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1971
Dynamic Programming Algorithm for Minimum
Cost Design of Multispan Highway Bridges.
Kambiz Movassaghi
Louisiana State University and Agricultural & Mechanical College
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MOVASSAGHI, Kambiz, 1940DYNAMIC PROGRAMMING ALGORITHM FOR MINIMUM COST
DESIGN OF MULTISPAN HIGHWAY BRIDGES.
The Louisiana State University and Agricultural
and Mechanical College, Ph.D., 1971
Engineering, civil
University Microfilms, A XEROX Company, Ann Arbor, Michigan
©
1972
KAMBIZ MOVASSAGHI
ALL RIGHTS RESERVED
DYNAMIC PROGRAMMING ALGORITHM FOR MINIMUM
COST DESIGN OF MULTISPAN HIGHWAY BRIDGES
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State University and
Agricultural and Mechanical College
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in
The Department of Civil Engineering
by
Kambiz Movassaghi
M.S. Louisiana State University, 1965
December, 1971
PLEASE NOTE:
Some pages may have
indistinct
F i l me d as
University Microfilms,
print.
received.
A X e ro x E d u c a t i o n Company
ACKNOWLEDGEMENT
The author is grateful and wishes to express his sincerest
appreciation to Dr. Rodolfo J. Aguilar who suggested the problem and
gave valuable guidance and assistance during the course of this study.
The author gratefully acknowledges the other members of the
advisory committee--Professors Mohamed Al-Awady, B. J. Covington,
F. J. Germano, Benjamin E. Mitchell, and especially H. T. Turner,
Committee Co-Chairman— for their review of the manuscript and their
constructive suggestions and criticisms.
Special thanks to professor John A. Brewer, III for his valuable
advice and assistance in the development of the computer program.
The Louisiana Department of Highways and the U. S. Department of
Transportation, Federal Highway Administration, are gratefully acknowl­
edged for providing the funds for the research project that made this
study possible.
Finally, the writer expresses his sincerest thanks to his wife,
Mazie, for her never ending faith and encouragement, and to his parents,
without whose dedication and aspirations his education would not have
been completed.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENT .....................................................
ii
LIST OF T A B L E S .....................................................
vi
LIST OF F I G U R E S ................................................... viii
ABSTRACT ...........................................................
x
1.
1
INTRODUCTION ...................................................
1.1
1.2
1.3
1.4
2.
Bridge Structures and Design Approach ....................
P u r p o s e ..................................................
Statement of the P r o b l e m ...............................
Survey of Literature......................................
1
3
3
6
METHOD OF S O L U T I O N .............................................
11
2.1
2.2
2.3
2.4
3.
Mathematical Programming Formulation ...................
Dynamic Progra m m i n g .....................................
Dynamic Programming Model ofthe B r i d g e .................
Basic Properties of the M o d e l ...........................
11
12
13
17
MODEL D E C O M P O S I T I O N ...........................................
22
3.1
3.2
Limits on State and Decision V a r i a b l e s .................... 22
........................................
26
Optimal Policies
4.
A L G O R I T H M .....................................................
52
5.
EXAMPLE P R O B L E M S ...............................................
56
5.1
5.2
5.3
Three Span B r i d g e ........................................
Six Span B r i d g e ..........................................
Five Span Constrained B r i d g e ...........................
57
84
98
6 . SUMMARY AND C O N C L U S I O N S ........................................ 101
7. R E C O M M E N D A T I O N S ................................................. 1Q4
8 . R E F E R E N C E S ....................................................... 1Q5
iii
TABLE OF CONTENTS (continued)
PAGE
9.
A P P E N D I C E S ...................................................... 109
9.1
Computer P r o g r a m ..........................................109
9.1.1
Main P r o g r a m ........................................ H O
9.1.1.1 P u r p o s e ................................... H O
9.1.1.2 Discussion of General Logic ............. 110
9.1.2
Deck Optimization R o u t i n e ......................... 128
9.1.2.1 Purpose .................................. 128
9.1.2.2 Discussion of General Logic ............. 128
9.1.3
Span Length Calculator R o u t i n e ..................... 135
9.1.3.1 Purpose .................................. 135
9.1.3.2
Discussion
of General Logic; Free Bridge.135
9.1.3.3
Discussion
of General Logic; Constrained
B r i d g e ................................... 135
9.1.3.3.1 Free Span and Case A ........... 137
9.1.3.3.2 Free Span and Case B ...........138
9.1.3.3.3 Free Span and Case C ...........140
9.1.3.3.4
Fixed Bent Span and Case A . . 143
9.1.3.3.5
Fixed Bent Span and Case B . . 143
9.1.3.3.6 Fixed Bent Span and Case C . . 144
9.1.3.3.7
Fixed Center Line Span and
Case A
. 144
9.1.3.3.8 Fixed Center Line Span and
Case B .........................146
9.1.3.3.9 Fixed Center Line Span and
Case C . . . . . . . . . . . .
149
9.1.4
Branch Absorption Routine .......................
9.1.4.1 Purpose ..................................
9.1.4.2 Discussion of General Logic .............
159
159
159
9.1.5
Table Optimization Routine I .....................
9.1.5.1 Purpose ..................................
9.1.5.2 Discussion of General Logic .............
166
166
166
9.1.6
Table Optimization Routine I I ..................... 172
9.1.6.1 Purpose .................................. 172
9.1.6.2 Discussion of General Logic ............. 172
9.1.7
Table Optimization Routine I I I ..................... 178
9.1.7.1 P u r p o s e ................. ................ 178
9.1.7.2 Discussion of General Logic ............. 178
9.1.8
Computer Program Listing .........................
iv
184
TABLE OF CONTENTS (c o n tin u e d )
PAGE
9.2 Notation
.............
227
9.3 List of Symbols foxr Program Macro flow C h a r t s ............. 230
10.
V I T A ........................
232
v
LIST O F TABLES
PAGE
3.1
A Partial Summary Table
for
Returns of
Stage (1,1), Span 1 • 31
3.2
A Partial Summary Table
for Returns of
Stage (1,2), Span 1 • 36
3.3
A Partial Summary Table
for Returns of Span 1
3.4
A Partial Summary Table
for Returns of Spans 1 and 2 . . . . 45
5.1
General Data for Bri-dge Decks
5.2
General Data for Pile Bents with Prestressed Concrete Piles.
66
5.3
Soil and Ground Profile
. . . .
67
5.4
Specified Optimization Parameters for 3 Span Bridge
. . . .
68
5.5
Deck Table . ..
5.6
Summary Table for Optimized
Returns of
Stage ( 1 , 1 ) ........69
5.7
Summary Table for Optimized
Returns of
Stage ( 1 , 2 ) ........70
5.8
Branch Absorption Summary Table
5.9
Summary Table for Optimized
Returns of
Stage 1 ............ 74
5.10
Summary Table for Optimized
Returns of
Stage (2,2) . . . . .
5.11
5.12
..
..
39
.....................65
Data fox Three Span Bridge
...
B ranch A b s o p r tio n T a b le
. . . . . . .
..
......................... 69
for Span 1 ...................72
75
fo r Span 2 .............................................................. 77
Summary Table for Optimized Returns of
Stage 2 ............79
5.13
Summary Table for Optimized Returns for Span 2 after
Optimization. Over Span L e n g t h ............................... 80
5.14
Summary Table for Optimized Returns of Stage (3,2) . . . . .
5.15
Branch Absorption Summary Table for Span 3 ...................82
5.16
Hanked Summary Table for the Entire B r i d g e ...................82
5.17
Optimum 3 Span Bridge C o n f i g u r a t i o n ......................... 83
vi
81
LIST OF TABLES (continued)
PAGE
5.18
Soil Data for Six Span B r i d g e .............................
86
5.19
Specified Optimization Parameters for the Six Span Bridge •
87
9.1
Main Program Variable Definition L i s t ...................... H 7
9.2
Subroutine DKOPT Variable Definition List ................
133
9.3
Subroutine SLC Variable Definition List ..................
154
9.4
Subroutine
TABABS Variable
Definition List
...............
163
9.5
Subroutine
TAB0P1 Variable Definition List
...............
170
9.6
Subroutine
TAB0P2 Variable
Definition List
...............
176
9.7
Subroutine
TAB0P3 Variable Definition List
...............
182
vii
LIST OF FIGURES
PAGE
Highway Bridge of N Simply Supported Spans...............
4
.............
14
Dynamic Programming Model of Span n .....................
27
Dynamic Programming Model of Span 1 .....................
28
Dynamic Programming Model of Span 2 .....................
41
Dynamic Programming Model of Span N .....................
49
Macroflow Chart for the Algorithm .......................
55
Layout and Deck Cross Section of the 3 Span Bridge
58
Dynamic Programming Model of N Span Bridge
. . .
.................
85
General Layout for the Five Span Bridge .................
99
General Layout
or the Six Span Bridge
Macroflow Chart for the Main Program
...................
112
Decks of Common Span Length .............................
131
Decks of Minimum Weight-Cost Boundary ...................
131
Decks for Prestressed Concrete Girders and Concrete Decks
131
Minimum Weight and Minimum Cost Deck Region .............
131
Test Point Region ........................................
131
Macroflow Chart for the Subroutine DKOPT
...............
132
Free Span and Case A
....................................
142
Free Span and Case B
....................................
142
Free Span and Case C
....................................
142
Fixed Bent Span and Case A
. . .........................
145
Fixed Bent Span and Case B
..............................
145
viii
LIST OF FIGURES (continued)
PAGE
9.13
Fixed Bent Span and Case C ...................................145
9.14
Fixed Center Line Span
and Case A
.......................... 147
9.15
Fixed Center Line Span
and Case B
.......................... 147
9.16
Fixed Center Line Span
and Case C
.......................... 147
9.17
Macroflow Chart for the Subroutine SI£
9.18
Macroflow Chart for the Subroutine T A B A B S ................. 161
9.19
Macroflow Chart for the Subroutine T A B 0 P 1 ................. 168
9.20
Macroflow Chart for the Subroutine T A B 0 P 2 ................. 174
9.21
Macroflow Chart for the Subroutine T A B 0 P 3 ................. 180
ix
........... 148
ABSTRACT
A theory for the systematic minimization of the total cost of an N
simple span deck and girder, highway bridge structure is presented.
Dy­
namic Programming, a mathematical technique applicable to the optimiza­
tion of serial systems is the method employed in developing
and its solution.
the model
The geometric allocation of total bridge length into
N spans (decisions concerning individual span lengths), and
the selec­
tion of structural components for each of the spans are implemented
simultaneously.
For this reason, the set of optimal decisions generates
a bridge design of minimum overall cost.
All parameters which affect
bridge structure economics, collectively or individually, can be readily
incorporated in the mathematical model.
The set of parameters includes,
but is not restricted to, bridge geometry, types of super-structure and
sub-structure components, material types and strengths, soil and ter­
rain conditions and construction costs.
The optimization procedure resulted in an algorithm suitable for
computer programming.
An extensive computer program was developed to
implement the optimization method.
The optimization theory and the
algorithm are independent of design procedures.
Thus, no particular
method of structural design for individual bridge components was em­
phasized.
The computer program is modular and any recent or established
structural design procedure can be incorporated with minimum program­
ming effort.
The program is capable of generating several optimum
bridge structures based on the user's specification of input data.
1.
INTRODUCTION
1.1 Bridge Structures and Design Approach
Modern bridge structures, In general, consist of several spans
where each span Is an assembly of structural elements of different ma­
terials with several strength characteristics.
The development of a
bridge structure can be categorized into the following stages:
1.
Planning and conceptual design.
2.
Preliminary and final design of components.
3.
Construction.
The preliminary and the final design of a bridge component is usually
a complex problem which requires a large number of computations.
De­
velopment of new design theories, improved construction materials, and
utilization of modern computational aids such as electronic computers
have facilitated the design task and have resulted in more efficient
and economical bridge structures.
However, in the planning stages,
selection of a geometric configuration; span lengths within the stated
physical and structural constraints, and of components for the super­
structure and sub-structure of a highway bridge is frequently a nonsystematic procedure.
That is, decisions are often reached based on
the incomplete consideration of the effects of all parameters entering
into the decision making process and influencing the overall cost of
the project.
Theoretically, absolute economy can be attained if an ex­
haustive search of all parameters affecting the cost of the project is
performed.
This approach, however, results in a multi-dimensional
problem with a large number of alternative solutions which include one
2
or more optima and it is not feasible even with the aid of large, fast
electronic computers.
Consequently, civil engineers involved in the
design of highway bridges must rely strongly on their knowledge of
material behavior, construction techniques and, most of all, on their
experience and engineering judgment in obtaining the most suitable
solution to a given design problem.
It is to the credit of these men
that, by and large, overall economy has generally been attained.
As more and more bridges are built in urban areas where a large
number of constraints are imposed on the bridge geometry, the need for
a systematic design procedure during the planning stages becomes of
the outmost importance.
To make the method meaningful, a large set of
significant parameters must be considered.
Solutions to these classes of multivariable problems have been
considered too complex in the past.
However, the recent advances in
the development of optimization techniques have provided the engineer­
ing profession with rational and efficient methods of design.
The
procedures start with a concise and detailed definition of the problem
and of the goals to be achieved.
Next, alternative solutions to the
problem, with full consideration of the set of established constraints,
are structured in a systematic fashion.
Finally, the merit of each
solution is evaluated through an objective function and the best
result or results are selected.
This general optimization procedure is intended to generate an
optimum value which here is defined to be the minimum cost of the bridge
structure.
An optimal solution is possible through the use of one or
more of a set of mathematical programming techniques.
Dynamic Program­
ming is one such technique and was extensively utilized in this work.
3
1.2
Purpose
The conceptual design stage of a bridge structure was studied in
this research.
A method to apply Dynamic Programming to the decision
making process involved in the minimum cost design of a hi.ghway bridge
of N simple deck and girder spans was developed.
Specifically, the
method, within the defined feasible region, seeks and finds local mini­
ma corresponding to optimum bridge configurations by generating opti­
mum decisions for:
1.
The allocation of total bridge length into N individual span
lengths.
2.
The selection of appropriate deck and bent components.
The end result is an optimization algorithm and an extensive computer
program to implement the method.
The results presented here are of theoretical and practical impor­
tance.
Theoretically, the results show, for the first time, the appli­
cation of Dynamic Programming to the overall planning and design of
bridge structures.
Practically, the algorithm and the computer program
provide the bridge design engineer with a fast and efficient tool to
compare alternative designs for economic selection purposes.
1.3
Statement of the Problem
A general highway bridge, as depicted in Figure 1.1, was selected
for this investigation.
The bridge structure consists of N simply
supported spans of total length L.
Each span consists of three prin­
cipal components.
One super-structure or deck component, which is the portion
of the bridge lying above the bents.
The deck can consist of
n+1
n-1
N-l
Span 1
Span 2
Span n-l
Span n
FIGURE 1.1 H IG H W A Y BRIDGE OF N SIMPLY SUPPORTED SPANS
Span N-1
Span N
5
several elements in a variety of materials.
Two sub-structure or bent components, formed by the right and
left bents, also consist of several structural elements of dif­
ferent materials.
The total bridge length, L, and the number of spans, N, must be es­
tablished a priori.
To conform to certain terrain geometry it may be
necessary to require that one or more of the spans in the bridge system
have specified location and/or length.
In addition, in a design problem
a restriction on the minimum span length is usually desirable.
fore, a lower bound span length is assumed for all spans.
There­
Theoretically,
the maximum span length can be established by the optimum policies in
the problem.
However, from the practical standpoint, an upper bound span
length is determined based on the structural capability of elements in
the deck components.
It must be emphasized that the specification of
upper and lower bound span lengths does not affect the generality of the
problem or of its solution.
To consider several policies for allocating the total bridge length
into N individual span lengths, an incremental span length, A , for all
spans must also be assumed.
A basic assumption made here is that the to­
tal bridge length, L, the upper bound span length, and the lower bound
span length, are all multiples of the incremental span length A
.
The
reasons for this assumption will become clear later.
A variety of deck and bent components must be considered as al­
ternative designs for a given span length.
A particular design method
for ascertaining the structural capacity of the deck or of the bent
components is not specified.
It is assumed that these components can
6
be designed and analyzed in accordance with some established rational
procedure, that the cost coefficients are known, and that the total cost
of each component can also be determined.
Terrain geometry and soil profiles must be known and, within the
boundaries of the bridge structure, soil data must be made available for
all locations.
1.4
Survey of Literature
Application of optimization theories to civil engineering struc­
tures has received moderate attention in the literature.
Several methods
for the minimum weight or cost design of frame and truss structures have
*
been reported (18-27).
In the area of bridge design, evidence of sig­
nificant research on the cost minimization of the total bridge structure
is non-existent.
On all the reported work the emphasis has been placed
on the optimization of a single member or a single component of the many
that constitute a total highway bridge structure.
The following is a
review of the reported work:
A method for cost minimization of a continuous constant depth plate
girder was developed by Razani and Goble (15).
The method is iterative.
It generates a minimum cost girder with optimum decisions for the lo­
cation of flange splices.
In addition, optimum thickness of flange seg­
ments along the length of the girder is also determined.
Their "smooth­
ing procedure" is based on the application of dynamic programming which
balances one type of cost against another and results in a minimum over­
all cost.
In 1966, Goble and DeSantis (16) reported on their development
*Numerals in parentheses refer to corresponding items in the list of
References
7
of a method for optimum design of mixed steel composite girders.
This
work was basically an extension of previous work reported by Razani and
Goble (15).
In 1967, Razani (13) published a paper in the same area.
In his latest work he discussed the analysis and optimization of con­
tinuous girders and bridge decks with floor beams subjected to standard
AASHO (1) loadings.
A procedure of minimum weight design of steel girders governed by
the specifications of the American Institute of Steel Construction (2)
has been provided by Holt and Heithecker (14).
In 1966, Sturman, Albertson, Cornell and Roeseset (12) reported on
the development of BRIDGE, a subsystem of ICES, the Integrated Civil
Engineering System.
They stated that no significant work was done on
the development and inclusion of an optimization package in the total
system.
This goal was to be reached in the future.
Then, in June of
1967, Cornell, Ho and Ehrlich (10) published a research report titled:
"A Method for the Optimum Design of Simple Span Deck and GirderHighway
Bridges."
The following direct quotations from Section 1.2 of this re­
port described the scope and limitations of their work:
...the study treats the optimum design of standard deck-and
girder highway bridges of a variety of structural crosssections. The non-prismatic girders are considered to be
simple supported.
...the number of girders is included as a variable to be
optimized. The span of the bridge is regarded as a speci­
fic input; no minimization with respect to individual span
length is considered.
...all design is by working-stress method.
...the method of optimization adopted (a cutting plane ap­
proach) has been modified and adapted to yield results in
a manner which showed itself to be both reasonably effi­
cient and generally reliable on the problems considered.
As stated in the report (10), the basic limitation of the method is
that a fixed span length must be specified.
According to section 1.4 of
this report, steel girder design was fully developed and is included in
the optimization packaged
Implementation of the method for other types
of girders was left to the interested users.
However, a point of much
interest is that the effect of the bridge sub-structures on the total cost
was not investigated.
Consequently, the generated minimum cost includes
only the deck portion of the bridge.
sions were modified and used.
Linear programming and its exten­
The resulting procedure is iterative in
nature.
In another published research report (11) from MIT, dated May of
1968, Raymond W. Lotona, under the supervision of C. A. Cornell, dis­
cussed a method for the optimization of bridge span arrangements.
The
method selects super-structure macro-geometry when steel stringers and
concrete decks are used.
For a given total bridge length, the number of
spans and the number of girders for each span are established by the op­
timization policy.
The method considers all "reasonable" alternative
span arrangements, optimizes the deck components for each arrangement,
and then selects the optimum bridge configuration.
This is accomplished
by:
a.
Defining support regions where a bridge abutment can be located.
b.
Determining which pairs of support regions can be connected by
a single span with the provision that the specified maximum
span length is not exceeded.
c.
Selection of all combinations of spans which require only one
support in any support region.
This step is carried out in a
manner that the selected spans will reach the abutments and
fit together without overlaps or gaps.
d.
Extending step c to include cases where more than one support
may fall within a support region.
The outlined procedure is restricted in use for highway bridges that are
straight in plan, have parallel edges, and have constant deck depth with
the exception of cover plates on the supporting girders.
Here again, the
cost of the individual bridge pier or abutment was not made a function of
span arrangements.
Instead, the cost of all sub-structure components
were lumped and considered as a fixed quantity.
ed
Like most of the report­
work, the optimization method is modified linear programming and it
is used in an iterative fashion.
In 1967, Aguilar (9) presented a new approach to structural opti­
mization in his paper titled:
"An Application of Dynamic Programming to
Structural Optimization.11 He deviated from the common approach of for­
mulating the design of a structural member by a particular design pro­
cedure in a mathematical programming format to attain optimum weight
or cost.
Rather, he represented the structure by its Dynamic Program­
ming model and sought the optimal structure having minimum total cost.
The structure considered was a simple platform consisting of a deck
slab, four columns, and four footings and subjected to a uniformly dis­
tributed load.
The approach did not involve itself with a specific
design procedure for any one of the structural elements.
Instead, it
considered all feasible varieties of components, designed by any of the
accepted methods in several available types of construction materials.
Information required by the optimization method was the cost and the
weight of each member considered.
Although the platform was a simple
structure, the procedure clearly demonstrated the feasibility and ef­
ficiency of structural optimization by Dynamic Programming.
Of course,
Dynamic Programming is only applicable to systems which are serial in
nature.
erty.
Fortunately, statically determinate structures possess this prop­
Needless
to say, the methodology for the procedure developed and
presented in the following pages was inspired by this work.
2.
2.1
METHOD OF SOLUTION
Mathematical Programming Formulation
-f
The first step in applying optimization theory to the performance
of any physical system of known characteristics is to represent the
system in an abstract or symbolic form known as a mathematical model.
The model generally consists of:
1.
An objective function which defines the total system utility
in terms of the set of independent parameters affecting its
behavior.
2.
A set of constraint equations which defines the range of varia­
tion for each of the parameters.
Solution of the problem through the use of an optimization technique
constitutes the second step.
The general approach outlined above is possible when the number of
independent variables in the problem is small and the mathematical equa­
tions relating them can be easily formulated.
In the case of a highway
bridge such as the one in Figure 1.1, there are too many parameters af­
fecting the total bridge cost.
In addition, the mathematical formula­
tion of the objective function and of the set of constraint equations is
extremely difficult if not impossible to attain.
Another class of optimization techniques generally known as direct
climbing can also be applied to problems of the type considered here.
However, as the dimensionality of the problem increases, the rate of
convergence of direct climbing methods tends to decrease and, consequently,
11
12
their application is limited to smaller models.
Problems which are serial in nature can be formulated and opti­
mized efficiently with Dynamic Programming.
The bridge structure con­
sidered here is of a serial type and, for this reason, Dynamic Program­
ming was employed as the method of solution.
Basic properties of this
technique are discussed in the next section.
2.2
Dynamic Programming
In the early 1950's, Richard Bellman "postulated his principle of
optimality" (3) which states:
"An optimal policy has the property that whatever the
initial state and initial decision are, the remaining
decisions must constitute an optimal policy with re­
gard to the state resulting from the first decision."
This principle was intended to apply to serial systems; those with com­
ponents or stages connected head to tail with no recycle.
In such a
system, behavior of the most downstream stage does not affect the per­
formance of any upstream stage.
In contrast, operation on any one of
the upstream stages affects the behavior of all downstream stages.
The most important property of Dynamic Programming is that it trans­
forms a complex, multistage, multidecision, serial problem with .many
interdependent variables, into a series of one decision problems, each
comprising only a few variables.
The transformation does not change
either the number of feasible solutions nor the corresponding values of
the objective function.
Dynamic Programming has proven useful in the
study of deterministic and stochastic, continuous and discontinuous,
linear and non-linear systems possessing a serial structure.
A variety
13
of serial problems such as initial value problems, final value problems,
boundary value problems, problems with converging and/or diverging
branches, and even problems with recycle loops can be formulated and
solved with Dynamic Programming.
In addition to published papers on its
application to the optimization of physical, real life systems, excel­
lent texts on dynamic programming theory are also available (3-8).
2.3
Dynamic Programming Model of the Bridge
The bridge structure of Figure 1.1 behaves as a serial system.
live load is applied by the moving traffic to the deck components.
The
The
deck components then transmit the live load plus their dead load to the
bent components, which in turn, transmit the applied load plus their weight
to the supporting soil.
As the bent component is the most downstream ele­
ment in the system, decisions concerning the type of bent do not affect
any other component.
Whereas, decisions about the length of a given span
influence all downstream elements (the deck and the bents in the span).
Figure 2.1 depicts the Dynamic Programming model of a serial struc­
ture of an N span, simply supported, highway bridge.
The model consists
of a system of deck stages to each of which two branched bent stages are
connected.
The information transmitted from one stage to another is
either length or load.
The nomenclature used is standard for Dynamic
Programming problems.
There are four state variables, two decision variables, and one re­
turn variable associated with each deck stage.
To define these variables,
consider deck stage n;
X
n
is the input state variable to stage n; it represents the
total length to be allocated to span n and to all spans
downstream of span n.
Di
kdi
11 11
*
Deck
1
n
N_1h
N-l A
N-l
W
Deck
xi
O
II
i> r
2
1
3
ro*1
W
ro
9
o
to
»
to
to
I
1
1
1
d
1.1
1,1
2,1
•
IO
a
1
1
1
2,1
n
n-l,2
*
Bent
n,2
n+1,1
Bent
n-l,2
n,l
ITT TT7
ri.i
03 9
9
0 0
0
a
1 a
1
h-» I— * H-* 1— »
a
a
a
a
tO to ro M
a
a
to I-1
Bent
1,2
2,1
Bent
1,1
0 0 nl
0 0 P
•
+
to »-»
a
a
to M M
a
M
01
0
9 9
M* M JO N>
a
• a
•
ro N»
» • a •
to 1— » ro
it
n-l,2
rn,2
dn,2
Bent
N,2
Bent
N-l,2
N, 1
rnr m
N-l, 2
d
N-l,2
N,2
A
2,1
n-l, 2
®n,2
Figure 2.1
Dynamic Programming Model of N Span Bridge
®N-1,2
N,2
N,2
15
Xn
is an
output state variable from stage n; it represents
the total length left over for allocation to all spans
downstream of span n.
s^ ^
is an
output state variable from stage n; it represents
the total load transmitted by deck stage n to bent stage
(n,1).
sn 2
i-s an
output state variable from stage n; it represents
the total load transmitted by deck stage n to bent stage
(n,2 ).
is a decision variable which allocates the available length
(X ) to span n.
n
d
n
is a decision variable which selects the type of deck for
a given span length in span n.
r^
is the return of stage n; it represents the cost of the
deck in span n.
Single numbers are used to designate the deck stages.
sistent with the bridge span numbering system.
ports the decks of two adjacent spans.
This is con­
Each interior bent sup­
Thus, each interior bent can be
considered part of one downstream or of one upstream span.
Consequently,
the intermediate bent stages have been designated by two sets of two
numbers (n,j).
The first number in each set identifies the span number
and the second, whether the bent is to the right or to the left of the
span.
As an example, for span n, the left bent is designated by (n,l)
and the right by (n,2).
The right bent of span n, called stage (n,2),
becomes the left bent of span (n+1), called stage (n+1,1).
stages have been identified by only one set of two numbers.
The end bent
16
Variables associated with a bent stage, say stage (n,l) = (n-l,2),
are defined as follows:
Sn 1 1
*‘S an *-nPut state variable which represents the dead
load contributed to the bent stage by the deck of span n.
The first two subscripts (n,l) of the variable represent
its relationship to the output state variable from deck
stage n, i.e., s
n,l
The last subscript 1, identifies
the dead load.
sn 1 2
*-s an input state variable which represents the live
load transmitted to the bent stage by the deck of span n.
The last subscript of the variable, 2, identifies the
live load.
s
i 9 i is an input state variable which represents the dead load
n-x,z ,i
contributed to the bent stage by the deck of span (n-l).
s
199
n-i,z ,z
is an input state variable which represents the live
load transmitted to the bent stage by the deck of span
(n-l).
*
Sn _-L 2
is an output state variable which represents total load
transmitted to the supporting soil.
d
, _
n-1,2
is a decision variable which selects the type of bent
for the input information.
rn ^ 2
*-s t*16 return variable which represents the cost of the
bent stage.
All other symbols are defined when they appear for the first time.
addition, a complete listing is given in Section 9.2.
In
17
2.4
Basic Properties of the Model
Associated with each stage of the dynamic programming model is a
set of functions which define its behavioral characteristics (transition
and return functions) and also its relationship to the adjacent stages
(incidence identities).
The convention used here to represent the func­
tional relationship of one variable,^
j ^An n^ or An n =
n
j^*
to another,
n > *s
^ith this convention in mind,
j =
for
deck stage n, the incidence identity is:
Xn
The transition
X„
n
and the
<2 -1)
function is given by
= ^(^,0
n nn
) = X" D
n
n
; n =1,2,...,N,
(2.2)
return function is,
r
n
Where,
= Xn+1; n =
—
p (X ,D ,d ); n = 1,2,...,N.
n
n
n n
(2.3)
is used to represent the functional relationship between rn>
X ,D , and d .
n n
n
Because one portion of the problem deals with the allocar
tion of length L into N spans, the serial branch of the model turns out
to be an allocation boundary value problem with values at the boundaries
defined by X^ = L, and X^ = 0.
This model characteristic requires
that a decision inversion be performed at stage 1 before any attempt is
made to establish optimal policies.
That is,
18
Xx = X L - D x = 0,
(2.4)
and therefore;
Xx = Dx .
(2.5)
There are other transition functions for deck stage n, as follows:
s
-I —
n, l
1 (D >d
n, l
n n
i)» n — 1,2,... ,N.
n-i
(2.6)
s
_ —
n ,2
n (D >d ,D ,. ); n - 1,2,... ,N.
n ,2
n n n+l
(2.7)
Equation (2.6) simply shows that the total load transmitted by the deck,
output state variable s* . , is a function of the decisions made about
n,l
the length and the type of deck of the span under consideration as well
as of the adjacent span for the live load portion of the load.
allel definition can also be given for equation (2.7).
A par­
For the bent
stages similar transition functions are developed.
Sn, 1,1 ~ ^i,l,l(8n,l) " ^i,l,l(Dn ,dn ); n “
(2 .8)
Equation (2.8) shows that the dead load contributed to bent stage
(n-l,2 ) = (n,l) by the deck of span n, is a function of its length and
type.
Also,
sXI j 2o y 1i = (TEll )ofaji1
9) =
Xl,2
9 1
Ily fa) i
11
H
* n =
Equation (2.9) shows that the dead load contributed to bent stage
(2.9)
19
(n,2 ) = (n+1 ,1) by the deck of span n, is a function of its length and
type.
The magnitude of the live load transmitted to an interior bent is
primarily a function of the
adjacent span lengths.
the deck configuration does
not affect the live load in the design of
an interior bent.
1 9 =
n , 1,2
It was assumed that
Thus,
7 9^n
n,i,2
n,l7> =
7 ?<Dn»Dn
n,i,2
n n-l7>* n = 2,3,...,N.
(2.10)
Equation (2.10) shows that the live load transmitted to bent stage
(n-l,2) = (n,l) by the decks of adjacent spans n and n-l, is a function
of their lengths.
Similarly,
Sn,2,2 = ^n,2,2(sn,2) = * n,2 J Dn ,Dn+l}; n =
shows that the live load transmitted to bent stage
(2.11)
(n,2 ) = (n+1,1) by
the decks of adjacent spans n and n+1, is a function of their lengths.
The return function (cost) for bent stage (n,2) = (n+1,1) is;
rn,2
^n,2 ^ n ,Sn,2,l,Sn,2,2,Sn+l,l,l,Sn+l,l,2,dn,2^*
(2.12)
It is necessary to introduce X^ in Equation (2.12) because, in general,
the soil profile changes within the boundaries of the bridge structure.
Thus, X , establishes the location of bent (n,2) = (n+1,1) in the
n
rererence plane that has been defined.
Substitution of Equations (2.8),
(2.9), (2.10), and (2.11) into Equation (2.12) results in;
rn,2 - Pn , 2 < V V dn-Dn+l’dn+l>dn,2); “ = d>2 .... "-1-
(2'l3)
20
For bent stage (1,1), one has
sl, 1,2 " % , 1 , 2 (S1,1) "
1,2^°1^’
(2.14)
and
rl,l
P l,l(Xl,s0,2,l,S0,2,2,Sl,l,l,Sl,l,2,dl,l)‘
(2.15)
But
S0 ,2 ,l
S0 ,2,2
°*
(2.16)
therefore, by recalling equation (2.5) and replacing s.. . . and s..
„
1,1,1
1,1,/
by their corresponding values, equation (2.15) becomes
rl,l
Pl,l(Dl ,dl,dl,l)*
(2.17)
Similarly, for bent stage (N,2)
SN,2,2
^N,2,2 ^SN,2^
^N,2,£V’
(2.18)
and since
SN+1,1,1 = SN+1,1,2 = °»
(2.19)
rN,2 = ^ , 2 (XN ,DN ’dN ,dN,2^ = ^i,2 ^L,DN ,dN ,dN,2^ *
(2.20)
then
because
= L.
It must be realized that equations (2.16) and (2.19) are not
21
necessarily valid.
In the case of an abutment, for example, the founda­
tion could also resist loads imposed by the approach structure.
However,
since the loads contributed by the approach structure would be constant
in the usual case, they can be taken directly into account and do not
affect the optimization procedure.
3.
3.1
MODEL DECOMPOSITION
Limits on State and Decision Variables
As stated previously, the Dynamic Programming problem of Figure
2.1
is composed of a serial system to which branch stages are connected.
The serial system represents a boundary value allocation problem.
Taking
advantage of the allocation nature of the serial system and with the
specified minimum and maximum span lengths, all upper and lower bound
values of those state and decision variables representing length can be
determined before any attempt is made to compute optimal policies and
returns.
Define the following notation:
(1)
X^
= Minimum permitted value of the state variable (lower-bound);
n “ 1,2 ,...,N-1.
(M)
X
= Maximum permitted value of the state variable (upper-bound);
n
n — 1,2 ,...,N—1•
= Specified minimum span length for all spans.
*»)
= Specified maximum span length for all spans.
At stage N, the state variable X jj, has a specified value which is the
total bridge length, L.
(1)
Therefore,
(M)
“ L-
(3a)
The lower-bound value of the state variable X
by;
22
n
entering stage n is given
23
(1)
n . a
(1)
X
n
= Max.
(2)
; n = 1,2,...,N-1.
(3.2)
L - (N-n) • £
The upper-bound value of X^ is given by;
(2 )
n
<M)
X
n
= Min.
; n “ 1,2,...,N-1.
(1)
L -
(3.3)
(N-n) • I
For each stage the superscript (M) is defined by
(M)
X
M a
(1)
-
n
X
n
+ 1; n => 1,2,... ,N-1.
(3.4)
The superscript (m ) at each stage n is used as a counter for values of
“
the state variable X .
n
„(1)
Therefore, it is important that L , £
, and
»(2)
£
be so specified that equation (3.4) results in an integer number, i.e.,
L,
(1)
£
, and
£
(2)
each should have a value equal to some multiple of A .
Any intermediate value of state variable X^ can now be computed as
follows,
X
(m )
(1)
n = X
+ (m-1) A ;
n
n
n
n = 1,2,... ,N-1;
m n — 1,2,...,M.
(3.5)
Equations (3.2) through (3.5) define the complete range of each state
variable at all stages except at stage N where the state variable has
only one value given by equation (3.1).
As in the case of state varia­
bles, upper and lower-bound values can also be computed for the decision
variable D
n
associated with stage n.
Define,
24
^ = minimum value of the decision variable at stage n (lower-bound),
for n = 2,3,... ,N.
(Kfl)
Dr
= maximum value of the decision variable at stage n (upper-bound),
for n = 2,3,...,N.
Because stage 1 is the end stage (Dynamic Programming convention) and
the problem is an allocation problem, X^ must have a value of zero as
specified in Figure 2.1.
Therefore a decision inversion is performed
at atage 1 and, consequently,
(ka)
D.J,
=
(mx)
; m^ = k.^ = 1,2,...,M.
(3.6)
For all other stages, the lower-bound value of
is given by the fol­
lowing equation:
i£
Dn
= Max.
- Max*
n — 2,3,... ,N;
\
i (m )
(2)*
X n - (n-l)- £
n
(3,7)
m = 1,2,. ..,M.
And the upper-bound value of the decision variable
/
£ 2)
Dn "
= M i n -{
X
is given by;
n =s 2 ^
N*
(m)
(1) ;
n - (n-l)- I
m » 1,2
n
n
M.
(3'8)
Values for the superscript (K ) can be determined from the following
n
relationship;
25
(K )
(1)
D n - D
= — ------
K
+ 1; n = 2,3,...,N.
(3.9)
Again, the superscript (k ) at stage n is used as a counter for the different values of
11
(mn )
that correspond to each X^
.
Any intermediate
value of the decision variable at stage n is given by
0 0
Dn "
(1)
= Dn
+ (kn - 1) • A ; n = 2,3,...,N; ^
= 1,2,...,!^.
(3 .10)
Equations (3.6) through (3.10) define the complete range of values of
(m )
the decision variable D associated with input state variable X
n
r
n
When optimal policies for a given stage e.g., n, are being deter­
mined (section 3.3), it is necessary to know the range of the decision
variable D
°f the adjacent span.
The lower-bound value of this de­
cision variable can be determined as follows:
n = 1,2,...,N-1;
l l)
(1)
D ,- = Max.
n+1
L - X
(m )
(2) 5
n - (N-n-1) • 1
m = 1,2,...,M.
n
n
(3.11)
The upper-bound value is given by;
(K
}
U
V f 1 -**■
(2)
n = 1,2 ,...,N-l;
(m)
(I) 5
(L - X n - (N-n-1)• ^
\
n
The value
of
relationship:
m
n
(3-12>
= 1,2,...,M.
the superscript (KQ+^) can be obtained from the following
26
K
N-l.
n+1
(3.13)
Any intermediate value of this decision variable is given by;
n = 1,2
N-l;
(3.14)
K
n+1*
The preceding discussion and development of equations were based
on the assumption
that
minimum
only specified constraints.
and
maximum span lengths were the
These relationships can be modified and
used also when, in addition, one or more of the spans has been specified
with a fixed length or location.
This application is demonstrated in
Section 9.1.3.3.
3.2
Optimal Policies
Decomposition is a usual approach in the solution of complex prob­
lems.
Decomposition breaks the problem into a series of smaller prob­
lems with more easily attainable solutions.
The total solution is ob­
tained by combining the results from the smaller problems.
This ap­
proach of multistage problem solving is implemented with Dynamic Pro­
gramming.
The model of Figure 2.1 was decomposed into N simpler models.
Each of them represents a given span in the bridge system of Figure 1.1
and each is a three stage (one deck and two bents) diverging branched,
Dynamic Programming problem.
ming model of span n.
Figure 3.1 depicts the Dynamic Program­
27
T—1
A
CM
A
r-H
d
CO
rH
*
CM t—4
A
#
CM
C>J I—1
• +
#»
d
£
d
09
CO
CO
CM
•>
H
CM r-H
CM
*
m
r-H
H
rH
*
*
1
d
d
d
CO
CO
CO
CM
•»
rH
*%
r-H
+
£
U>
n+1,1
n-l, 2
n, 1
n ,2
Figure 3.1
Dynamic Programming Model of Span n.
It must be pointed out that decomposition does not alter the basic
properties of the total model of Figure 2.1 and equations (2.1) through
(2 .20) still hold.
Figure 3.2 depicts the dynamic programming model of span 1„
Adopting the conventional approach and terminology one can proceed to
determine the minimum cost return function for stage (1,1) as follows:
28
fl,l(Dl ,s0 ,2 ,l,S0 ,2 ,2 ,8l,l,l,sl,l,2)
m *n
{ P1,1(D1,S0 ,2 ,1’
dl.l
S0 ,2 ,2 ,Sl,l,l,Sl,l,2 ,dl,l) ‘
(3.15)
'
1=0
CM
<N
rH
CM
rH
10
CM
CM
CM
rH
CM
1,2
‘1,1
Figure 3.2
Dynamic Programming Model of Span 1
Note that the state variable
This is because X^ =
does not appear in equation (3.15).
due to the decision inversion performed at stage
29
1 as indicated by equation (2.5).
Recalling equation (2.16), equation (3.15) is modified with the
following results,
fl,l(Dl ’sl 5
d l,l 1
'
(3.16)
Equation (3.16) represents a three state one decision optimization
stage.
To set up summary tables for each stage, the following conven­
tion will be adopted:
Different values of state and decision variables
will be designated by the symbols already defined with the addition of
( V
superscripts, e.g.,
, is defined to represent the k^th value of D^,.
Equation (2.8) gives s. . , as a function of D . , and d...
(k ^
i,i,i
i
i
each value of D- = D,
1
1
For
, T, different values of d. can be considered.
* k]L
1
That is, there exist T, different types of bridge decks which can be
l
used for the given span length.
It has been assumed that for each value
(kx)
of
= D^
, the configurations and also the costs of these T^
ferent types of decks are known.
Consequently, the different values of
sn . ■, can be obtained from equation (2.8).
1,1,1
can now be labeled as
dif­
These values of s. . .
1,1,1
follows:
(k^tj)
sl,l,l = Sl,l,l
’ fcl =
k
= 1,2,...,Kr
(kj.tj)
s., , .
is now defined as the dead load supported by the
1 ,1,1
1
1,1
(3.17)
left bent
of span 1, and produced by the t^th type deck when the k^th value of
30
(kx)
span length
is being
as a function of D..
1
considered. Equation (2.14)
represents s^ ^ ^
The values of s. . , are designated by,
1 , 1, z
(kx)
Sl,1,2 = Sl,l,2 J k l =
*
(3 '18>
Then, knowing a particular value of D.., the corresponding value of s. . ,
1
1,1,/
can be readily evaluated.
The values of
can be determined from
equation (3.2) through (3.5), and (3.6).
(k1 ,t1)
For each value of
j
different types of bents can be used.
(kx)
and al l 2 = a1>l>2 ,
Again it is assumed that either
these bents have already been designed or can be designed and that
their individual costs are known.
Thus, the different values of the
decision variable d^ ^ are represented by,
(kj j ^
d i,i = d i,i
jU j )
; ui
k
“ 1,2 ,,**,uk 1,t1; ti =
= 1 ,2 ,...,^
.
(3.19)
Finally, the stage return (bent cost) has been represented by
an(i the stage's best return by f^k ^ ,t:i’ \
where the aste­
risk (*) is meant to indicate that the return has been optimized over
the variable u^.
Table 3.1 is a partial summary table for stage (1,1).
By definition a summary table includes only the optimum values of
component return.
Table 3.1 gives only T^ minimum returns corresponding
to the T. values of s. .
which in turn correspond to the first value
1
1,1,1
TABLE 3.1
A Partial Summary Table for Returns of Stage (1,1)) Span 1
1.1.1
(1,1)
,«>
1
Sl,l,l
(1,2) -<1 .2)
’1,1,1
,(D
* \
1*1.
1.1^
5(1)
j (1»1»Uj ).
dl,l
’
a (1)
j(l,2,u.).
dl,l
» U1
1,1,2
1,1,2
j(1,Tj)
Hd , ^ )
a( D
,(1,T ,u )
dl
*1,1,1
*1.1.2
dl,l
■ 1,2 ,...,K^ .
o
.i .d
__Li__
i «
it
a
u
1,2,...,U1j2
m
1.1
f (l,Tx,*)
!
1
and
£l,l
■ 1,2,... ,U,
I
d l ' d lkl,tl>i h
(1 , 2 ,*)
(l,Ti,tti)
1,2..... 1,Xj rl.l
(kl,ti,U!)
d
1.1
.(1,2,u x)
1,1
i
* "l
(i,i,*)
r (l.l,ux)
rl,l
1
,
K x»cl
(3.1.5)
all kj, and t1#
” 1 .2 .---.Tk .
r1 1 ■ r^k J*tl,Ul\
and all k^ .
si,i,i “ ®iki*il)» for a11
for all
and
(3.1.6)
(3.1.2)
h * and
kl* (3,1*3)
s(ki, ti,*)
■
•1 .1.2 ' *ik i!2 ' £or al1 k r
1,1
u>
(3.1.4)
for
all t1, and k 1 .
(3.1.7)
32
of D .
Equations given at the bottom of Table 3.1 can be used to gene­
rate all other minimum returns, for every value of D^.
The minimum cost return function for stage (1,2) with respect to
the feed it receives can be written as;
;n
K Pl,2(D1’S1,2,
:
£1,2(D1’S1,2,1,S1,2,2,S2,1,1,S2,1,2) “ m^n
{
V
1,2 <
Sl,2,2,S2,l,l,S-2,l,2,dl,2) '
From equations (2.9), (2.11),
*
(3.20)
(2.8), and (2.10), recall that,
1 ,2,1 = ^l,2 ,l^D lsdl^’
1 ,2,2 = ^L,2,2 ^D 1,D2^ *
(3.21)
(3.22)
(3.23)
2 ,1,1 =
2 ,1,2 = $2 ,1 ,2 (D2 ,D1)#
(3.24)
The values of s, _ , have been labeled similarly to those of s. - 1 .
1,Z ,1
I,1,X
,
It is anticipated that, in most cases, corresponding values of s. 1 1
1 1,1
and s. _ 1 will be identical.
1 , Z , 1
To determine values for
to know the different values of
The values of
2 2* S2 1 1* and S2 1 2 *
£fc £s necessary
and the corresponding values of d
can be determined from equations (3.11) through (3.14).
These equations, by their nature, generate the largest possible decision
set.
That is, if one moves to the next span and with equations (3.2)
33
through (3.5) calculates each possible value of the state variable X,
then, if each decision set corresponding to every value of X is calcu­
lated, close scrutiny of these decision sets reveals that every calcu­
lated decision set is a subset of the decision set generated by equa­
tions (3.11) through (3.14) for this span.
Therefore, by using equa­
tions (3.11) through (3.14), one is always assured that all values of
the decision variable D for the span adjacent to the one being consi­
dered have been investigated for any increment in the value of the state
variable X of the adjacent span.
Now let the values of the decision set
be represented by:
(k s)
°2 = D2
5 k2 = 1,2 9 •
(3.25)
• • 9
Then, as in the case of d^ and with similar assumptions, different values
of d2 can be represented by
(k 2>ta)
d2 - d2
;
1 ’2
k2 - 1,2 9 • • • 9 K,
2 '
(3.26)
Now the values of s1 „ 9 are given by
o v V
sl,2,2 " S l,2,2 ! k2 "" 1,2 9
\
The values of s9 . ,, by
Z , 1 51
- 1.2
• • • 9
(3.27)
34
<k 2»t2)
S2 ,l,l = s2 ,l,l ; *2 " 1>2 »*“ »Tk S
3
1&2
and those of s. ..
192 ,• • •
(3• 28)
by
Z 9 ±jZ
(k3,lc1)
S2,l,2 = S2,l,2
’ k2 = 1>2 >*--»K2 5
k x = 1,2,...,!^.
As in the case of
1,2 "
(3.29)
^ let the values of d^ ^ be represented by
1,2
’ T1
k 1,tl ,ks,t2
and
all t2 ,k2 ,t^, and k^.
(3.30)
Where vn designates distinct values for decision variable dR ^ .
Also,
let the values of r1 9 be designated by
i.9Z
G
rl ,2 " rl ,2
W
^
’V V i )
• 17 = I_V _____ V
,t
5 vl
1,2 ,**,,Vk 1,t1,k3,t2
all t2 »k2 ,t^, and k^.
and
(3.31)
Equation (3.31) gives r^ ^ as a function of D^,d^,D2 »d2 > and d^ ^ .
Equation (2.13) represents r^ ^ also as a function of X^.
of
locates the position of bent (1,2).
Each value
But because a decision in­
version was performed at stage 1, by equation (2.5) these bent lo­
cations are known for every value of D^.
has not been included in equation (3.31).
Therefore, the variable X^
35
Now let the values of f-
,
1 2.
be represented by
(k^t^kgjtg,*)
2 = fi 2
’ for a11 t2 ,k2 ,tl’ and k l*
Table (3.2) is a partial summary table for stage (1,2).
(3.32)
The equations
given in the table can generate all other minimum values of the stage
return.
The return of stage 1 in Figure 3.2 is given by
rl = Pl (Dl»d l) *
(3*33)
Let R^, the total return for span 1, be defined by the following equa­
tion after branch absorption has been effected:
Substituting for each state variable in equation (3.34) the correspond­
ing decision variables from equations (2 .8 ) through (2 .20), the following
results:
£l,2(Dl,dl’D2*d2)-
(3,35)
TABLE 3.2
A Partial Summary Table for Returns of Stage (1,2), Span 1
X 1=D 1
dl
Df >
d (l,l)
•
•
o<»
•
Sl,2,2
S2, 1,1
,(!,!)
sa , D
1,2,1
(1,1)
1,2,2
,D
1,1
sa , D
1,2,1
•
•
a (1»1)
1,2,2
•
•
) ’a.i)
1,2,1
'u ,d
1.2,2
d a,2)
•
;a,i)
D<»
D l * D f 1); k l
II
O
d2 "
S1.2.1
•
•
dl =
d2
D2
d f i ’V ;
"
sl,2 ,1
t2 =
g(l»2)
2,1.1
•
•
d i,2;vr 1,2,**,vi,1,1,1
d i,2 ;vr 1J2’--’v i,i,i,2
••
•
•(l.l.l.Ti.v^
3 d ■*i>
2,]L,1 d l,2;Vl=ls2,',,Vl,l,l,T1
(3.2. 1)
1,2,... ,Tfc , and all k^ (3.2 .2)
D ^ 2^- k » 1, 2,...,K2
2
* 2
2
rl,2
(l»l»l»2,v1)
= 1, 2,...
h
4
d l,2
(1,1,1,1,Vl)
(3.2 .3)
(3.2 .5)
v x)
(l,l,l,l,v )
1,2
(1,1,1,1,*)
1,2
(l,l,l,2,v )
1,2
•
•
(1,1,1,2,*)
1,2
•
•
^(l,l,l,T1,vl)
1,2
’(1,1,1,T ,*)
1,2
sl,2,2 =
£or a U V
s2,l,l ‘
f°r 311 t2 ’ a“d k2
and kl
(3.2.6)
(3.2.7)
d
1,2
= d^i.ti.ka.ta.vi) »
1,2
- 1,2,...,Vk
and all t2 >k2 ,ti , and k^
1 , 2 , . ,Tk , and all k2 (3.2 .4)
ti)
= s ^k i’
for all t^, and k.
L
1,2, 1 »
/
i»k 2»*-3» *)
| (kjjt
= mxn
1,2
1,2
V1
f 1,2
1,2
»t i»ka»ts»
(3.2.8)
(ka jtj ,ks ,t2 jV j )
, for all v 1,t2 >k2 ,tx, and k L
1,2
(3.2.9)
, for all t2 ,k2 ,t1, and ^
(3.2.10)
OJ
O'
37
Then, the total minimized return for span 1 is
fl (Dl ’D2 ’d2) =
n | R1 (D1,d1,D2 ,d2 ) |
.
(3.36)
d
Let the values of r^ be designated by
— ^(0^,d^) — r^
; t^ — 1,2,...,T^ ;
k x = 1,2,...,^ .
Also let those of
(3.37)
be given by
(k ,t ,k ,t )
Rj_
= R1 (D1 ,d1,D2 ,d2) = ^ 1 1 2
2 ,
for all t2 ,k2 ,t^, and k^.
(3.38)
Then the values of R^ can be obtained from the following equation
(k1,t1,ks ,t8)
®1
(kl,t1,*)
= rl
+ *1,1
(k1,t 1,kaSts ,*)
+ *1,2
for all t2 ,k2 >t1 and k^.
(3.39)
The values of f^ are given by;
(kjj^jkgjtg)
f^
= min
*
j (kj,tjjkg,t3) )
^ R^
^
t^ = 1,2,...,T^ , and
I
all t2 >k2> and k^.
(3.40)
Equation (3.40) yields all minimum values of the returns for span 1;
They are functions of decision variables D^,D2> and d2 *
Table 3.3 is a
38
partial summary table for the returns of span 1., Equations given at
the bottom of Table 3.3 can be used to generate all other minimum re­
turns.
The selection of span 1 minimum return values yields the cor-
responding optimal decisions (d^).
This enables the minimum stage re­
turns for all stages of span 1 to be determined as shown below;
K
k x = 1,2
(k,1*
,*,k_,t_,*)
t2 - 1,2 9 •
• •
(3.41)
1•
y T,
, and all
2
k2 , and ^
(3.42)
(kx,*)
r^ = r^
; for all k^.
Let f
(3.43)
represent the total
minimized span 1 returns.
Although these returns are functions of
and one might assume at this point that another minimization over
is necessary, it must be remembered that a decision inversion was per­
formed and that in reality the span 1 minimized returns are functions
of state variable
rather than of decision variable D^.
Therefore,
no additional minimization of the total cost isvneeded as the only other
variables associated with this span are D2 and d^ and they are part of
span 2 which will be considered when the sum of the returns from spans
1 and 2 is minimized.
Figure 3.3 depicts the dynamic
programming model of span 2.
Stage
(2,1) of Figure 3.3 has already been considered as stage (1,2) in span
1.
The minimum returns of this stage are given by equation (3.42).
TABLE 3.3
A Partial Summary Table for Returns of Span 1
V
°2
D1
D<1>
tl -
1,2....T l
Tl
D*1*
dj1,tl); tL - 1,2...
D 11J
di1,tl)5 fci “ 1,2,...,^
D 1 “ “l ^
l-
D<1}
d2
d^1,1)
rj1 *6 ^
d*1,2)
r^'V
fj1^!'**
f
d*1,Ti* rj1’*!*
f*1^!***
jU.t^l.Tj,*) gU.t^l.Tx)
r
(3.3.1)
5 kl "
d ^ i * * ^ ; tx « 1,2,...,Tk , and all ^
(3.3.2)
R
*1**!*1 ’1*
fU,*,l,D
(l,t1,l,2)
f (l,*,l,2)
f(l,*,l,Tj)
(3.3.6)
^ ■ f p ^ ,t:i* * \ for all t^, and kj
fl,2 * fl,2*
3 * *' ^
,7)
for a11 t2»k2 ,tl* (3,3*
and k,
Dj " D ^ 8 » kj ■ 1*2,... ,K2
(3.3.8)
R 1 “ rl + fl,l + f1,2
(3.3.3)
(3.3.9)
R1 " R^k i*£l*k 2,t2), for all t2 ,k2 ,t1,
d2 - d^ka»ta>; ^
(k
rl “ rl 1
» l,2,...,Tk , and all k2
8
(3.3.4)
)
1 » for a11 fcl* and kl
and k,
1
f (^i»*»k a»tg)_ min j nOCx.tj.kg.tg) j
(3.3.5)
1
t
j
1
1
f
j
all t2 »k2 » and k i
(3.3.10)
w
VO
This equation can now be written as follows:
(kjjkgjtg)
f2 1 “ f2 1
5 t2 “ 1»2 »,**>Tk * and a11 k 2 and k l*
The minimum return function of stage (2,2) is given by
f2 ,2 (X2 ’S2 J2,l,S2,2,2’S3,l,l,S3,l,2) ** m *n I P2,2(X2 ,S2,2,1
2 ,2 *
S2,2,2,S3,l,l,S3,l,2,d2,2) ^ *
(3.45)
Recalling equations (2.8) through (2.20), the minimum return function
for stage (2 ,2) can also be written as follows:
f2 ,2 (X2 ’V
’W
d2 '
W
[".in { P2
,
= mdn
P2,2(X2'D2
’d2-D3 ’d3 ’d2,2) > ‘ (3-46)
9.9 *
2,2
Adhering to the convention defined for designating values of
and d^,
equation (3.46) becomes
_
2,2
K
- W
W
* *
2,2
■ C3
’
k 3’
k^ 3 1,2 ,...,K^ ,
and all t^ and k^;
m^ = 1,2,...,M.
(3.47)
Equation (3.47) gives the minimum returns of stage (2,2) as functions
of X2 >D2 ,d2 ,D2 , and d^ where the optimum decisions concerning the types
ic
of bents to be selected, (d_ 9), have been determined.
41
For the return of stage 2 one writes,
(m2 ,k2 ,ts)
r2 ” ^2 ^x2 ’^ 2 ,d2^ e r2
>
—
and all
>
and
.
(3.48)
The total return for spans 1 and 2 is given by the following equation j
®2 "" R2^X2 ,Dl,D2 ,d2 ,D3 ,d3^ “ ^2^X2 ,D2 ,d2^ + f2,2^X2 ,D2*d2 ,D3 ,d3^ +
(3.49)
£l(D l,D2 ,d2 ).
CM
CM
CM
CM
CM
CM
CM
CM
CM
CO
CO
2,2
2,2
2,2
2,1
Figure 3.3
Dynamic Programming Model of Span 2
42
Combining equations (2.1) and (2.2) results in
“i
-
W
Therefore,
V
-
(3.50)
- V
x2
can be eliminated from equation (3.49), thus
R2 ■ W
W
W
W
W
, t 3
W
>
+ f2,2(X2>D2>d2 ’D3 ’d3) +
(3.51)
Now the values of
R ( m a j k a
= W
can be represented by
, k g
, t 3 )
_
r
2
(m a > k 2
, t g )
,
( m g , k s
2
, t 2 , k 3
, t
)
2,2
f (m2 ,k3 ,t2) ^ for aii
and m.;.
(3.52)
The minimized returns for two spans are given by the following equations;
f2 (X2 ,D2 ,D
d )= min
R2^X2 ,D2 ,d2 ,D3 ,d3)
(3.53)
2
and
f(ms ,ks ,*,k3 ,ta ) _ r|lln
(mg,kg,tg,k3 ,t3 )
2
t2
(3.54)
1 j2 j• *• jTj^ j find all
jk^ j1&29
^2 *
s
Equation (3.54) yields the optimum decisions for stage 2 , ^ 2), as well
as the minimum returns from spans 1 and 2.
Consequently, the return
from span 1 becomes;
f
1
= f (D ,D ) =
lK V
2J
1
= f
1
for all k jkg, and n^.
(3.55)
Also the returns from stages 2 and (2,2) become;
r
for all k0 and m„,
2
2
(3.56)
£
_ £ (ma>kg>*>k3 ,tg,*)
■2j2
-2j2
, for all t3 ,k3 ,k2 , and m,
(3.57)
2
= r^m8 ’R2 ’ \
2
Substituting equations (3.55) through (3.57) into equations (3.54) and
deleting the asterisks (*), one obtains;
and equation (3.53) becomes
f2 (X2 ’D2-D3-d3) " P2 (X2 ’D2 > + f2,2(X2 ’D2 ’D3-d3) +
fl<X2 > V ’
<3 '59)
Equation (3.59) represents the total minimized return function of spans
1 and 2 , where f^(X2 ,D2) is the minimized return function of span 1 and
P2 (X2>D2^ + ^2 2^X2 ,D2 ,D3 ,(*3^ *S t*ie contr^ ut^on °f span 2.
Concerning
the decision variable D2 > equation (3.59) represents the sum of the re­
turn functions of the two adjacent spans which has not been minimized
over D2 as yet.
Thus, to minimize the return function over the variable
D2 one must write;
F2 * P2 « 2 - D3-d3 ) = "J”
j W
W
V
j
“ ”^ n j
£2,2CX2>I)2'D 3'd3 ) + fl
<
W
P2 <X2 - V
j-
<3 ‘60>
and in terms of values, one has;
I
F2
- min
Cm 3"k s)
r
(m 3 .ka .k3.t3)
C » v V
+ f
+ fj
I
;
2
k2 = 1,2,...,K2 > and all
and
m 2 = 1,2,...,M.
The optimum decision variable (D2) is determined for each value of
(3.61)
44
ic
from equation (3.61).
Once (D^) is known, equation (3.50) yields (1)^)
for each value of X^.
Consequently, equation (3.55) becomes,
(m2,*,*,*)
; m2 53 1,2,...,M;
f1 - f 1
(3.62)
equation (3.56) results in
(m2,*,*)
* m2 “
r2 ~ r2
(3.63)
equation (3.57) becomes
(ma,*,*,k3 ,ta,*)
f2 2 = ^2 2
* for
*"3 and k3 ’ and m2 ’
(3.64)
and, finally, equation (3.58) becomes,
(m2 ,*,k3,t3)
f2 ” f2
’ f°r a11
and m 2"
(3.65)
(ms)
For a specified value of X^= X2 , equation (3.62) results in a single
minimum return value for span 1.
Similarly, equation (3.65) gives
minimum return values for spans 1 and 2 as functions of the length and
the type of deck in span 3.
Table 3.4 is a partial summary table of
returns from spans 1 and 2.
Equations given at the bottom of Table 3.4
can be used to generate all other minimum returns.
Notice that from
equations (2.1) and (2.2) X2 = X^ = X^ - D^, therefore
(3.66)
TABLE 3.4
A Partial Summary Table for Returns of Spans 1 and 2
3 __________ ^2_________
X2(1)
d
'V;
V
P3
1.2.....^
d3
fl
r2
_______ f2.2
D<1) d f " 1* fO.*.k..*) ra.k,.*)
x f > »*•>; kj-1.2
d (D d (1.2) fa,*,ka,*) r(l.k„.)
f2
^2.
f(l.k,.l.D , < W . 1 >
£(l,ks,*,l,2,*)
f(I.k..l.2) F U,*,1,2)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
x a> D o*>i V 1 > 2 .... ^
Xj = X ^ V ; nij « 1,2
D a > d a,Tx )£a,*,k2,*)
M.
(k )
°2 = °2 2 J k2 = 1,2,...,K2 ,
ra,k2,*)
•
£(i,ka,*,i,ii ,*) ,<1,^,1^ )
^i.m.i,)
(3.4.1)
r£ = r faa’k a ’* \
(or all k2> and m2 .
(3.4.6)
(3.4.2)
^m a»^ca**»^3»t3»*^
f2 2 = f2 2
* f0r a11 t3 ,k3 ,k2 ’ and
m 2 . (3*4 '7)
D3 =
d
k3 = 1>2 >---»k3 -
= d^k 3,t:3^* t = 1 2
3
3
* 3
(3.4.3)
T
3
and all k_.
3
® f^” 2’ *k2> \
for all k2 , and i y
(3.4.4)
f2 =
for all t ^ k ^ k ^
and n y
F (ma >*»k3,t3) _ min | f (m2 ,ka,k3,t3) | fQr
2
.
rC,
k2
'
,, . ,
,
all tgjkg* and i y
(3.4.8)
(3 .4 .9)
(3.4.5)
4
L?
n-
46
Proceeding to span n (Figure 3.1) according to the method out­
lined above, the total minimum return function of all downstream spans,
including stage (n,l) of span n, is obtained as follows;
f - (X . ,D ,,D ,d ) = min
P
(Xn •,,D
,d
) +
n-i n-i n-i n-i
n-1 n-1’ n-1* n ’ n
,
n-1
£n-l,2<Xn - l - V l ’dn - l " V dn> + F„-2<X„ - l ' V l ’dn-l)
>
(3.67)
from which
F
n-i
(X
n-i
,D
n
d ) = mm
n
D
f
n-i
(X
n-i
,D
n-i
D ,d )
n
(3.68)
n
n-1
but X
, = X = X - D from equations (2.1) and (2.3), therefore,
n-i
n
n
n
F
, (X
n-i
n-i
,D , d ) = F
n
n
n-i
(X jD ,d ),
n
n
(3.69)
n
and its values are given by,
(m ,k ,t )
F . = F ,
; t ® 1,2,...,T
n-1
n-1
n
k
, for all k
n
n
amd m .
n
(3.70)
The minimum return function of stage (n,2) can be written as follows;
fn , 2 < V V d„-IW
d,rH)
min
Pn ,2(Xn ’Dn ’dn ’Dn+1 ’dn+l ’dn ,2)
d
n,2
(3.71)
and its values as,
47
f
f(mn>kn ’tn-kn+l>tn+l-*)
n,2
n,2
• f°r a11 tn+l *kn+l*tn *kn ’
(3.72)
and m .
n
The return function of stage n can be written ;
r
n
(3.73)
= Pn (X ,D ,d ),
n n n n
and its values as,
(m ,k ,t )
r
n
= r
n
(3.74)
, for all t ,k , and m .
n n
n
Now define
R
n
= R (X ,D ,d . D , d
) = P (X ,D ,d ) +
n n n n n+i n+l
n n n n
(3.75)
f
.(X ,D ,d ,D
,d
) + F
(X ,D ,d ),
n,2 n n n n+i n+1
n-l n n n
and its values as,
n =
D (mn ,kn ,tn ,kn+l,tn+l)
n
» f°r a U
n+1 * n+1 * n * n *
(3.76)
and m .
n
The minimum retura function of all n spans is given by
£n < W V l ’dn+l>
min
d
n
and its values by
Rn (Xn ,Dn ’dn ,Dn+l,dn+l)
*
(3,77)
48
£n - W
-
V
W
W
- f„
• f°r a U
k
n+i
,k , and m .
n
n
Now the minimization of equation (3.78) over
minimum return from n spans.
W
W
F
= F
W
“ "Jn
n
‘n+l-
(3.78)
results in the total
Thus ;
W
-
V
W
W
j -
(3-79)
also ,
n
<
V
W
W
n
, for all t ..,k ,.,and m .
*
n+1
n+1*
n
(3.80)
Finally, Figure 3 . 4 depicts the dynamic programming model of span
N.
Proceeding to this span according to the method outlined; the mini­
mum return function of all downstream spans, including stage (N,l) of
this span, is given by
FN - 1 = ^ - l ^ - l ^ N ’ V
= FN - 1 (XN ,DN ,(V '
(3.81)
For stage ( N , 2 ) one has,
fN,2 ^XN ,SN,2,1,SN,2,2,SN+1,1,2,SN+1,1,2^ “ “j11 J N,2 ® N *
N,2'
SN , 2 ,1,SN,2,2 *SN+1,1,1,SN + 1 ,1,2,dN,2^
j
'
(3.82)
49
H
CM
CM
f-4
rH
CM
N,1
N,1
CM
N,2
N,2
N,2
Figure 3.4
Dynamic Programming Model of Span N
But, by recalling equations (2.19) and (2.20), equation (3.82) can be
written
£N,2(L>Dr V
“
"7
j PH^ L-DN-dN"dN,2)j-
H,2'
and the values of this return are given by
<3'83>
50
S
*N,2
-
W
*
0
fH,2
! ‘n
1,2
TkN
kjj = 1,2,... ,K^
;
;
mjj ® 1.
(3 .8 4 )
The return for stage N is given by
rN -
P
(L ,^ ,^ )
,
(3 .8 5 )
and its values by
<V
rN
kH’tH)
rN
, for all t ^ k ^ , and ra^ = 1.
(3.86)
Define;
= RH ® ‘,DH ’dlP “ % ( L >DH )dN ) + fN,2 *L,DN ’dN^
fn -i (l ’dh > V
•
(3'87)
= 1.
(3.88)
and the values of R^, as
Rjj “ Rj,
; for all tN ,kN> and
Then, the minimum return function can be obtained, thus
fN(L’V
“ ”ln j V ^ W - j
(3-89)
^n
The values of this function are given by
fN = fN
; for all 1^ and
« 1.
(3.90)
51
liquation (3.89) gives the minimum total return from N spans as
a function of the length of span N,(D^), where the optimum values of
k
the decision variable d^,(d^) have been ascertained.
f^ over the decision variable
Minimization of
results in the minimum bridge cost.
Mathematically,
F„(L ) = min
jf
(L,DN ).
°N /
j
(3.91)
'
*
Equation (3.91) also yields optimum decision (D^) .
Then, from equa-
*k
tions (2.1) and (2.2),
is determined.
downstream and obtain all decision variables:
n
Optimum span
lengths
*
■k
D
Now one can easily move
, for n = 1,2,...,N-1; decks types d
•k
types d^ ^ > f°r n = 1»2,...,N.
n
, for n = 1,2,...,N; and bent
4.
ALGORITHM
The developments of the previous sections can now be presented in
concise form as a set of rules to carry out the optimization procedure.
This set of rules, when arranged in an iterative form, is also called an
algorithm.
To condense the explanation of each step of the algorithm, the fol­
lowing definitions must be stated:
The left bent in each span is cal­
led bent No. 1, and the right bent is bent No. 2.
The term "geometrical
properties of the bent" is used to refer to some of the data needed for
the design of a bent structure.
For example, data such as bent height,
soil properties, types of bents which can be used at a given location,
and available construction materials, are all considered part of the geo­
metrical properties of the bent.
The algorithm is presented in two different forms:
In step by step
mode where the particular operation in each step is fully explained and
in a macroflow chart form (Figure 4.1) where the interconnections of the
algorithm steps are clearly depicted.
The step by step description
follows:
1.
For each value of span length in the specified range (increment­
ed from minimum to maximum span length) design as many bridge
decks as possible.
Suboptimize these decks for each span length
(See Section 9.1.2)and enter the resulting information in a table
called Deck Table.
2.
Consider span No. 1.
Determine the values of the state
52
53
variable, X, (equations (3.2) through (3.5)), and the corres­
ponding set of the decision variable, D, (equation (3.6)).
Also determine the set of values for the decision variable, D,
of the adjacent span (equations (3.11) through (3.14)).
3.
For the fixed location of bent No. 1 in this span, determine
the geometrical properties of the bent.
4.
For every value of span length and corresponding deck types
from the Deck Table, design as many bents as possible.
Select
minimum cost bents (optimization over the decision variable
which represents bent types).
5.
Set up a summary table for this bent.
6.
Consider bent No. 2 in the span being designed.
Determine the
geometrical properties of the bent for every value of X.
7.
For every value of X, and corresponding span lengths, adjacent
span lengths, and corresponding deck types from the Deck Table,
design as many bents as possible.
Select minimum cost bents
(optimization over the decision variable which represents bent
types).
8.
Set up a summary table for this bent.
9.
Perform branch absorption for the span under consideration,
i.e., set up a summary table where the cost column represents
the combined cost of all downstream components including the
deck component for the span under consideration.
10.
Optimize over the decision variable, d, that represents the
deck types.
Set up a new summary table.
11.
Consider the next span.
12.
Determine values of X (equations (3.2) through (3.5)),
54
corresponding values of D (equations (3.7) through (3.10), and
D's of the adjacent span (equations (3.11) through (3.14)).
13.
Repeat steps 6 through 10.
14.
In the resulting table of step 13, perform an optimization over
the decision variable, D, representing the span length of the
span under consideration.
Set up a new summary table to be used
in the next branch absorption.
15. Repeat steps 11 through 14 until the last span is the next span
to be considered.
16. Consider the last span.
For the fixed value of the state varia­
ble, X, determine the corresponding set of values for the deci­
sion variable, 0.
17. Consider bent No. 2.
For the fixed value of X, determine the
geometrical properties of the bent.
18.
For every value of span length and corresponding deck types from
the deck table, design as many bents as possible.
Select mini­
mum cost bents.
19.
Repeat steps 8 and 9.
20.
Select the minimum total bridge cost from the resulting table of
step 19.
to
21.
Also select the last span configuration corresponding
this cost.
With the selected last span properties, trace through the sum­
mary tables and determine the optimum bridge configuration cor­
responding to the minimum total cost.
The application of the algorithm to some example problems is demon­
strated in Section 5.
55
START
CREATE DECK TABLE
BRANCH ABSORPTION
CONSIDER
DETERMINE X, D, AND
SPAN NUMBER 1
THE ADJ. SPAN D's
CREATE
SUMMARY TABLE
FOR BENT NO. 2
IS THIS SPAN
NUMBER 1 ?
YES
IS THIS THE
OPTIMIZATION
IAST SPAN?
OVER DECK TYPES
CREATE
SUMMARY TABLE
FOR BENT NO. 1
YES
SELECT OPTIMUM
IS THIS SPAN
BRIDGE STRUC.
NUMBER 1?
YES
CONSIDER THE
NEXT SPAN
END
OPTIMIZATION
OVER SPAN LENGTH
FIGURE 4.1 MACROFLOW CHART FOR THE ALGORITHM
5 EXAMPLE PROBLEMS
In this section the application of the optimization method
developed and the resulting algorithm are demonstrated.
To use
the algorithm, it is necessary to adopt sets of criteria for the
structural design of the deck and the bent components of the bridge.
In the following examples the design specifications of the Louisiana
Department of Highways (28) are adhered to.
To implement the structural design of the bridge components
efficiently, two computer programs developed at the Louisiana
Department of Highways were utilized.
The deck design program is
used in the analysis, preliminary design, and cost evaluation of
composite bridge decks, consisting of AASHO prestressed concrete
girders and reinforced concrete slabs.
The bent design program gene­
rates minimum cost pile or column bents for the total applied dead
and live load.
The required data by both design programs include
information such as properties of structural elements to be considered,
strength levels, terrain geometry, bridge geometry, soil conditions
and unit costs.
The design theory for both programs is based on the
American Association of State Highway Officials (1) and the Louisiana
Department of Highways (28) specifications.
The design theory and
the details of the deck design computer program are fully explained
in a publication by the Louisiana Department of Highways (29).
The first example problem considered is a small bridge of three
spans.
With the exception of the structural design of the deck and the
56
57
bent components, the problem is worked by hand.
Every step of the
algorithm such as construction of Dynamic Programming tables, branch
absorption, and optimization of each stage over deck types and span
lengths is demonstrated.
In the subsequent two example problems,
implementation of the algorithm is accomplished through the use of
the computer program developed.
In these problems, only the problem
statement, selected parameters and the final results are presented.
5.1
Three Span Bridge
Consider the three span bridge of Figure 5.1.
The general data
for the prestressed concrete girders and the reinforced concrete slab
for the deck components are summarized in Table 5.1.
the data required in the design of pile bents.
Table 5.2 gives
Table 5.3 summarizes
the information on ground profile and soil data obtained from a typical
boring.
The ranges for the parameters considered that affect bridge
economy are given in Table 5.4.
The algorithm starts by designing and suboptimizing all bridge
decks in the specified range of minimum to maximum span length.
Table 5.5 summarizes the information on the suboptimized decks.
For span number 1, equations (3.2) through (3.5) generate all
values of state variable
as given below:
Beginning of Bridg
End of Bridg
180'
sta. 24+76.
Bent No. 1
Span 3
Span 2
Span 1
Compacted Fill
sta. 26+55.
Bent No. 2
Bent No. 3
Compacted Fill
Bent No. 4
26'-6
l'-8
l'-3
Clear Roadway
Elevation 50.0
prestressed concrete girders
—
I
I
FIGURE 5.1 LAYOUT AND DECK CROSS SECTION OF THE 3 SPAN BRIDGE
pile bent
59
50
-CD
_X^
= Max.
= 50,
\
180 - (3-1)(80) = 20
80
(M)
X1
= Min.
= 80,
\
j
180 - (3-1)(50) = 80
M = _8^ r _ 50_
M
10
=
’
(1)
Xx
= 50,
(2)
Xx
= 50 + (2-1)(10) = 60,
(3)
X
= 50 + (3-1)(10) = 70,
(4)
Xx
= 80.
As the problem is an allocation boundary value problem, performing a
decision inversion results in;
(1 )
Dx
(1)
= Xx
(2 )
Dx
(2 )
= Xx
(3)
(3)
D,
= XL
(4)
D1
= 50,
- 60,
= 70,
(4)
" X,
- 80.
Using equations (3.11) through (3.14) and for every value of the state
variable X^; the following sets of values for the decision variable
are obtained;
60
(1)
for Xj_
= 50;
50
(1)
D2
= Max.
= 50,
180 - 50 - ( 3 - 1 - 1 X 8 0 ) = 50
80
(k2)
D2
= 80.
= Min.
180 - 50 - ( 3 - 1 - 1 X 5 0 )
80 - 50
2=
. ,
IT- + 1 = 4’
(1)
D2
= 50,
(2 )
D2
o
50 + ( 2 - 1 ) ( 1 0 ) = 60,
(3 )
D2
= 50 + ( 3 - 1 ) ( 1 0 ) = 70,
(4)
D2
= 8 0 ..
= 60;
For
50
(1)
D2
= 50,
= Max.
180 - 60 - ( 3 - 1 - 1 ) ( 8 0 ) « 40
80
(Ka)
D2
- 70,
= Min.
180 - 60 - ( 3 - 1 - 1 ) ( 5 0 )
_
v
2
70 ..-„.50,.. + i = 3
10
3 ’
61
(1 )
D2
= 50,
(2 )
D2
= 50 + (2-1)(10) = 60,
(3)
D2
= 70.
It is obvious that as the value of X^ increases the number of values
and the maximum value of decision variable D2 decrease.
for
= 70;
= 50,
Therefore,
= 60, and for X ^ ^ = 80; there is only
one value for D2 which is;
= 50.
With the values generated for X^,
and D2> Table 5.6, which is
the Dynamic Programming Summary table of Stage (1,1), bent number 1
in Span 1, is constructed.
The values of the return variable r1 1
1,1
(bent cost) represent optimized returns of stage (1,1); that is, opti­
mization over d., 1 has already been performed as the bent design pro1,1
gram selects minimum cost bents.
*
The d^ ^ column in Table 5.6
represents properties of the selected bents in the following format;
pile diameter/ no. of piles/ penetrated length of pile/ height of bent
above ground.
Table 5.7 is the summary table for the optimized returns of bent 2
in span 1, stage (1,2).
Absorption of stages (1,1) and (1,2) into
Stage 1 (deck stage) results in Table 5.8.
Table 5.9 is then obtained
by selecting the minimum cost entries and the corresponding rows from
Table 5.8 (optimization over deck types or decision variable d^).
As
Stage 1 is the most downstream stage and a decision inversion has been
performed, no optimization over span length is necessary.
62
For span number 2, the values of state variable
are calculated
by using equations (3.2) through (3.5) again, as shown below.
2(50) = 100
(1)
= 100,
= Max.
180 - (3-2)(80) = 100
(M)
X2
= Min.
2(80) = 160
= 130,
180 - (3-2)(50) = 130
M-
-13-Q - 100
+1
10
= 4
(1)
X2
= 100,
(2)
X2
= 100 + (2-1)(10) = 110,
(3)
X2
= 100 + (3-1)(10) = 120,
(4)
X2
= 130.
For every value of state variable X^, the decision sets D2 and
are determined by using equations (3.7) through (3.14).
For X ^ ^ = 100,
the following results;
50
(1)
D2
= Max.
= 50,
100 - (2-1)(80) = 20
80
»»)
D2
= Min.
= 50.
100 - (2-1)(50) = 50
Obviously there is only one value for D2 corresponding to X^(1 ) = 100.
For this value of X2> the set of values for decision
as follows;
is calculated
63
50
(1)
= 80,
= Max.
180 - 100 - (3-2-1)(80) = 80
i80
(K )
D3
= Min.
180 - 100 - (3-2-1)(50) = 80
_ 100.
corresponding to X^(1) =
Here again there is only one value for
Considering
(2 ) =
110, one obtains,
(1 )
(1)
»2
k3
= 60,
= 50’
K2 = 2,
°3
(Ka)
= 70’
D,
= 70,
= 1.
(3)
For X2
= 120, the following is obtained;
(2 )
(1)
d2
=50,
D2
(1)
(3)
= 60,
= 70,
K2 = 3,
D3
= 60,
(K3)
60,
*3-1.
(U)
And, finally, for X2
(1 )
D,
= 50,
>2
(1)
D3
= 50,
= 130, one obtains,
(2 )
D2
=60,
(K )
D,
= 50,
(3)
D,
=70,
k3
(4)
D2
= 80,
K2 = 4,
= 1.
The optimized returns of stage (2,2), bent 2 in span 2, are given
in Table 5.10.
The results of branch absorption, optimization over
deck types (decision variable d2) and optimization over span length
(decision variable D2 ) are presented in Tables 5.11 through 5.13.
Proceeding to span 3, state variable X3 has a specified value equal
to the length of the bridge.
The values for the decision set D3 are
given as;
= 50,
D^2) = 60,
D^3) = 70,
D f
= 80.
Table 5.14 is the summary table for the optimized returns of stage
(3.2).
Results of branch absorption are presented in Table 5.15.
As
stage 3 is the last stage of the model, instead of optimization over
decision variables d^ and D^» Table 5.15 is ranked in order of ascend­
ing cost and the results are given in Table 5.16.
The first entry in
the cost column of Table 5.16 represents the optimum bridge cost and
all other cost entries correspond to the near optimum bridges.
The optimum bridge configuration can now be determined by tracing
through the optimized summary tables.
the best decisions from Table
D* = 60, d* = 2, X2 For
To demonstrate the procedure;
5.16 are:
X3 = X3-D3 = 120, d * 2 = 16/4/38.0/0.0.
= 120, the corresponding values of best decisions from Table
5.13 are;
D* = 60, d* = 2, X 1 = X2 = X2-D2= 60, d* 2 = 16/6/35.0/4.7.
with X^ = 60, the following is obtained from Table 5.9:
D* = 60, d* = 2, X^ =
Finally from Table 5.6 for
= 0, d* 2 =
16/6/35.0/4.7.
= 60, the best bent type for bent
1 in
span 1 is obtained as,
d* 1 = 16/4/38.0/0.0.
A summary of the preceding information on the optimum bridge
is presented in Table 5.17.
In addition, the total optimum bridge cost
and the calculated individual component costs are also given.
Table 5.1
General Data For Bridge Decks
Concrete Slab Data
f'c = 3000 psi
unit
cost of concrete
fs
unit
cost of reinforcingsteel = 0.16 $/lb.
= 20,000 psi
Min. Main Reinforcing Steel Size
Unit weight of curb and
=
=
#5 Bars
hand rail
=
90 $/cu. yd.
Max. Main Reinforcing Steel Size
=
#6 Bars
303lb./L.F.
Prestressed Concrete Girder Data
AREA OF
GIRDER
(in2)
WIDTH OF
TOP FLANGE
(in)
WIDTH OF
BOTTOM
FLANGE
(in)
DEPTH OF
GIRDER
(in)
MOMENT
OF
INERTIA
(in4)
SEC. MOD.
FOR BOTTOM
FIBERS
(in3)
I
276
12
16
28
22,750
1807.0
288
14.00
II
369
12
18
36
50,980
3220.5
384
18.00
III
560
16
22
45
125,390
6186.0
583
23.00
IV
789
20
26
54
260,730
10543.1
822
29.00
AASHO
GIRDER
TYPE
WEIGHT OF
GIRDER
(lb/LF)
UNIT COST
OF GIRDER
($/LF)
CT>
Ln
75.00
75.00
0 0 'Si
75.00
o
o
% Steel In
Pile Cap
m
m
m
m
m
m
CM
CM
CM
CM
CM
CM
Pile Cap
Thic. (Ft.)
O
CM
o•
o•
O
CM
CM
•
CM
2.25
2.25
Pile Cap
Width (Ft.)
2.33
m•
3.16
m
o•
CM
o
o
o
o
oo
O
O
o
o
o
o
O
O
CM
i—l
cn
l-l
M t
m
o
m
00
o
M l-
M l
Allow. Unsupp­
orted Pile
Length (Ft.)*
00
r“4
o
cn
CM
m
Pile Unit
Weight (lb/LF..)
o
r-»
VO
CM
00
cn
nt-4
on
CM
oo
i-i
CM
M t
M l-
VO
Pile Length
(Ft.)
M l
O
on
oo
o
00
vo
Pile Perimeter
(In.)
vo
m
Pile Area
(In.)
vo
OV
i— i
Pile Diameter
(In.)
M li —i
CM
00
m
n-
CM
OV
O
CM
in
r4
r4
O
on
cn
cn
o
1-4
M lVO
CM
r>»
o
00
VO
OV
m
CM
i—i
o
CM
CM
VO
1-1
00
o
rH
CM
diameter
VO
M l-
m
CM
m
on L/d
m
M l-
o
*Based
VO
400
r-4
of pile/pile
CM
Recommended
Max. Pile Load
(Tons)
1-4
u~i
M l
height
l—1
1^.
Freestanding
Is-.
•
cn
•
=
Pile Unit
Cost ($/LF.)
324
to
tu
i-i
i-i
P4
a)
■u
a)
M
o
a
o
o
'd
a)
CO
to
a)
M
4J
CO
a)
14
CM
00 'SL
CM
•
in
i
—0)i
£>
tfl
H
X!
•U
•H
s
CO
4J
a
<u
pq
a)
i—i
•r4
P-(
»4
O
h
to
4J
CO
a
i-i
tO
M
a)
a
a)
o
Unit Cost of
Cone, in Pile
Cap ($/cu.yd.)
2.16
66
cn
CM
cn
Table 5.3
Soil and Ground Profile Data
for 3 Span Bridge
Ground Profile Data
Station
24
25
25
25
26
26
26
27
+
+
+
+
+
+
+
+
50.00
0.00
50.00
80.00
0.00
50.00
80.00
0.00
Elevation
50.0
44.5
4o.o
40.0
41.8
46.3
49.0
49.0
Assume the following soil data are typical for the site
Boring station = 25 + 50.00
Boring Elevation = 20.6
Factor of safety on soil = 3
Soil Data
Tip Bearing
Coefficient
Type
of
Soil
Thickness
(Ft.)
QU 2
(Tons/Ft )
5
0.25
0
Clay
10
0.50
2.5
Clay
30
1.0
4.0
Clay
20
0.5
20.0
Sand
68
Table 5.4
Specified Optimization Parameters
for 3 Span Bridge
Total Bridge Length = 180 feet.
Number of spans = 3,
Miminum span length = 50 feet,
Maximum span length = 80 feet,
Incremental span length = 10 feet,
Minimum slab thickness = 6.5 inches,
Maximum slab thickness = 8 inches,
Incremental slab thickness = 0.5 inches,
Minimum number
of girders = 4,
Maximum number
of girders = 7,
Types of girders = AASHO Type II, III, and IV,
Minimum number
of piles in bent
=2,
Maximum number
of piles in bent
=
8,
Types of piles = 16, 18, and 24 inch diameter prestressed concrete piles.
Table 5.5
Deck Table
Span
Length
Deck
Type
No. of
Girders
Types of
Girders
Slab
Thickness
Deck
Reaction
Deck
Cost
50
50
60
60
70
70
80
1
2
1
2
1
2
1
4
4
5
5
4
4
5
2
2
2
2
3
3
3
7.0
7.5
6.5
7.0
7.0
7.5
6.5
142.74
147.41
175.59
181.18
226.63
233.15
276.98
8249.05
7920.56
10153.38
10062.82
12700.71
12337.93
15322.46
Table 5.6
Summary Table for Optimized Returns of Stage (1,1)
X1
D1
dl
S 0,2,l
s l,l,l
S l,l,2
50
50
60
60
70
70
80
50
50
60
60
70
70
80
1
2
1
2
1
2
1
0
0
0
0
0
0
0
142.74
147.41
175.59
181.18
226.63
233.15
276.98
117.1
171.1
121.6
121.6
124.8
124.8
127.2
S 0,2,2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
d*
1,1
16/3/42.0/0.0
16/3/42.5/0.0
16/4/37.5/0.0
16/5/38.0/0.0
16/4/42.0/0.0
16/4/42.5/0.0
16/5/39.5/0.0
f l,l
1847.62
1862.62
2067.62
2087.62
2247.62
2267.62
2522.62
Table 5.7
Summary Table for Optimized Returns of Stage (1,2)
50
50
50
50
50
50
50
50
50
50
50
50
50
50
60
60
60
60
60
60
60
60
60
60
60
60
70
^1
_1
1 1
50
50
50
50
50
50
50
50
50
50
50
50
50
50
60
60
60
60
60
60
60
60
60
60
60
60
70
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
h
_2
d
sM , i
S2.1.2
S1.2.2 S2.1.2
1*1
50
50
50
50
60
60
60
60
70
70
70
70
80
80
50
50
50
50
60
60
60
60
70
70
70
70
50
1
1
2
2
1
1
2
2
1
1
2
2
1
1
1
1
2
2
1
1
2
2
1
1
2
2
1
142.74
147.41
142.74
147.41
142.74
147.41
142.74
147.41
142.74
147.41
142.74
147.41
142.74
147.41
175.59
181.18
175.59
181.18
175.59
181.18
175.59
181.18
175.59
181.18
175.59
181.18
226.63
142.74
142.74
147.41
147.41
175.59
175.59
181.18
181.18
226.63
226.63
233.15
233.15
276.98
276.98
142.74
142.74
147.41
147.41
175.59
175.59
181.18
181.18
226.63
226.63
233.15
233.15
142.74
121.6
121.6
121.6
16/5/36.0/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.0/3.8
16/5/38.5/3.8
16/5/38.5/3.8
16/5/39.0/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/37.0/3.8
16/6/39.0/3.8
16/6/39.5/3.8
16/5/37.5/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/6/34.5/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/37.5/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/35.0/5.6
121.6
124.6
124.6
124.6
124.6
128.8
128.8
128.8
128.8
135.2
135.2
124.6
124.6
124.6
124.6
128.8
128.8
128.8
128.8
135.2
135.2
135.2
135.2
128.8
1*1
'
2535.12
2560.12
2560.12
2560.12
2635.12
2660.12
2660.12
2685.12
2942.62
2942.62
2942.62
2972.62
3092.62
3122.62
2655.12
2680.12
2680.12
2680.12
2876.62
2906.62
2906.62
2906.62
3056.62
3086.62
3086.62
3086.62
2960.62
o
Table 5.7 (Continued)
Summary Table for Optimized Returns of Stage (1,2)
h
^1
d
_l
70
70
70
70
70
70
70
80
80
70
70
70
70
70
70
70
80
80
2
1
2
1
2
1
2
1
1
V
D1
0
0
0
0
0
0
0
0
0
h
50
50
50
60
60
60
60
50
50
1
2
2
1
1
2
2
1
2
Sl,2,l
S2.1,2
S1.2.2~S2,1,2
233.15
226.63
233.15
226.63
233.15
226.63
233.15
276.98
276.98
142.74
147.41
147.14
175.59
175.59
181.18
181.18
142.74
147.41
128.8
128.8
128.8
135.2
135.2
135.2
135.2
135.2
135.2
h
d
16/6/35.0/5.6
16/6/35.0/5.6
16/6/35.5/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/6/38.0/6.0
16/6/38.0/6.0
fl,2
2960.62
2960.62
2990.62
3080.62
3080.62
3080.62
3110.62
3167.62
3167.62
Table 5.8
Branch Absorption Summary Table for Span
x,
1
Dn
1
d.1
50
50
50
50
50
50
50
50
50
50
50
50
50
50
60
60
60
60
60
60
60
60
60
60
60
60
70
70
70
50
50
50
50
50
50
50
50
50
50
50
50
50
50
60
60
60
60
60
60
60
60
60
60
60
60
70
70
70
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
X, -I
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
n
_2
d„
_2
50
50
50
50
60
60
60
60
70
70
70
70
80
80
50
50
50
50
60
60
60
60
70
70
70
70
50
50
50
1
1
2
2
1
1
2
2
1
1
2
2
1
1
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
d?1,2
o
16/5/36.0/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.0/3.8
16/5/38.5/3.8
16/5/38.5/3.8
16/5/39.0/3.8
16/5/36.5/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/37.0/3.8
16/6/39.0/3.8
16/6/39.5/3.8
16/5/37.5/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/6/34.5/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/37.5/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/35.0/5.6
16/6/35.0/5.6
16/6/35.0/5.6
12631.79
12343.30
12656.79
12343.30
12731.79
12443.30
12756.79
12468.30
13039.29
12725.80
13039.29
12755.80
13189.29
12905.80
14876.13
14830.57
14901.13
14830.57
15097.63
15057.07
15127.63
15057.07
15277.63
15237.07
15307.63
15237.07
17908.95
17566.18
17908.95
Table 5.8 (Continued)
Branch Absorption Summary Table for Span 1
X,
_1
D
_!
d,
_1
X -D,
1 1
D
_2
d„
_2
70
70
70
70
70
80
80
70
70
70
70
70
80
80
2
1
2
1
2
1
1
0
0
0
0
0
0
0
50
60
60
60
60
50
50
2
1
1
2
2
1
2
d, .
Ij2
16/6/35.5/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/6/38.0/6.0
16/6/38.0/6.0
f, ,+ r. + f, _
1.1
1
1.2
17596.18
18028.95
17686.18
18028.95
17716.18
21012.70
21012.70
-s--------«----«---
"4
Table 5.9
Summary Table for Optimized Returns of Stage 1
X
11
D,
_i
_i
X, -D,
1 1
D„
_2
d„
_2
50
50
50
50
50
50
50
60
60
60
60
60
60
70
70
70
70
80
80
50
50
50
50
50
50
50
60
60
60
60
60
60
70
70
70
70
80
80
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50
50
60
60
70
70
80
50
50
60
60
70
70
50
50
60
60
50
50
1
2
1
2
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
d*
,
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.5/3.8
16/5/39.0/3.8
16/6/36.5/3.8
16/6/37.0.3.8
16/6/39.5/3.8
16/5/38.0/4.7
16/5/38.0/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/35.0/5.6
16/6/35.5/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/6/38.0/6.0
16/6/38.0.6.0
f,
1
12343.30
12343.30
12443.30
12468.30
12725.80
12755.80
12905.80
14830.57
14830.57
15057.07
15057.07
15237.07
15237.07
17566.18
17596.18
17686.18
17716.18
21012.70
21012.70
Table 5.10
Summary Table for Optimized Returns of Stage (2,2)
^2
A
100
100
110
110
110
110
110
110
110
110
120
120
120
120
120
120
120
120
120
120
120
120
130
130
130
130
130
130
130
50
50
50
50
50
50
60
60
60
60
50
50
50
50
60
60
60
60
70
70
70
70
50
50
50
50
60
60
60
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
V
D2
50
50
60
60
60
60
50
50
50
50
70
70
70
70
60
60
60
60
50
50
50
50
80
80
80
80
70
70
70
A *
°a
fa
S2,2,l
S3,l,l
S2,2,2 S3,1,2
2,2
80
80
70
70
70
70
70
70
70
70
60
60
60
60
60
60
60
60
60
60
60
60
50
50
50
50
50
50
50
1
1
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
142.74
147.41
142.74
147.41
142.74
147.41
175.59
181.18
175.59
181.18
142.74
147.41
142.74
147.41
175.49
181.18
175.59
181.18
226.63
233.15
226.63
233.15
142.74
147.41
142.74
147.41
175.59
181.18
175.59
276.98
276.98
226.63
226.63
233.15
233.15
226.63
226.63
233.15
233.15
175.59
175.59
181.18
181.18
175.49
175.59
181.18
181.18
175.59
175.59
181.18
181.18
142.74
142.74
147.41
147.41
142.74
142.74
147.41
135.2
135.2
128.8
128.8
128.8
128.8
135.2
135.2
135.2
135.2
124.6
124.6
124.6
124.6
128.8
128.8
128.8
1 8.8
135.2
135.2
135.2
135.2
121.6
121.6
121.6
121.6
124.6
124.6
124.6
16/6/38.0/6.0
16/6/38.0/6.0
16/6/35.0/5.6
16/6/35.0/5.6
16/6/35.0/5.6
16/6/35.5/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/5/37.5/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/6/34.5/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/37.5/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/5/36.0/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.0/3.8
16/5/38.5/3.8
16/5/38.5/3.8
f2,2
3167.62
3167.62
2960.62
2960.62
2960.62
2990.62
3080.62
3080.62
3080.62
3110.62
2655.12
2680.12
2680.12
2680.12
2876.62
2906.62
2906.62
2906.62
3056.62
3086.62
3086.62
3086.62
2535.12
2560.12
2560.12
2560.12
2635.12
2660.12
2660.12
Table 5.10 (Continued)
Summary Table for Optimized Returns of Stage (2,2)
X
_2
D
_2
d
_2
130
130
130
130
130
130
130
60
70
70
70
70
80
80
2
1
2
1
2
1
1
X -D
2 2
70
60
60
60
60
50
50
D„
_3
d
_3
s_ „ ,
2.2,1
s„ , ,
3,1,1
50
50
50
50
50
50
50
2
1
1
2
2
1
2
181.18
226.63
233.15
226.63
233.15
276.98
276.98
147.41
142.74
142.74
147.41
147.41
142.74
147.41
S_
-
2,2,2
*"s _ _ _
3.1.2
124.6
128.8
128.8
128.8
128.8
135.2
135.2
d*
2,2
16/5/39.0/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/37.0/3.8
16/6/39.0/3.8
16/6/39.5/3.8
f
2,2
2685.12
2942.62
2942.62
2942.62
2972.62
3092.62
3122.62
•v l
cr>
Table 5.11
Branch Absorption Summary Table for Span 2
h
d
_2
50
50
50
50
50
50
60
60
60
60
50
50
50
50
60
60
60
60
70
70
70
70
50
50
50
50
60
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
X -D„
2 2
50
50
60
60
60
60
50
50
50
50
70
70
70
70
60
60
60
60
50
50
50
50
80
80
80
80
70
D„
_3
d
_3
80
80
70
70
70
70
70
70
70
70
60
60
60
60
60
60
60
60
60
60
60
60
50
50
50
50
50
1
1
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
1
d~
o
2,2
16/6/38.0/6.0
16/6/38.0/6.0
16/6/35.0/5.6
16/6/35.0/5.6
16/6/35.0/5.6
16/6/25.5/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/5/37.5/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/5/38.0/4.7
16/6/34.5/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/25.0/4.7
16/6/37.5/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/6/28.0/4.7
16/5/36.0/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.0/3.8
f,+ r + f
1
2
2
23759.97
23431.48
26040.23
25711.75
26040.23
25741.75
25677.30
25611.75
25677.30
25641.75
28470.34
28196.86
28495.34
28196.86
28087.07
28026.51
28117.07
28026.51
28483.13
28180.36
28513.13
28180.36
31796.87
31493.38
31821.87
31493.38
30737.68
Table 5.11 (Continued)
Branch Absorption Summary Table for Span 2
x
__2
D„
_2
_2
130
130
130
130
130
130
130
130
130
60
60
60
70
70
70
70
80
80
2
1
2
1
2
1
2
1
1
X -D„
2 2
70
70
70
60
60
60
60
50
50
D„
_3
d_
_3
2,2
50
50
50
50
50
50
50
50
50
1
2
2
1
1
2
2
1
2
16/5/38.5/3.8
16/5/38.5/3.8
16/5/39.0/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/36.5/3.8
16/6/37.0/3.8
16/6/39.0/3.8
16/6/39.5/3.8
f,+ r + f.
1 2
2
30439.12
30499.68
30464.12
30880.40
30517.62
30880.40
30547.62
31320.88
31350.88
Table 5.12
Summary Table for Optimized Returns of Stage 2
h
^2
S
V D2
Ha
Ha
100
110
110
110
110
120
120
120
120
120
120
130
50
50
50
60
60
50
50
60
60
70
70
50
50
60
60
70
70
80
80
2
50
60
60
50
50
70
70
60
60
50
50
80
80
70
70
60
60
50
50
80
70
70
70
70
60
60
60
60
60
60
50
50
50
50
50
50
50
50
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
130
130
130
130
130
130
130
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
JLO.
16/6/38.0/6.0
16/6/35.0/5.6
16/6/35.5/5.6
16/6/37.0/5.6
16/6/37.5/5.6
16/5/38.0/4.7
16/5/38.0/4.7
16/6/35.0/4.7
16/6/35.0/4.7
16/6/38.0/4.7
16/6/38.0/4.7
16/5/36.5/3.8
16/5/36.5/3.8
16/5/38.5/3.8
16/5/39.0/3.8
16/6/36.5/3.8
16/6/37.0/3.8
16/6/39.0/3.8
16/6/39.5/3.8
f2,2
23431.48
25711.75
25741.74
25611.75
25641.75
28196.86
28196.86
28026.51
28026.51
28180.36
28180.36
31493.38
31493.38
30439.12
30464.12
30517.62
30547.62
31320.88
31350.88
Table 5.13
Summary Table of Optimized Returns for Span 2
After Optimization Over Span Length
x„
2
D*
d*
2
100
110
110
120
120
130
130
50
60
60
60
60
60
60
2
2
2
2
2
2
2
X.-D*
2 2
50
50
50
60
60
70
70
D_
3
80
70
70
60
60
50
50
3
1
1
2
1
2
1
2
d*
2,2
16/6/38.0/6.0
16/6/37.0/5.6
16/6/37.5/5.6
16/6/35.0/4.7
16/6/35.0/4.7
16/5/38.5/3.8
16/5/39.0/3.8
2
23431.48
25611.75
25641.75
28026.51
28026.51
30439.12
30464.12
Table 5.14
Summary Table for Optimized Returns of Stage (3,2)
*
x
3
D
_3
d
Z3
X -D
3 3
180
180
180
180
180
180
180
50
50
60
60
70
70
80
1
2
1
2
1
2
1
130
130
120
120
110
110
100
s
3,2,1
s.
4,1,1
142.74
147.41
175.59
181.18
226.63
233.15
276.98
0.0
0.0
0.0
0.0
0.0
0.0
0.0
s„ „ „
3,2,2
117.1
117.1
121.6
121.6
124.8
124.8
127.2
S, , „
4,1,2
0.0
010
0.0
0.0
0.0
0.0
0.0
d„ „
3,2
16/3/42.0/0.0
16/3/42.5/0.0
16/4/37.5/0.0
16/4/38.0/0.0
16/4/42.0/0.0
16/4/42.5/0.0
16/5/39.5/0.0
„
f
3.2
1847.62
1862.62
2067.62
2087.62
2247.62
2267.62
2522.62
oo
Table 5.15
Branch Absorption Summary Table for Span 3
h.
_3
d„
_3
X -D„
3 3
180
180
180
180
180
180
180
50
50
60
60
70
70
80
1
2
1
2
1
2
1
130
130
120
120
110
110
100
d*
3,2
16/4/38.0/0.0
16/3/42.5/0.0
16/4/37.5/0.0
16/4/38.a/0.0
16/4/42.0/0.0
16/4/42.5/0.0
16/5/39.5/0.0
F„+ r.+ f„
2
3
3
40535.79
40247.30
40247.52
40176.96
40560.08
40247.30
41276.56
Table 5.16
Ranked Summary Table for the Entire Bridge
h
180
180
180
180
180
180
180
V
i
60
50
70
60
50
70
80
2
2
2
1
1
1
1
D3
120
130
110
120
130
110
100
A*
3,2
16/4/38.0/0.0
16/3/42.5/0.0
16/4/42.5/0.0
16/4/37.5/0.0
16/3/42.0/0.0
16/4/42.0/0.0
16/5/39.5/0.0
h
40176.96
40247.30
40247.30
40247.52
40535.79
40560.08
41276.56
Table 5.17
Optimum 3 Span Bridge Configuration
Deck
Type
No. of
Girders
SPAN NO. 1
60
2
5
AASH02
7.0
$10062.82
SPAN NO. 2
60
2
5
AASH02
7.0
10062.82
SPAN NO. 3
60
2
5
AASH02
7.0
10062.82
No. of
Piles
Penetrated
Pile Length
Pile
Diameter
Types of
Girders
Cost
of Deck
Span
Length
Slab
Thickness
Height of Bent
Above Ground
Cost
of Bent
Bent No. 1
16
4
38.0
0.0
$ 2087.62
Bent No. 2
16
6
35.0
4.7
2906.63
Bent No. 3
16
6
35.0
4.7
2906.63
Bent No. 4
16
4
38.0
0.0
2087.62
Optimum Bridge Cost
=
$40176.96
84
5.2
Six Span Bridge
Consider Figure 5.2 depicting the general layout of a six span
bridge.
Information on the bridge and terrain geometry is given in
the figure.
Tables 5.1 and 5.2 give data for the sub-structure and
super-structure components. Four soil borings were obtained and the
soil data are summarized in Table 5.18.
The bridge is to accommodate
two lanes of traffic with deck cross-section as given in Figure 5.1.
Table 5.19 gives the range of variation for the parameters considered
which affect the bridge economy.
In addition to the optimum bridge configuration, all near optimum
bridges having total cost within five percent of the optimum bridge are
also requested for this problem.
Final computer output results for the optimum and near optimum
bridges are presented in the following pages.
The Central Processing
Unit (CPU) time of execution was 3 minutes and 12 seconds in an IBM
360/65.
300'-0
COMPACTED
COMPACTED
FILL
SPAN NO. 2
SPAN NO. 1
SPAN NO. 4
SPAN NO.3
FIN. GRADE EL. 52.04
SPAN NO. 6
SPAN NO. 5
WATER EL. 36.70
EXCAVATE
EXCAVATE
M e ®
TOP BERM EL. 46.80
TOP BERM EL 46.80
150'-0‘
[CHANNEL BOTTOM
CO
332
+25
CO
+46 +62 +75
CM
333
oo
CO
+25
+50
+75
+97 334
FIGURE 5.2 GENERAL LAYOUT FOR THE SIX SPAN BRIDGE
+30
+63.5
FILL
Table 5.18
Soil Data for Six Span Bridge
3.0
1.0
4.0
2.0
4.0
Clay
Clay
Clay
Clay
Clay
Type of
Soil
0.75
0.25
0.95
0.65
1.00
CM
.
■U
pH
O' co
C
o
H
x_/
Tip Bearing
Coefficient
19.0
12.0
8.0
3.0
85.0
Thickness
(Ft.)
Type of
Soil
Boring Sta. = 332 + 8 0 . 0
Elev. Top of Boring = 30.0
Factor of Safety = 2.5
Tip Bearing
Coefficient
Boring Sta. = 332 + 10.0
Elev. Top of Boring = 45.0
Factor of Safety = 2.5
QU
2
(Tons/Ft. )
Boring No. 2
Thickness
(Ft.)
Boring No. 1
15.0
4.0
21.0
65.0
0.05
0.50
0.90
1.00
0.0
2.0
4.0
4.6
Clay
Clay
Clay
Clay
1.0
3.5
4.0
Clay
"Clay
Clay
^
g
!=>
o'a
o
wH
Type of
Soil
0.25
0.80
1.00
t=>
o '
4J
Pn
Tip Bearing
Coefficient
15.0
23.0
60.0
o
H
/— N
CNJ
«
Thickness
(Ft.)
Boring Sta. = 334 + 3 5 . 0
Elev. Top of Boring = 43.0
Factor of Safety = 2.5
Type of
Soil
Boring Sta. = 333 + 75.0
Elev. Top of Boring = 27.0
Factor of Safety = 2.5
Tip Bearing
Coefficient
Boring No. 4
Thickness
(Ft.)
Boring No. 3
19.0
13.0
65.0
0.25
0.60
1.00
1.0
2.0
4.0
Clay
Clay
Clay
CM
.
4-1
87
Table 5.19
Specified Optimization Parameters for the Six Span Bridge
Total Bridge Length = 300 feet,
Number of Spans = 6,
Minimum Span Length = 40 feet,
Maximum Span Length = 100 Feet,
Incremental Span Length = 5 feet,
Minimum Slab Thickness = 6.5 inches,
Maximum Slab Thickness = 8.0 inches,
Incremental Slab Thickness = 0.5 inches,
Minimum Number of Girders = 4,
Maximum Number of Girders = 7 ,
Type of Girders = AASH0 Type II, III and IV,
Minimum Number of Piles in Bent = 2,
Maximum Number of Piles in Bent = 8,
Types of Piles = 16, 18and 24 inch diameter prestressed concrete piles.
*
FINAL OUTPUT
*
OPTIMUM BRIDGE STRUCTURE
*
* NUMBER 1 FOR 6 SPAN BRIDGE
ft******************•*•*«*#*••
I
I
I
I
I
1
I
I SPAN 1 I SPAN 2 t SPAN 3 t SPAN 4 I SPAN 5 I SPAN 6 I
I
1
I
1
I
I
I
I
I
1
I
I
I
I
BENT 1
BENT 3
. BENT S
BENT 6
BENT 7
BENT 2
BENT 4
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
TOTAL COST OF
OECK
SPAN NO. 1
DECK NO. 1
50.0 FT.
4
AASHO 2
7.50 IN.
*
7920.56
SPAN NO. 2
OECK NO. 2
45.0 FT.
4
AASHO 2
7.50 IN.
8
7128.50
SPAN NO. 3
DECK NO. 3
50.0 FT.
4
AASHO 2
7.50 IN.
8
7920.56
SPAN NO. 4
OECK NO. 4
45.0 FT.
4
AASHO 2
7.50 IN.
8
7128.50
SPAN NO. S
OECK NO. 5
50.0 FT.
4
AASHO 2
7.50 IN.
8
7920.56
SPAN NO. 6
OECK NO. 6
60.0 FT.
5
AASHO 2
7.00 IN.
8
10062.82
STATION
OF BENT
PILE
OIAMETER
BENT NO. 1
331470.00
IB.00 IN.
s
21.0 FT.
0.0 FT.
BENT NO. 2
332*20.00
18.00 IN.
4
36.5 FT.
BENT NO. 3
332465.00
18.00 IN.
4
BENT NO. 4
333415.00
18.00 IN.
BENT NO. S
333460.00
BENT NO. 6
BENT NO. 7
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
8 '
2070.83
12.5 FT.
S
3184.52
38.5 FT.
20.9 FT.
8
3687.95
4
36.5 FT.
22.8 FT.
8
3678.95
18.00 IN.
4
39.5 FT.
20.9 FT.
8
3735.95
334410.00
18.00 IN.
5
31.0 FT.
16.3 FT.
8
3650.96
334470.00
18.00 IN.
4
43.5 FT.
0.0 FT.
8
2922.83
SPAN NO. t
SPAN NO. 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. 5
SPAN NO. 6
TOTAL COST OF THE BRIDGE
71013*44
NEAR OPTIMUM
6 SPAN BRIDGE
NUMBER
1
TOTAL BRIDGE COST IS MITHIN
5*0 PERCENT OF OPTIMUM
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
length
‘
of
TOTAL COST OF
OECK
SPAN NO. 1
OECK NO. 1
50.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. 2
OECK NO. 2
43.0 FT.
4
AASHO 2
7.50 IN.
S
7128.50
SPAN NO. 3
OECK NO. 3
30.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. 4
OECK NO. 4
43.0 FT.
4
AASHO 2
7.50 IN.
S
7128.50
SPAN NO. s
DECK NO. 5 ■
30.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. 6
DECK NO. 6
60.0 FT.
5
AASHO 2
6.50 IN.
S
10153.38
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
TOTAL COST OF
BENT
STATION
OF BENT
PILE
DIAMETER
BENT NO. I
331.70.00
18.00 IN.
5
21.0 FT .
0.0 FT.
S
2070.83
BENT NO. 2
332.20.00
18.00 IN.
4
36.S FT •
12.5 FT.
*
3184.52
BENT NO. 3
332*65.00
18.00 IN.
4
38.5 FT •
20.9 FT.
S
3687.95
BENT NO. 4
333.13*00
18.00 IN.
4
36.5 FT ■
22.8 FT.
S
3678.95
BENT NO. 5
333*60.00
18.00 IN.
4
39.5 FT •
20.9 FT.
S
3735.95
BENT NO. 6
334*10.00
18.00 IN.
5
30.5 FT .
16.3 FT.
S
3620.96
BENT NO. 7
334.70.00
18.00 IN.
4
43.5 FT .
0.0 FT.
S
2922.83
•
HEIGHT OF
BENT
SPAN NO. 1
SPAN NO. 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. 5
SPAN NO. 6
TOTAL COST OF THE BRIOGE -
71074.00
NEAR OPTIMUM
6 SPAN 8RIOGE
NUMBER
2
TOTAL BRIOGE COST IS WITHIN
5.0 PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
OTAL COST OF
DECK
span
NO* I
OECK NO. 1
60.0 FT.
5
AASHO 2
7.00 IN.
S
10062.62
span
NO. 2
OECK NO. 2
50.0 FT.
4
AASHO 2
7.50. IN.
6
7920.56
SPAN NO. 3
OECK NO. 3
45.0 FT.
4
AASHO 2
7.50 IN.
*
7126.50
SPAN NO. 4
OECK NO. 4
45.0 FT.
4
AASHO 2
7.50 IN.
*
7128.50
SPAN NO. S
OECK NO. S
50.0 FT.
4
AASHO 2
7.50 IN.
6
7920.56
SPAN NO. 6
OECK NO. 6
50.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
STATION
OF BENT
•
PILE
DIAMETER
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
BENT NO. 1
331+70.00
24.00 IN.
S
16.5 FT.
0.0 FT.
S
2223.67
8ENT NO. 2
332+30.00
t e . o o IN.
S
33.5 FT.
15.7 FT.
S
3763.62
BENT NO. 3
332+00.00
i b .o o
IN.
4
38.5 FT.
20.9 FT.
S
3687.95
BENT NO. 4
333+25.00
16.00 IN.
4
36.5 FT.
22.0 FT.
*
3644.15
BENT NO. 5
333+70.00
16.00 IN.
4
39.5 FT.
20.9 FT.
•
3735.95
BENT NO. 6
334+20.00
16.00 IN.
4
36.5 FT.
13.0 FT.
6
3213.07
BENT NO. 7
334+70.00
18.00 IN.
4
41.5 FT.
0.0 FT.
5
2826.63
SPAN NO. I
SPAN NO. 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. S
SPAN NO. 6
TOTAL COST OF THE BRIOGE -
71176.88
NEAR OPTIMUM
6 SPAN BRIOGE
NUMBER
3
TOTAL BRIOGE COST IS WITHIN
5.0 PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLA8
TOTAL COST OF
OECK
50.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
2
OECK NO. 2
50.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
3
OECK
•
u.
4
AASHO 2
7.50 IN.
S
7128.50
4
OECK NO. 4
50.0 FT.
4
AASHO 2
7.50 IN.
S
7920.56
S
OECK NO. 5
60.0 FT.
5
AASHO 2
7.00 IN.
*
10062.62
6
OECK NO. 6
45.0 FT.
4
AASHO 2
7.50 IN.
*
7128.50
•
•
z
3
o
OECK NO. 1
o
1
NUM8ER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
STATION
OF BENT
PILE
OIAMETER
BENT NO. 1
331470.00
18.00 IN.
5
21.0 FT.
0.0 FT.
*
2070.83
BENT NO. 2
332420.00
10.00 IN.
4
37.5 FT.
12.5 FT.
S
3232.52
BENT NO. 3
332470.00
10.00 IN.
4
30.5 FT.
20.9 FT.
S
3687.95
BENT NO. 4
33341S.00
10.00 IN.
4
36.5 FT.
22.S FT.
S
3676*95
BENT NO. 5
333465.00
10.00 IN.
5
36.5 FT.
20.9 FT.
S
4257.23
BENT NO. 6
334425.00
10.00 IN.
5
34.0 FT.
11.4 FT.
S
3534.96
BENT NO. 7
334470.00
10.00 IN.
4
41.0 FT.
0.0 FT.
%
2802.83
*
71346.75
1
2
3
4
5
6
TOTAL COST OF THE BRIDGE -
**«****»****«•****»•»»******•*•*«»**»*
NEAR OPTIMUM
6 SPAN BRIDGE
NUMBER
4
TOTAL BRIDGE COST IS VITHIN
9*0 PERCENT OF OPTIMUM
LENGTH OP
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
TOTAL COST OF
OECK
SPAN NO* 1
OECK NO* 1
50.0 FT.
4
AASHO 2
7.50 IN.
8
7920.56
SPAN NO* 2
OECK NO* 2
SO.O FT.
4
AASHO 2
7.50 IN.
%
7920.56
SPAN NO* 3
DECK NO. 3
SO.O FT*
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO* A
OECK NO* 4
45.0 FT.
4
AASHO 2
7.50 IN.
*
7128.50
SPAN NO* 5
OECK NO* S
SO.O FT*
4
AASHO 2
7.50 IN.
*
7920.56
SPAN NO. 6
OECK NO. 6
55.0 FT.
S
AASHO 2
7.00 IN.
S
9224.25
STATION
OF BENT
•
PILE
OIAMETER
NUMBER OF
PILES IN BENT
PENETRATEO
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT -
BENT NO. 1
331470*00
18.00 IN.
5
21.0 FT.
0.0 FT.
•
2070.83
BENT NO. 2
332420.00
18.00 IN.
4
37.5 FT.
12.5 FT.
S
3232*.52
BENT NO. 3
332470.00
18.00 IN.
5
34.S FT.
20.9 FT.
*
4137.23
BENT NO. 4
333420.00
18.00 IN.
4
37.0 FT.
22.4 FT.
S
3685.55
BENT NO* 5
333465.00
18.00 IN.
4
39.5 FT.
20.9 FT.
S
3735.95
BENT NO. 6
334415.00
18.00 IN.
S
31.5 FT.
14.7 FT.
*
3582.30
BENT NO* 7
334470.00
18.00 IN.
4
43.0 FT.
0.0 FT.
S
2898.83
SPAN NO* >
SPAN NO* 2
SPAN NO* 3
SPAN NO. 4
SPAN NO. 5
SPAN NO* 6
TOTAL COST OF THE BRIOGE
71376*19
NEAR OPTIMUM
6 SPAN 8R10GE
NUMBER
5
TOTAL BRIOGE COST IS WITHIN
S.O PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
TOTAL COST OF
DECK
SPAN NO. 1
OECK NO. 1
SO.O FT.
4
AASHO 2
7.50 IN.
S
7920.S6
SPAN NO. 2
OECK NO. 2
SO.O FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. 3
OECK NO. 3
SO.O FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. A
OECK NO. 4
45.0 FT.
4
AASHO 2
7.50 IN.
S
7128.50
SPAN NO. 3
OECK NO. 5
SO.O FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO. 6
OECK NO. 6
SS.O FT.
S
AASHO 2
6.50 IN.
*
9307.27
NUMBER OF
PILES IN BENT
PENETRATEO
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
STATION
OF BENT
PILE
OtAMETER
BENT NO. 1
331470.00
18.00 IN.
S
21.0 FT.
0.0 FT.
S
2070.83
BENT NO. 2
332420.00
18.00 IN.
4
37.5 FT.
12.5 FT.
S
3232.S2
BENT NO. 3
332470.00
18.00 IN.
5
34.5 FT.
20.9 FT.
S
4137.23
BENT NO. 4
333420.00
18.00 IN.
4
37.0 FT.
22.4 FT.
S
3685.55
BENT NO. S
333465.00
18.00 IN.
4
39.S FT.
20.9 FT.
S
3735.95
BENT NO. 6
334415.00
18.00 IN.
5
31.0 FT. '
14.7 FT.
S
3552.30
BENT NO. 7
334470.00
18.00 IN.
4
42.5 FT.
0.0 FT.
S
2874.83
•
SPAN NO. 1
SPAN NO. 2
span
NO. 3
SPAN NO. 4
SPAN NO. S
SPAN NO. 6
TOTAL COST OF THE BRIOGE -
71407.19
•**********•••*•*••*•*••**.«.**«•**»*.
NEAR OPTIMUM
6 SPAN BRIOGE
NUMBER
6
TOTAL BRIOGE COST IS WITHIN
5.0 PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
TOTAL COST OF
DECK
OECK NO* I
30*0 FT.
5
AASHO 2
7.00 IN.
S
10062.82
2
OECK NO* 2
50*0 FT.
4
AASHO 2
7.50. IN.
S
7920.56
3
OECK
3
45.0 FT*
4
AASHO 2
7.50 IN.
S
7128.50
4
OECK NO* 4
45.0 FT.
4
AASHO 2
7.50 IN.
S
7128.50
9
OECK NO* 5
50.0 FT.
4
AASHO 2
7.50 IN.
*
7920.56
6
OECK
50.0 FT*
4
AASHO 2
7.00 IN.
S
8249.05
2
O
•
z
o
•
1
6
STATION
OF BENT
PILE
DIAMETER
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
331*70.00
24.00 IN.
5
16.5 FT.
0 .0 FT.
*
2223.67
BENT NO* 2
332*30.00
18.00 IN.
5
33.5 FT.
15 .7 FT.
*
3763.82
BENT NO* 3
332*80.00
18.00 IN.
4
38.5 FT.
20 .9 FT.
S
3687.95
4
333*25.00
18.00 IN.
4
36.5 FT.
22 .0 FT.
S
3644.15
BENT NO* 5
333*70.00
18.00 IN.
4
39.5 FT.
20 .9 FT.
S
3735.95
BENT NO* 6
334*20*00
18.00 IN.
4
36.0 FT.
13.0 FT.
S
3189.07
BENT NO* 7
334*70.00
18.00 IN.
4
41.5 FT.
0.0 FT.
S
2826.83
S
71481.38
Z
o
•
1
SENT
>
2
3
•
o
z
BENT
4
5
6
TOTAL COST OF THE BRIOGE -
VO
*****•*»*»**»»»*»•*»»»»*»*»»*»»***•«**
NEAR OPTIMUM
6 SPAN BRIOGE
NUMBER
7
TOTAL. BRIOGE COST IS WITHIN
S.O PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
I
OECK NO* I
30*0 FT.
4
AASHO 2
7.50 IN.
z
OECK NO* 2
30*0 FT.
4
AASHO 2
7.SO IN.
3
OECK NO* 3
43.0 FT.
4
AASHO 2
7.50 IN.
4
OECK NO* 4
SO.O FT.
4
AASHO 2
7.50 IN.
5
OECK NO* S
60.0 FT.
5
AASHO 2
7.00 IN.
6
OECK NO* 6
43.0 FT.
4
AASHO 2
7.00 IN.
NUMBER OF
PILES IN BENT
PENETRATEO
LENGTH
HEIGHT OF
BENT
STATION
OF BENT
PILE
DIAMETER
BENT NO* 1
331+70*00
IB.00 IN.
3
21.0 FT.
0 .0 FT.
BENT NO* 2
332420.00
18*00 tN.
4
37.5 FT.
12.5 FT.
BENT NO* 3
332470.00
18.00 IN.
4
38.5 FT.
20 •9 FT.
BENT NO* 4
333415.00
IB.00 IN*
4
36.5 FT.
22 .8 FT.
BENT NO* 3
333465*00
16.00 IN.
3
36.5 FT.
20 •9 FT.
BENT NO* 6
334425.00
18.00 IN.
3
34.0 FT.
It .4 FT.
BENT NO* 7
334470.00
18.00 IN.
4
40.5 FT.
0• 0 FT.
I
£
3
4
5
6
TOTAL COST OF THE BRIOGE -
NEAR OPTIMUM
6 SPAN BRIOGE
NUMBER
8
TOTAU BRIDGE COST IS WITHIN
5.0 PERCENT OF OPTIMUM
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIROERS
THICKNESS OF
SLAB
length of
TOTAL COST OF
OECK
SPAN NO. 1
DECK NO. I
SO.O FT.
4
AASHO 2
7.50 IN.
*
7920.56
SPAN NO. 2
OECK NO. 2
SO.O FT.
4
AASHO 2
7.50 IN.
8
7920.56
SPAN NO. 3
DECK NO. 3
SO.O FT.
4
AASHO 2
7.50 IN.
8
7920.56
SPAN NO. A
OECK NO. 4
60.0 FT.
S
AASHO 2
7.00 IN.
8
10062.82
SPAN NO. 5
OECK NO. 5
SO.O FT.
4
AASHO 2
7.SO IN.
8
7920.56
SPAN NO. 6
DECK NO. 6
40.0 FT.
4
AASHO 2
7.50 IN.
8
6336.45
•
STATION
OF 8ENT
PILE
DIAMETER
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
BENT NO. I
331470.00
18.00 IN.
5
21.0 FT.
0.0 FT.
8
2070.83
BENT NO. 2
332420.00
18.00 IN.
4
37.5 FT.
12.5 FT.
8
3232.52
BENT NO. 3
332470.00
18.00 IN.
S
34.5 FT.
20.0 FT.
8
4137.23
BENT NO. 4
333420.00
18.00 IN.
5
34. 5 FT.
22.4 FT.
8
4224.23
BENT NO. 5
333480.00
18.00 IN.
5
36.5 FT.
20.9 FT.
8
4257.23
BENT NO. 6
334430.00
18.00 IN.
4
37.5 FT.
9.8 FT.
8
3103.20
BENT NO. 7
334470.00
18.00 IN.
4
40.0 FT.
0.0 FT.
8
2754.83
SPAN NO. 1
SPAN NO. 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. 5
SPAN NO. 6
TOTAL COST OF THE BRIDGE -
71861.56
*
NEAR OPTIMUM
*
•
•
6 SPAN BRIOGE
NUMBER
9
TOTAL BRIOGE COST IS WITHIN
5*0 PERCENT OF OPTIMUM
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
TOTAL COST OF
OECK
SPAN NO* t
OECK NO* 1
SO.O FT.
4
AASHO 2
7.50 IN.
S
7920.56
SPAN NO* 2
OECK NO* 2
50*0 FT.
4
AASHO 2
7.30 IN.
9
7920.56
SPAN NO. 3
OECK NO. 3
SO.O FT.
4
AASHO 2
7.SO IN.
S
7920.56
SPAN NO* 4
OECK NO* 4
60*0 FT.
S
AASHO 2
7.00 IN.
S
10062.02
SPAN NO* S
OECK NO* S
SO.O FT.
4
AASHO 2
7.S0 IN.
S
7920.56
SPAN NO* 6
OECK NO. 6
40.0 FT.
-4
AASHO 2
7.00 IN.
•
6599.23
STATION
OF BENT
•
PILE
DIAMETER
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
TOTAL COST OF
BENT
BENT NO. 1
3 3 1 + 7 0 .0 0
ta .o o
IN.
5
2 1 .0
FT.
0 .0
FT.
•
2 0 7 0 .6 3
BENT NO. 2
3 3 2 + 2 0 .0 0
IB. 00 IN.
4
3 7 .5
FT.
1 2 .5
FT.
*
3 2 3 2 .5 2
BENT NO. 3
3 3 2 + 7 0 .0 0
ta .o o
IN.
S
34.S FT.
2 0 .9
FT.
S
4 1 3 7 .2 3
BENT NO. 4
3 3 3 + 2 0 .0 0
ta .o o
IN.
5
34.S FT.
2 2 .4
FT.
s
4 2 2 4 .2 3
BENT NO. S
3 3 3 + 8 0 .0 0
1 8 .0 0
IN.
5
3 6 .5
FT.
2 0 .9
FT.
s
4 2 5 7 .2 3
BENT NO. 6
3 3 4 + 3 0 .0 0
ta .o o
IN.
4
3 7 .5
FT.
9 .8
FT.
9
3 1 0 3 .2 0
BENT NO. 7
3 3 4 + 7 0 .0 0
ta .o o
IN.
4
3 9 .5
FT.
0 .0
FT.
9
2 7 3 0 .8 3
S
72100*31
SPAN NO. 1
SPAN NO. 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. S
SPAN NO. 6
TOTAL COST OF THE BRIOGE -
VC
98
5.3
Five Span Constrained Bridge
The bridge structure considered here is the one presented in
Section 5.2 with the exception that it has 5 spans and is constrained
as follows:
Spans 1 and 5 are specified to have a 40 Ft . length;
that is the location of bent number 2 for spans 1 and 4 is specified.
In addition, to minimize the channel obstruction, the center line of
span number 3, the center span, is specified to be 150 feet from
either end of the bridge, with a minimum span length of 60 feet.
the other spans, the specified minimum length is 40 feet.
For
Figure 5.3
presents the general layout of the bridge with the imposed constraints.
Other necessary data are given in Tables 5.1, 5.2, 5.18 and 5.19.
computer output is presented in the following pages.
execution was 19 seconds in an IBM 360/65.
The
The CPU time of
300'-0
150'
40'
40'
COMPACTED
COMPACTED
SPAN NO. 1
SPAN NO. 3
SPAN NO. 2
FILL
SPAN NO. 5
SPAN NO. 4
FILL
FIN. GRADE EL. 52.04
WATER EL. 36.70
EXCAVATE
F. L. EL. 24.6
TOP BERM EL. 46.80
TOP BERM EL. 46.80
150'-0'
(CHANNEL BOTTOM)
*o
CO
332
+25
CO
+46 +62 +75
CM
333
oo
CO
CM
+25
CO
+50
CO
co
+75 +97 334
FIGURE 5.3 GENERAL LAYOUT OF THE FIVE SPAN BRIDGE
OO
+30
+63.5
VO
VO
•
FINAL OUTPUT
*
*
OPTIMUM BRIOGE STRUCTURE
* NUMBER I FOR S SPAN BRIOGE
****»»*••***♦*•**»»*»*****«*»
1
I
I
I
I
I
I
I
I SPAN 2 I SPAN 3 t SPAN 4 I SPAN S I
I
I
1
I
I
I
I
I
I
I
BENT 6
BENT S
BENT 4
BENT 3
BENT 2
SPAN I
t
BENT I
LENGTH OF
SPAN
NUMBER OF
GIRDERS
TYPE OF
GIRDERS
THICKNESS OF
SLAB
OTAL COST OF
OECK
SPAN NO* 1
OECK NO* t
40*0 FT.
4
AASHO 2
7.50 IN.
6336.45
SPAN NO* 2
OECK NO. 2
70.0 FT.
4
AASHO 3
7.50 IN.
12337.93
SPAN NO* 3
DECK NO* 3
80.0 FT.
S
AASHO 3
6.50 IN.
15322.46
SPAN NO* 4
OECK NO* 4
70.0 FT.
4
AASHO 3
7.50 IN.
12337.93
SPAN NO* 5- OECK NO* S
40.0 FT.
4
AASHO 2
7.50 IN.
6336.45
NUMBER OF
PILES IN BENT
PENETRATED
LENGTH
HEIGHT OF
BENT
OTAL COST OF
BENT
STATION
OF BENT
PILE
DIAMETER
BENT NO. I
331470*00
18.00 IN.
S
17.5 FT.
0.0 FT.
1860.83
BENT NO. 2
332*10.00
18.00 IN.
S
36.0 FT.
9.2 FT.
3522.05
BENT NO* 3
332*80.00
24.00 IN.
4
43.5 FT.
20.9 FT.
4712.30
BENT NO* 4
333*60.00
24.00 IN.
4
44.5 FT.
20.9 FT.
4768.30
BENT NO* S
334*30.00
18.00 IN.
S
37.0 FT.
9.8 FT.
3616.30
BENT NO* 6
334*70.00
18.00 IN.
4
40.0 FT.
0.0 FT.
2754.83
SPAN NO* I
span
NO* 2
SPAN NO. 3
SPAN NO. 4
SPAN NO. 5
TOTAL COST OF THE BRIOGE -
7390S.S1
6.
SUMMARY AND CONCLUSIONS
This investigation has dealt with a study of the minimum cost
design of simple span deck and girder highway bridges.
The optimization
method employed was Dynamic Programming.
As a first step in the investigation, the Dynamic Programming model
of a general highway bridge structure was developed.
All bridge com­
ponent characteristics were preserved in the corresponding stages of
the model.
Return functions representing component costs were derived
for every stage.
The interrlationships between stages were defined by
transition functions and incidence identities.
The model's structure turned out to be a branched, boundary value
allocation Dynamic Programming system.
The mathematical equations to
evaluate upper and lower bounds and all intermediate values of the state
and decision variables representing length, were subsequently derived
for every stage of the model.
The solution of the problem was obtained through model decomposi­
tion to generate optimal policies by minimizing stage returns over the
following two decisions variables:
1.
A decision variable representing component types with sufficient
structural integrity to withstand the superimposed loads.
2.
A decision variable representing allocatable lengths to each
span of the bridge structure.
Detailed procedures for model decomposition, branch absorption,
stage and branch optimization and construction of Dynamic Programming
101
102
Summary tables were described.
The solution method developed was structured as a computational
algorithm which was described in step by step and macroflow chart forms.
To demonstrate the application of the algorithm and the procedure de­
veloped, a three span bridge was given as an example problem.
Every
step of the algorithm was carried out in detail and a bridge structure
of a minimum cost was generated.
An extensive computer program was developed for fast and efficient
implementation of the algorithm.
The program is extremely flexible in
that it permits a complete economic investigation of bridge structures
with a variety of externally imposed constraints.
The two most impor­
tant capabilities of the program were demonstrated in the example prob­
lems of sections 5.2 and 5.3.
In conclusion, this investigation has shown that a systematic pro­
cedure for minimum cost design of highway bridges is possible with Dy­
namic Programming.
The procedure permits efficient and comprehensive
economic studies of bridge structures based on alternative cost evalu­
ations of both super-structure and sub-structure components simul­
taneously considering bridge geometry and soil and terrain conditions.
The method and the resulting computer program can be used:
1.
To study the economics of bridge configurations to meet finan­
cial, esthetic, and other requirements.
2.
To study the economics of newly developed structural components
and to compare them with those presently in use.
3.
To study thecost trade off between bridge openings across
waterways and the damages to upstream property or the
103
acquisition of innundation rights resulting from backwater
conditions which can develop from the contraction imposed on
the natural channel by the bridge embankments.
4.
To study bridge geometry when constrained by physical obstruc­
tions .
7.
RECOMMENDATIONS
It is recommended that the study of bridge optimization be
continued with special emphasis placed on the following:
1.
Revision of the procedure developed to include an efficient
search technique for the selection of the optimum number of
—
spans for an unconstrained bridge.
As an example, Fibonacci
Search can be used if it were ascertained that the relationship
of bridge cost to number of spans is given by a unimodal
function.
2.
Modification of the general theory to accept msss production
cost factors.
That is, as the quantity of structural elements
of the same type and size increases, a reduction factor could
be applied to the unit cost.
This would result in a more
accurate computation of total bridge cost.
3.
Development of techniques to include long span structures such
as trusses, in the total bridge optimization procedure.
4.
Dynamic Programming Optimization of continuous span bridge
structures.
The computer program should be expanded to include design modules
for specialized bridge components such as, composite steel girderconcrete slab bridge decks.
A full man-machine interactive system can
be the result of incorporating the program in a system such as ICES or
TIES.
Many other problems related to bridge structural optimization, such
as suspended and orthotropic deck bridges, merit a considerable amount
of detailed study.
104
8.
1.
References
"Standard Specification for Highway Bridges," AASHO (American
Association of State Highway Officials). 1970.
2.
"Specification for the Design, Fabrication and Erection of
Structural Steel for Building," Manual of Steel Construction,
6th ed., American Institute of Steel Construction, New York,
N.Y. 1967.
3.
Bellman, R.E. Dynamic Programming, Princeton University Press,
Princeton, New Jersey. 1957.
4.
Dreyfus, S.E. Dynamic Programming and Calculus of Variations,
Academic Press, New York, N.Y. 1965.
5.
Bellman, R.E., and S.E. Dreyfus. Applied Dynamic Programming,
Princeton University Press, Princeton, New Jersey. 1962.
6.
Aris, R. Discrete Dynamic Programming, Blaisdell Publishing Co.,
New York, N.Y. 1964.
7.
Nemhouser, G.L. Introduction to Dynamic Programming, John Wiley
and Sons, Inc., New York, N.Y. 1967.
8.
Wilde, D.J., and C.S. Beightler. Foundations of Optimization,
Prentice-Hall, Inc., Englewood Cliff, New Jersey. 1967.
9.
Aguilar, R.J. "An Application of Dynamic Programming to Structural
Optimization," Bulletin No. 91, Division of Engineering
Research, Louisiana State University, Baton Rouge, La. 1967.
105
106
10.
Cornell, C.A., P.K. Ho, and R.A. Ehrlich. "A Method for the Optimum
Design of Simple Span Deck and Girder Highway Bridges,11
Report R67-22, Department of Civil Engineering, M.I.T.
June, 1967.
11.
Latona, R.W. “Optimization of Span Arrangement for Highway
Bridges," Report R68-54, Department of Civil Engineering,
M.I.T. May, 1968.
12.
Sturman, G.M., L.C. Albertson, Jr., C.A. Cornell, and J.M.
Roesset. "Computer-Aided Bridge Design," Journal of the
Structural Division, ASCE, vol. 92, No. ST6. December, 1966.
13.
Razani, R. "Computer Analysis of Highway Bridge Girder," Journal
of the Structural Division, ASCE, vol. 93, No. ST1. February,
1967.
14.
Holt, E.C., and G.L. Heithecker. "Minimum Weight Proportions for
Steel Girder," Journal of the Structural Division, ASCE,
vol. 95, No. ST10. October, 1969.
15.
Razani, R . , and G.G. Goble.
Plate Girders," Journal
ASCE, vol. 92, No.
16.
"Optimum Design of Constant-Depth
of the StructuralDivision,
ST2. April, 1966.
Goble, G.G., and P.V. DeSantis.
"Optimum Design of Mixed Steel
Composite Girders," Journal of the Structural Division,
ASCE, vol. 92, No.
17.
Jacques, F.J. "Study of
ST6. December, 1966.
Long Span Prestressed Concrete Bridge
Girders," Journal of the Prestressed Concrete Institute,
vol. 16, No. 2. March-April, 1971.
107
18.
Ayer, F.A., and C.A. Cornell. ’'Grid Moment Maximization by Mathe­
matical Programming," Journal of the Structural Division,
ASCE, vol. 94, No. ST2. February, 1968.
19.
Bigelow, R.H., and E.H. Gaylord. "Design of Steel Frames for
Minimum Weight," Journal of the Structural Division, ASCE,
vol. 93, No. ST6. December, 1967.
20.
Hill, Jr., L.A. "Automated Optimum Cost Building Design,"
Journal of the Structural Division, ASCE, vol. 92, No. ST6.
December, 1966.
21.
Reinschmidt, R.F., C.A. Cornell, and J.F. Brotchie. "Iterative
Design and Structural Optimization," Journal of the Structural
Division, ASCE, vol. 92, No. ST6. December, 1966.
22.
Brown, D.M., and A.H.,-S. Ang. "Structural Optimization by Non­
linear Programming," Journal of the Structural Division,
ASCE, vol. 92, No. ST6. December, 1966.
23.
Moses, F. "Optimum Structural Design Using Linear Programming,"
Journal of the Structural Division, ASCE, vol. 90, No. ST6,
Part 1. December, 1964.
24.
Romstad, K.M., and C.K. Wang.
"Optimum Design of Framed Structures,"
Journal of the Structural Division, ASCE, vol. 94, No. ST12.
December, 1968.
25.
Dobbs, M.W., and L.P. Felton. "Optimization of Truss Geometry,"
Journal of the Structural Division, ASCE, vol. 95, No. ST10.
October, 1969.
108
26.
Johnson, D., and D.M. Brotton. "Optimum Elastic Design of Redun­
dant Trusses," Journal of the Structural Division, ASCE,
vol. 95, No. ST12. December, 1969.
27.
Barnett, R.L. *T)esign of Statically Determinate Trusses for
Minimum Weight and Deflection," IXT Research Institute
Project No. M6152, IIT Research Institute Technology Center,
Chicago, Illinois. May, 1967.
28.
Louisiana Department of Highways,
Bridge Design Section, 'detail
Standards," Baton Rouge, Louisiana, Unpublished Manual of
Design.
29.
Porter, James, C. "A Program for the Design and/or Analysis of
Prestressed Girders," Bridge Design Section, Louisiana
Department of Highways, Baton Rouge, Louisiana, January 1970.
9
9.1
APPENDICES
Computer Program
To implement the algorithm of section 4 an extensive computer pro­
gram was developed.
in form.
The program is written in Fortran IV and is modular
It consists of a main program and several subroutines, each of
which performs a particular task.
The program is designed to permit full
man-machine interaction not only through the input of data but also
through structural design module interchangeability, which allows the
user to incorporate his own procedures as subroutines.
The program is capable of accepting constraints on the bridge geo­
metry by specifying locations of bents, or span center lines, or both to
account for special conditions such as bridge crossings over existing
roadways and other physical obstructions.
In addition to the lowest cost
bridge configuration, the program can also generate all other bridge
structures that fall within any specified percentage of the minimum cost
(1% or 5% for example).
The range and the increment are specified by the
user.
The average Central Processing Unit (CPU) time required for a typi­
cal bridge is in the order of one minute
figuration.
per span on an IBM 360/65 con­
As the number of spans increases, the CPU time per span
also increases.
The program operates in an environment of approximately
200k of core when the search is specified at 5 foot intervals.
The core
requirements decrease to approximately 150k when search interval is
increased to 10 feet.
In the following sections the general logic of the main program
109
110
and all subroutines directly related to the algorithm are discussed.
The other subroutines in the program are either input-output or
design routines.
9.1.1
Main Program
9.1.1.1
Purpose.
The basic function of the main program is to call
on individual subroutines to perform specific tasks at appropriate
times.
It also provides a data pool for the exchange of information
between the subroutines.
9.1.1.2
Discussion of General Logic.
are specified by the main program.
table B (array TABB).
Basically two work areas
They are table A (array TABA) and
Table B represents a Dynamic Programming table
constructed for each stage of the model.
It includes six columns, each
of which contains values of state or decision variables.
A separate
cost column designated as array BCST is associated with table B.
Re­
sults of branch absorption, optimization over deck type and optimiza­
tion over span length are sequentially recorded in this table.
Sub­
sequently, the pertinant data, needed when branch absorption for the
next upstream stage is performed, is transferred to table A.
Then the
content of table B, which corresponds to a Dynamic Programming summary
table of all downstream stages, is written on the disk for further
reference.
The information on the disk is used for selecting compo­
nents of the optimum or near optimum bridge.
Table B is then reused
for construction of a Dynamic Programming table for the next upstream
stage.
Ill
Tabic A includes three columns; each column contains different
values of state or decision variables.
The cost column associated with
this table is designated as array ACST.
Figure 9.1 represents the general logic flow chart for the Main
Program.
Definitions of all the variables used in the Main Program
are given in Table 9.1.
112
START
subroutine (IDENT)
IDENTIFY THE BRIDGE
CALCULATE BEGINNING AND END OF
BRIDGE STATIONS. ESTABLISH
WHETHER THE BRIDGE IS
CONSTRAINED OR FREE. ADJUST
MIN. SPAN LENGTH IF NECESSARY
subroutine (DKDSN)
DESIGN DECK FOR EVERY SPAN
LENGTH IN THE SPECIFIED RANGE OF
MINIMUM TO MAXIMUM SPAN LENGTH.
subroutine (DKOPT)
SELECT SUBOPTIMUM DECKS FOR
EVERY SPAN LENGTH. SET UP DKTB.
SET UP EXTREME END SPANS
PROPERTIES.
CONSIDER SPAN NO. 1.
subroutine (SLC)
DETERMINE ALL BENT NO. 2
LOCATIONS AND SPAN LENGTHS FOR
SPAN NO.I DETERMINE ALL SPAN
LENGTHS FOR THE ADJACENT
UPSTREAM SPANS
r subroutine (BTDSN)
DESIGN AND OPTIMIZE BENT NO. 1
IN SPAN NO.1 FOR EVERY SPAN
LENGTH AND DECK TYPE.
Figure 9.1
Macroflow Chart for the Main Program
CONSTRUCT TABLES A AND B FOR
BENT NO. 1 IN SPAN NO. 1
subroutine (WTDSK) |
WRITE TABLE B ON DISK
subroutine (BTDSN)
DESIGN AND OPTIMIZE BENT NO. 2
IN THE SPAN UNDER CONSIDERATION
FOR EVERY BENT LOCATION, LENGTH
OF SPAN UNDER CONSIDERATION,
TYPE OF DECK FOR THE SPAN UNDER
CONSIDERATION, LENGTH OF
ADJACENT UPSTREAM SPAN* TYPE
OF DECK IN THE ADJACENT
UPSTREAM SPAN
1'
CONSTRUCT TABLE B FOR BENT
NO. 2 IN THE SPAN UNDER
CONSIDERATION
subroutine (TABABS) |
ADD CORRESPONDING COSTS FROM
TABLES A,B,AND DECK TABLE.
CONSTRUCT A NEW TALBE B.
(ABSORPTION OF STAGES)
IS
THIS
SPAN NO. 1
YES
subroutine (TABOPI)
NO
SELECT MIN. COSTS
AND CORRESPONDING
ROWS FROM TABLE B
AND WRITE IN TABLE B
CONSTRUCT TABLE A.
(OPTIMIZATION
OVER DECK TYPES)
Figure 9.1 (Continued)
subroutine (TAB0P1)
SELECT MIN. COSTS AND
CORRESPONDING ROWS FROM TABLE B.
WRITE THE SELECTION IN TABLE B.
(OPTIMIZATION OVER DECK TYPE)
subroutine (TABOP2)
SELECT MIN. COSTS AND THE
CORRESPONDING ROWS FROM TABLE B.
WRITE THE SELECTION IN TABLE B.
CONSTRUCT TABLE A.
(OPTIMIZATION OVER SPAN LENGTH)
subroutine (WTDSK)
WRITE TABLE B ON DISK
CONSIDER THE NEXT SPAN
subroutine (SLC)
DETERMINE ALL BENT NO. 2
LOCATIONS AND SPAN LENGTHS FOR
THE SPAN UNDER CONSIDERATION.
DETERMINE ALL SPAN LENGTHS FOR
THE ADJACENT UPSTREAM SPAN.
IS
THIS
THE LAST
SPAN
NO
subroutine (BTDSN)
DESIGN AND OPTIMIZE BENT NO. 2
IN THE LAST SPAN FOR EVERY SPAN
LENGTH AND DECK TYPE.
CONSTRUCT TABLE B FOR BENT
NO.2 OF THE LAST SPAN
i
Figure 9.1 (Continued)
subroutine (TABABS)
ADD CORRESPONDING COSTS FROM
TABLES A,B,AND DECK TABLES.
CONSTRUCT A NEW TABLE B.
(ABSORPTION OF STAGES)
subroutine (TABOP3)
11
______________
RANK TABLE B ACCORDING TO
TOTAL BRIDGE COST. DETERMINE
NUMBER OF NEAR OPTIMUM BRIDGES
IF REQUESTED
subroutine (WTDSK)
WRITE TABLE B ON DISK
CALCULATE NO. OF REPETITIONS
CORRESPONDING TO NO. OF
CALCULATED NEAR OPTIMUM
BRIDGES
SELECT OPTIMUM OR NEAR OPTIMUM
COST FROM TABLE B. SELECT
CORRESPONDING INFO. ABOUT THE
LAST SPAN.
RECORD THE SELECTED
INFO. AT THE BOTTOM
OF TABLE B.
TRACE THROUGH
TABLE B AND SELECT
SPAN CONFIGURATION
CORRESPONDING TO
THE OPTIMUM OR
NEAR OPTIMUM BRIDGE
subroutine (RDDSK)
READ TABLE B
OFF THE DISK
HAVE
ALL TABLES
ON THE DISK BEEN
READ
?
REMEMBER SPAN
LENGTH, DECK TYPE,
AND BENT LOCATION
FOR THIS SPAN TO
BE USED AS TEST
VALUES.
Figure 9.1 (Continued)
116
subroutine (OUTPUT)
TRANSLATE THE INFO. AT THE
BOTTOM OF TABLE B INTO FINAL
DOCUMENT FORM.
DOCUMENT
^routine (RDDSK)
ARE
NEAR
OPTIMUM BRIDGES
REQUESTED
ARE
OPTIMUM
BRIDGES FOR
DIFFERENT NO. OF
SPANS
REQUESTED
READ THE LAST TABLE
OFF THE DISK
SELECT THE NEXT NUMBER
OF SPANS.
STOP
Figure 9.1 (Continued)
Table 9.1
Main Program Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
DESCRIPTION
UNITS
ACST
Type:Real.Array
Global:Common.In/Out
Dollars
A one dimensional array containing total cost of
all considered bridge components
BBS
Type:Real
Global:Common.Out
:Argument. Out
Stations
Beginning Bridge Station
BBS1
Type:Real
Global:Argument.Out
Data:Read in
Stations
Number of whole stations in Beginning Bridge Stations
BBS 2
Type:Real
Global:Argument.Out
Data:Read in
Feet
Fraction of whole station in Beginning Bridge
Station
BCST
Type:Real.Array
Global:Common.In/Out
Dollars
A one dimensional array containing total cost of
all considered bridge components
BNTLCN
Type:Integer.% word
Feet
Bent Location
BRH
Type:Integer
None
Beginning Row. Designates beginning row of a section
in the deck table
BRHADJ
Type‘
.Integer
None
Beginning Row Adjacent. Designates beginning row
of a section in the deck table
CNST
Type:Real
None
Constant number used for calculating section
numbers in the deck table
Table 9.1 (Continued)
VARIABLE
CHARACTERISTICS
UNITS
CTLB
Type:Real
Feet
Calculated Total Length of the Bridge based on
minimum span length for all spans
D
Type:Real .Array
Global:Common.In/Out
Feet
A two dimensional array containing values of span
length corresponding to a particular bent location
DADJ
Type:Real.Array
Global:Common.In/Out
:Argument.Out
Feet
A one dimensional array containing values of span
length for the adjacent span
DCKTYP
Type:Integer:h word
None
Deck Type
DELL
Type:Real
Global:Common.Out
Data:Read in
Feet
Specified incremental span length
DKPR
Variable
Type:Real .Array
Global:Argument.In/Out
Deck Properties. A two dimensional array containing
information about different types of decks for a
given span length
DKTB
Type: Real.Array
Global:Common.In
:Argument.Out
Variable
Deck Table. A two dimensional array of suboptimized
deck information. For details see subroutine DKOPT
Variable Definition List
DPOB
Type:Real
Global:Argument.In
Inches
Diameter of Pile in the Optimum Bent
EBS
Type: Real
Global:Argument.Out
Station
End of Bridge Station
VARIABLE
NAME
DESCRIPTION
118
Table 9.1 (Continued)
VARIABLE
CHARACTERISTICS
UNITS
EBS1
Type:Real
Stations
Number of whole stations in the End of Bridge Station
EBS2
Type: Real
Feet
Fraction of whole stations in the End of Bridge
Station
ELSL
Type: Real
Global:Common.Out
:Argument.Out
Data:Read in.
Feet
Extreme Left Span Length. Length of approach span
resting on the first bent of the bridge
ELSR
Type:Real
Global:Common.Out
:Argument.Out
Data:Read in
Kips
Extreme Left span dead load Reaction
ERSL
Type:Real
Global:Common.Out
:Argument. Out
Data:Read in
Feet
Extreme Right Span Length. Length of approach
span resting on the last bent of the bridge
ERSR
Type:Real
Global:Common.Out
:Argument. Out
Data:Read in
Kips
Extreme Right Span dead load Reaction
ERW
Type:Integer
None
End Row. Designates the last row of a section in
the deck table
VARIABLE
NAME
DESCRIPTION
Table 9.1 (Continued)
VARIABLE
CHARACTERISTICS
UNITS
EBWADJ
Type:Integer
None
End Row Adjacent. Designates the last row of a
section in the deck table
FIX
Type:Logical
Global:Common.Out
Data: Alphameric
None
FIX - TRUE designates a constrained bridge
FIX 83 FALSE designates an unconstrained bridge
FIXED
Type:Alphameric
None
Designates a constrained bridge
FIXTES
Type:Alphameric
Data:Data Statement
None
Used to establish if the bridge is constrained
GUTT
Type: Real
Feet
Global:Argument.In/Out
Gutter distance of bridge decks
HTOB
Type:Real
Global:Argument.In
Height of Optimum Bent above ground line
IDKTB1
Type:Function Statement None
Used to calculate section numbers in the deck table
IDKTB2
Type:Integer.Array
Global:Common.In
None
A two dimensional array which describes the
structure of the deck table. See subroutine DKOPT
Variable Definition List, Table 9.2
INDES
Type:Integer
Data:Read in
None
Indicator for Extreme End Spans. INDES = 1
designates approach spans are specified at both
ends of the bridge. INDES = 0 means no extreme
end spans are specified
VARIABLE
NAME
Feet
DESCRIPTION
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
K
Type:Integer.Array
Global:Common.In
None
A one dimensional array representing the number
of values for D corresponding to each value of
bent location
KADJ
Type:Integer
Global:Common.In
None
Represents number of Values for DADJ
KS
Type:Integer
None
A Counter
of different values of D
KSADJ
Type:Integer
None
A Counter
of different values of DADJ
LSTROW
Type:Integer
Global:Common.In
None
Number of starting row for optimum ornear optimum
bridge data which have been placed atthe bottom
of TABB and BCST
M
Type:Integer
Global:Common.In
None
Counter, designating number of values for bent
locations
MSGR
Type:Integer
Global:Common.Out
Data.Read in
None
Message level indicator; MSGR = 0 causes no inter­
mediate data to be printed. MSGR = 1 causes all
intermediate data to be printed. MSGR = 2 causes
partial intermediate data to be printed
N
Type:Integer
None
Number of the
span under condiseration
NBLK
Type:Integer
None
Number of the
Block of generated data on the disk
NCHK
Type:Integer
Global:Common.In
None
Check indicator used in allocating total available
length to bent location and adjacent upstream
span length
UNITS
DESCRIPTION
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
NDK
Type:Integer
None
Global:Argument.In/Out
Number of rows (Decks) in array DKPR
NDT
Type:Integer
Global:Common.In
None
Absolute row number in DKTB
NFBLK
Type:Integer.Array
Global:Common.In/Out
None
Number of First Block of generated data on the
disk in a specified section
NLBLK
Type:Integer.Array
Global:Common.In/Out
None
Number of Last Block of generated data on the
disk in a spcified section
NLSEC
Type:Integer
None
Number of Last Section of generated data on the
disk
NNBK
Type:Integer
Global:Common.In/Out
None
Number of the Block of input data on the disk
NOB
Type:Integer
Global:Common.In/Out
None
Number of Optimum Bridges
NPOB
Type:Integer
Global:Argument.In
None
Number of Piles in the Optimum Bent
NREPS
Type:Integer
None
Number of Repetitions. Counter for number of
optimum and near optimum bridges generated
NROW
Type:Integer.Array
Global:Common.In/Out
None
Number of Rows for each section of generated data
on the disk
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
NRTABB
Type:Integer
Global:Common.Out
Data:Data Statement
None
Total Number of Rows in Table B
NS
Type:Integer
Global:Common.Out
None
Number of spans in the bridge
NSEC
Type:Integer
Global:Common.In/Out
None
Number of section of data on the disk
NSMAX
Type:Integer
Data:Read in
None
Maximum Number of spans in the bridge
NSMIN
Type:Integer
Data:Read in
None
Minimum Number of spans in the bridge
PCINC
Type:Real
Global:Common.Out
Data:Read in
None
Incremental percentage for near optimum bridges
PCMAX
Type:Real
Global:Common.Out
Data:Read in
None
Maximum percentage for near optimum bridges
PERCNT
Type:Integer.Array
Global:Common.In/Out
None
An array representing number of near optimum
bridges for each percentage range
PLPOB
Type:Real
Global:Argument.In
Feet
Penetrated Length of pile in the Optimum Bent
Table 9.1 (Continued)
VARIABLE
CHARACTERISTICS
UNITS
PSL
TyperReal
Feet
RDWY
Type: Real
Feet
Global:Argument.In/Out
Roadway width
ROWA
Type:Integer
Global:Common.In/Out
None
Number of occupied rows in arrays TABA and ACST
ROWB
Type:Integer
Global:Common.In/Out
None
Number of occupied rows in arrays TABB and BCST
SEC
Type.Integer
None
Section number in the deck table, used concurrently
with the length of span under consideration
SECADJ
Type:Integer
None
Section number in the deck table, used concurrently
with the length of adjacent upstream span
SL
Type:Real
Global:Argument.Out
Feet
Span Length
SMNSL
Type: Real
Global:Common.Out
Data:Read in
Feet
Specified Minimum Span Length for all spans
SMXSL
Type:Real
Globa1:Common.Out
Data:Read in
Feet
Specified Maximum Span Length for all spans
SPNLGH
Type:Integer.\ word
Feet
Span Length
VARIABLE
NAME
DESCRIPTION
Partial Span Length, used concurrently with NCHK
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
SYS IN
Type:Integer
Global:Common.Out
Data:Data Statement
None
Card reader unit number
SYSOT
Type:Integer
Global:Common.Out
Data:Data Statement
None
Line printer unit number
TABA
Type:Integer.Array.\ Word
Variable
TABB
Type:Integer.Array.% word
Variable
TALA
Type:Real
Global:Common.In
Feet
A two dimensional array representing a condensed
Dynamic Programming summary table for a par­
ticular span. Column 1 represents the adjacent
upstream span length in feet. Column 2 represents
the adjacent upstream span deck types. Column 3
represents bent locations in feet for the span
under consideration
A two dimensional array representing a Dynamic
Programming table of a particular stage. Column
1 contains the values of the adjacent upstream
span length in feet. Column 2, deck types for
the adjacent upstream span. Column 3, bent
locations, in feet, for the span under considera­
tion. Column 4, values of span length, in feet,
for the span under consideration. Column 5, deck
types for the span under consideration. Column 6
contains left over lengths, in feet, to be allo­
cated to downstream spans
Total Available Length for Allocation
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
TCOB
Type:Real
Global:Argument.In
Dollars
Total Cost of the Optimum Bent
TLB
Type:Real
Global:Common.Out
Data:Read in
Feet
Total Length of the Bridge
TOTAL
Type:Real
Feet
Sum of the values of bent locations for the span
under consideration and adjacent upstream span
length
UCCB
Type:Real
Global:Argument.Out
Data:Read in
$/cu.yd.
Unit Cost of Concrete in Bent
UCCD
Type:Real
Global:Argument.Out
Data:Read in
$/cu.yd.
Unit Cost of Concrete in Deck
UCS
Type:Real
Global:Argument.Out
Data:Read in
$/lb.
Unit Cost of Reinforcing Steel
WKF1
Type:Integer
Global:Common.Out
Data:Data Statement
None
Work File 1. Name of the input data file on
the disk
WKF2
Type:Integer
Global:Common.Out
Data:Data Statement
None
Work File 2. Name of the generated data file
on the disk
Table 9.1 (Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
X
Type:Real.Array
Global:Common.In
Feet
A one dimensional array representing values of
bent location for each span
XCON
Type:Real.Array
Global:Common.In
:Argument.Out
Stations
A one dimensions1 array representing values of
bent location for each span
128
9.1.2
Deck Optimization Routine
9.1.2.1
Purpose.
The deck design routine (subroutine DKDSN) de­
signs and generates necessary data defining the properties of bridge
decks of a given length.
Information on all decks of common span length
are then transferred to the main program in an array called DKPR.
Deck
optimization routine (subroutine DKOPT) is subsequently supplied with
this information and deck suboptimization is performed.
Finally, the
pertinent information on the selected decks are transferred to the main
program in an array called DKTB.
9.1.2.2
Discussion of General Logic.
Subroutine DKDSN generates
a substantial number of decks for each span length.
As the number of
decks increases, the number of decisions to be made and the amount of
information to be saved and searched in the optimization process grow
rapidly.
Therefore, it became necessary to devise a method to reduce
the number of decks considered for a given span length.
The philosophy
in reducing the number of decks is based on the following simply con­
cept:
"A deck which is both of greater cost and greater weight than
another deck for the same span length can be omitted from consideration
without altering the optimization policy.11
To demonstrate this policy consider Figure 9.2 which depicts rela­
tive positions of decks of common length in a cost versus weight plot.
The result of eliminating decks of both greater cost and weight is
shown in Figure 9.3.
The set of decks among which there exists no deck
of both smaller weight and cost constitutes the "minimum weight-cost
boundary".
Single decks can be arbitrarily chosen as test points and
any deck of both greater weight and cost omitted, thus obtaining the
minimum weight-cost boundary.
However, in devising a systematic method
to obtain the minimum weight-cost boundary, it was observed that the
test runs of subroutine DKDSN for prestressed concrete girders with
concrete slabs often produced decks as represented in Figure 9.4.
In
this case, the minimum cost deck and the minimum weight deck eliminate
all other decks from consideration and are the only two decks on the
minimum weight-cost boundary.
It was therefore deemed desirable for
computational efficiency to determine the minimum weight and the mini­
mum cost decks first.
Then all decks having lesser weight than the
minimum cost deck and lesser cost than the minimum weight deck were
selected (decks that lie inside the shaded region, such as shown sche­
matically in Figure 9.5).
The properties of the selected decks were
placed in an array called DKTB which is referred to as deck table.
The
minimum weight and the minimum cost decks were also included in the deck
table.
If only one or no deck lay inside the shaded region of Figure
9.5 no additional work is necessary.
However, if more than three decks
were added to the deck table further suboptimization becomes necessary
as the decks in the shaded region may eliminate each other.
For further
suboptimization each point is selected as a test point in order (first
to last).
The minimum cost and minimum weight decks are not used as
test points.
A test point eliminates from further consideration any
deck with greater cost and weight (see Figure 9.6).
decks are not considered in any way.
Once eliminated,
Voids left in DKTB by eliminating
decks are filled by immediately moving up in the table the remaining
130
information.
In a special case, it is possible for a single deck to be both the
minimum cost and the minimum weight deck and as such eliminate all
other decks from further consideration.
mum deck for the given span length.
If so, this deck is the opti­
131
minimum weight deck
minimum weight deck
IO
O
min. weight-cost boundary
o
w
min. cost deck
minimum cost deck
Weight
Weight
FIGURE 9.3
FIGURE 9.2
MINIMUM WEIGHT COST BOUNDARY
DECKS OF COMMON SPAN LENGTH
minimum weight deck
^ minimum cost deck
Weight
FIGURE 9.4
DECKS FOR PRESTRESTRESSED CONCRETE GIRDERS AND CONCRETE SLAB
, minimum weight deck
\
minimum weight deck
eliminated by
current test point
test point
o
minimum cost deck
u
^ m i n . cost deck
selected decks
Weight
FIGURE 9.5
MINIMUM WEIGHT AND MINIMUM COST DECK
REGION
Weight
FIGURE 9.6
TEST POINT REGION
ENTER:
INPUT - A SPAN LENGTH
AND PROPERTIES OF ALL DECKS
DESIGNED BY DKDSN WITH THAT
SPAN LENGTH
DETERMINE MINIMUM COST DECK
REGARDLESS OF WEIGHT
DETERMINE MINIMUM WEIGHT DECK
REGARDLESS OF COST
PLACE IN DECK TABLE ONLY THOSE DECKS WITH
SMALLER COST THAN THE MINIMUM WEIGHT DECK
AND SMALLER WEIGHT THAN THE MINIMUM COST
DECK; INCLUDE MINIMUM COST DECK
AND MINIMUM WEIGHT DECK
HAVE
MORE THAN
DECKS BEEN
ADDED TO
DKTB
?
NO
OF THOSE DECKS ADDED TO DKTB CHOOSE EACH AS
A TEST POINT AND ELIMINATE FROM FURTHER
CONSIDERATION ALL DECKS WHICH COST AND WEIGH
MORE THAN A TEST POINT; READJUST DKTB BY
MOVING INFO. UP IN ORDER TO CLOSE VOIDS
MADE BY ELIMINATING DECKS
FOR FUTURE REFERENCE RECORD THE BEGINNING
ROW OF NEW SECTION AND NUMBER OF DECKS PLACED
IN DKTB COMPRISING THE SECTION
RETURN
Figure 9.7
Macroflow Chart for Subroutine DKOPT
TABLE 9.2
Subroutine DKOPT Variable Definition List
UNITS
DISCRIPTION
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
DKPR
Type:Real.Array
Global:Argument.In
Variable
Deck Properties; a two-dimensional array of deck pro­
perties.
DKTB
Type:Real .Array
Global:Common.Out
Variable
Deck Table; a two dimensional array of suboptimized
deck information; each row in DKTB describes a single
deck; Col. 1 gives span length in feet; Col. 2, deck
type; Col. 3, no. of girders, Col. 4, type girder, Col.
5, thickness of slab in inches, Col. 6, deck reaction
in Kips, and Col. 7, cost in dollars.
FIRST
Typerlnterger
None
Number of first row to be added to array DKTB for
given call to subr.DKOPT.
IDKTB2
Type:Integer.Array
None
A two dimensional array which describes the structure
of the deck table (DKTB); DKTB is ordered in sections
of common span length; Col. 1 in IDKTB2 contains the
initial row no. of a section in DKTB and Col. 2 contains
the no. of rows in that section.
JH
Type:Integer
None
Row indicator of minimum cost deck in array DKPR
JHS
Type:Integer
None
Temporary row indicator of minimum cost deck in DKPR
Table 9.2
(Continued)
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
JK
Type:Integer
None
Row indicator of minimum weight deck in array DKPR
JKS
Type:Integer
None
Temporary row indicator of minimum weight deck in array
DKPR
MSGR
Type:Integer
Global:Common.In
None
Message level indicator; MSGR=0 causes no intermediate
messages to be printed; MSGR=1 causes intermediate
messages to be printed.
NDK
Type:Integer
Global.Argument.In
None
Number of rows (decks) in array DKPR.
NDST
Type:Integer
None
Relative row number in a section (decks of common span
length comprise a section) in deck table (DKTB).
NDT
Type:Integer
Global:Common.In/Out
None
Absolute row number in deck table (DKTB).
NDT1
Type‘
.Integer
None
Absolute row number plus one in DKTB.
NSL
Type:Integer
None
Sequence number corresponding to span lenght; minimum
span length corresponds to NSL=1; the deck table (DKTB)
is ordered such that decks of common span length com­
prise a section, NSL indicates section number.
TES6
Type:Real
Kips
Weight of a deck being used as a test point
TES7
Type:Real
Dollars
Cost of a deck being used as a test point
SYSOT
Type:Integer
Global:Common.In
Data:Data Statement
None
Unit number of line printer; initialized to 6 in
BLOCK DATA.
DESCRIPTION
135
9.1.3
Span Length Calculator Routine
9.1.3.1
Purpose.
The span length calculator routine (subroutine
SLC) is designed to calculate bent locations (state variable X), cor­
responding span lengths (decision variabl D) and span lengths for the
adjacent upstream span (decision variable D).
SLC generates all neces­
sary data for every span of both free and constrained bridges.
9.1.3.2
Discussion of General Logic: Free Bridge.
A free bridge is
defined as one with no restrictions placed upon the lengths of indi­
vidual spans or the locations of bents with the exception of the speci­
fied single minimum and single maximum span lengths required for all
spans.
The program logic to generate the necessary data for each span
of a free bridge follows very closely the development presented in Sec­
tion 3.3.
The general logic flow chart for the free bridge is given
in Figure 9.17.
9.1.3.3
Table 9.3 gives subroutine SLC variable definition list.
Discussion of General Logic; Constrained Bridge.
A con­
strained bridge is defined as one where some of the bents and/or span
center lines are specified to fall at prescribed locations.
Every
span of a constrained bridge can be of one of the following three
types:
free span; span with location of its right bent specified; span
with fixed center line.
To develop methods for calculating bent loca­
tions and span lengths for each of the above types, assume span n is
under consideration.
Define,
_(D
X
83 specified center line location for span n
136
_(2)
= specified right bent location in span n
(1)
n
specified minimum span length for span n
(2 )
specified maximum span length for all spans.
A basic assumption made here is that the minimum and the maximum values
of bent location
for the adjacent downstream span
n is being considered, i.e., X
(1)
n-1
and X
(M)
...
n-1
are known when span
To calculate the values
for the bent locations and the adjacent upstream span lengths, know­
ledge of the type of constraint placed upon the closest upstream span to
span n is also necessary.
Three possible cases ensue:
case A :
All upstream spans are unconstrained
case B :
Upstream span number i has a specified location for its
right bent and all other spans bounded by spans n and i
are unconstrained
case C :
Upstream span number i has a specified location for its
center line and all other spans in between are free.
With these assumptions and definitions, one can now proceed to consider
nine basic possibilities.
Special cases for each of these possibili­
ties also exist.
The program logic is based on the efficient utilization of the
equations developed in the proceeding sections.
The program generates
all necessary data for any specified number of combinations of the
constraints already discussed.
Figure 9.17 depicts the general logic
137
of the program in flow chart form.
9.1.3.3.1
Free Span Case A .
under consideration, is free.
In addition, no constraints on any of
the upstream spans are specified.
tation of this case.
This case occurs when span n, the one
Figure 9.8 is a schematic represen­
The minimum value of bent location G^"^) is given
by;
(l)
x
n-1
„ (1)
+ £n
X (1) = Max.
n
(2)
L - (N-n) •
^
The maximum value of bent location
(M)
x
n-1
(M)
(X^ ) is given by;
(2)
+
X<M) = Min.
n
£
N
L -
(9.1)
(1)
T
j=n+l
(9.2)
I.
J
The lower and upper bound values for the span length corresponding to
/ \
a given bent location, e.g., X n , can now be calculated in the foln
lowing manner;
(1)
n
= Max.
(m )
X
n
(M)
- X .
n-1
(9.3)
.( 2 )
D^Kn^ = Min.
n
(m )
n
(1)
- X
n-1
(9.4)
138
The lower and upper bound values for the span length of the adjacent
upstream span are given by the following equations;
(1)
(1)
Dn+1
I ^n+l
= MaXm
{
(M)
L - X
(2)
(9'5)
- (N-n-1) • I
n
(2 )
«
i)
'
1
Dn «
(1 )
L
9.1.3.3.2
- Xn
■
-
(1 )
j = g+2
Free Span Case B .
t
(1 )
+
V l
•
This case consists of an N span bridge
with free span n and with bent location specified for span i (i>n).
Figure 9.9 depicts this case.
The lower and upper bound bent location
values for span n are given by;
(1)
(1)
Xn
\ Xn-1
= H*-
Xn
I
= Min*
+ l n
i _(2)
Xi
- (i-n) •
(M)
(M)
(1)
J
Xn-1
_(2)
X,
(2)
I
(9’7)
(2)
+ ^n
i
(1)
E
I.
j=n+l
2
The minimum and the maximum values of span length corresponding to
specific bent locations are represented by equations (9.3) and (9.4).
139
The lower and upper bound values for the span length of the adjacent
upstream span are given by the following equations;
*(1)
n+1
(1)
D
n+1
= Max.
- X
X.
1
(9.9)
(2)
(M)
_(2)
(i-n-1)
?(2 )
(X ..)
D J r 1 = Min.
n+1
_(2 )
CD
X,
n
i
■ ,o
j=n+2
(1)
(1)
j
J
n+1
(9.10)
A special case of this general case develops when i = n+1.
the constrained upstream span is adjacent to span n.
That is,
As a consequence,
equations (9.7) through (9.10) reduce to the following:
(1)
X
(1)
n
= Max.
.
n-1
(1)
+ I
n
(2)
_(2 )
(2)
(M)
(M)
n
Xn-1
n-1
= Min.
.(2)
(9.11)
+
1
(i)
(9.12)
X.
(1)
(1)
D
n+1
^n+l
Max.
_(2 )
X,
(M)
(9.13)
140
(2)
(M)
I
D , = Min.
n+1
9.1.3.3.3
(9.14)
_(2)
(1)
- X
X.
n
1
Free Span Case C .
This case considers the condition
when span n is free and center line of span i is specified to fall at
a prescribed location (See Figure 9.10).
The following relationships
generate the necessary data.
(1)
Xn-1i
(1 )
X
n
(1)
= Max.
n
- (i-n- .5)
X.
l
X
(2 )
X
n
= Min.
(9.15)
(2)
_(D
(M)
.
n-1
(2)
+
I
(1)
_(D
i
S
j=n+l
X.
l
(1)
I. +
J
(9.16)
The minimum and the maximum values of span length are given by equa­
tions (9.3) and (9.4).
To compute the lower and upper bound values of
the adjacent upstream span
length use the following equations,
(1)
^n+l
(1)
}n+l
= Max.
_ (D
X,
(9.17)
(M)
- X
n
- (i-n-1.5) • I
(2)
141
(2)
(K .,)
Dn+1
(1)
=
l
E
j=n+2
_(1)
(1)
X.
- X
l
n
(1)
(1 )
(9 .1 8 )
n+1
As in the preceding section, a special case of this general case de­
velops when i = n+1, that is, the constrained upstream span is adjacent
to span n.
The following equations result;
X
(1)
X
n
n
+
I
n
= Max.
X
(M)
n-1
(9 .1 9 )
(2)
(
_(D
X.
-
(M)
X
(1)
(1)
n
n-1
-
I
(2)
I
+
(1)
= Min.
_(D
X.
(9 .2 0 )
*1
(1)
l n+l
(1)
D
n+1
= Max.
_(D
X,
(M)
- X
(2)
(9 .2 1 )
n
(2)
D
(IW
n+1
Min.
(1)
_(1)
X.
- X
(1)
(9.22)
________________I
x(n»n)
N
n+1
FIGURE 9 .8
FREE SPAN AND CASE A
L
x!2)
XK
)
1
n
n + 1
i
N
FIGURE 9.9
FREE SPAN AND CASE B
L
xV>
tori
v (n v )
A n
1
1
1
I
1
1
n
n + 1
FIGURE 9.10
FREE SPAN AND CASE C
i
N
143
9.1.3.3.4
Fixed Bent Span and Case A .
Now consider the case where
the location for the right bent of span n is specified and the values
of span length for spans n and n+1 are sought.
As before, the charac­
teristics of upstream spans should be investigated.
is assumed that all upstream spans are free.
case.
In this case it
Figure 9.11 depicts this
As span n has a specified right bent location,
(1)
(M)
_(2)
The values of minimum and maximum span length
equations (9.3) and (9.4).
for span n are given by
As a special case if X ^ ^ =
n-1
n-I
, that
is the right bent location of span n-1 was also fixed, then
(1)
D
n
(K )
= D
n
(1)
=
£
n
.
(9.24)
The lower and upper bound values of the adjacent upstream span length
are given by equations
9.1.3.3.5
case.
(9.5) and (9.6).
Fixed Bent Span and Case B .
Equation (9.23) still holds.
applicable for the span length.
Figure 9.12 depicts this
Equations (9.3) and (9.4) are
If span n-1 has also a fixed bent
position, special case, equation (9.24) applies.
The bounds for the
length of the adjacent upstream span are given by equations (9.9) and
(9.10).
If i = n+1, special case, then the only value of this span is
144
given by;
(9.25)
9.1.3.3.6
Fixed Bent Span and Case C .
of this case is given by Figure 9.13.
and (9.4) are applicable.
A schematic representation
Again, equations (9.23), (9.3)
For the special case when span n-1 has a
fixed bent position, equation (9.24) applies.
The bounds for the ad­
jacent upstream span lengh are given by equations (9.17) and (9.18).
It should be noted that for span i no special case exists, that is, i
cannot be equal to n+1.
If i = n+1, then it will not be possible to ad­
here to the specified constraint on span n+1.
9.1.3.3.7
Fixed Center Line Span and Case A.
This is the case
where span n is specified with a fixed center line location and all the
upstream spans are free.
Figure 9.14 depicts this case.
The lower and
upper bound values of bent location are given by,
_(D
X
(1)
X
n
n
+
2
= Max.
(2)
(9.26)
L - (N-n) • I
_(D
X
+
(M)
X
n
l (2)
— ---(9.27)
= Min
L
FIGURE 9.11
FIXED BENT SPAN A N D CASE A
L
x f 2>
y <2)
n
n+1
i
N
FIGURE 9.12
FIXED B E N 7.S P A N AN D CASE B
y(2)
n+1
FIGURE 9.13
FIXED BENT SPAN AND CASE C
N
146
The lower and upper bound values of the span length and of the adjacent
upstream span length are given by equations (9.3) through (9.6).
9.1.3.3.8
this case.
Fixed Center Line Span and Case B .
Figure 9.15 depicts
The minimum and maximum values of bent location
are given
by;
- (1>
n
+
(1)
X
n
= Max.
- r - (i-n)
X
X
n
n
= Min.
I
.( 2 )
_(D
(M)
(9.28)
(2)
_«>
X.
1
+
(2)
(9.29)
l
X.
x
s
j-n+1
I
(1)
j
Equations (9.3) and (9.4) generate the values of span length.
Equations
(9.9) and (9.10) are applicable for the adjacent upstream span.
Also, for the special case (i=n+l), equations (9.13) and (9.14) can be
used.
9.1.3.3.9
FjLxed-..Qenter.,.liinel.S.pan and Case C.
sents this case schematically.
cable for the bent locations.
values of span length.
Figure 9.16 repre­
Equations (9.28) and (9.29) are appli­
Equations (9.3) and (9.4) generate the
For the adjacent upstream span, lower and upper
bound values are given by equations (9.17) and (9.18).
FIGURE 9.14
FIXED CENTER LINE SPAN A N D CASE A
X
(2)
y ( l" n )
A n
y(l)
An
n+1
N
FIGURE 9.15
FIXED CENTER LINE SPAN A N D CASE B
X (nin)
n
V (l)
A n
n+1
FIGURE 9.16
FIXED CENTER LINE SPAN AND CASE C
N
148
C
Key to Abriviations
ENTER
IS
THIS
A FREE
BRIDGE
CL
BL
SL
ASL
=
=
=
=
Center Line of Span
Bent Location
Span Length
Adjacent upstream Span Length
IS
THIS
THE FIRST
SPAN
IS
THIS
THE FIRST
SPAN
READ DATA
FOR CONSTRAINED
BRIDGE
CHECK FOR
SPECIFIED
MIN. AND
MAX SPAN
LENGTHS
CALCULATE
ADJACENT
UPSTREAM
SPAN
LENGTHS
CHECK FOR
SPECIFIED
MIN. SPAN
LENGTH AND
SPECIFIED
INDIVIDUAL
SPAN
LENGTHS
IS
THIS
THE LAST
SPAN
ESTABLISH
BOUNDS FOR
BENT
LOCATION
SET UP
LOOP TO
CALCULATE
BENT
LOCATIONS
FOR THE
FIXED BENT
LOCATION
CALCULATE
BOUNDS FOR
SPAN
LENGTH
Figure 9.17
Macroflow Chart For The Subroutine SLC
149
ESTABLISH
BOUNDS FOR
SPAN
LENGTH
CALCULATE
SPAN
LENGTHS
/
YES
Is\
BL OF
PREVIOUS
SPAN FIXED
NO
SL EQUALS
SPECIFIED MIN
SPAN LENGTH
CALCULATE
BENT
LOCATIONS
CALCULATE
ALL SL'S
WITHIN THE
BOUNDS
CALCULATE
TOTAL
AVAILABLE
LENGTH FOR
ALLOCATION
TO THE
ADJACENT
SPAN
RETURN
RETURN
Figure 9.17 (Continued)
150
IS
CL OF
THIS SPAN
FIXED
?
CALCULATE
BOUNDS FOR
BL BASED ON
DOWNSTREAM
SPANS
IS
BL OF
THIS SPAN
FIXED
?
CALCUALTE
BOUNDS FOR
BL BASED ON
DOWNSTREAM
SPANS
BOUNDS FOR
BL
ARE ESTABLISHED
SEARCH FOR
CLOSET CONSTRAINED
UPSTREAM
SPAN
IS
CL OF
THIS SPAN
FIXED
NO
’
YES
THIS
THE FIRST
SPAN
CALCULATE
BOUNDS FOR
BL BASED ON
UPSTREAM
SPANS
IS
BL OF
THIS SPAN
FIXED
is
BL OF
PREVIOUS
SPAN FIXED
?
9
Figure 9.17 (Continued)
151
2
CALCULATE
BOUNDS FOR
SL. CALCULATE
ALL SL'S
WITHIN
BOUNDS
IS
THIS
THE LAST
SPAN
SL IS SPECIFIED AS
MIN. SL
FOR THIS
SPAN
CONSIDER
THE NEXT
UPSTREAM
SPAN
CALCULATE
BOUNDS FOR
BL BASED ON
UPSTREAM
m u z ____
IS
BL OF
THIS SPAN
FIXED
SELECT
BOUNDS FOR
BL
?
NO"
MYES s
?
SINGLE
VALUE FOR
BL
E
CALCULATE
TOTAL
AVAILABLE
LENGTH FOR
ALLOCATION
TO THE
ADJACENT
SPANS
Q
RETURN
IS
THIS
THE LAST
SPAN
^
Figure 9.17 (Continued)
SEARCH FOR THE
CLOSEST
CONSTRAINED
UPSTREAM
SPAN
CALCULATE
NO. OF BL'S,
THE VALUE OF
EACH BL, AND
CORRESPONDING
SPAN LENGTHS
^
CALCULATE
UPPER
BOUNDS
FOR ASL
YES
IS x
CL OF
THIS SPAN
FIXED
NO
CALCULATE
UPPER BOUND
FOR ASL
CALCULATE
UPPER BOUND
FOR ASL
YES
BL OF
THIS SPAN
FIXED
NO
CALCULATE
UPPER BOUND
FOR ASL
YES
THIS
THE LAST
SPAN
IS
CL OF
THIS SPAN
FIXED
?
LAST SPAN
IS REACHED
IS
BL OF
THIS SPAN
FIXED
?
Figure 9.17 (Continued)
153
CALCULATE
LOWER BOUND
FOR ASL
/
IS
\
THIS SPAN
ADJACENT TO
THE SPAN UNDER
CONSIDERATION
NO
CALCULATE
LOWER BOUND
FOR ASL
CALCULATE
LOWER BOUND
FOR ASL
CALCULATE
UPPER BOUND
FOR ASL
SELECT UPPER AND
LOWER BOUNDS FOR
ASL
CALCULATE ALL
VALUES OF ASL
CALCULATE TOTAL
AVAILABLE LENGTH
FOR ALLOCATION
TO THE ADJACENT
SPANS
RETURN
Figure 9.17 (Continued)
Table 9.3
Subroutine SLC Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
BBS
Type:Real
Global:Common.In
Stations
Beginning Bridge Station
CMNSL
Type:Real
Feet
Calculated Minimum Span Length for the uncon­
strained bridge
CMXSL
Type:Real
Feet
Calculated Maximum Span Length for the uncon­
strained bridge
D
Type:Real.Array
Global:Common.Out
Feet
A two dimensional array representing span lengths
for the span under consideration
DADJ
Type:Real.Array
Global:Common.Out
Feet
A one dimensional array representing span lengths
for the adjacent upstream span
DADJMN
Type:Real
Feet
Minimum value in the
array DADJ
DADJMX
Type:Real
Fegt
Maximum value in the
array DADJ
DELL
Type:Real
Global:Common.In
Feet
Incremental Span Length
DMAX
Type:Real
Feet
Maximum value in the
DMIN
Type:Real
Feet
Minimum value in array D
array D
Table 9.3 (Continued)
Subroutine SLC Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
FIX
Type:Logical
Global:Common.In
None
FIX = TRUE designates that the bridge includes
constrained spans. FIX = FALSE mean the bridge
is unconstrained
FIXL
Type:Logical
Feet
As defined by the equations which it appears in
FIXV
Type: Real
Data:Read in
Feet
The value of constraint
K
Type:Integer.Array
Global:Common.Out
None
One dimensional array representing the number of
values for D corresponding to each value of bent
location
KADJ
Type:Integer
Global:Common.Out
None
Represents the number of values for DADJ
KSADJ
Type:Integer
None
A counter on different values of DADJ
M
Type:Integer
Global:Common.Out
None
Counter designating number of values for bent
location
N
Type:Integer
Global:Common.In
None
Number of the span under consideration
NCHK
Type:Integer
Global:Common.Out
None
Check indicator used in allocating total avail­
able length to bent location and adjacent upstream
span length.
Table 9.3 (Continued)
Subroutine SLC Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
NCS
Type:Integer
Data:Read in
None
Number of the constrained span
NN
Type:Integer
None
Number of the span downstream to the span
under consideration
NNBK
Type:Integer
Global:Common.In
None
Number of the block of input data on the disk
NS
Type:Integer
Global:Common.In
None
Number of spans in the bridge
SMLES
Type:Real.Array
Data:Read in
Feet
Specified Minimum Length
SMNSL
Type:Real
Global:Common.In
Feet
Specified Minimum Span Length
for all spans
SMXSL
Type:Real
Global:Common.In
Feet
Specified Maximum Span Length
for all spans
SUMSL
Type:Real
Feet
Sum of Span Lengths
SYS IN
Type:Integer
Global:Common.In
None
Card reader unit number
SYSOT
Type:Integer
Global:Common.In
None
Line printer unit number
for Each Span
Table 9.3 (Continued)
Subroutine SLC Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
TALA
Type:Real
Global:Common.Out
Feet
Total available length for allocation
TLB
Type:Real
Global:Common.In
Feet
Total Length of the Bridge
TC
Type:Integer
Data:Read in
None
Type of Constraint. TC = 0 means unconstrained
span. TC = 1 means span center line is specified.
TC = 2 means bent location is specified
X
Type:Real.Array
Global:Common.Out
Feet
A one dimensional array representing values of
bent location for each span
XCON
Type:Rea1.Array
Global:Common.Out
Stations
A one dimensional array representing values of
bent location for each span
XFIX
Type:Real.Array
Data'.Data statement
Feet
A two dimensional array representing the type and
the value of the constraint for each span
XMAX
Type:Real
Feet
Calculated maximum value of X based
downstream spans
on the
XMAXA
Type:Real
Feet
Calculated maximum value of X based
upstream spans
on the
XMIN
Type:Real
Feet
Calculated minimum value of X based
downstream spans
on the
Table 9.3 (Continued)
Subroutine SLC Variable Definition List
VARIABLE
CHARACTERISTICS
UNITS
XMIINA
Type:Real
Feet
Calculated Minimum value of X based on the
upstream spans
XMND
Type:Real
Data:Data statement
Feet
Selected minimum value of X to be used in next
call to SLC to calculate span lengths
XMXD
Type:Real
Data:Data statement
Feet
Selected maximum value of X to be used in next
call to SLC to calculate span lengths
WKF1
Type:Integer
Global:Common.In
None
Work File 1. Name of the input data file on the
disk
WKF2
Type:Integer
Global:Common.In
None
Work File 2, Name of generated data file on the
disk
VARIABLE
NAME
DESCRIPTION
159
9.1.4
Branch Absorption Routine
9.1.4.1
Purpose.
The branch absorption routine (sub-routine
TABABS) is designed to generate a Dynamic Programming table for perform­
ing optimization over deck types and span lengths.
The cost column asso­
ciated with this table includes best costs for all downstream stages in
addition to the cost of the deck stage in the span under consideration.
The procedure used in obtaining this table is called branch absorption.
9.1.4.2
Discussion of General Logic.
Subroutine TABABS is supplied
with arrays TABA, TABB and their associate cost columns ACST and BCST.
Table A constins deck types and span lengths for the span under con­
sideration and bent locations for the adjacent downstream span.
The cost
column ACST represents total cost of all downstream stages excluding the
costs for the bent and deck stages associated with the span under con­
sideration.
The columns of Table B represent different values of bent
location, span length and deck types for the span under consideration,
as well as values for span length and deck type for the adjacent up­
stream span.
The values of leftover length to be allocated to the
downstream spans are also included in TABB.
The associated cost column
BCST gives the costs for the selected bents in the span under consider­
ation.
The program logic is to incorporate ACST, BCST and the deck cost,
of the span being considered, into a single cost column.
This is
accomplished by selecting rows from TABB and finding corresponding
rows from TABA and adding associated costs.
corresponding rows is based on the following:
The criteria for selecting
For the span under con­
sideration, the deck type and span length entries in the selected rows
of both tables should match.
In addition, the leftover length entry in
TABB should also match the value of bent location in the selected row of
TABA.
Once corresponding rows in both tables are determined, with the
given values of span length and deck type, the deck cost is obtained
from the deck table and is added to the selected costs from BCST and
ACST.
It must be noted that branch absorption does not affect the entries
in TABB, only the cost column BCST is revised to represent the new total
costs.
ENTER
SET UP LOOP TO CONSIDER
ALL OCCUPIED ROWS OF
TABLE B
CONSIDER A ROW
FROM TABLE B
SET UP LOOP TO CONSIDER
ALL OCCUPIED ROWS OF
TABLE A
CONSIDER A ROW
FROM TABLE A
IS
LEFT OVER
LENGTH FOR THIS
SPAN EQUAL TO THE
BENT LOCATION FOR THE
JAGENT DOWNST
SPAN
9
IS
DECK TYPE
FOR THIS SPAN
EQUAL TO THE DECK
TYPE WHEN THIS SPAN
WAS AN ADJACENT
UPSTREAM
PAN
Figure 9.18
Macroflow Chart for Subroutine TABABS
162
/
is x
^ S P A N LENGTHX
FOR THIS SPAN \
EQUAL TO THE SPAN
LENGTH WHEN THIS SPAN
WAS AN ADJACENT
\
UPSTREAM SPAN X
YES
FOR THIS SPAN LENGTH
AND DECK TYPE SELECT
DECK COST FROM DECK
TABLE
ADD COST COLUMN OF TABLE B
AND CORRESPONDING COST
COLUMN OF TABLE A AND
DECK COST. ENTER THE
RESULTS IN THE PROPER ROW
OF COST COLUMN OF TABLE B
X
NO
HAVE X ^
ALL ROWS OF
TABLE B BEEN
CONSIDERED
YES
RETURN
Figure 9.18 (Continued)
NO
Table 9.4
Subroutine TABABS Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
DESCRIPTION
UNITS
ACST
Type:Real.Array
Global:Common.In
Dollars
A one dimensional array containing total cost of
all bridge components considered
BCST
Type:Real.Array
Global:Common.In/ out
Dollars
A one dimensional array containing total cost of
all bridge components considered
CNST
Type:Real
Global:Common.In
None
Constant number used for calculating sections
numbers in the deck table
DELL
Type:Real
Global:Common.In
Feet
Specified incremental span length
DKTB
Type:Real .Array
Global:Common.In
Variable
Deck table. See Table 9.1
IDKTB1
Type:Function
statement
None
Used to calculate section numbers in the deck
table.
IDKTB2
Type:Integer .Array
Global:Common.In
None
See Table 9.2
LSTROW
Type:Integer
Global:Common.In
None
See Table 9.1
MSGR
Type:Integer
Global:Common.In
None
See Table 9.1
163
Table 9.4 (Continued)
Subroutine TABABS Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
DESCRIPTION
UNITS
N
Type:Integer
Global:Common.In
None
Number of the span under consideration
NDT
Type:Integer
Global:Common.In
None
Absolute row number in DKTB
NNDK
Type:Integer
Global:Common.In
None
Number of the block of input data on the disk
NRTABB
Type:Integer
Global:Common.In
None
Total number of rows in Table B
NEW
Type:Integer
None
Number of
a row inDKTB
NS
Type:Integer
Global:Common.In
None
Number of
spans inthe bridge
ROMA
Type:Integer
Global:Common.In
None
Number of
occupiedrows in
TABA and ACST
ROWB
Type:Integer
Global:Common.In
None
Number of
occupiedrows in
TABB and BCST
SEC
Type:Integer
None
Section number in DKTB. See Table 9.1
SL
Type:Real
Feet
Span length
t
164
Table 9.4 (Continued)
Subroutine TABABS Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
UNITS
DESCRIPTION
SYS IN
Type:Integer
Global:Common.In
None
Card reader unit number
SYSOT
Type:Integer
Global:Common.In
None
Line printer unit number
TABA
Type:Integer.1/2 word
Global:Common.In
Variable
See Table 9.1
TABB
Type:Integer.1/2 word
Global:Common.In
Variable
See Table 9.1
WFKl
Type:Integer
Global:Common.In
None
Work file 1. Name of input data file on the disk
WFK2
Type:Integer
Global:Common.In
None
Work file 2. Name of the generated data file on
the disk
166
9.1.5
Table Optimization Routine I
9.1.5.1
Purpose.
Table optimization routine I (subroutine TAB0P1)
is designed to select the best deck types for all values of span length
for the span under consideration.
The optimization is accomplished
by a systematic selection of minimum cost entries from array BCST and
the corresponding rows from TABB.
9.1.5.2
Discussion of General Logic.
The main program, through
a common area, supplies subroutine TAB0P1 with arrays TABB and BCST.
The fifth column of TABB contains deck types corresponding to every
span length for the span under consideration.
If there is more than
one deck type available for a given span length, the corresponding
number of entries also exist in the fifth column of TABB with sequen­
tially increasing numbers for each deck type.
For example, if four
deck types are available for a given value of span length, numbers 1,2,
3 and 4 appear in the fifth column of TABB.
It should be noted that
there is a one to one correspondence between the deck types in TABB
and the deck table for a particular span length.
The optimization over deck types starts by searching along the
fifth column of TABB to determine how many deck types are recorded for
a given value of span length.
If only one deck type is found, no opti­
mization is performed and that row with its related cost entry is
selected.
However, if more than one deck type is found, then by
searching the associated cost entries in BCST, the minimum cost value
and its corresponding row from TABB are selected.
This procedure is
continued until all occupied rows in TABB are checked.
167
As the optimization over deck types is being performed, the
selected rows are moved to the top of TABB and BCST such that the final
rearranged arrays contain only the selected data representing the
optimum values.
For span number one, the recorded final information from TABB and
BCST are copied in TABA and ACST, which are referenced when the next
branch absorption is performed.
c
ENTER
START WITH THE FIRST
ROW OF TABB
SELECT THE NEXT ROW
IN TABB
IS
THE
DECK TYPE
ENTRY IN THIS
OW EQUAL TO DECK
TYPE 1
?
SET UP A SECTION WITH
THE FIRST ROW AS
ESTABLISHED AND THE
LAST ROW - NO. OF THE
ROW UNDER
CONSIDERATION-1
I
SELECT MINIMUM COST
ENTRY FROM BCST IN
THIS SECTION
MOVE THE ROW
CORRESPONDING TO THE
MINIMUM COST ENTRY
IN THIS SECTION TO
THE TOP OF TABB
O
Figure
9.19
Macroflow Chart for the Subroutine TABOP1
A
THE FIRST ROW OF
THE NEW SECTION
IS THE LAST ROW OF
PREVIOUS SECTION
PLUS ONE
NO
.
IS
THE LAST
ROW IN TABB
CONSIDERED
?
IS
SPAN
NUMBER ONE
UNDER
CONSIDERATION
COPY THE NEW TABB
AND BCST INTO TABA
AND ACST
RETURN
Figure 9.19
(Continued)
Table 9.5
Subroutine
Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
AGST
Type:Real
Global:Common.Out
Dollars
A one dimensional array containing total cost of all
bridge components considered.
BCST
Type:Real
Global:Common.In/Out
Dollars
A one dimensional array containing total cost of all
bridge components considered.
FIRST
Type:Integer
None
Number of the first row of the section for selecting
the minimum cost.
LAST
Type:Integer
None
Number of the last row of the section for selecting the
the minimum cost.
LIT
Type:Integer
None
Index indicating smallest value of BCST in the designated
section.
N
Type:Integer
Global:Common.In
None
Number of current span being considered.
ROW
Type:Integer
None
Number of current row in arrays TABB and BCST.
ROWA
Type:Integer
Global:Common.Out
None
Number of occupied rows in arrays TABA and ACST.
ROWB
Type:Integer
Global:Common.In/Out
None
Number of occupied rows in arrays TABB and BCST.
UNITS
DESCRIPTION
Table 9.5 (Continued)
Subroutine TAB0P1 Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
TABA
Type:Integer.Array.
1/2 word
Global:Common.In/Out
Variable
A two dimensional array representing a condensed
dynamic programming summary table. For details see
Table 9.1.
Type:Integer.Array.
1/2 word
Global:Common.In/Out
Variable
A two dimensional array representing a dynamic programming
summary table. For details see Table 9.1.
TABB
UNITS
DESCRIPTION
172
9.1.6
Table Optimization Routine II
9.1.6.1
Purpose.
The main program, through the common areas,
supplies this routine (subroutine TAB0P2) with arrays TABA, TABB,
ACST and BCST.
The primary function of subroutine TAB0P2 is to opti­
mize span lengths for the span under consideration.
performed in TABB and BCST.
This operation is
Upon completion of the optimization, the
final results are transferred into TABA and ACST which are used in the
next call to subroutine TABABS.
9.1.6.2
Discussion of General Logic.
The results of the
optimization over deck types, corresponding to a given span length for
the span under consideration, are recorded in TABB and BCST.
Sub­
routine TAB0P2 accesses these two arrays through a common area.
The third column of TABB contains all values of bent location
(state variable X).
Starting with the first row in TABB, a search
along this column results in the securing of a number of rows having a
common value of bent location.
This constitutes a section in TABB.
The fourth column of TABB contains the values of span length
corresponding to every value of bent location.
A search for different
values of span length along the fourth column of the established sec­
tion results in one of the following possibilities:
If there is no
change in the value of the span length, no optimization can be per­
formed as there are no choices.
moved to the top of TABB.
Thus, the section in its entirety is
However, if more than one value of span
length is found, the section is subdivided into a number of subsections,
one for each value of span length.
Optimization over the span length
173
is performed by selecting the minimum cost entry among the first rows
of all subsections, the second rows of all subsections, and so on.
These minimum cost values and their corresponding rows are then trans­
ferred to the top of arrays BCST and TABB.
Returning to the third column of TABB, the next section having
common value for the bent location is determined and the procedure out­
lined before is followed until all the rows of TABB are exhausted.
Finally, the now condensed TABB and BCST are copied in TABA and
ACST to be accessed by subroutine TABABS.
enter
SEARCH ALONG THE THIRD
COLUMN OF TABB TO
LOCATE A CHANGE IN THE
VALUE OF BENT LOCATION.
SET UP A SECTION
SEARCH ALONG THE FOURTH
COLUMN OF TABB TO
DETERMINE HOW MANY SPAN
LENGTHS CORRESPOND TO
THE CURRENT VALUE OF
BENT LOCATION.
CALCULATE THE NUMBER OF
SUBSECTIONS BASED ON
THE DIFFERENT VALUES OF
SPAN LENGTH FOR THE
CURRENT BENT LOCATION
/
NO
ARE
THERE
ANY SUBSECTIONS
FOR THE CURRENT
VALUE OF BENT
LOCATION
MOVE THE ENTIRE
SECTION TO THE
TOP OF TABB AND
BCST
YES
SELECT THE MINIMUM COST
VALUES BY COMPARING THE COST
ENTRIES OF CORRESPONDING
ROWS FROM ALL SUBSECTIONS.
Figure
9.20
Macroflow Chart for the Subroutine TABOP2
175
MOVE THE SELECTED ROWS
TO THE TOP OF TABB AND BCST
CONSIDER THE
NEXT VALUE OF BENT
LOCATION
IS
^
THE
LAST ROW
IN TABB
CONSIDERED
YES
COPY THE NEW TABB AND
BCST INTO TABA AND ACST
RETURN
Figure 9.20
(Continued)
NO
Table 9.6
Subroutine TABOP:
Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
ACST
Type:Real.Array
Global:Common.In/Out
Dollars
A one dimensional array containing the total cost
of all bridge components considered.
BCST
Type:Real.Array
Global:Common.In/Out
Dollars
A one dimensional array containing the total cost
of all bridge components considered.
BRW
Type:Integer
None
Designates the number of the first row in a section.
ERW
Type:Integer
None
Designates the number of the
NRPS
Type:Integer
None
Designates number of rows per subsection.
NRW
Type:Integer
None
Designates number of rows in a section.
NSEC
Type:Integer
None
Designates number of subsections.
OPCST
Type:Real
Dollars
Optimum cost. Selected minimum cost corresponding
to the best span length for the given value of bent
location.
ROW
Type:Integer
None
A counter for the number of rows in the rearranged
TABB and BCST.
ROWA
Type:Integer
Global:Common.Out
None
Number of occupied rows in arrays TABA and ACST.
ROWB
Type:Integer
Global:Common.In/Out
None
Number of occupied rows in arrays TABB and BCST.
UNITS
DESCRIPTION
last row in asection.
176
Table 9.6 (Continued)
Subroutine TAB0P2 Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
TABA
Type:Integer.Array.
1/2 word
Global:Common.In/Out
Variable
A two dimensional array representing a condensed
dynamic programming summary table. For details
see Table 9.1.
Type:Integer.Array.
1/2 word
Global:Common.In/Out
Variable
A two dimensional array representing a dynamic
programming summary table. For details see
Table 9.1.
TABB
UNITS
DESCRIPTION
178
9.1.7
Table Optimization Routine III
9.1.7.1
Purpose.
Subroutine TAB0P3 is supplied by the main
program through the common area with two arrays; TABB which is a summary
table containing information representing several optimum and/or near
optimum bridge designs, and BCST which contains the cost associated with
each bridge.
TAB0P3 performs two functions.
First, it reorders the summary table
and the cost array so that the cost array is ranked in order of ascending
cost.
Second, it subdivides the ranked cost array into sections.
Each
section consists of the near optimum bridges that fall within a user
specified range of percentages of the optimum bridge.
9.1.7.2
Discussion of General Logic:
BCST proceeds as follows:
The reordering of TABB and
Starting with the last element in BCST the
routine compares each element to the one immediately above it.
When­
ever a low entry is numerically less than the entry above it, the entries
exchange position, as do the associated lines of information in TABB.
This procedure moves the information about the optimum bridge structure
to the top of TABB and its corresponding cost to the top of BCST.
The
same procedure is again repeated until tables BCST and TABB are recon­
structed in the order of ascending cost.
The second objective of TAB0P3 is to count the number of near
optimum costs that are within each specified percentage range.
The per­
centage ranges are specified in the form of a percentage maximum (PCMAX),
and an incremental percentage (PCINC) which will be incremented up to the
maximum.
The subroutine first calculates the number of increments
(NINC).
Then, the highest possible cost for the specified value of
PCINC is evaluated.
Starting with the lowest non optimum cost in BCST,
the routine compares each entry to the calculated maximum cost.
When
an entry exceeds this cost the routine ceases checking and places the
number of bridges in the first element of the array PERCNT.
The pro­
cedure then starts again, this time calculating the highest possible
cost for twice PCINC.
Again starting from the first non optimum, the
routine counts the number of bridges that cost less than the recently
calculated maximum cost and enters that number in the second element of
PERCNT.
This process continues until all percentage increments have
been accounted for.
Finally, starting with the last entry in the array
PERCNT, the routine takes each element, subtracts from it the entry
immediately preceding it, and enters the result in its former location.
This final manipulation causes each element of PERCNT to represent only
the number of bridges which fall within the each specified range.
180
C
ENTER
COMPARE EACH ELEMENT
IN BCST TO THE ONE
ABOVE IT STARTING
WITH THE LAST ELEMENT
IS
THE
ENTRY LESS
THAN THE ONE
ABOVE IT
?
EXCHANGE THE POSITION
OF THE TWO ELEMENTS
AND THE CORRESPONDING
ROWS IN TABB
IS
THE TABLE
COMPLETELY
REORDERED
CALCULATE THE
MAXIMUM COST FOR
A PERCENTAGE
INCREMENT
COMPARE EACH
ENTRY IN BCST
TO THE MAXIMUM
IS
THE ENTRY
GREATER THAN
MAX.
YES
?
B
Figure 9.21
Macro Flow Chart for Subroutine TABOP3
^
HAVE
\
ALL ENTRIES
BEEN CHECKED
NO
YES
^HAVE \
^
ALL
^
INCREMENTS
1EEN ACCOUNTED
\
FOR
^
YES
RETURN
Figure 9.21 (Continued)
WRITE THE NO.
OF ENTRIES LESS
THAN THE
MAXIMUM IN THE
ARRAY PERCNT
Table 9.7
Subroutine TABOP3 Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
BCST
Type:Real Array
Global:Common.In/Out
Dollars
A one dimensional array containing total cost of
all considered bridge components
CHECK
Type:Real
Dollars
Highest possible cost of near optimum bridge in
a given percentage range
CKNI
Type:Real
None
Check for number of increments
JTES
Type: Integer
None
Counter for rows being compared
NINC
Type:Integer
None
Number of increments
NLAS
Type:Integer
None
Number of rows in TABB minus one
NOB
Type:Integer
Global:Common.Out
None
Number of optimum bridges
PERCNT
Type:Integer.Array
Global:Common.Out
None
An array representing number of near optimum
bridges for each percentage range
PCINC
Type:Real
Global:Common.In
None
Incremental percentage for near optimum bridges
PCMAX
Type:Real
Global:Common.In
None
Maximum percentage for near optimum bridges
UNITS
DESCRIPTION
Table 9.7 (Continued)
Subroutine TABOP3 Variable Definition List
VARIABLE
NAME
VARIABLE
CHARACTERISTICS
RAT
Type:Real
None
Ratio of PCMAX and PCINC used to determine NINC
ROWB
Type:Integer
Global:Common.In
Dollars
Number of occupied rows in array TABB and BCST
TABB
Type:Integer.Array.
1/2 word
Global:Common.In/Out
Variable
A two dimensional array for definition see
Table 9.1
Type:Integer. Array
Variable
A word array used in reordering TABB.
WORK
UNITS
DESCRIPTION
9.1.8
Computer Program Listing
N P R O G R A M FOR BRIDGE O P T I M IZATION
00000100
1NTEGER42 T A B A •T ABB •S P N L G H .O C K T Y P •B N T L C N
00000200
00000300
COMMO N / T A B L E S / T A B A ( 600 * 3 > • TABBI6 0 0 0 * 6 ) • A C S T (6001•B C S T (6000)
1
.ROWA.ROWB »NR T ABB.LSTROW.MSGR
00000400
COM M O N / O E C K / D K T B ! 5 0 . 7 ) • 1DKTB2I2S.2).CNST.NOT
00000500
OCOMMON/ L E N G T H / X ( 100>.X C O N (1001.K ( 1 0 0 ) • 0 < 1 0 0 . 3 0 ) »DA0J(30>
00000600
00000700
1
.FIX.BOS* T L B •S M X S L •S M N S L •M . K A O J •T ALA.NCHK
C OM M O N / O I S K / N R O W ( 1 0 0 ) *NFBLK(I0 0 1 • N L B L K ( 1 0 0 ) .NSEC
00000800
00000900
C O M M O N / E X S P A N / ELSL. E L S R . E R S L . E R S R
00001000
COMMON / V A L U E / O E L L . N S . N
00001 too
COMMON/OPT 10/ P E R C N T (20 ) .NOB.PCINC.PCMAX
C O M M O N / 10/ S Y S I N •SYSOT *W K F 1»W KF2.NNBK
00001200
0 IMENSION O K P R (50•51
00001300
INTEGER S Y S I N •S Y S O T •W K F 1• WKF2
00001400
00001500
INTEGER R O W . R O W A •ROWB
00001600
INTEGER S E C . B R W . E R M * S E C A O J # B R W A O J * E R W A D J
00001700
INTEGER PERCNT
DEFINE FILE 1 U 2 1 0.600.U.IBLKI
00001800
00001900
DEFINE FILE I 2 ( 2 0 0 * 8 0 * E . N N B K >
00002000
LO G I C A L FIX
1DK T B 1 ( 0 1S T )s o I S T/DELL+CNST
00002100
DATA FIXTES/3HFI X/
00002200
00002300
MR ITEC 8.1000)
■ FORMAT 1 5 X . ' S T A R T OF E XECUTION OF BRIO G E OPT I M I Z A T I O N - ZAP - ZAP* >00002400
00002500
C ALL IDENT
R E A D I W K F 2 *N N B K • t >O Q S I •B B S Z •E B S I . E B S 2 •T L B *S M X S L •SMNSL* OELL
00002600
R E A O (W K F 2 *NNBK »2 )F IXED.NSMIN »N S M A X . P C 1N C •P C M A X • INOES* MSGR
00002700
RE A O (WKF 2•N N B K •3 IUCC 0• U C C B •UC S •USPP
00002800
00002900
FI N O (W K F 2 *NNBKI
00003000
F IX».FALSE*
IF(FIXEO*E O * F I X T E S I F I X * * TRUE*
00003100
B B S = 100*A B B S 1+BBS2
00003200
00003300
EB S = I 0 0 . * E 8 S I * E B S 2
IF!NSMAX.EO.O> NSMAX a N S M I N
00003400
00003500
IF(FIX) GO TO 9
CTLU=S M N S L * N S M A X
00003600
00003700
IF(C T L 8 * L E * T L B > GO TO 9
00003800
W R I T E (S Y S O T « 500 ) SMNSL.NSMAX
00003900
SMNSL=*SMNSL— OELL
00004000
CTLB>SMNSL*NSMAX
00004100
IF (CTLBaLE* T L B ) GO TO 8
00004200
GO TO 7
00004300
W R I T E ! S Y S O T . 501) SMNSL
00004400
CNST*-SMNSL/DELL+1.0
00004500
S L = SMNSL— OELL
00004600
SL=SL*DELL
CALL D K D S N I S L *UCS.UCCO.RDtfY.GUTT.OKPR.NDK)
00004700
00004800
CALL OKOP TISL . D K P R . N D K •MSGRI
IF (SL*LT * S M X S L ) G OTO 10
00004900
0 0005000
I F ( M S G R a G E * 1 )CALL WTTABI3.0)
0 0005100
I F I M S G R * E O * 1) WRITE!SYS0T.5)
00005200
IF ( INDE S . E Q . 1) G O TO 12
0 0005300
ELSL^O*
00005400
ELSR=0.
0 0005500
ERSL°0*
00005600
ERSR«0.
00005700
GO TO 15
00005800
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187
MAIN
160
170
tao
190
200
210
215
220
230
240
250
00017500
OCALL BTDSNIEDS.DIMS.KS1 .ERSL.0KT B C J . 6 1 . E R S R . U C S . U C C B . U S P P .
00017600
1
RDMV.GUTT.0P0 8.HT0 0 . P L P 0 B . N P 0 B . T C a B >
00017700
JJ3JJ41
0 0017800
ROMaROM.l
OIFCMSCR.EO.il MR1TEC S Y S O T .61 R O M . E B S . D C M S . K S I . E R S L . D K T B C J .6) .ERSR.00017900
0 0018000
I
DPOB.NPOB . P L P O B . H T O B
0 0018100
T A B B C R O M . 11=0
00018200
T A B BCROM.2>aO
0 0018300
T ABBCROW ,3>aXCMS)
00018400
T ABBC R O W . 41=0 CMS »KS!
00018500
T ABBCROM.51aJJ
0 0018600
T A B B C R O M . 61 3 X CMS I— DC MS.KS)
00018700
B CST C R O M l ^ T C O B
00018800
CONTINUE
00018900
CONTINUE
0 0019000
CONTINUE
0 0019100
R O M BaROW
0 0019200
I FIMSCR.EO.I1CALL MTTABC2.2)
0 0019300
CALL TABABS
00019400
I FCMSGR.EO.tICALL MTTABC2.31
0 0019500
CALL TAB0P3
00019600
I FCMSCR.GE.lICALL MTTABC2.BI
0 0019700
CALL MTOSK
0
0019800
N L S E C=NSEC
00019900
N REPS=NOB
00020000
DO 190 I S U M * | ,20
00020100
NREPSaNH.EPS+PERCNTC ISUMI
00020200
NSECaNSEC-1
00020300
K REPSaO
0 0020400
K REPSaKREPS.I
00020500
J TESaKREPS
00020600
L S T ROMaNRTABB+l
00020700
IF CNSEC.EO.OIGO TO 215
00020800
NDLKaNFBLKCNSEC1
00020900
F I N D I M K F 1* N B L K 1
00021000
L S T R OMaLSTROM-l
00021100
00 220 Ja|,6
00021200
TABBCLSTROM,J1«TABBCJTES.J1
00021300
BCSTCLSTROMl=BCSTCJTES1
00021400
IF INSEC.EQ.O1G0 TO 240
00021500
SPNLGHaTABBC JTES.4)
00021600
OCKTVPaTABBC JTES.51
00021700
BNTLCNaTABB IJTES.6)
00021800
CALL ROOSK
0 0021900
IFCMSGR.EO.il CALL MTTABC2.6I
00022000
00 230 JTESal.ROMB
00022100
O I F C T ABBCJTES.31.E0.BNTLCN.AND.
00022200
1
T A B B I J T E S . 21 .EQ.OCKTYP.AND.
00022300
2
T A B B I J T E S . 11.E0.SPNLGH1G0 TO 210
00022400
CONTINUE
00022500
1FCKREPS.EO.NREPS1GO TO 250
0 0022600
N S E C aNLSEC
00022700
NBLKaNFBLKCNSEC)
0 0022800
FINO CM K F I * N O L K )
0 0022900
CALL OUTPUT IBBS1.80S2.BBS.UCS.UCCB.USPP.ROMV.GUT T1
00023000
IF I K REPS.EQ.NREPS1G0 TO 260
00023100
CALL RDDSK
00023200
IFCMSGR.EO.il CAL L MTTA8C2.6I
188
MAIN
GO TO 200
CONTINUE
WRITEIB.2000)
STOP
F O R M A T I 8 F 10*01
FO R M A T I A3.7X.2II0.2FI0.0.2I10)
FORMAT 14F 10*01
F O R M A T I 4 F 10*0 I
O F O R M A TIIHI.45X.*BENT P R O P ERTIES*.//.
1
7 X »•B N T .STA LGH.SPN.LFT. LGH.SPN.RGT. R C N . S P N . L F T . *.
2
• R C N . S P N . R G T • PILE OIA. NO.OF PI L E S PENT.LGH. B N T . H G T .* I
O F O R M A T I I X . 1 4 . 1 X.F9.2.4X.F4.0.9X.F4.0.7X.FB.2.5X.F8 .2.
I
6X.F5.2.SX.I2.8X.F6.2.4X.F5.I)
TA OA.
ZAP - ZAP ZAP •>
2000 FORMATISX.*IT RAN.
500 OFORMA T I ( H I . 2 9 X . 4 4 1 • ♦ • ) . / . 30X.
1
••
SPEC IF IEO MINIMUM SPAN L E N G T H ^ •. F 3 . 0 .• FEET
••/
2
3 0 X . *• IF THIS L E N G T H WAS USEO F O R ALL SPANS OF ••/
3
3 0 X . •• T H E * . 13.* SPAN BRIDGE. TOTAL BRIDGE L E N G T H
••/
4 3 0 X .•4 WOULD E X C E E D S PECIFIED BRIOGE LENGTH*
*•)
501 FORMATI 3 0 X . •• S PECIFIED MIN I M U M SPAN L E NGTH IS C H A N G E D •*./.
1
30X.*4 TO *. F 4 . 0 . • FEET TO MEET R E Q U I R E M E N T S . •. B X . •••/
2
30X.44I***!!
510 OFORMATIIHI.29X.39I***>./.30X.
1
•* MAXIMUM TABB ARRAY SIZE IS E X C E E D E D «*./.30X.
2
•• INCREASE TABB DIMENSION
**./.30X.
3
3 9 1 * 4 * II
520 F ORMATISX.*M0. OF R OWS IN TABLE B =*.IS>
END
260
00023300
00023400
00023500
00023600
00023700
00023800
00023900
00024000
00024100
00024200
00024300
00024400
00024500
00024600
00024700
00024800
00024900
00025000
0002SIOO
00025200
00025300
0002S400
00025500
00025600
00025700
00025800
00025900
00026000
189
BLK DATA
BLOCK OATA
1NTEGER42 TABA.TABB
INTEGER PERCNT.SVSIN.SVSOT.UKFt.WKF2
COMMON/TABLES/ TABAI600•3).TABUI6000.6>.ACST1600>.BCST(6000>
1
.ROWA•ROWB.NRTABB.LSTROW.MSGR
COMMON/OECK/ OKTBISO.7 I.I0KT82(25.2).CNST.NDT
COMMON/OPT 10/ PERCNT(20).NOB.PC INC.PCMAX
COMMON/10/ SVSIN.SVSOT.WKFI.WKF2.NNBK
OATA NRTABB/oOOO/.NDT/O/.PEHCNT/2040/.SVSIN/5/.SVSOT/6/.WKF1/11/
OATA WKF2/I2/
END
00026100
00026200
00026300
00026400
00026500
00026600
00026700
00026S00
00026000
00027000
00027100
190
SLC
10
IS
20
25
30
35
40
43
50
SS
60
61
62
65
SUBROUTINE SLC
C O M M O N / L E N G T H / X (100)• X C O N <100>* K <100> » 0 < 100*30> . D A D J (30)
1
.FIX.B0S.TL0»SMXSL»S M N S L . M » K A D J . T A L A * N C H K
C O M MON/VALUE/ O ELL»NS*N
C O M M O N / 10/ SYSIN* S Y S O T *M K F 1•WKF2*NNBK
D I MENSION S M L E S I 1 0 0 ) *XFIX(100*21
L OG I C A L FIX
INTEGER T C •S Y S I N * S Y S O T •MKF1*MKF2
DATA X F 1X/20040*/* XMND/0*/*XMXD/0*/
NCHKaO
IF IF IX > GO TO 60
I F I N * N E * 1) GO TO 25
CMNSLaTLB-SMXSL«<NS-l)
IF ISMNSL*GT * C M N S L 1 CMNSL * S M N S L
C M X S L * T L B ~ C M N S L * I N S - 11
IFIS M X S L * L T * C M X S L ) CMXSL * S M X S L
KSAOJ«0
KSAOJ»KSAOJ41
0 A D J ( K S A 0 J ) 3 C MNSL*<KSA0J-1)«0ELL
IFCDAOJ(KSAOJ)-CMXSL) 10*20*13
KSADJ*KSA0J-1
KADJaKSAOJ
x m i n * n *c m n s l
XMIN A * T L B — 1N S — N)4CMXSL
1F<XMINA*GT*XMIN> XMIN=XMINA
XMAX=N*CMXSL
X M A X AaTLB-lNS-Nl«CMNSL
i f < x m a x a *l t * x m a x > x m a x =*x m a x a
M S a0
MS=MS»1
XlMS)»XMIN+<MS-lI40ELL
X C O N ( M S > 3 HBS+X(MS)
DM IN=*X (MS )- CN- 1 )4CMXSL
IFI DMI N*L T * C M N S L ) 0M|N=CMNSL
DMAX = X (MS >-(N - 1)4CMNSL
1F(DMAX.GT#CMXSL) DMAX*CMXSL
KS^ O
KS»KSM
O l M S , K S ) = D M I N f ( K S - l )«DELL
1F(0(MS»KS>-DMAX) 35*45*40
KS^KS-1
K(MS)*KS
IF(XIMS)-XMAX) 30*55*50
MS=*MS-1
M»MS
IFIN.LT.NS) T A LA»TLB-INS-N-1|*CMNSL
GO TO 240
I F ! N * N E * 1) GO TO 65
DO 62 1 = 1 *NS
READ I W K F 2 *NN8K*1 )NCS * S M L E S < 1 1 .TC.FIXV
FIND <WKF2«NNBK)
IF ! S M L E S ( I 1* G E * S M N S L 1 GO TO 61
W R I T E (S Y S O T •2) S M N S L * S M L E S < I )
STOP
IF(T C * E Q * 0 ) TC*1
XF IXINCS* TC)*FIXV
CONTINUE
IF(N*NE«NS> GO TO 85
00027200
00027300
0 0 0 27400
00027500
00027600
00027700
00027800
00027900
00026000
0 0028100
00028200
00028300
00028400
00028500
0002B600
00028700
00028800
00028900
00029000
00029100
00029200
0 0 0 29300
00029400
00029500
00029600
0 0 0 29700
00029800
00029900
00030000
00030100
00030200
00030300
00030400
00030500
00030600
00030700
00030800
00030900
0 0 0 31000
00031100
0 0 0 31200
00031300
00031400
0 0 0 31500
00031600
00031700
00031800
00031900
00032000
00032 1 0 0
00032200
00032300
00032400
00032 3 0 0
00032 6 0 0
00032 7 0 0
00032800
00032 9 0 0
00033000
*o o o o o o © © o e o o G o o o o o o o © o o o o o e
io o o o o o o o o o o o O o o o o o o o o o o o o o o
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00044700
IF INCHK #GT* 0) GO TO 235
00044750
00044800
KSADJ30
KSADJ=KSADJ+I
00044900
D A D J (K S A O J )s D A D J M N A I K S A D J — I>*OELL
00045000
IFI D A D J (K S A D J )—D A D J M X ) 215.2 2 0 . 2 2 5
00045200
KA D J = K S A O J
00 0 4 5 3 0 0
00045400
GO TO 230
K A D J 3 KSADJ-I
00045500
XMND ^ X M I N
00045600
X MX0*XMAX
00045700
00045800
IF C XFIX(N#2)# G T •0) GO TO 235
TALA=sXMAXA+OAOJMN
00045900
GO TO 240
00046000
TALAaXMIN
+DADJMX
00046100
I F (D<t• 1 > .GE.SMNSL) GO TO 245
00046200
MR I TECSYSOT #2 ) SMNSL.DCl.I)
00046300
STOP
00046400
IF(0A D J ( 1 )•GE#SMNSLI GO TO 250
00046500
MRITEC SYSOT #2 I S M N S L . D A D J 1 11
00046600
00046700
STOP
RE TURN
00 0 46800
F O R M A T ( II0#FI0#0« 1 10.F10.0)
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00047000
1
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00047300
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00047600
END
DKOPT
6 000
C MIN
to
20
30
C MIN
40
SO
60
62
64
65
70
2000
1000
SUBROUTINE OKOPT !SL,OKPR,NDK,MSGR>
0 IMENS ION DKPR(50'S)
COMMON/DECK/ D K T B I 5 0 . 7 ) . I0KT82I2 5 * 2 1 .CNST.NOT
COMMON/VALUE/ OELL o N S v N
C O M M O N / 10/ S Y S I N * S Y S O T •W K F 1 *W K F 2 *NNBK
DATA NSL/0/
INTEGER F I R S T •S Y S I N * S Y S O T •WKFI#WKF2
NSLaNSL+1
IF ( N S L . E U * 1. A N D . M S G R #E Q . 1)M R 1T E 1S Y S O T ,6000)
FORMAT! IH1.20X. •SU80PTI M I Z I N G D E C K TABLES IN DKOPT*)
NOTI=*NOT+l
NSUB=*NDT * 1
IO KTH2INSL.1 )xNSUB
NDST»0
FIRST^NSUB
IMUM COST DECK R E G A R O L E S S OF R E A CTION
JH* 1
DO 30 1*2*N0K
IFI O K P R ( I.2 1 - O K P R ! J H . 2 ) )10.20.30
JH*I
GO TO 30
IF I O K P R ! 1,1) « L E « D K P R ! J H . 1 I>GOTO 10
CONTINUE
IMUM REACTION DECK R E G A R O L E S S OF COST
JK*I
DO 60 1=2*NDK
IFIOKPR( 1 * 1 l - O K P R C J K . l ))40.50*60
JK*I
GOT O 60
IFIOKPR( I*2)*LE»0KPR!JK.2)IGOTO 40
CONTINUE
DO 70 l*l«NDK
IFI I«EQ« J H ) GO TO 62
IF! I.EQ«JK> GO TO 64
IF!OKPRI 1*1)» L E a O K P R i J H • 1 )•A N D * D K P R ( I .2>. L E . O K P R !J K .2> >GOTO 65
GOTO 70
JHS=>NDT+1
GO TO 65
J K S = N D T ♦1
NOT=NDTM
NDST*NOST*l
D KTBINOT.l)*SL
O K T B I N O T •2)=NDST
D K T B I N O T •3)= D K P R ( I • 3 )
DKTBlNOT ,4 )*DKPR ( 1.4)
0 K T 8 ( N D T . 5 ) = D K P R < 1,5)
DK T D I N O T • 6 ) * D K P R ! 1*1)
O K T BINOT *7)= O K P R 11*2)
CONTINUE
IF1MSGR * E 0 « I ) W R ITE!SYSOT,20 0 0 ) J H , J K . J HS.JKS*NOST
FORMAT! *0CALL TO OKOPT • *3X •• JH * • • 1 3 . 3 X ,*JK =• • 13 • 3X , *JHS «*•
I
13* 3 X •* JKK
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FORMAT!IX.7F10.2)
IFINOST*LE* 3) GO TO 100
JH*JHS
JK=*JKS
I*F1RST
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00051500
00051600
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00051900
00052000
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00052200
00052300
00052400
00052500
00052600
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S U B R OUTINE TABOPI
S U B R O U T I N E TO OPTIMIZE OVER THE DECK TYPES C SETUP SUMMARY TABLE
I NTEGER*2 TA0A*TAB8
C O M M O N / T A 8 L E S / T A 8 A I 600* 3 ) • TABBI6000*61 *A C S T ( 6 0 0 ) *BCST(6000)
1
*R0WA*R0WB*NRTA88* LSTROW . M S G R
C O M M O N / V A L U E / DELL . N S . N
INTEGER R O W A •ROWB#ROW*FIRST
RUW*0
FIRST*!
1*2
70
1FITABBII*5>*NE*1 ) GO TO 65
75
ROtfsROW+l
L A S T *1— 1
LI T = F I R S T
DO 80 J*F1RST«LAST
IFCBCSTIJI
«LT•BCSTILITI
I LiT*J
80
CONTINUE
DO 82 K K * I •6
T A B B 1R O W * K K I * TAUBILIT«KKI
62
B C S T I R O W )aBCSTlLITI
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as
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IFCI-ROW8) 70*90*100
IF IT ABB I I*5)* N E * 1 I GO TO 95
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JJal-l
00 92 J*JJ*R0W8
R OW=ROW*l
OO 91 K K * 1*6
91
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BCST(ROW1»BCSTI J)
92
CONTINUE
GO TO 100
95
100
105
ItO
1*1*1
G O TO 75
ROWB*ROV
IF (N * N £ * 1) GO TO 110
00 105 1*1* ROWO
A C S T 1 I ) * B C S T ( 1)
OO 105 J * t •3
TABAI I*J)=>TABB(I*J)
ROwAsROWB
RETURN
END
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ENO
00070100
00070200
200
TAB0P3
s
20
25
35
40
45
SO
55
60
65
70
75
SUBROUTINE TA60P3
INTEGER42 TABA.TABB*WORK
INTEGER ROWA* ROWB*PERCNT
COMMON/TABLES/ TABA(600.3)•TABB<6000»6).ACSTC600>•BCSTC6000)
1
*ROWA *ROWB «NRTABB •i.STROW* MSGR
COHMON/OPTIO/ PERCNTC20)•NOB*PCINC•PCMAX
DIMENSION W0RKC6)
NLAS»ROWB-l
OO 25 KI^I.NLAS
K2»R0WB-K|
00 20 I»1*K2
JTES=ROWU~tI-t>
1FC0CSTCJTES).GE«BCSTC JTES-'I ) ) GO TO 20
WORKlsQCSTCJTES-tI
BCSTCJTES-1>=BCSTCJTESl
8CST (J TES )3 WORK 1
OO 5 J=I«6
WORKCJ )*TABBCJTES-1•JI
TAUBCJTES-1•J)=TABB<JTES.J)
TABBIJTES*J »*WORKIJ)
CONTINUE
CONTINUE
N0B*1
DO 35 l*l«ROttB
IFCBCSTCll«LT«OCSTCt+l>)GO TO 40
N0B=N0BA1
IF IPC INC aEQ*0*IG0T045
IFIPCMAX«LT #PCINCIPCMAX s PCINC
GOT050
IF CPCMAX«EQ*0»IGO TO 7S
PCINCsPCMAX
RATaPCMAX/PCINC
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CKNI s NINC
IF tCKNI *L.T*RAT )N INC*N 1N C+1
00651^1•NINC
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CHECKsBCSTI 1)•(1•♦(1 * IPC INC/100*) ))
NOBtsNOBM
0055JsNOB1*ROWB
IFC8CSTCJ)*GT.CHECK)GO TO 60
PERCNTCI)3PERCNTCI)♦!
GO TO 65
CONTINUE
CONTINUE
IF(NINC«LT«2IG0T075
NINCtsNINC-1
OO 7013 1«NINC1
NINCBsNINC+l-I
PERCNT CN INCH )sPERCNT (NINCB>—PERCNTCNINCB-1 >
RETURN
ENO
00070300
00070400
00070500
00070600
00070700
00070800
00070900
00071000
00071100
00071200
00071300
00071400
0007IS00
00071600
00071700
00071000
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00072000
00072100
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00072300
00072400
00072500
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00072700
00072800
00072900
00073000
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00073300
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00074000
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00074300
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00074600
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O O O O O O O O O O O O O O O O O O O O P O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O O O O O O O O O O O O Q O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
pppb
N>
N5
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BTDSN
00162700
OFORMATI1H 1• 29X.47I•*•)./•
00162800
1
30X ••4 BENT AT STATION* #F9*2*• WAS BEING OES1GNED *••/.
2
30X* •* BORING NUMBER * •12** WAS BEING USED*• I4X*•4**✓.
00162900
3
30X. U **F3* 0 * * INCH 01Aa PILE WAS BEING USED*•1IX*•4•*/*
00163000
4
30X* •* BORING NOT DEEP ENOUGH* DESIGN NOT COMPLETE 4*./.
00I63100
00163200
5
30X ••4 JOB ABORTED* *•32X•* 4 * */•30X *471 *4*>.//>
OFORMATI1H 1• 29X.61I•*•)♦/.
00163300
1
30X* •4 BENT AT STATION**F9*2*• WAS BEING OESIGNEO*•15X .•4• *00163400
2
/•30X* *4 **F3* 0•* INCH OIAMETER PILE WAS BEING CONSIDERED**
00163500
00163600
3
I5X ••4 *•/•
4
30X» •4 ALLOWABLE UNSUPPORTED HEIGHT OF PILE***F5*I•• FEET* •00163700
5 I1X.•**./•
00163800
6
30X. •4 HEIGHT OF PILE ABOVE GROUND LINES=**F6*2* * FEET*•14X00163900
7*•4* */*JOX* •4 UNUSABLE THICKNESS OF TOP SOIL FOR PILE SUPPORT***
00164000
6
F4.1 f t FEET4•)
00164100
OFORMATC30X. •4 NO ADDITIONAL PILE SIZE SPECIFIED* JOB ABORTED***
00164200
1
1 IX. •4 *•/*30X*6I1*4*)>
00164300
OFORMATIIH1•29X.S5I•4•) */ *
00164400
1
30X *•4 BENT AT STAT1O N •«F9*2* * WAS BEING OESIGNEO*•9 X *•4* * 00164S00
2
/• 30X••4 LENGTH OF SPAN LEFT a**F4*0»* FEET**22X.• 4 *
00164600
3
30X • •4 LENGTH OF SPAN RIGHT** *FA*0** FEET•*22X*•4**✓.
00164700
4
30X* •4 BORING NUMBER*•12•• WAS BEING USED••22X* *4 **/*
00164800
5
JOX. *4*«F4*0.* INCH OIAMETER PILE WAS BEING CONSIDEREO* *9X *00164900
6 •4 *./•JOX. •4 ALLOWABLE MAXIMUM NUBMBER OF PILES IN BENT** * 12*7X• 00165000
7 •**./.30X. •4 AVAILABLE LENGTH OF PILE FOR PENETRATION***F5*1*
00165100
8
00165200
• FEET 4»)
00165300
OFORMAT130X • •4 NUMBER OF PILES OEING CONSIDEREO** *12*17X*•♦•#/*
00165400
1
30X. •4 CALCULATED OES1GN L O A D * •*FB*1 * * KIPS*•I6X*•4*./.
2
30X. • 4 TOTAL PENETRA FEO LENGTH OF PILE***FS*&*• FEET**I0X* 00165500
3 * 4•*/* 30X • • 4 CALCULATED BENT CAPACITV***F8*I•• KIPS*•I4X*•4 * */*
00165600
4
30X . • 4 NO ADDITIONAL PILE SIZE IS SPEC IF1ED*••15X••4 * */.
00165700
S
30X, •4 LACK OF SOIL CAPACITY* JOB ABO R tED*••I7X*•4 •./.
00165800
6
30X. S S I »♦•))
00165900
00166000
FORMATII5.3FI0*2* I5.F10*2*//*7F10*2)
00166100
END
GEON
SUBROUTINE GEOMCSTA.GELB.HPAG)
00166200
C VERTICAL CONTROL PROGRAM
00166300
COMMON/10/ SYS1N.SVS0T.WKF1.WKF2.NNBK
00166400
DIMENSION G R S ( 100 ) .GREC1001
00166500
DIMENSION P C S ! 10)*PIS(10).PTS!10).PCEC10).PIE I10).PTE(10)
00166600
01MENSION P C T 1C10).PCT2 C10).LVC<10).KL1(10).KL2(10).CP1(10)
0 0 I 66700
REAL LB.LVC.KLI*KL2
00166600
INTEGER SYSIN.SYSOT»VKF1.WKF2
00166900
00167000
LOGICAL CUR V C 10)
DATA IGEY/0/
00167100
IFC 1GEY.NE.0) GO TO 39
00167200
IGEY3 1GE Y ♦ 1
00167300
REAOIWKF2 BN N 8 K #1 JNC.NSTA.ADT
00167400
FINDCWKF2"NN8K)
00167500
OO 5 I3 1•NC
00167600
RE AOCWKF 2 *NN8 K •2 )P1S 1.PIS2 *PIEC 1).LVC(I)*PCT1(1).PCT2 CI).PCS)<PCS2 0 0 1 6 7 7 0 0
FINO!WKF2 aN N 8 K )
00167600
PI SC I)3PISt*l00e«-PlS2
00167900
PCS!I)*PCSl*100•*PCS2
00166000
IF( I«EQ«t) GO TO 5
00168100
ERR* ABSC P IE ( I )-PlEC I-I >-PCTl C I l M P t S C 11—PISI I-t >)>
00168200
IF(ERR.GT*0•001) GO TO 120
00168300
5 PCECDsO*
00166400
OO 10 13 1.NST A
00168500
READ!WKF2*NNBK.3)GRS1.GRS2.GRECI)
00168600
FIND!VKF2*NNBK>
00166700
10 GRSC I)3GRS1*100.*GR$2
00168600
OO 30 13 1•10
00168900
IFC I.GT.NC) GO TO 30
00169000
00169100
CURVCI)=*TRUE«
IF(PCS!II.NE.O) KLlCI)3 IP1SCI)— PCS ! I ))/LVC(II
00169200
IFCPCSCIJ.EQ.O) KL1CI>=*5
00169300
IFCPCSC I )»EQ« 0 ) PCSI I )=«PISC I )-KLI(I)*LVC(I)
00169400
K L 2 C I )3 1— K L 1C 1 )
00169500*
00169600
PTSCI)=LVCC I)+PCSC1 )
PCECI)=PIEC Il-PCTI tI )*KL1(I>*LVCCI)
00169700
PTEI 1)3P I E ( I )*PCT2<1 )*KL2CI)4LVCC1)
00169600
C P U I)=.S*(PIE<I)-(PTE(I)—PCE C I ))*KL1CI)—PCECI)I
00169900
IF( I.EO.l) GO TO 30
00170000
00170100
IFCPTSC1-1).GT.PCSiI)) GO TO 120
IFCPCSCI)-PTSCI— 1 ).£0«0) GO TO 30
00170200
00 20 U 3 I.NC
00170300
I 2-NCM-I1
00170400
PISI 1 2 M )~P1S( 12)
00170500
00170600
P I E ! 12*1)=PIE!12)
00170700
L V C C 12*1)3L V C C 12)
00170600
PCT IC 12*1)*PCTI(12)
00170900
PCT 2 C 12*1)=PCT2I12)
00171000
PCSI I 2 M )3PCS( 12)
00171100
PCEC 124-1 >=PCEC 12)
00171200
20 CONTINUE
00171300
NC3NC*1
CURVCI>*=.FALSE.
00171400
00171500
PCSCI)3PTSC I-l)
00171600
PTSCI)3P CSCl+l)
00171700
PISC I >*PTSCI)
00171600
PCECI)3P T E C 1-1)
00171900
PTECI)=PCE<!♦!) •
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00175600
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226
LtVEi.
10
15
20
25
30
35
40
45
SUBROUT|NE LlVEL (ROWY»LSR«LSL*TLL)
REAL LSR.LSL
IF ILSR«EO*0•) GOTO 25
IFlLSL*Ea*0«l GOTO 20
IFCLSR.LE.LSL) GOTO 10
C=4«/LSR+I»/LSL
SPAN=LSL
GOTO 15
C24./LSLM*/LSR
SPAN=LSR
P2=2«M36«-S6.*C)
GOTO 35
SPANaLSR
GOTO 30
SPAN-LSL
P2=>2.*< 36.-336./SPAN)
PI«•32*1LSR +L SL >426*
P3®2« * I24*™48«/SPAN}
IFIP1#GT«P21 GOTO 40
CLL3P2
GOTO 45
CLL=*P1
IF(CLL «LE*P31 CLL»P3
IF(ROWY*LT«20* 1 NL»1
IF<R0WY*GTo20.«AND«RDWY.LE«30« » NL*2
IF<RDMY*Gr.30«.AND«RDWY*LE*42*) NL*3
IF(R0MY*GT«42e»ANO«RDWY«LE* 54 • I NL-4
IF(ROMY*GT*S4« •ANO*ROYY»LE•66« 1 NL=*5
IF1ROWY«GT *66••ANOaRDMY«L£•70*> NL»6
IF <PO h Y«GT•78* «AND*ROWY«LE« 90*1 NL«7
1F(ROMY*GT«90••AND«RDMY*LE*102*) NL»8
TLL»CLL*NL
RETURN
END
00183200
00 K83300
00183400
00183500
00183600
00183700
00183800
00183900
00184000
00184100
00184200
00184300
00184400
00184500
00184600
00184700
00184800
00184900
00185000
00185100
00185200
00185300
00185400
00185500
00185600
00185700
00185800
00185900
00186000
00186100
00186200
00186300
00186400
00186500
227
9.2
Notation
The following is a complete list of the definitions and notations
used:
d^
= stage decision variable; decision concerning type of
deck to be selected.
dn j = stage decision variable; decision concerning type of
bent to be selected; j = 1,2.
= stage decision variable; decision concerning allocation
of available length
f
to span n.
= minimized return function; represents the minimized
cost of stage n and all downstream stages when the
minimization is performed over the decision variable dn
^n,j = minimized return function; represents the minimized
cost of stage (n,j) when the minimization is performed
only over the decision variable d
.; j = 1,2.
n »J
F
n
= minimized return function; represents the minimized
cost of stage n and of the downstream stages when the
minimization is performed over the decision variable D
kn
= 1,...,K , designate distinct values of decision
variable D .
n
= specified minimum span length for all
spans.
= specified minimum span length for span n.
i2)
=
specified maximum span length for all
L
=
specified total bridge length.
spans.
n
228
N
= specified total number of spans.
QU
= unconfined compressive strength of soil.
rn
= stage return; cost of the deck component in
r^
= stage returns; cost of the bent component in
span.
span n; j = 1,2.
= total return; return from span n plus the minimized
returns from all of the downstream spans.
sT .
n, 1
= output state variable from stage n; magnitude of
live and dead load transmitted to the right bent by
the deck of span n.
~
Sn,2
= output state variable from stage n; magnitude of
live and dead load transmitted to the left bent by
the deck of span n.
s
io t =
n-l,2,1
input state
variable;
dead load contributed by the
downstream span deck (stage n-1) to bent (n-1,2) =
(n,1).
s
n-1,2,2
= input state variable;
live load contributed by the
downstream span deck (stage n-1) to bent
n-1,2) =
(n,1).
s
n ,
n,l,l
= input state
variable;
dead load contributed by the
upstream span deck (stage n) to bent (n-1,2) = (n,l).
s
, _
n, 1,2
= input state
variable;
live load contributed by the
upstream span deck (stage n) to bent (n-1,2) = (n,l).
§
,
n, 1
output state variable from stage (n,1); total load
transmitted to the soil by the right bent stage in
span n.
229
§
. = output state variable from stage (n,2); total load
n,2
transmitted to the soil by the left bent stage in
span n.
t
n
u
v
-
I,...,!1
, , designate distinct values of decision
k .,
n
variable d •
n
= 1,...,U, ,t , designate distinct values of decision
k
n
°
n
variable d ..
n,l
n
= 1.....V,
, t , k
k
n
n+1
n
decision variable d
n
t
, designate distinct values of
n+1
°
n,2
= input state variable to stage n; total length which
will be allocated to span n and to all spans downstream
of n.
Xr
= output state variable from stage n; totallength
left
over to be allocated to spans downstream n.
X (1) = specified center line location for span n.
n
i
-(2)
Xr
= specified right bent location in span n.
A
= specified incremental span length.
^
= represents a functional relationship between
two or
more variables.
= represents the functional relationship between the cost
Pn
of the deck component in span n and the variables affect­
ing it.
n
. = represents the functional relationship between the cost
J
of the bent component in span n and the variables affecting it; j = 1,2.
230
9.3
List of Symbols for Program Macroflow Charts
Symbols used in the flowcharts conform to the International
Organization for Standardization (ISO) Draft Recommendation on Flow­
chart Symbols for Information Processing.
Following is a listing
of the symbols used:
Input/Output function in card medium
Process
input/
output
Decision
Predefined
Process
Document
Any processing function; defined operation
causing change in value, location of information
General input/output function; information
available for processing
A decision switching type operation that
determines which of the number of alternative
path followed
One or more named operations or program steps
specified in a subroutine or another set of
flowcharts
Final output document
A terminal point in a flowchart: Start, Stop,
Halt, Delay, or Interrupt; may show exit from
a closed subroutine.
O D
Conector: Exist to, or entry from another part of chart
Offpage Conector: For entry to or exit from a page
10.
VITA
Kambiz Movassaghi was born in Mashad, Iran, on May 20, 1940.
He
received his elementary and secondary education at several schools in
Iran.
In May of 1959 he graduated from Tarbiat Secondary School in
Tabriz, Iran.
In February of 1960 he arrived in the United States and enrolled
as a freshman in the Department of Civil Engineering at the University
of Southwestern Louisiana.
In May of 1963, upon obtaining a B. S. in
Civil Engineering from U.S.L., he was awarded a research assistantship
by the Department of Civil Engineering, Louisiana State University from
where he obtained the M.S. in Civil Engineering in May 1965.
In June of 1965 he was employed as a Structural Engineer by
Fromherz Engineers, Consulting Engineers, and he attended Tulane Univer­
sity as a part time graduate student.
In September of 1967 he was
awarded a graduate fellowship by Tulane University where he entered as
a full time student.
In February of 1969 he reentered the graduate school at Louisiana
State University as a research assistant, where he is now a candidate
for a Doctor of Philosophy degree.
232
EXAMINATION AND THESIS REPORT
Candidate:
Kambiz Movassaghi
Major Field:
Civil Engineering
Title of Thesis:
Dynamic Programming Algorithm For Minimum Cost Design
of Multispan Hgihway Bridges
Approved:
and Chairman
Dean of the Graduate School
EXAMINING COMMITTEE:
7
Date of Examination:
November 19, 1971
7
]
^
^