length was reached where the
maximum moment was found to occur simultaneously at the
girder and the floor beam midspans. As the span length
continued to increase, the maximum moments occurred at the
floor beam midspans only. The critical span length was
found to be independent of the loading conditions and
influenced primarily by the geometrical arrangement of the
structural members within the bay.
The maximum deflection values were always found to
occur at the interior floor beam midspans. These were
measured with respect to the beam supports, in order to
comply with the assumption that the beams were simply
supported at the girders. Since the supports will deflect
along with the girders, the total deflection calculated
with respect to the true simple supports (column supports)
would have to be modified to account for the simply
supported beam assumption. Equation (5.1) was used to
perform the modification, and Fig. 6 illustrates how this
modification was made.
Db = Dt - Dg (5.1)
34
Column Support Level
Figure 6 Total Interior Floor Beam Deflection
The maximum floor beam deflections calculated by Eg.(5.1)
are the ones used in the design curves.
Using Eg (5.1) actually decouples the girder and
beam deflections, even though this is done following the
analysis considering the girder iteration. Therefore,
using Eq.(2.1), which relates the girder and beam moments
of inertia, should have no effect on the validity of the
design curves which consider the floor beams only. If Eq
(5.1) was not used, and the deflections of the beams were
measured with respect to the column supports, the girder
iteration would have to be considered, and Eq (2.1) should
not be used. Instead, the girders should be designed using
standard procedures each time the characteristics given in
Chapter 4 are changed. It is obvious that the amount of
time involved in producing each floor composition would
35
have increased substantially, to the point of making this
study impractical if the girders had to be redesigned each
time the floor composition was varied. This is the reason
why Eg (2.1) and Eg (5.1) had to be developed.
Using Eg.(5.1) also allows the perimeter floor
beams to be incorporated into the design curves, since the
only difference between the deflections of the interior
and perimeter beams is the value of the support deflection
of the interior beams.
Finally, the results have been used to develop the
following recommendations for achieving a leveled slab
surface:
1. If the floor structure has already been built and
the first placement of concrete has been made,
the midspan deflections of the structural members
must be checked. If the midspan deflections are
within 3% of the maximum allowable value, then
other alternative to achieve a leveled slab sur
face should be used (such as the ones given in
Chapter 1). Otherwise, the slab releveling proce
dure can be used without violated allowable de
flection values. Next, maximum dead load moments
must be checked. Enter the corresponding design
curve with the beam "I" value and the computed
36
maximum moment. If the actual maximum moment
falls above the point where the increase on the
maximum moment due to slab releveling is above
3%, compute the actual increase by the following
formula:
1% = (Cv - Iv) x 100 / Cv (5.2)
Where:
Cv = Converged Data Point Value.
Iv <= Initial Data Point Value.
If the increase obtained from Eg (5.2) is more
than the percentage difference between the actual
maximum moment and the maximum allowable dead
load moment, then the slab releveling procedure
must not be used and other alternatives must be
considered. Otherwise, using the releveling pro
cedure should not violate maximum allowable dead
load moment values. On the other hand, if the ac
tual moment value falls below the point where in
creases under 3% are predicted, slab releveling
can also be used, provided that the percentage
difference between the actual and allowable mom
ents is not below 3%.
must be equal). Then a region between the "I"-va-
lue curves can be chosen and floor beam sections
within the region can be picked. The coorespon-
ding maximum moment and deflection values can be
determined from the curves and checked to see if
they violate established design values. If the
chosen sections violate the design values, then
the beam span can be changed or a new section can
be chosen until all beams are adequate. The sec
tions can also be checked for the slab releveling
procedure, along with the standard checks an the
deflection and moment values. The sections that
are chosen must also be tested agaist live load
moments and deflections before the girders are
designed. Once the design of the girders is com
pleted, the whole floor structure can be reche-
cked with program GRIDS for the final design.
When using the design curves to see if the slab
releveling procedure is a feasible alternative for the
floor structure, it is important to recognize that these
curves were developed on the basis of specific loading
conditions as given by the numerical values of the
different w-loads (see Tables 4.3 (a), (b), and (c) ). If
the actual w-load is different from the assumed value that
39
pertains to the case under consideration, then the nature
of this difference will define the applicability ot the
curves. If the actual value is lower, then the resulting
structure may be on the conservative side. If the actual
value is higher, then the resulting structure may be
unconservative. If the values differ significantly in any
way, the use of the design curves is not recommended.
However, program GRIDS will still provide the means of
testing the structure for the feasibilty of the slab
releveling procedure, since it allows the user to use the
actual loading conditions and the actual properties of the
structural members.
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 Summary
Slab releveling is one of the commonly used
procedures for floor structures for achieving a leveled
slab surface. The procedure is iterative in nature and
consists of adding concrete to level the uneven surface
caused by the deflecting floor. This is continued until an
equilibrium position is reached, and further deflection
increases are negligible.
The purpose of this study was to investigate the
strength and behavior of the floor beams when subjected to
the load and deflection increments caused by the slab
releveling procedure.
An analytical model was developed for the floor
and the loading, and the model configuration was varied to
accommodate many structural configurations. The analyses
were carried out by a computer program that was developed
40
41
for this specific purpose. The results of the analysis are
given in the form of a set of design curves.
Finally, from the results of the study,
recommendations are given for achieving a leveled slab
surface.
6.2 Conclusions
The following conclusions can be drawn:
1. All structural configurations converged to an
equilibrium position, which assures the possibi
lity of obtaining a leveled slab surface.
2. The convergence rate was found to be good, and
only 3 to 5 iterations were needed to reach an
equilibrium position.
3. Only a maximum increase of 3% was found for the
dead load deflections and moments when the equi
librium position was reached. This was true for
deflection values of 3 in. and less, and for mo
ment values of about 4000 kip-in for moment of
inertia of 4000 in.4 and below, and 9000 kip-in
for moments of inertia above 4000 in.4. After
reaching these approximate limits, the percentage
42
increase of the moment values can be predicted by
using the design curves.
Due to the small increments in load and defle
ction, overstressing of the structure will rarely
occur, and the slab releveling procedure is
therefore a feasible approach to obtaining a le
veled slab surface.
APPENDIX A
DESIGN CURVES
Included in this appendix is the set of design
curves developed from the results of the computer analysis
of the different floor structure compositions considered
in this study.
The curves define the relationship between the
floor beam stiffness and the maximum dead load moments and
deflections. Although a modulus of elasticity of 30,000
ksi was used in the analysis, a value of unity (E = 1.0)
was used to construct the curves. Therefore, only the
moment of inertia and the span length are needed to enter
the curves. However, these values must be entered in
inches since the maximum dead load momemts and deflections
are in Lbs.-in. and in. respectively. There are 15 sets of
curves, each being defined by the beam spacing and slab
thickness given on the upper right corner of the curve
sets. Each set consists of 4 graphs, which include the
curves for the different "I"-value groups. The solid
43
44
curves represent the behavior of the floor beans after the
first placement of concrete, and the dashed curves
represent the behavior of the beams after an equilibrium
position has been reached.
Due to the assumption that the system is perfectly
elastic, the curves are asymptotic to the vertical axis.
In reality, the curves should have a cutoff point where
the plastic moment is reached for the maximum moment
curves, and at the corresponding deflection value for the
maximum deflection curves. To avoid any confusion, the
user is adviced to draw a cutoff line at the maximum
allowable dead load moments and deflections. In general,
the curves are asymptotic to the axes as long as the
structure is within the elastic range.
Finally, it is important to understand the limitations
and applicability of these curves. Therefore, the user is
advised to read Chapters 4 and 5 before using these
curves.
9.00
5.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
o o i o jz
o c
6.00
a a> CD O
3.00 E u
E X a 2
0.00 0.00 3.00
Stiffness (El/L) Bea
Figure A1 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=5.5 in.; Sb=5.0 ft.)
9.00 - o o o o
o o 9o i o
i o | o oJ o
5.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
CM
6.00 -
o 3.00 -
0.00 0.00 5.00 10.00 15.00
Beam Stiffness 20.00
(El A) 25.00 30.00
Figure A2 Maximum Deflection vs Beam Stiffness C1=2000 to 20000 in.*; Ts=5.5 in.; Sb=5.0 ft.)
9.00 -i
CD * * O .
5.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
x 6.00 - o
o o o
o o
<v> CO 3.00 -o o
<? CM
X
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A3 Maximum Moment vs Beam Stiffness CI=200 to 2000 in.4*-; Ts=5.5 in.; Sb=5.0 ft.)
25.00 -i
20.00 -j * O
5.5 in. Slab 5.0 ft. Beam Sp.
Initial Value o Converged Value
M5.00 -
10.00 -
5.00 h
0.00 0.00
M i i i i 5.00 10.00
Beam
i M 11 1 5.00
Stiffness
i M 11 11 M 20.00
(ei A) I I II | I I M I J 1 I 1 | 25.00 30.00
Figure A4 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.1*-; Ts=5.5 in.; Sb=5.0 ft.)
9.00 -
o o
o o 6.0 in. Slab
5.0 ft. Beam Sp. * Initial Value o Converged Value
o o CM
O _ O Q O /n T
© — I c
6.00 -
O " 3.00 -
x
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A5 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.0 in.; Sb=5.0 ft,)
o o o o eg
o o o
9.00 -
5.0 ft. Beam Sp. * Initial Value o Con-verged Value
6.00 -
a 3.00 -
x
0.00 0.00 5.00 10.00 15.00 20.00
Beam Stiffness (El/L) 25.00 30.00
Figure A6 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=5.0 ft.)
9.00
to * * o
6.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o x 6.00 -
o o
o JL 9 to
_Q
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A7 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.t-; Ts=6.0 in.; Sb=5.0 ft.)
25.00
^ 20.00 * o
o o o
? ° . CSJ :Q JL
6.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
o o o o
CO
10.00 o Q Ti
cs O O CM
5.00
0.00 0.00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A8 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=5.0 ft.)
Ul to
9.00 -
CO CD J= (J CZ
6.00 -
c o CJ w CJ O
3.00 -E
E X o
0.00 0.00
6.5 in. Slab 5.0 ft. Beam Sp.
Initial Value o Converged Value
Beam 3.00
Stiffness (EI/L) 6.00
Figure A9 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.5 in.; Sb=5.0 ft.)
•9.00 -
6.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
o o o o
o 1 o -.o t o ?01j O 1 CM Td-
o o
c
6.00 -
Q 3.00 -
x
o.oo 4t-.0.00 5.00 10.00 15.00 20.00 25.00 30.00
Beam Stiffness (El/L)
Figure AlO Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=6.5 in.; Sb=5.0 ft.)
9.00
CD 6.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
O o o o x 6.00
c:
o o o
3.00 £Z a>
x o
0.00 6.00 0.00 3.00
Beam Stiffness (El/L)
Figure All Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=6.5 in.; Sb=5.0 ft.)
25.00
o o o
• 8 il
6.5 in. Slob 5.0 ft. Beam Sp. * Initial Value o Converged Value
* 20.00 * o
o o o o
>15.00
_Q
10.00
o o
5.00
x
0.00 25.00 30.00 0.00 5.00 15.00 20.00 10.00
Beam Stiffness (El/L)
Figure A12 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.5 in.; Sb=5.0 ft.)
9.00 -
CO CD
sz o
6.00 -
c o o _QJ
a.i 3.00 -
E Z3
E X o
0.00 0.00
7.0 in. Slob 5.0 ft. Beam Sp. * Initial Value o Converged Value
Beam 3.00
Stiffness (EL/L) 6.00
Figure A13 Maximum Deflection vs. Beam Stiffness CI=200 to 2000 in.*; Ts=7.0 in.; Sb=5.0 ft.)
25.00
^ 20.00 * o
7.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
o o o >15.00
o o
9 5 10.00 o o CM
5.00
X
0.00 0.00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A14 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4-; Ts=7.0 in,; Sb=5.0 ft.)
9.00 n
to * * o
X 6.00
d * I (A -O
c <u E o
3.00 -
x a
7.0 in. Slab 5.0 ft. Beam Sp.
Initial Value o Converged Value
0.00 6.00 0.00
t r 3.00
Beam Stiffness (El/L)
Figure A15 Maximum Moment vs. Beam Stiffness CI=200 to 2000 in.1*-; Ts=7.0 in.; Sb=5.0 ft.)
o o o
9.00 - o o a o
Oa o o O 1 o CM , Tj-
-M Ji 7.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
to <u
6.00 -c o a <D QJ O
3.00 -E
£ X a
0.00 0.00 5.00 10.00 30.00 15.00 20.00 25.00
Beam Stiffness (El/L)
Figure A16 Maximum Moment vs. Beam Stiffness CI=2000 to 20000 in.*; Ts=7.0 in.; Sb=5.0 ft.)
o o o CM
O O O
to 7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value
CM to
6.00 -
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A17 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=5.0 ft.)
9.00 -o o o o
o a 7.5 in. Slab
5.0 ft. Beam Sp. * Initial Value o Converged Value
6.00 -
a 3.00 -
x
0.00 5.00 10.00 Beam Stiffness (El/L)
15.00 20.00 25.00 30.00
Figure A Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.5 in.; Sb=5.0 ft.)
9.00 n
7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o
o © o . oj
x 6.00 -
o o o
jD o
3.00 -
X
0.00 3.00 Beam Stiffness (El/L)
6.00
Figure A19 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=5.0 ft.)
25.00
^ 20.00 * o
7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o o o
9 o
x
>15.00 t=
CO -O
o o
9 ° 10.00
o o o CM
CD
5.00
X
0.00 0.00 5.00 10.00 25.00 30.00 15.00 20.00
Beam Stiffness (El/L)
Figure A20 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.5 in.; Sb=5.0 ft.)
9.00 -
o o o
i 8 CM
5.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
to
9 -!!•
6.00 -
O 3.00 -
0.00 0.00 3.00
Stiffness (El/L) 6.00
Bea
Figure A21 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=5.5 in.; Sb=10.0 ft.)
3.00 -
to CD
SZ o c
6.00 -
c: Q
O _QJ
O 3.00 -
£
E "x o
5.5 in. Slob
* Initial Value o Converged Value
0.00 0.00 5.00 10,00 15.00
Beam Stiffness
11 pi 11 20.00
(El/L)
I fl I 1 I I I I 25.00
T-n 30.00
Figure A22 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=5.5 in.; Sb=10.0 ft.)
I 9.00 -i i 0 L ° CD o
5.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
CD * * o o
9 £ x 6.00
o o , o 1 CO
9 o i ° ' CM
3.00
0.00 6.00 3.00
Beam Stiffness (El/L) o.oo
Figure A23 Maximum Moment vs. Beam Stiffness (1 = 200 to 2000 in,1*-; Ts=5.5 in.; Sb=J0.0 ft.)
-j
<?
25.00 o o o o . CN o
o
o ** 20.00 * o
5.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
>15.00
CO a o
<?c3 10.00
5.00
x
0.00 0.00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A24 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=5.5 in.; Sb=10.0 ft.)
o o o
o o o
CO (f> II
CM 6.0 in. Slob 10.0 ft. Beam Sp. * Initial Value o Converged Value
6.00 -
tr
a 3.00 -
ZJ
0.00 r* i—i— ii—r P i n=* 3.00
Beam Stiffness (El/L) 0.00
6.00
Figure A25 Maximum Deflection vs. Beam Stiffness (1 = 200 to 2000 in.f-; Ts=G,0 in.; Sb=10.0 ft.)
I 1
o o o
o o
o o o 9.00 - o
CN
6.0 in. Slab 10.0 ft. Beam Sp. • Initial Value o Converged Value
6.00 -
a 3.00 -
0.00 0.00 5.00 10.00 15.00 20.00
Stiffness (El/L) 25.00 30.00
Bea
Figure A26 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=10.Q ft.)
o o 9.00
CSJ
CO * * O
6.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
o o
x 6.00 9 o i ° 1 to 1 II *i —
<?o 1 O . CN
3.00
0.00 3.00 Beam Stiffness (El/L)
6.00
Figure A27 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.0 in.; Sb=10.0 ft,)
25.00 o o o
o o
20.00 * o
6.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value o o o >15.00
o o CM 10.00
5.00
0.00 0.00 5.00 10.00
Beam Stiffness ' (El/L) 15.00 20.00 25.00 30.00
\
Figure A28 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.f-; Ts=6.0 in.; Sb=10.0 ft.)
9
o o o
o o o CM
O o to 9 — I Ji.
6.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
CO
6.00 -
£Z
Q 3.00 -
x
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A29 Maximum Deflection vs. Beam Stiffness (1 = 200 to 2000 in.*-; Ts=6.5 in.; Sb=10.0 ft.)
-4 U
o o o o o
o o o o o o CM _ -51-
9 A 6.5 in. Slob 10.0 ft. Beam Sp. • Initial Value o Converged Value
6.00
a 3.00
x
0.00 0,00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A3 0 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in>; Ts=6.5 in.; Sb=10.0 ft.)
9.00 -i
o o o
CO 6.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
O
** o l o , CO
x 6.00 -
;1 —
IS -Q
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A31 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.5 in.; Sb=10.0 ft.)
25.00 o Q O Y O I O
q> JJ_
o o CM
6.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
* 20.00 * o
o Q O T r-s x
>15.00
CO _Q O o o CSI 10.00
5.00
x
0.00 0.00 5.00 10.00 15.00 20.00
Beam Stiffness (El/L) 25.00 30.00
Figure A32 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.t-; Ts=6.5 in.; Sb=10.0 ft.)
9.00 - o o o
o o CM
O o CO
o o CM
<? JL 7.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
JL CO a)
6.00
c o u (D m— CD o
3.00 -E
E X a
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A33 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.0 in.; Sb=10.0 ft.)
© o 1 § H o
'9.00 - o 0*0 O T O CM I Tf
> O 1 CM
-i 7.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
to
6.00 -
o 3.00 -
x
0.00 0.00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A34 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=7.0 in.; Sb=10.0 ft.)
9.00 -i o
o CM
?8 t o CO
7.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
O ( § \ j J!
x 6.00 -
o o
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A35 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.0 in.; Sb=10.0 ft.)
25.00 o o o o
9 R o
^ 20.00 # o
7.0 in. Slab 10.0 ft. Beom Sp. * Initial Value o Converged Value
« ° 9 o , o x
>15.00
o o o CN
0.00
5.00
0.00 0.00 5.00 10.00
Beam Stiffness (El/L) 15.00 20.00 25.00 30.00
Figure A36 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.*-; Ts=7.0 in.; Sb=10.0 ft.)
9.00 -
7.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
to o o o o to CN
9 ~ 6.00 -
o 3.00 -
x
0.00 H— 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A37 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=10.0 ft.)
9.00 - o o
o o o o
o o
<?CM o o
7.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
6.00 -
O 3.00 -
x
0.00 0.00 5.00 10.00 15.00 20.00
Beam Stiffness (El/L) 25.00 30.00
Figure A38 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.f; Ts=7.5 in.; Sb=10.0 ft.)
9.00 -i o o
Q O » I *
CD 7.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value
i JL
O i to
X 6.00 -
00 _Q
3.00 -
0.00 0.00 6.00 3.00
Beam Stiffness (El/L)
Figure A39 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=10,0 ft.)
25.00
o o CM
O
: V
^ 20.00 * o 10.0 ft. Beam Sp.
* Initial Value o Converged Value
9 o
x
>15.00
o o o CN 10.00
5.00
0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00
Beam Stiffness (El/L)
Figure A40 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in,*; Ts=7.5 in.; Sb=10,0 ft.)*
9.00 -
o o o
o o o
o o CM 15.0 ft. Beam Sp.
* Initial Value o Converged Value
CO CD _c o d
6.00 -
c: o CJ QJ
a> Q
3.00 -E ZD .i "x a
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A41 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*;Ts=5.5 in,; Sb=15.0 ft.)
o o o o CNI
o o o o
o o o 9.00 -
5.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
6.00
O 3.00
x
0.00 0.00 5.00 10.00
Beam Stiffness (El/L) 15.00 20.00 25.00 30.00
Figure A43 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=5.5 in.; Sb=15.0 ft.)
9.00 n o o
o Q O T /—i
V — 16 to
* * o
5.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value x 6.00 - 9 o
i ° 1 CN
CO -Q
3.00 -
0.00 0.00 3.00
Beam Stiffness (Ef/L) 6.00
Figure A43 Maximum Moment vs. Beam Stiffness CI=200 to 2000 in.*; Ts=5.5 in.; Sb=15.0 ft.)
25.00 o o o o CM
O O o
^ 20.00 * O
5.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
o
>15.00
10.00
J 5.00
x
0.00 0,00 10.00 20.00
Beam Stiffness (El/L) 30.00 40.00
Figure A44 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.1*-; Ts=5.5 in.; Sb=15.0 ft.)
CO 09
9.00 -
6.0 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
o o o
o o CM
9 -c
6.00 -
3.00 -
x
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A45 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.0 in.; Sb=15.0 ft.)
' 9.00 o o o o
o o o CM
o o o
o o 6.0 in. Slab
15.0 ft. Beam Sp. * Initial Value o Converged Value
9
6.00 -
O 3.00 -
x
0.00 0.00 5.00 10.00 15.00
Beam Stiffness (El/L) 20.00 25.00 30.00
Figure A46 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=15.0 ft.)
9.00 o o o
o o o C\|
o
CO 6.0 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
O o
x 6.00 -
3.00 -
x
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A47 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.0 in.; Sb=15.0 ft.)
u> H
25.00 o o o o CM
O
O _ o <? o * "51-
^ 20.00 * o
6.0 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
>15.00
o o o CM
to -Q
10.00
0.00 0.00 10.00 20.00
Beam Stiffness (El/L) 30.00 40.00
Figure A48 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=15.0 ft.)
9.00 -
o o o o o o CM
O o CN
O „ O 9 co 6.5 in. Slab
15.0 ft. Beam Sp. * Initial Value o Converged Value
co
9 ~
6.00
a
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A49 Maximum Deflection vs. Beam Stiffness [1=200 to 2000 in.*; Ts=6.5 in.; Sb=15.0 ft.)
9.00 - o o o CN
O O O o o o
o o
6.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
CO a> _ir o c
6.00 -
u a) CD o
3.00 -E 13
E X o 2
0.00 0.00 5.00 10.00 15.00
Beam Stiffness 20.00
(El/L) 25.00 30.00
Figure A50 Maximum Deflection ys, Beam Stiffpess (1=2000 to 20000 in.*; Ts=6.5 in.; Sb=15.0 ft.)
9.00 o o
o CM
CO 6.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
* O o
o x 6.00
CO
3.00
0.00 0.00 3.00
Beam Stiffness .(El/L) 6.00
Figure A51 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.5 in.; Sb=15.0 ft.)
25.00 o o o o
o o o o CM
o
^ 20.00 * o
6.5 in. Slob 15.0 ft. Beam Sp. * Initial Value o Converged Value
>15.00
10.00
5.00 -
x
0.00 0.00 10.00 20.00
Beam Stiffness (El/L) 30.00 40.00
Figure A52 Maximum Moment vs, Beam Stiffness (1=2000 vs. 20000 in.*; Ts=6.5 in.; Sb=15.0 ft.)
9.00 -
CO CD
sz o d
6.00 -
c: o
u CD
"cd Q
3.00 -
E
E X o
0.00 0.00
Beam 3.00
Stiffness
7.0 in. Slab 15.0 ft. Beam Sp-• Initial Value o Converged Value
(El/L) 6.00
Figure A53 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.0 in.; Sb=15.0 ft.)
9:oo -
6.00 -
3.00 -
7.0 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
0.00 0.00 5.00 10.00
Beam Stiffness (El/L)
T !"l I j TT rTI I I I I iT I A I M 1 I I I I I l i I I I I 15.00 20.00 25.00 30.00
Figure A54 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.0 in.; Sb=15.0 ft.)
9.00 -i
o o o
o o CM
9 - 7.0 in. Slob 15.0 ft. Beam Sp. * Initial Volue o Converged Value
O
x 6.00 -
to JO
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A55 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.t-; Ts=7.0 in.; Sb=15,0 ft.)
25.00 o o o
o o o o
o o
92
^ 20.00 * o
7.0 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
x
>15.00
10.00
5.00
x
0.00 0.00 10.00 20.00 40.00 30.00
Beam Stiffness (El/L)
Figure A56 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.0 in.; Sb=15.0 ft.)
o o o CM
o o o CO
9.00 - o CM
o
7.5 In. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
to CD
_£=
O c
6.00
c; o
u QJ CD o
3.00
E c X O
0.00 6.00 3.00 0.00
Beam Stiffness (El/L)
Figure A57 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5in.; Sb=15.0 ft.)
9.00 -
CO 0) _C u a.
6.00 -t= o
<_> _QJ M— QJ o
3.00 -
E 13 E X O
7.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
0.00 7^ I I I I [
's!6o 10.00 15.00 20.00
Beam Stiffness (El/L)
11 In I I I |
25.00 30.00
Figure A58 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.5 in.; Sb=15.0 ft.)
o to
i q
9.00 o o o CN
O O
9 —
7.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
o
x 6.00
3.00 -
0.00 0.00 3.00
Beam Stiffness (El/L) 6.00
Figure A59 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=15.0 ft.)
25.00
o o o o o CN
CD O O O O O CM Q
o
V 20.00 * o
7.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value
x
>15.00 c:
CO -O
10.00
a>
5.00
0.00 0.00 10.00 20.00 30.00 40.00
Beam Stiffness (El/L)
Figure A60 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.5 in.; Sb=15.0 ft.)
APPENDIX B
PROGRAM GRIDS
B.l Introduction
Program GRIDS is a finite element computer program
which performs a structural analysis on a floor system
whose members are modeled by grid elements.
The program was written for an IBM-XT
microcomputer, using a Microsoft FORTRAN compiler. Care
was taken to assure maximum portability of the program.
Therefore, the program should run on any IBM compatible
microcomputer as well as any mini or main frame computer.
Comments are provided throughout the program to make it
more readable and understandable to the user.
Program GRIDS uses an iterative procedure which
accounts for the increase in load and deflection due to
the ponding of the concrete during the slab releveling
procedure. The purpose of the iterative procedure is to
investigate the convergence of the structural system to an
equilibrium position so that a leveled slab surface can be
achieved.
105
106
The program computes the linear and rotational
nodal displacements as well as the bending and torsional
end moments of all grid elements of the model under
consideration, each time an iteration is completed. In
this fashion, initial and converged values for the force
and deflection values can be obtained.
Finally, it must be understood that the solutions
obtained from program GRIDS are not guaranteed to be the
"optimum" solutions for the problem under consideration.
In finite element analyses, the solutions of a given
problem are only as good as the finite element model
representing such a problem.
107
B.2 User Manual
Program GRIDS prompts to the terminal screen for
the name of the input and output file. The input file
under the name given by the user must be in existence at a
memory location readibly available for the executable
program. The output file is created by the program under
the name given by the user.
The input file must be written in free format
style, with all data entries separated by commas. All data
entries must be filled. If a data entry is not relevant
for a particular problem to be analyzed, then a "zero"
must be entered at that location.
The input file must be created in the following
format:
PART I: Problem Identification Number. Any integer
may be entered
PART II:General Parameters for the Problem.
Enter in order: Number of nodes, Number
of elements, Number of element property
types, Number of concentrated loads, Num
ber of Iterations, Concrete Unit Weight.
108
PART III:Structure Geometry.
Enter the X and Y nodal coordinates. Use
one line per node starting with the
coordinates corresponding to the first
node and continuing in sequence until
the last nodal coordinates are reached.
The node number is automatically assig
ned by the program.
Note: A right handed coordinate system
must be used.
Maximum number of nodes allowed: 100
PART IV: Element Property Types.
Enter : Elastic Modulus, Rigidity Modu
lus, Torsional Constant, Moment of Iner
tia. Enter this information for each
element, starting with the first proper
ty type and proceeding in sequence until
the last property type is reached. The
property type number is automatically
assigned by the program.
Maximum number of property types allo
wed: 100.
109
PART V: Element Connectivity and Distributed Load
Position.
Enter : Node i, Node j, Element Property
Type, DLV^, DLVj, Atw. Enter this
information for each element in the mo
del, starting with the first element and
proceeding in sequence until the last
element is reached. The element number
is automatically assigned by the program.
The i-th and j-th nodal positions must
be assigned from left to right.
DLV^ stands for Distributed Load Value
at the i-th node.
Atw stands for Tributary Area Width of
the element.
z
j
Figure B1 Load Distribution on the Element
110
PART VI:Concentrated Loads.
Enter : Node Number, Load Intensity. Do
this for all concentrated loads applied
to the model. If no concentrated loads
are present, ignore this part. Do not
fill the data entry corresponding to this
part with "zeros".
PART VII:Support Conditions.
Enter : Support in the 2-direction, rota
tional support in the x-direction and y-
direction. Enter this information for
each node in the model, starting with the
first node and proceeding in sequence un
til the last node is reached. The program
automatically assigns the node number.
Support Coding:
Support = 0 : Mode is free to move in the
specified direction.
Support = 1 : Node is restricted to move
in the specified direction.
Ill
PART VIII:End of File Flag.
Enter a "zero" followed by a comma to
signal the end of the file. Program GRIDS
can handle more than one input file in a
single run. To achieve this, replace the
"zero" with the problem ID number of the
next file to be run. Continue this proce
dure until the last file is typed in. The
"zero" must still appear on the last line
of the last file to be run.
112
B.3 Program Listing c********************************************************************** C*************************** PROGRAM GRIDS ***************************
C PROGRAMER: CARLOS E. PEflA RAMOS C UNIVERSITY OF ARIZONA C CIVIL ENGINEERING DEPT.
C THIS IS A FINITE ELSMENT PROGRAM THAT PERFORMS A STRUCTURAL ANALYSIS C ON A FLOOR SYSTEM WHOSE MEMBERS ARE MODELED BY GRID ELEMENTS. C THE PROGRAM HAS AN ITERATIVE PROCEDURE WHICH ACCOUNTS FOR THE INCREA C SE IN LOAD AMD DEFLECTION CAUSED BY PONDING OF THE CONCRETE DECK DU-C RING SLAB RELEVELING PROCEDURES. THE PURPOSE OF THE ITERATIVE PROCE-C DURE IS TO INVESTIGATE THE CONVERGENCE OF THE FLOOR STRUCTURE TO AN C EQUILIBRIUM POSITION SO THAT A LEVELED SLAB SURFACE CAN BE ACHEXVED.
PROGRAM GftlDS IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100)/NODJ(IOO).EMOD(IOO),RG(100), STC(IOO),XMI(100),MAT(100),T(6,6),SL(6,6),SE(6,6),SC(6,6),R(300) , &U(300),C(300),8(300,160),UI(100),UJ(100),BX(2,6),SG(6),SD(6), &ST(2),P(2),SP(6),13(300),UP(100),B(300),ATW(100) CHARACTER*15 IFNAME CHARACTER*15 OFNAME
C WRITE NAME OF INPUT FILE TO THE SCREEN
WRITE(*,10) 10 FORMAT(IX,"NAME OF INPUT FILE :•)
R£AD(*,30) IFNAME
C WRITE NAME OF OUTPUT FILE TO THE SCREEN
WRITE(*,20) 20 FORMAT(IX,'NAME OF OUTPUT FILE :')
READ(*,30) OFNAME 30 FORMAT(A)
C OPEN THE INPUT AND OUTPUT FILES UNDER THE NAMES GIVEN BY THE USER
OPEN(10,FILE-IFNAME) OPEN(11,FILE-OFNAME,STATUS-•NEW')
C OPEN FILES FOR DISPLACEMENT VECTOR STORAGE
OPEN(12,FILE-'CVEC.DAT',STATUS-1NEW•,FORM- • UNFORMATTED•) OPEN(13,FILE-1BVEC.DAT',STATUS-•NEW•,FORM-•UNFORMATTED')
C READ PROBLEM NUMBER
READ(10,*) NPROB IF(NPROB.EQ.O) GO TO 1000 WRITE(11,40) NPROB
40 FORMAT(//,5X,'PROBLEM NO.',13,//)
C ENTER GENERAL PARAMETERS FOR THE PROBLEM
C NODEN - NUMBER OF NODES. C NUMEL - NUMBER OF ELEMENTS. C NMAT - NUMBAR OF ELEMENT PROPERTY TYPES. C NC - NUMBER OF CONCENTRATED LOADS. C NIT - NUMBER OF ITERATIONS. C WC - CONCRETE UNIT WEIGHT.
READ(10,*) NODEN,NUMEL,NMAT,NC,NIT,WC WRITE(11,50) NODEN,NUMEL,NMAT,NC,NIT,WC
50 FORMAT(>//,2IX,'GENERAL PARAMETERS',//,2X,*NO. OF NODES -',13,/, &2X,'NO. OF ELEMS -',I3,/,2X,'NO. OF Ellf TYPS -*,13,/,2X,'NO. OF SCON. LOADS ,I3,/,2X,'NO. OF ITERATIONS -',I3,/,2X, £'CONCRETE UNIT WEIGHT -',F4.3,/)
113 WRITE(11,195) HEQ.HBAND
195 FORMAT(///,10X,•NUMBER OF EQUATIONS «•,14,//,10X,'BANDWIDTH «3,//) WRITE(11,240)
240 FORMAT (////72 ('*') ,/,31(1 *1) , 1 RESULTS 32 (1 *'),/, 72 (•*'),///)
C******************** START ITERATIVE PROCEDURE *******************
DO 500 N-1,NIT IF(N.EQ.l) GO TO 225 DO 215 K-1,NEQ
215 R(K)-0.
C NOTE:IF INITIALIZATION OF THE R(I) VECTOR IS MADE WHEN CONCENTRATED C LOADS ARE PRESENT, THE POSITIONS OF THESE LOADS MUST BE FILLED C AGAIN WITH THE VALUES GIVEN IN SUBROUTINE "INPUT". THIS MUST BE C DONE INSIDE LOOP 500.
C INITIALIZE THE SYSTEM MATRIX OF THE STRUCTURAL MODEL.
225 DO 210 J~1,NEQ DO 200 K«1,MBAND
200 S(J,K)-0. 210 CONTINUE
C CALCULATE THE GLOBAL STIFFNESS MATRIX FOR EACH GRID ELEMENT AND C STORE IT IN THE SYSTEM MATRIX. C MODIFY ALSO THE LOAD VECTOR TO REFLECT DISTRIBUTED LOADING CONDI-C TIONS.
CALL GRID(NUMEL,X,Y,NODI,NODJ,EHOD,RG,TC,MAT,UI,UJ,XMI,T,SC,SL, £SE,S,SP,SQ,R,MBAND,NEQ)
C MODIFY THE SYSTEM MATRIX AND LOAD VECTOR TO REFLECT SUPPORT CONDI-C TIONS.
DO 230 1-1,NEQ IF(IS(I).EQ.O) GO TO 230 S(I,1)»(S(I,1)+1.0)*1.0E+20 R(I)-0.
230 CONTINUE
C CHECK TO SEE IF THE MAXIMUM NUMBER OF NODES ALLIED BY THE PROGRAM C IS NOT EXCEEDED.
IF(NODEN.GT.IOO) GO TO 998
C NUMBER OF SIMULTANEOUS EQUATIONS REPRESENTING THE STRUCTURAL MODEL.
NEQ-0 NEQ-NODEN*3
C INITIALIZE THE PARAMETERS TO BE USED IN THE PROGRAM.
CALL INITIAL(X,Y,NODI,NODJ,MAT,EMOD,RG,TC,XMI,UI,UJiUP,IS,C,R,U, &B,NODEN,NEQ,ATW)
C READ INFORMATION FROM INPUT FILE.
CALL INPUT(NODEN,X,Y,NMAT,EMODfRG,TC,XMX,NUMEL,NODI,NODJ,MAT,UI, «UJ,NC,R,IS,ATH)
C CALCULATE THE BANDWIDTH.
MHAND-0 DO 190 I-1,NUMEL MB»(IABS(NOD(I)-NODJ(1))+l)*3 IF(KB.GT.MBAND) MBAND-MB
190 CONTINUE
c***************** RESULTS OF THE PROBLEM SET FOLLOWS 114
C SOLVE FOR THE MODAL DISPLACEMENTS.
DO 250 I-1,NEQ C(I)-RCI)
250 CONTINUE CALL GEQSOL(NEQ,MBAND,NODEN, S, C) REWIND 12 WRITE(12) (C(I),I-1,NEQ)
C CALCULATE TORQUE AND BENDING MOMENT FOR EACH GRID ELEMENT.
DO 280 I-1,NEQ U(I)-C(I)
2BO CONTINUE CALL GREAC(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,XMI,MAT,UI,UJ,SG,T,SD, &BX,ST,U,P)
C MODIFY DISTRIBUTED LOADING CONDITIONS TO ACCOUNT FOR THE INCREASE C LOAD DUE TO THE SLAB RELEVELING PROCEDURE.
REWIND 12 READ(12) (C(I),I-1,NEQ) IF(N.GT.l) GO TO 310 DO 300 J-l,NODEN UP(J)-C(3*J-2)
. 300 CONTINUE GO TO 325
310 REWIND 13 READ(13) (B(I),1-1,NEQ) DO 320 J-l,NODEN UP(J)-C(3*J-2)-B(3*J-2)
320 CONTINUE 325 DO 330 I-1,NUMEL
UI(I)-UP(NODI(I))*ATW(I)*WC + UI(I) UJ(I)-UP(NODJ(I))*ATW(I)*WC + UJ(I)
330 CONTINUE DO 340 I—1,NEQ BCI)-C(I)
340 CONTINUE REWIND 13 WRITE(13) (B(I),I-1,NEQ)
C SET UI(X)-0. t UJ(I)-0. FOR ALL GIRDER ELEMENTS FOR CONSISTENCY IN C ASSUMED LOADING CONDITIONS. IF GIRDERS ARE ASSUMED TO CARRY ANY C DISTRIBUTED LOADING, THEN UI(I) AND UJ(I) MUST NOT BE INITIALIZED C AND THE FOLLOWING TEN FORTRAN LINES MUST BE DELETED:
UI(5)-0. UJ(5)-0. DO 345 J—10,14 UI(J)-0. UJ(J)-0.
345 CONTINUE UI(19)-0. UJ(19)-0. UI(24)-0. UJ(24)-0.
C THE POSITIONS IN THE ARRAYS UI(I) AND UJ(I) GIVEN ABOVE (5,10 TO C 14,19,24) DEFINE NODAL POSITIONS WITHIN A GIRDER LENGTH. THEREFORE, C IF A MODEL DIFFERENT THAT THE ONE GIVEN IN THE THESIS STUDY IS USED C THESE NODAL POSITIONS MAY CHANGE AND THE NODAL POSITIONS GIVEN C ABOVE MAY HAVE TO BE CHANGED ACCORDING TO THE USER'S OWN MODEL.
500 CONTINUE GO TO 35
C FLAGS
998 HRITE(*,999) 999 FORMAT(//,XX,'MAXIMUM NUMBER OF MODES EXCEEDED.',/,IX,
6'MUST MODIFY THE PROGRAM TO INCREASE CAPABILITY TO HANDLE MORE (NODES')
1000 STOP END
C******************************************************************** C************************** SUBROUTINES *****************************
SUBROUTINE INITIAL(X,Y,NODI,NODJ,HATtEMOD,RG,TC,XHI,UI,UJ,UP(IS, fiC,R,U,B,NODEN,NEQ) IMPLICIT DOUBLE PRECISION (A-H,0-2) DIMENSION X(100),Y(100),NODI(100).NODJ(IOO),MAT(100),EMOD(100) , tRG(lOO),TC(100),XMI(100),UI(100),UJ(100),UP(100),IS(300),C(300), &R(300),U(300),B(300)
C THIS SUBROUTINE INITIALIZES THE PARAMETERS USED UN THE MAIN PROGRAM
DO 60 1-1,NODEN X(I)-0. Y(I)-0. NODI(I)-0. NODJ(I)-0. MAT(I)-0. EMOD(I)"0. RG(I)-0. TC(I)-0. XMI(I)»0. UI(I)-0. UJ(I)-0. UP(I)-0. ATW(I)«0.
60 CONTINUE DO 70 J-1,NEQ IS(J)-0. C(J)»0. R(J)-0. U(J)-0. B(J)-0.
70 CONTINUE RETURN END
SUBROUTINE INPUT(NODEN,X,Y,NMAT,EMOD,RG,TC,XMI,NUMEL,NODI,NODJ, £MAT,UI,UJ,NC,R,IS) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),EMOD(100),RG(100),TC(100),XMI(IOO),
(NODI(100),NODJ(100),MAT(100),UI(100),UJ(100),R(300),IS(300)
C THIS SUBROUTINE READS THE INFORMATION FROM THE INPUT FILE.
C READ NODAL POINT COORDINATES.
WRITE(11,B0) 80 FORMAT(////,21X,'NODAL POINT COORDINATES',//)
WRIT£(11,90) 90 FORMAT(5X,'NODE NO. ' ,9X,'COORDINATES' ,//,20X,'XM3X,'Y»,//)
DO 100 1-1,NODEN READ(10,*) X(X),Y(X) WRITE(11,95) X,X(I),Y(I)
95 FORMAT(8X,I2,2(SX,F8.2)) 100 CONTINUE
C READ ELEMENT PROPERTY TYPES.
WRITE(11,110) 1X0 FORHAT(///,30X,'ELEMENT PROPERTIES',//,IX,'ELTYP',5X,'ELASTIC
SHOD',6X,'RIGID MOD*,8X,'TORSIONAL C.',5X,*H. OF INERTIA',//) DO 120 I-1,NMAT READ(10,*) EHOD(I),RG(I),TC(I),XHI(I)
. WRITE(11,115) I,EMOD(I),RG(I),TC(I),XMI{I) 115 F0RKAT(1X,I2,5X,F15.2,1X,F15.8,3X,F15.8,6X,FB.2,/) 120 CONTINUE
C READ ELEMENT CONNECTIVITY AND DISTRIBUTED LOADING CONDITIONS.
WRITE(11,130) 130 FORMAT(////,3OX,'ELEMENT CONNECTIVITY1,//,IX,'ELEMENT',6X,
S'NQDEI',6X,'NODEJ',6X,'ELTYP',7X,'DLV I1,9X,1DLV J',/) DO 140 I-1,NUMEL READ(10,*) NODI(I),NODtf(I),MAT(I),UI(I),UJ(I),ATW(I) WRITE(11,135) I,NODI(I),NODJ(I),KAT(I),UX(I),UJ(I)
135 FORMAT(IX,15,IX,110,1X,I10,9X,I2,6X,F9.4,6X,F9>4) 140 CONTINUE
C READ CONCENTRATED LOADING CONDITIONS.
IF(NC.EQ.O) GO TO 165 WRITE(11,150)
150 FORMAT(///,30X,'CONCENTRATED LOADING CONDITIONS',//,5X,'NODE &NO. MIX, *Z LOAD',/) DO 160 I-1,NC KEAD(10,*) NODE,ZLOAD K-NODE*3-2 R(K)-R{K)+ZLOAD WRITE(11,155) NODE,ZLOAD
155 FORMAT(4X,I5,12X,E10.2) 160 CONTINUE
C READ SUPPORT CONDITIONS.
165 WRITE(11,170) 170 FORMAT(///,2IX,'SUPPORT CONDITIONS',//,12X,'UNSUPPORTED-O',/,12X
ft' SUPPORTED" 1', //, 10X, ' NODE NO.',3X,'Z SUPP',5X,'ROTX SUPP',5X, S'ROTY SUPP',//) DO 180 J-l,NODEN READ(10,*) IS(3*(J-1)+1),IS(3*(J-l)+2),IS(3*(J—l)+3) WRITE(11,175) J,IS(3*(J-l)+1),IS(3*(J-l)+2),IS(3*(J-l)+3)
175 FORMAT(11X,13,3(9X,12)) RETURN END
117
SUBROUTINE GRID(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,MAT,UI,UJ,XMI,T,SC 6,SL,SE,SP,SQ,R,MBAHD,NEQ) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100),NODJ(100),EMOD(1CIO),RG(IOO), fiTC(lOO),MAT(100),UI(100),UJ(100),XMI(100),T(6,6),SC(6,6),SL(6,6) , 6SE(6,6),8(300,160),SP(6),SQ(6),R(300),ID0F(12)
C THIS SUBROUTINE CALCULATES THE GLOBAL STIFFNESS MATRIX FOR THE GRID C ELEMENTS AND STORES IT IN THE SYSTEM MATRIX. C THE LOAD VECTOR IS MODIFIED TO REFLECT DISTRIBUTED LOADING CONDI-C TIONS.
DO 200 I-1,NUMEL XL-X(N0DJ(I))-X(NODI(X)) YL-Y(NODJ(I))-Y(NODI(I)) EL-SQRT(XL**2+YL**2) COSX-XL/EL SINX-YL/EL
C POPULATE THE TRANSFORMATION MATRIX.
DO 20 J-1,6 DO 10 K-1,6
10 T(J,K)-0. 20 CONTINUE
T(l,l)-1.0 T(4,4)-1.0 T(2,3)-SINX T(5,6)-SINX T(3,2)— SINX T(6,5) — SINX T(2,2)-COSX T(3,3)-C0SX T(5,5)—COSX T(6,6)-C0SX
C POPULATE THE LOCAL STIFFNESS MATRIX.
DO 40 J-1,6 DO 30 K-1,6
30 SL(J,K)-0. 40 CONTINUE
K-MAT(I) SL1-(12.0*EMOD(K)*XMI(K))/(EL**3) SL2-(RG(K)*TC(K))/EL SL3-(4.0*EM0D(K)*XHI(K))/EL SL4-(6.0*EMOD(K)*XMI(K))/(EL**2) SL(1,1 -SL1 SL(4,4 -SL1 SL(1,4 —SL1 SL(4,1 —SL1 SL(2,2 -SL2 SL(5,5 -SL2 SL(5,2 —SL2 SL(2,S —SL2 SL(3,3 -SL3 SL(6,6 -SL3 SL(6,3 -SL3/2.0 SL(3,6 -SL3/2.0 SL(4,3 -SL4 SL(3,4 -SL4 SL(6,4 -SL4 SL(4,6 -SL4 SL(3,1 —SL4 SL(1,3 —SL4 SL(6,1 —SL4 SL(1,6 —SL4
C CALCULATE THE GLOBAL STIFFNESS MATRIX.
DO 70 N-1,6 DO 60 M-1,6 SC(N,H)-0.
60 CONTINUE DO 65 H-1,6 DO 64 ]>1,6
64 SC(N,M) ",SC(N,H)+SL{N,L) *T(L,H) 65 CONTINUE 70 CONTINUE
DO 100 N-1,6 DO 90 M-1,6 SE(N,H)-0.
90 CONTINUE DO 95 M-1,6 DO 94 L-1,6
94 SE(N,M)-SE(N,M)+T(L,N)*SC(L,M) 95 CONTINUE 100 CONTINUE
C STORE THE GLOBAL STIFFNESS MATRIX IN THE SYSTEM MATRIX.
DO 300 J-1,2 NR-3*N0DJ(I)-3 IF(J.EQ.l) NR-3*NODI(I)-3 DO 290 M-1,3 NR-NR+1 IX-(J-1)*3+H DO 260 K-1,2 N9-3*NODJ(I)-3 IF(K.EQ.l) N9-3*NODI(I)-3 DO 270 N-1,3 LL-(K-1)*3+N KK-N9+N+1-NR IF(NK.LE.O) GO TO 270 S(NRfNK)-S{NR,NK)+SE(II,LL)
270 CONTINUE 280 CONTINUE 290 CONTINUE 300 CONTINUE
C MODIFY THE LOAD VECTOR TO REFLECT DISTRIBUTED LOADING CONDITIONS.
DO 140 J-1,6 SP(J)-0.
140 CONTINUE SP(1)—7 »0*UI (I) *EL/20.0 + 3.0*TJJ (I) * EL/20.0 SP(3J«—(UI(I)/20.0 + UJ(I)/30.0)*EL*EL SP(4)—3.0*UI(I)*EL/20.0 + 7.0*UJ(I)*EL/20.0 SP(6)-(UI(I)/30.0 + UJ(I)/20.0)*EL*EL DO 160 K-1,6 SQ(K}»0. DO 150 J-1,6
150 SQ(K)-SQ(X)+T(J,K)*SP(J) 160 CONTINUE
IN-3*NODI(I)-3 JN-3*NODJ(I)-3 R(IN+1)-R(IN+1)+SQ(1) R(IH+2)-R(IN+2)+SQ(2) R(IN+3)-R(IN+3)+SQ(3) R(JN+1)-R(JN+1)+SQ(4) R(JN+2)-R(JN+2)+SQ(5) R(JN+3)-R(JM+3)+SQ(6)
200 CONTINUE RETURN END
SUBROUTINE GEQSOL(NEQ,HBAHD,NODEN, S , C ) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION S(300,160),C(300)
C THIS SUBROUTINE SOLVES FOR THE NODAL DISPLACEMENTS FROM THE SYSTEM C MATRIX AND THE LOAD VECTOR.
WRITE(11i200) 200 FORMAT(////,SOX,'DISPLACEMENTS',//,5X,'NODE NO.',10X,'Z-DISPL'
ft,'X-ROT',8X,'Y-ROT',/) DO 100 I-1,NEQ IK-X DO 90 J»2,MBAND IK-IK+1 CN-S(I,J)/S(I,1) JK-0 DO BO K-J,HBAND JK-JK+1 S(IK,JK)-S(IK,JK)-CN*S(I,K)
80 CONTINUE S(I,J)-CH C(IK)-C(IK)-CN*C(I)
90 CONTINUE C(I)-C(I)/S(I,1)
100 CONTINUE DO 120 N-2,NEQ I-NEQ-N+1 DO 110 K-2,MBAND J-I+K-l C(I)-C(I)-S(X,K)*C(J)
110 CONTINUE 120 CONTINUE
DO 260 K-l,NODEN XC-1.0E-10 IF(ABS(C(3*(K-1)+1).LE.XC) C{3*(K-l)+l)-0. IF(ABS(C(3*(K-l)+2).LE.XC) C(3*(K-1J+2)-0. IF(ABS(C(3*(K-l)+3).LE.XC) C(3*(K-l)+3)-0. WRITE(11,255) K,C(3*(K-1)+1),C(3*(K-l)+2),C(3*(K-l)+3)
255 FORMAT(7X,12,12X,3(E10.4,3X)) 260 CONTINUE
RETURN END
SUBROUTINE GREAC(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,XMI,MAT,UI,UJ,SG, &T,SD,BX,ST,U,P) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100),NODtf(100),EMOD(100),RG(100) ,
HC(IOO),XMI(100),MAT(100),UI(100),UJ(100),SG(6),T(6,6),SD(6),P(2) , ftBX(2,fi),ST(2),U(300)
C THIS SUBROUTINE CALCULATES THE TORQUE AND BENDING MOMENTS OF EACH C GRID ELEMENT ON THE FLOOR STRUCTURE.
WRITE(11,270) 270 FORMAT(///,3OX,'GRID REACTIONS',//,5X,'ELEMENT',12X,'TORQUE',6X,
&'MOMENT(I)',4X,'MOMENT(J)',/)
DO 200 I-1,NUMEL XI>X(N0DJ(I))-X(N0DI(Z)) YI Y(HODJ(IJ)-Y(HODI(I)) EL-SQRT(XL**2+YL**2) cosx-xl/el sihx-yl/el IN-3*NODI(I)-3 JN~3*N0DJ(I)-3 DO 10 K-1,3 MM-IN+K NM-JN+K SGCK)-U(HM) SG(K+3)-U(NM)
10 CONTINUE T(l,l)«1.0 T(4,4)-1.0 T(2,3)-SINX T(5,6)-SINX T(3,2)—SIHX T(6,5)— SIHX T(2,2)-COSX T(3,3)-COSX T(5,5)-COSX T(6,6)-COSX DO 50 J-1,6 SD(J)-0. DO 40 K-1,6
40 SD{J)-SD(J)+T(J,K)*SG(K) 50 CONTINUE
DO 110 H-1,2 XI-1.0 IF(H.EQ.l) XI-0. DO 70 J-1,2 DO 60 K-1,6
60 BX(J,K)-0. 70 CONTINUE
K-MAT(I) BX(1,2)—RG(K) *TC(K)/EL BX(1,5)—BX(1,2) BX(2,1)—(EMOD(K)*XMI(K)*(6*0 - 12.0*XI)/<EJ>*2) BX(2,3)—(EMOD(K)*XHI(K)*(-4•0 + 6.0*XI)/EL BX(2,6)—(EMOD(K)*XHI(K)*(-2.0 + 6.0*XI)/EL BX(2,4)--BX(2,1) DO 90 J-1,2 ST(J)-0. DO 80 K-1,6
80 ST(J)-ST(J)+BX(J,K)*SD(K) 90 CONTINUE
IF(XI.EQ.O) P(1)-ST(2) - (UI(I)/20.0 + UJ(IJ/30.0)*EL*EL IF(XI.EQ.l) P(2)-ST(2) - (UI(I)/30.0 + UJ(I)/20.0)*EL*EL
110 CONTINUE XC-1.0E-5 IF(ABS(ST(1)).LE.XC) ST(l)-0. IF(ABS(P(l)).LE.XC) P(l)-0. IF(ABS(P(2)).LE.XC) P(2}-0. WRITE(11,100) I,ST(1),P(1),P(2)
100 FORM&T(7X,I2,12X,3(E10.4,3X)) 200 CONTINUE
RETURN END
NOTATION
Atw = Tributary Area Width
Cv = Converged Data Point Value
d = Deflection of the Floor Beam
Db = Deflection of the Floor Beam Measured with Respect
the Its Supports
Dg = Deflection of the Floor Beam Supports
DL = Additional Dead Loads (Construction Loads, Parti
tions, etc.)
DLV = Distributed Load Value
Dt .= Total Deflection of the Floor Beams Measured with
Respect to the Column Supports
1% = Percentage Difference Between Initial and Converged
Dead Load Moments
lb = Moment of Inertia of the Floor Beams (in.4)
Ig = Moment of Inertia of the Girders (in.4)
Iv = Initia Data Point Value
Lb = Span Length of Beams
Lg = Span Length of Girders
Sb = Beam Spacing
Ts = Slab Thickness
121
122
w = Uniform Load Applied on the Floor Beam
Wb = Beam Self-Weight
Wc = Unit Weight of Concrete
LIST OF REFERENCES
(1) Marino, Frank, "Ponding of Two-Way Roof Systems", AISC
Engineering Journal, Vol. 3, July, 1966, (pp.93-100).
(2) Ruddy, John L., "Ponding of Concrete Decks", AISC En
gineering Journal, Third Quarter 1986, (pp.107-115).
(3) American Institue of Steel Construction, "AISC Steel
Construction Manual", Eighth Edition, American Insti
tute of Steel Construction Inc., Chicago 111., 1980.
(4) Ewell W.W., Okubo S., and Abrams J.I., "Deflections in
Gridworks and Slabs", Journal of the American Society
of Civil Engineers, Paper No.2520, 1952, (pp.872-913).
(5) Taraporewalla, K.Y., "Design of Grid and Diagrid Sys
tems on the Analogy of Design of Plates", Journal of
the Institution of Structual Engineers, Vol. XXXVI,
No. 4, April, 1958, (pp.121-128)
(6) Martin, I., and Hernandez, J., "Orthogonal Gridworks
Loaded Normally to Their Planes", Journal of the Stru
ctural Division, ASCE, Vol. 106, No. STI, January,
1980, (pp.1-12).
(7) Kardestuncer, H., "Elementary Matrix Analysis Of Stru
ctures", McGraw-Hill, 1974.
123
124
(8) Cook, R.D., "Concepts and Applications of Finite Ele
ment Analysis", John Wiley and Sons, 1981.
(9) Elwin G. Smith Division, Cyclops Corporation, Bowman
Construction Products, "Catalog of Steel Decks for
Floor and Roofs", Pittsburgh, PA, 1976.
(10)Ross, C.T.F., "Computational Methods in Structural and
Continuum Mechanics", John Wiley and Sons, 1982.