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L I B R A R Y
Michigan State
University

.1
3.6;

This is to certify that the

thesis entitled
COMPUTER-AIDED DESIGN OF A

TWO-SPAN CANTILEVER HIGHWAY BRIDGE

presented by

KULKARNI SUDHAKAR R.

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Wine er ing

 

m \

Major professor

l
Damifi i

5.9.- 3. I, ' . i”;-
0-7539

 

 

3% H W3

APRoQ E 22905 6

SEP 18 2006
812! 06

 

ABSTRACT

COMPUTER—AIDED DESIGN OF
A TWO—SPAN CANTILEVER HIGHWAY BRIDGE

By
Kulkarni Sudhakar R.

Computer—aided designs of two—span cantilever highway
bridges are investigated. The designs are governed by the
current AASHO Specifications. Three methods: linear pro—
gramming, grid search, and the method of bounds are studied
to obtain designs that are nearly optimal (minimum cost).

The linear programming approach was formulated by a lineariza—
tion of the nonlinear cost or objective function and of the
nonlinear cOnstraints. As some of the design variables are
discrete, the algorithm of Land and Doig for mixed linear
programming problems was used. The grid search method involved
a search in the design space of the four independent design
variables: the number of girders, the length of cantilever,
the depth of web plate, and the width of flange plate. The
method of bounds differs from the grid search method in regard
to the termination criteria. While the former search termi-
nates upon locating a (local) minimum, the latter ends when the
latest feasible design reaches within a prescribed neighborhood

on an "effective lower bound."

Kulkarni Sudhakar R.

Three bridges that had been designed by the traditional
method and built in Michigan were studied. The costs of the
'computer-aided designs were found to be 4% less than the cost
of the deSigns obtained by the traditional method.

Of the three optimization methods considered the linear
programming approach seems least efficient. The method of
grid search offers a straight forward procedure for an
automated design. The method of bounds would further reduce
the computational work without appreciable loss of accuracy.
A complete design by this method can be obtained by a single

run of the computer program prepared.

COMPUTER-AIDED DESIGN OF

A TWO-SPAN CANTILEVER HIGHWAY BRIDGE

By

Kulkarni)Sudhakar R.

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Civil Engineering

1973

a? ACKNOWLEDGMENTS

The author is deeply indebted to his major academic
advisor, Dr. Robert K. L. Wen, Professor of Civil Engineer—
ing, for his continuous guidance and assistance during the
course of this study. The author also wishes to thank the
other members of his guidance committee, Dr. Charles E. Cutts,
Dr. William A. Bradley, and Dr. J. S. Frame.

Sincere appreciation is extended to the Michigan
Department of the State Highways for granting the permission
to the author to pursue the graduate study while employed by
the department, and to use the computer facility of the
department for this investigation. The author wishes to
thank his supervisors Mr. O. E. Mace, and Mr. R. G. Montgomery
of the-Design Division, and Mr. VanAuken, and Mr. Kellerman of
the Computer Co—ordinating Section of the Design Division for
their assistance.

Thanks are due to the relatives and the friends who

helped the author to pursue the graduate study abroad.

ii

LIST OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . . . . . .’. . . . . ii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . vi

CHAPTER I: INTRODUCTION . . . . . . . . . . . . . . 1

1.1 Objective . . . . . . . . . . . . . . . 1

1.2 Sc0pe . . . . . . . 2

1.3 Statement of. the Design Problem . . . 6

1.4 Notation . . . . . . . . . . . . . . . 14

CHAPTER II: DESIGN BY LINEAR PROGRAMMING . . . . . . 16

2.1 Application of Linear Programming

to A Nonlinear Programming Problem . . . . 16

2.1.1 Linearization . . . . . . 17

2.1.2 A Linear Programming Problem . . . . 18

2.2 Application to Design . . . . . . 20
2.2.1 Statement of the Linear Programming

Problem . . . . . . . . . . . . . . 20

2.3 Iterative Procedure . . . . . . . . . . . . 2n

2.H Implementation and Examples . . . . . . . . 27

2. H. 1 Example 1 . . . . . . . . . . . . . 27

2. H. 2 Example 2 . . . . . . . . . . . . . 28

2. ”.3 Example 3 . . . . . . . . . . . . . 28

2.”.H Discussion . . . . . . . . . . . . . 28

CHAPTER III: DESIGN BY THE GRID SEARCH METHOD . . . 29

3.1 Introduction . . . . . . . . . . . . . . . 29
3.2 Problem Solution . . . . . . 30
3.2.1 Design Space and The Size of Grid . 31
3. 2. 2 Details of the Procedure . . . . . . 32
3.3 Practical Applications . . . . . . . . . . 33
3.3.1 Example 1 . . . . . . . . . . . . . 33
3.3.2 Example 2 . . . . . . . . . . . . . 3”
3.3.3 Example 3 . . . . . . . . . . . . . 3”

CHAPTER IV: DESIGN BY THE METHOD OF BOUNDS . . . . . 36

1 Introduction . . . . . . . . . . . . . . 36
2

u
H. The Method of Bounds . . . . . . . . . . . 37

iii

iv

Page
4.3 Practical Applications . . . . . . . . . . 38
4.3.1 Example 1 . . . . . . . . . . . . . 39
4.3.2 Example 2 . . . . . . . . . . . . . 39
”.303 Example 3 I O O O 0 O O O ., O O C O O 39

CHAPTER V: COMPARISON OF THE METHODS OF DESIGN
AND CONCLUSIONS . . . . . . . . . . . . . 41

5.1 General . . . .'. . . . . . . . . . . . . . 41
5.2 Comparison of the Results . . . . . . . ... 42
5.3 Conclusions . . . . . . . . . . . . . . . . 43
LIST OF REFERENCES. . . . . . . . . . ~ 70

APPENDIX I: USER'S MANUAL . . . . . . . . . . . . . 71

APPENDIX II: COMPUTER PROGRAM . . . . . . . . . . . 84

LIST OF FIGURES

Photographs of A Two- -Span Cantilever
Highway Bridge . . . . . . . .

Two-Span Cantilever Bridge Details
Two—Span Cantilever Girder Details
Standard HS 20 Truck Loading . .
Flow Chart of the Iterative Method
The Grid Search Method . . .

Flow Chart of the Grid Search Method

The Method of Bounds . . . . . .

Page

us
47
48
50
51
52'
53

55

LIST OF TABLES

Data for Design of the Bridges

Results of Cost-Optimization of Example
(using Linear Programming)

Results of Cost—Optimization of Example
(using Linear Programming)

Results of Cost-Optimization of Example
(using Linear Programming)

Summary of the Results of Example 1 . .
(using the Grid Search Method)

Summary of the Results of Example 2
(using the Grid Search Method)

Summary of the Results of Example 3
(using the Grid Search Method)

Summary of the Results of Example 1
(using the Method of Bounds)

Summary of the Results of Example 2
(using the Method of Bounds)

Summary of the Results of Example 3
(using the Method of Bounds)

Comparison of the Results . .

Comparison of Computer Time .

vi

Page
56

57

59

61

62

63

64'

65

66

67

68

69

CHAPTER I

INTRODUCTION

1.1. Objective

 

Minimum-cost design of structures is an important problem
in the field of structural engineering. It is particularly
significant for structures that are constructed frequently.
For these structures a modest reduction in the cost of one
structure would mean a substantial saving in expenditure when
a number of such structures are considered together. Two-span
cantilever bridges (see photographs in Figure 1.1) fall into
this category as they are frequently used for grade separation
in highway systems.

The objective of the present study is to develop a
computer program that can be used to produce complete designs
of two—span cantilever bridges and such designs should be
Optimal or, at least nearly optimal, and, of course, satisfy
the requirements of the current Standard Specifications for
Highway Bridges adopted by the American Association of State
Highway Officials (abbreviated as AASHO).

In recent years, since circa 1960, many researchers have
attempted to develop various structural optimization methods.
While a great many fundamental works have been published,

most have been concerned with methodology and principles.

2
Applications to day—to—day structural designs as practiced by
the engineering profession have been rather limited.

The author has been working as a highway bridge engineer
with the Michigan Department of State Highways for the last
six years. The selection of this particular problem is made
with a view to increasing the effectiveness of highway bridge
engineering as well as to providing an example of bridging

the gap between engineering theory and application.

1.2. Scope

The "traditional method" to obtain a design of a highway
bridge is largely based upon the engineer's intuition and
certain empirical rules developed in the design office. Usually,
after the values of certain key design variables are assumed
the design of the structural components of the bridge such as
the reinforced concrete slab and the welded steel plate girders
proceeds by using design tables, slide rule methods, or by
using computer programs.

The necessary input information for the design of a two—
span cantilever bridge usually consists of the geometrical
data, design loads, prOperties of construction materials, and
cost (material and labor) data. A design of the bridge may be
carried out after choices are made for the number of girders,
length of the cantilever, depth of web plate, and width of the
flange. These variables are defined as the "independent design
variables." With the independent design variables known all

other variables such as the slab thickness, thickness of the

3
flange plates, defined as "dependent design variables," may
be obtained by the conventional elastic design method.

From a mathematical point of view, the problem of
optimal design may be defined as one of minimizing the total
material and labor cost of the structure——a function of the
dependent and independent design variables——subject to certain
constraints composed mainly of the specifications. (Specific
design requirements are described in Section 1.3.) The cost
function and most of the constraints are nonlinear. Thus,
the minimum~cost design of the bridge is a nonlinear program—
ming problem. An additional complication lies in the often
required condition that the variables can take on only certain
discrete values. An exact solution of the nonlinear program-
ming problem is very difficult, and only approximate solutions
'of the problem can be attempted. In this thesis, three methods
for optimization will be considered. They are the "linear
prOgramming method," the "grid search method," and the "method
of bounds."

The use of linear programming is one approach to obtain
approximate solutions of nonlinear programming problems. This
approach has been used to solve structural optimization
problems in References (2)*, and (8).

The nonlinear programming problem may be solved
approximately as a sequence of linear programming problems by

linearization of the nonlinear problem at certain eXpansion

 

* number in parentheses refer to entries in the list of
references.

u
points. Linearization can be obtained as the linear terms
of the Taylor series expansion of the nonlinear objective
function and/or nonlinear constraints of the problem. Some
of the variables of the problems such as the number of girders
and the depth of the web plate are required to take on dis—
crete values. Solutions of such "mixed" linear programming
problems in which some variables are required to take on
discrete values, may be obtained by the Land and Doig algor-
ithm described in Reference (7). An iterative procedure
solving a sequence of mixed linear programming problems can
be constructed to obtain the values of the design variables
which would minimize the nonlinear objective function.

Various direct search methods are well suited to the
solution of a structural optimization problem in which the
discrete design variables are few in number and the ranges
of the values of such design variables are small. A direct
search method uses simple strategy easily adaptable for
computer solutions. A structural optimization_problem using
a direct search methodangrid search——has been considered in
Reference (4).

The minimum—cost design of the bridge can be formulated
as a four dimensional minimization problem, by considering
the cost of the bridge as a function of the independent
design variables. -A solution of the problem can be obtained
by successive applications of a two—dimensional grid search
which examines the discrete design space to determine the
values of the independent design variables which would

minimize the cost of the design.

5

The "method of bounds" is suggested in Reference (5).

The basic idea of the method of bounds as adopted herein is
that instead of finding the optimal design, a near optimal
design is determined by terminating the search when the cost
of a current design is within a specified tolerance of an
"effective lower bound" (to be defined later). The procedure
generally reduces the computational work provided that the
values of tolerance and the effective lower bound are properly
specified.

As it will be shown in later chapters, near optimal designs
of the three bridges have been obtained by using the above
described methods. These bridges designed according to the
traditional method had actually been built in the State of
Michigan. Comparisons of the designs by the various approaches
indicate that (a) computer-aided designs are more economical
than the traditional designs, and (b) the method of bounds
using a direct search method together with certain judiciously
chosen criteria to terminate the search provides an efficient
method of automated design.

The designs by using linear programming, the grid search
merthod and the method of bounds are presented in Chapters II,
.YEI, and IV, respectively. Comparisons of these approaches
and conclusions based upon numerical results and computer
tines are presented in Chapter V. User's Manual for a
Commuter program named "BRIDGE" which has been prepared to
implement the method of bounds is given in Appendix I, and

the program is presented in Appendix II.

6

 

1.3 Statement of the Design Problem
The type of bridge under consideration here consists of
steel plate girders with a reinforced concrete slab. Two
photographs of a typical bridge are shown in Figure 1-1, and
the usual engineering details of the bridge are shown in
Figure 1—2, and those of a single girder in Figure 1—3.
.The information necessary for design such as the geometrical
data, loading, material properties, and cost data is shown in
Table 2—1. The details of live load are shown in Figure 1—4.
The expression cf the total cost of the bridge which is
to be minimized, is a function of the design variables, and
is written as follows.

Z : ClBLBWTS + C2BLBwAs

+0.5C3N(l — Kl) L (t + t + t + t ) b

1 fl f3 f6 f8 f

+C3N K1 1 (tf2 + tf7) bf + C3N LC (tf3 + tf8) bf

+C3N (1 —K2) (LB ~ LC) (tf4 + tfg) bf

+C3N K2 (LB - LC) (tf5 + tflO) bf

+C3N (Ll + LC) htwl + C3N (LB — LC) htw2 (l—l)

Iwhere.Z = total cost of the bridge ($),

AS = cross—sectional area of steel reinforcement (in.2/ft-)a
BL = length of bridge (ft.),
BW = width of bridge (ft.)
bf = width of flange plate (in.),

The
Standard
American

ments of

.7

cost co—efficient derived from unit cost of
concrete in $/cu. yd.,

cost co-efficient derived from unit cost of steel
reinforcement in $/lb.,

cost co-efficient derived from unit cost of
structural steel in $/lb.,

depth of web plate (in.),

cut-off ratio in anchor span,

cut—off ratio in suspended span,

length of
length of
length of

number of

thickness

thickness
i = 1,2,.

thickness

thickness

anchor span (ft.),
span (ft.),
cantilever (ft.),

girders,

of slab (in.),

of flange plates (in.),
., 10,

of web plate in anchor span (in.),

of web plate in suspended span (in.).

design of the bridge is controlled by the current
Specifications for Highway Bridges adopted by the
Association of State Highway Officials. The require-

the specifications which are applicable to the

design are as follows:

The reinforced concrete slab is designed according to

the conventional elastic design procedure which is described

in most text books on reinforced concrete design. The design

 

8

of slab is such that the concrete and reinforcing steel are

stressed to their maximum allowable limits at the same time.

The main reinforcement is perpendicular to the direction of

traffic. Practical considerations would set the minimum

thickness of the slab and the minimum reinforcement to be

7.5 inches, and 0.66 in.2/ft., respectively. The mathematical
‘\

expression for the thickness of slab is given by the follow—

ing equation

T8 = VM/K + c (1—2)
where TS = thickness of slab (in.),
c = cover for main reinforcement (in.),
M = maximum total external moment (ft.—lb. per ft.

width of slab) which is the sum of the moment
due to live load with impact and moment due

to dead load of slab and future wearing Surface
(25 lbs. per sq. ft. of slab). The moment due
to live load is determined according to Section
1.3.2 of the Specifications which states the
following. The live load moment for simple
spans shall be determined by the following
formula (impact not included):

S + 2 P

ML.L. : “Tab—— 20 (1'3)

live load moment (ft.-lb.),

1n wh1ch ML.L.

S = girder spacing minus 1/2 the
width of flange plate (ft.),
P = load due to HS 20 truck loading

20
(lbs.).

9

In slab continuous over three or more supports,
a continuity factor of 0.8 shall be applied to
the above formula (1-3) for both positive and
negative moment. The value of impact is to be
0.30 (max.).

The positive and negative moments due to

dead load are determined by the following

formulas:
+ 2
MD.L. — (1/14) w Sg (1-4)
M‘ = (1/10) w s2 (1—5)
D.L. g
in which M; L = positive moment (ft.—lbs.),
MB L = negative moment (ft.-1bs), and
w = uniform dead load (lbs. per

sq. ft. of slab).

Refer to Section 904 of the ACI Building code
(ACI 318-63) for explanation of formulas (1—4)
and (1—5).

design constant determined from the elastic

design as follows.

K (1/2) fcjk (1—6)

allowable compressive stress of

in which fC

concrete (psi.),

1 — 1/3 k,

L_J-
ll

10

k : 1/ [1 + fS/ (nfc)]
f8 = allowable tensile stress-of steel (psi.),
n = ratio of modulus of elasticity of steel to that

of concrete.

The crossesectional area of steel reinforcement is determined

from the following equation.

.AS = M (in. — lbs.) / (fsjd) (1-7)

where d = TS— c

The actual bending stresses in the top and bottom flanges
must be less than cu" equal to the allowable bending stress.
This requirement is usually checked at the points of the
maximum bending moments and at the cut—off points (i.e.,
points at which the thickness of the flange plates is changed.)
of the flange plates (Section 1.7.1 of the Specifications) by

the following:
f .. g F (1-8)

 

where fb = actual bending stress (psi.),
Fb = allowable bending stress (psi.),
C = distance from neutral axis (in.),
I = moment of inertia (in.u),
Mg = maximum total external moment (in.—lbs.).

At the cut—off points the flange plates of the girder are
connected by butt welds. The actual bending stress in the weld
metal and the base metal adjacent to the butt welds must be
less than the allowable fatigue stress. Refer to Section

1.7.3 of the Specifications.

ll

fb 5 fr (1-9)

where fr 2 allowable fatigue stress (psi.).

The current design practice is to use steel plate
girders without transverse intermediate stiffeners. The
thickness of web plate, when transverse intermediate stiff—
eners are omitted, shall not be less than the thickness
determined by the following formula: (Section 1.7.72 of the

Specifications)

h/fv
tw : 7500 (1—10)

the thickness of web plate (in.),

where t
w

h = the depth of web plate (in.),
fV = the average calculated shear stress (psi.) in
the gross section of the web plate at the point

considered. Thus,

f = V/(ht )
V w

: +
V VD.L. (VL.L.) (Sg/WL) (1-11)
VD.L. = shear due to dead load (lbs./girder),
VL L = maximum shear due to H8 20 truck, or HS 20

lane loading (lbs./1ane),

W = width of lane loading (ft.).

In no case, the web plate thickness shall be less than
h/150. Refer to Section 1.7.72 of the Specifications.

According to prevalent practice tw should not be less than

0.375 inches. Thus, the following inequalities hold:

l2

_ h/lSO (1—12)

(‘1'
V

W

t 0.375 (1-13)

W

IV

For composite girders, the ratio of the depth of the
steel girder alone to the length of the Span shall not be
less than 1/30. (Section 1.7.11 of the Specifications).
According to prevalent practice the depth of girder shall
not be less than 42 inches. Thus, the following inequalities
hold:

h/L 1/30 (1—14)

IV

h 3 42 (1—15)

where L = L1 or L

The ratio of the compression flange width of the thickness
shall not exceed the value determined by the following formula.

(Section 1.7.70 of the Specifications).

bf 3250 I
__ : .__—— (1-16)

tf ’fb

but in no case, shall bf/tf exceed 24. (Section 1.7.70 of

the Specifications), i.e.,

bf

tf

24 (l-l7)

IA

The preceding constraint (1-17) also applies to the
tension flange. (1—18)
The width of the top and bottom flanges should be the

same, and should be equal to or greater than 14 inches.

 

l3

bf 2 14 (1-19)

The ratio of the live-load—plus—impact deflection at
mid span to the length of the span shall be less than
1/1,000, and the ratio (If the live—load-plus—impact deflec—
tion at the end of cantilever to its length shall be less

than 1/350. Thus, the following constraints apply:

A < l
“L:— " #1000 (1-20)
33 5 1 (1—21)
LC 3 0
where A = deflection at mid span, and
Ac = deflection at the end of cantilever.

The factor of safety against lateral buckling of the
steel girders, during transportation and erection must be

equal to, or greater than 1.25, i.e.,

FS = for /fb 3 1.25 (1-22)
where FS = factor of safety against lateral buckling, and
for = critical bending stress. (See Reference (1) for

the calculation of fcr')'

According to prevalent practice, the minimum length of

the: cantilever is to be 5 feet. Thus,

L z 6 (1—23)
c

The girder spacing should be within the range of 6.5
feei: to 12.0 feet. Thus,

6.5 < S 5 12.0 (1-24)

14

The above described constraints are based upon the

current AASHO specifications and prevalent design practice.

It is recognized that from time to time, the requirements

.would be changed to reflect new research and/or economic

conditions.

These changes, however, can be incorporated in

the computer programs prepared for this study without major

difficulty.

1.4 Notation

A list of important symbols used in this thesis is given

as follows:

A

As

+4

[aijJ’ matrix derived from constraints;

cross—sectional area of steel reinforcement
. 2 ‘

(1n. /ft.);

[bi]’ matrix derived from constraints;

length of bridge (ft.);

width of bridge (ft.);

width of flange plate (in.);

{cj} matrix of cost co-efficients;

cost co—efficient derived from unit cost of
concrete in $/cu. yd.;

cost co-efficient derived from unit cost of steel
reinforcement in $/lb.;

cost co—efficient derived from unit cost of
structural steel in $/1b.;

depth of web plate (in.);

cut—off ratio.:hi anchor span;

15

cut-off ratio in suspended span;

length of anchor span (ft.);

length of suspended span (ft.);

length of span (ft.);

length of cantilever (ft.);

number of girders;

girder spacing (ft.);

thickness of slab (in.);

thickness of flange plate (in.);

1, 2, . . . , 10;

thickness of web plate in anchor span (in.);
thickness of web plate in suspended span (in.);
{xj} vector of design variables;

nonlinear objective function;

effective lower bound of Z;

linearized objective function;

square or rectangular matrix;

column vector.

CHAPTER II

DESIGN BY LINEAR PROGRAMMING

2.1 Application of Linear Programming to A Non-Linear

Programming Problem

 

A cost-optimization problem in structural design can
usually be cast in the form of a problem of nonlinear pro-
gramming. The general mathematical statement of a nonlinear
programming problem is as follows.

x

Find x xn so as to minimize

l:2>°°°9

Z = f(xl, x2, , xn) (2-1)
subject to the constraints:

gl (x1, x2, . . . , xn) < bl

g2 (x1, x2, . . . , xn) 5 b2

(2—2)

gm (x1, x2, . . . , xn) 5 bm
and

xj 2 0 for j = l, 2, . . . , n,
where f (xl, x2, , xn) and the gi (x1, x2, . . . , Xn)

are given functions of the n variables.

16

17

The above problem can be re—written as follows. Find a
vector X = {X1’ X2, , xn} so as to minimize:

Z = f(X)
subject to gi (X) 5 bi for i = l, 2, . . . , m,
and xj 3 0 for j = l, 2, . . . , n.

For the bridge design problem the x's are the design variables
such as the number of girders and the web depth, Z is the cost
of the structure, and gi (X) represents a constraint such as a
requirement Of the AASHO specifications. When f (X) and/or
any of the gi (X) are nonlinear, the problem is one of non-
linear programming. Generally, an exact solution of a non—
linear programming problem is very difficult, and only
approximate solutions can be obtained. The use of the linear
programming method is one of the methods that may be used to
obtain an approximate solution of the nonlinear programming

problem. This approach is described in the following sections.

2.1.1 Linearization

 

The first step of the approach involves a linearization
of the nonlinear problem. The step consists of the approxi—
mation of the nonlinear function by the linear terms of the
Taylor series expansions. The linear terms of the Taylor

series of a nonlinear function in n variables expanded about

0 O O
the point X0 = (x1, x2, , Xn)’ are as follows:
0 3f(XO) 0 o
f(X) = f(X > + _.___.___ (x1 — x) + . . . + af(x > (x — x°>(2_3)
3X1 1 ESE—— I1 I1

I]

0
where 3f(X ) : 3f(X)
5x. 8x. 0
J 3 X=X

By a linearization of the objective function and the constraints

the problem becomes one of linear programming.

2.1.2 A Linear Programming Problem

 

The-mathematical statement of the general form of the
linear programming problem is as follows.

Find x x xn which minimize the linear objective

1:29°-°9

function,

X + . . . + c x (2—4)

+
allxl + a12X2 + aln n 5 bi
+ + +
a21X1 a22x2 a2n n 5 b2
+ +
amlxl am2X2 ° ' ’ + amnxn S bm

with xj > 0,

in which aij’ bi’ and cj are all known constants.

The above can be written in matrix notation as follows.

Find X = {XI’ x2, , Xn} so as to minimize

Z = CX (2-5)

19

subject to the constraints

AX g B
X z 0
where
A : [aij], B = {bi}
C = {cj}, X = {xj}
for i = l, 2, . . . , m, and j = l, 2, . . . , n.

IE1 contrast 'to the case of non-linear programming problem,
a solution of a linear programming problem can usually be
obtained by a specific algorithm.

A linear programming problem, in which some variables
are required to take on only discrete values, is defined as
a "mixed" problem. A solution of a mixed problem may be
obtained by using the Land and Doig algorithm described in
Reference (7), if an approximate solution obtained by round—
ing off the continuous solution to satisfy the discreteness
requirements is deemed too crude.

The Land and Doig algorithm can be briefly described as
follows. First, a continuous solution of a mixed problem is
obtained. Then, two additional solutions of the problem are
obtained by forcing a discrete variable to take on successively,
the discrete values on either side of its value from the
continuous solution, while all the other constraints remain
unchanged. The solution with the smaller value of the linear
objective function is selected, and the corresponding discrete
variable is held constant. In this manner, other discrete

variables are forced to take on discrete values, and each time

20

a solution with the smaller value of the linear objective
function is selected. Proceeding this way, the vector of
design variables in which all the variables required to take
on discrete values have done so is considered to be a solu-
tion of the mixed linear programming problem.

An iterative procedure involving the solution of a
mixed linear programming problems can be constructed to obtain
an approximate solution of the non-linear programming problem.

This is illustrated in the following.

2.2 Application to Design,

 

The application of the above mentioned method is explained

in the following.

2.2.1 Statement of the Linear Programming Problem. The
linearized objective function ZL to be minimized is obtained
by a linearization of the expression of Z as given by Equation
(1—1) by a direct application of equation (2—3), and is

presented as follows.

0 O O
ZL = ClBLBWTS + C2BLBWAS + 0.5C3N (l-Kl) leftfl

+

O O O O O O
C N K le t + C N [0.5 (l-Kl) L + LC] bft

3 l f f2 3 f3

1

+

O 0 TO 0 OO O O
C3N (l-K2) (LB-EC) 13],:ch4 + C3N K2 (LB—LC) bftf5

O OO O
1bftf6 + C3N Kllefo7

+

o o
0.5C3N (l—Kl) L

+

o o o o o o o
C3N [0.5 (l—Kl) L + LC] bftf8 + C3N (l—K2)(LB-LC) bftfg

l

+ O O O O
C3N K2 (LB—LC) bftflo

O O O O O O O O
+ C3N [-0.5 (tfl + th + tf3 + tf8) + tf2 + tf7] lele
O O O O O O O
+ C3N [-LB (tfu + tfg) + L (tfUr + tfg - L (tf5 + tflo
O O
+ LB (tf5 + tflo)] be2

O O O O O
+ C3N [(Ll + LC) twl + (LB — LC) th] h
0 O O O O O O
+ C3N [tf3 + tf8 — (1—K2) (tfH + tfg) — K2 (tf5 + tflo)
O O O
+ h (t l - tw2)] LC

0 O
+ C N [0.5 (1 - Kl) Ll (tfl + tf3 + tf6 + tf8)

O O O O O O
+ KlL1 (tf2 + tf7) + LC (tf3 + tf8)
O O O O
+ (1 — K2) (LB — LC) (tfu + tfg)
O O O O
+ K2 (LB — LC) (tf5 + tflo)] bf
O O O O O
+ C3{ [0.5 (1 — Kl) Ll (tfl + tf3 + th + tf8)
O O O O O O
+ KlLl (tf2 + tf7) + LC (tf3 + tf8)

O

o o o
+ (l - K2) (LB — LC) (tf4 + tfg)

O O O O O
+ K2 (LB - LC) (tf5 + tf10)] bf

+ (LB — Lo) ho to } N (2-6)
C

o o o
+ (Ll + LC) h tw W2

1

The superscript "o" denotes the known values of the design
variables of the initial design for the first cycle, or, for
subsequent cycles, the known values of the design variables
of the preceding cycle.

Some of the constraints imposed by the specifications

are nonlinear. A nonlinear constraint can be linearized as

22

follows. All the variables of a nonlinear constraint are
transferred to the left side of the equality, or inequality
representing the constraint. The linear terms of Taylor
series expansion of that side, about the given expansion
point, yields a linearized constraint. 3

Consider a non—linear constraint ¢i(X) 5 bi' The linear
terms of the Taylor series expression of ¢(X) at a expansion

0
point X is as follows.

_ o 3¢i (X') o
¢1(X) - ¢1(X ) +532: (X1 - X1) +
o
+axn (Xn — Xn) S bi (2-7)

A rearrangement of the equation (2—7) yields

 

O O
a¢i(x ) X + +3¢i(x ) < b_
3x1 1 axn ' l
+ 331‘)” . x0 + . . . +-:3£(X) . x0 (2-8)
3x1 1 axn n

For purposes of calculation the following approximation

was used.

a¢i(xo) ¢i<x° + ij> — ¢i<x°) (2-9)

3x. Ax.
J J

 

The treatment of the nonlinear constraints will be
illustrated by a consideration of the nonlinear

constraint Equation (1-10):

thv
7500

w = (2—10)

23

Substituting fv = V/(h tw), in which V is the total

shear, the above is written as follows.

t3
33- = 1 (2 11)
hV Z7500)2 “

An expression of V at any point along the girder can
be written in terms of the design variables. For
example, the expression of V at the simple support of
the suspended span L2, is written as follows.

V = VD.L. + (VL.L.) (Sg/WL) (2—12)
The expressions to determine VD.L. and VL.L. can

be written as follows.

 

 

 

VD.L = w L2 (2-13)
2
— 32(L — 14) 8(L — 28)
2 + 2
vL L = I 32 +
. . - L2 L2 .
or I 26 + 0.640 L2/2] in kips (2—14)

 

which ever is larger.

where I impact factor [SO/(L2 + 125), or maximum 0.30] ,

and w the total dead load including the bridge railings
plus concrete slab and girder.
The quantity w is a function of all the design variables.

Thus, the constraint (2-11) can be written as

t3 l
X = W : b : _
¢< > h__v 7500 2 <2 15)

Since, as explained above, the quantity V can be calculated for
a given set of design variables, so can the function 0.
Therfore, by use of equation (2—9) the derivatives can be
computed. The nonlinear constraint can be thus linearized by

a direct application of equation (2—8).

24

The co—efficients of the design variables in the
linearized as well as linear constraints are the elements
aij of the matrix A in Equation 2—5. The constant terms
'on the right hand side of the equality or inequality are
the elements bi of the matrix B in Equation 2—5.

The nonlinear constraints which check fatigue, local
buckling of compression flanges, live load deflections, and
factor of safety against lateral buckling of the girder, are
not considered in the formulation of the problem using
linear programming. Usually, these constraints are inactivei
when the other constraints are satisfied, and the inclusion
of these constraints greatly increases the complexity of the
problem. It will be shown later that even without these

constraints the linear programming approach is rather time

consuming as compared to the other methods.

2.3 Iterative Procedure

 

In this section will be described a procedure for an
approximate solution of the nonlinear programming problem.
The procedure involves a number of cycles of iterations.
Each cycle consists the solution of a series of linear
programming problems. A cycle of iterative procedure
consists of: (a) the selection of the expansion point in
the space of design variables, (b) the solution of a mixed
linear programming problem obtained by linearizing the non-
linear programming problem at the expansion point, and (c)
a check of the convergence of the iterative procedure. The

details of the procedure are described in the following steps,

25
and a flow chart of the iterative procedure is shown in
Figure 2-1.

1. For the first cycle, assume a set of feasible
values for the independent design variables, and the
dependent design variables are determined by using the
conventional elastic design procedure. The vector of the
design variables-—dependent and independent——is taken as the
initial expansion point, and the cost of the bridge (based
upon the nonlinear function) is evaluated.

2. Obtain a linear programming problem by linearizing
the nonlinear programming problem. This involves the deter-
mination of the elements of the matrices A, B, and C in
Equation 2—5, as explained in Section 2.2.1.

3. Solve the mixed linear programming problem using
the Land and Doig algorithm. Obtain a continuous solution
of the linear programming problem. Denote the solution
vector by X3 = Xi,

cycle of iteration, and p the index of the linear programming

in which q stands for the index of the

problems for a given q (for a fixed expansion point XE).
If value of N in Xi is not an integer, obtain two solutions
of the linear programming problem by subjecting N to take on

integer values of either side of its value in the continuous

solution, while all other constraints remain unchanged. For
example, if N = 5.36, obtain first solution with N = 5, and
second N = 6, while all other constraints remain unchanged.

These two vectors of design variables can be indicated by

i, and Xi. Select the solution vector (Xi, or X%) with the

smaller value of the linearized objective function, and hold

X

26
the corresponding value of N constant for the linear
programming problems that are to follow. Similarly,
discreteness requirements of the variables h and bf are to
be satisfied, successively, by using a solution vector with
the smaller value of the linearized objective function
(2L); where p = 4, 5,

The vector of the design variables with discrete values
of N, h, and bf, and the smallest value of (ZL)g, is con—
sidered as a solution of the mixed linear programming program
for qth cycle of iteration, and it is denoted by £3. Compute
the value of the nonlinear objective function ZS, correspond-
ing to the solution vector X3. Note that the values of
(ZL)g, and Zq corresponding to X: are different. (Numerical
examples indicated that (ZL)E were greater than Z3.)

4. Obtain the vector X2+l by using the independent
design variables from X3, and then determining the dependent
design variables by using the conventional elastic design
procedure. The vector X3+l is taken as the expansion point
for the (q+1)th cycle of iteration, if it is necessary.

5. The iterative procedure is considered as "converged"

if the following requirement is satisfied.

zq — 2q 5 8 (2—16)
0 p

where Z2 is the value of the nonlinear objective function

corresponding to Xg, and s is of the order of 1% of 23.

If the iterative procedure converges in the qth cycle of

+ . .
iteration, the vector X3 1 18 accepted as the de81gn.

27

2.4 Implementation and Examples

 

A computer—aided design of the bridge using linear
programming consists of the following steps: (a) Assume
initial feasible values for the independent design variables,
and determine the values of dependent design variables.

A computer program named GIRDER was written for this purpose.
(b) The initial vector of design variables is used to deter-
mine the elements of matrices A, B, and C. A computer program
named MATRIX was written for this step. (c) The linear
programming problem is solved by using the OPTIMA program
which is available on the CDC 6500 Computer System at Michigan
State University. Each step was completed separately. The
entire procedure was semi—automatic, and did not prove to be
quite efficient.

Three examples were considered. The data are given in
Table 2—1. The initial designs of these bridges had been
obtained by the conventional method, and they were actually

built for use in the State of Michigan.

2.4.1 Example 1 The results of cost—Optimization of this

 

example are given in Table 2—2. The cost of initial design
of the bridge, at the first cycle of iteration, is $194,800.
Two cycles of iteration requiring five solutions of linear
programming problem are needed to determine a near minimum—
cost design. The cost of the final design, $184,900, is

5.1% less than the cost of the initial design.

 

28

2.4.2 Example 2 The results of cost-optimization of this

 

example are given in Table 2-3. The cost of initial design
of the bridge at the first cycle of iteration is $173,500.

Two cycles of iteration, requiring six solutions of linear

programming problem are needed to determine a near minimum—
cost design. The cost of the final design, $161,200, is

7.1% less than the cost of the initial design.

2.4.3 Example 3 The results of cost-optimization of this

 

example are given in Table 2—4. The cost of initial design
of the bridge, at the first cycle of iteration, is $46,440.
One cycle of iteration, requiring three solutions of linear
programming problem is needed to determine a near minimum—
cost design. The cost of the final design, $47,860, turns
out to be greater than that of the initial design.

For this example, the iterative procedure converged to
a design which costs more than the initial design. This is
not entirely unexpected since linearization of a non—linear

problem does introduce errors.

2.4.4 Discussion

 

The linear programming method, as outlined in this
chapter, seems to work for Example 1, and 2, but not for
Example 3. Improvements in the solution of Example 3 could
be attempted by selecting a different initial design point.
However, such an approach is deemed time consuming and
expensive. Therefore, alternative methods of optimization
which would consider the nonlinear problem without lineariza—
tion are investigated. These methods are described in the

following chapters.

CHAPTER III

DESIGN BY THE GRID SEARCH METHOD

3.1 Introduction

 

A successful application of linear programming to a
minimum cost structural design problem must overcome the
difficulties due to the nonlinearities of the problem and
the discreteness of the design variables as illustrated in
the preceding chapter. The procedure using linear program—
ming as described earlier did appear somewhat unwieldy. As
an alternative, a more direct search method suggests itself
because the independent design variables are usually discrete.
A straight forward examination of all the possibilities to

determine the optimal design would involve 3 mi designs,

i=1
where mi is equal to number of possible values that the ith
independent design variable can take on, and n is equal to
number of independent design variables. Of course, it is not
efficient to investigate all the possibilities in order to
determine the Optimal design. However, a suitable search
method such as the grid search method could enable the
engineer to reduce the number of possibilities to be
investigated.

For the problem at hand, a two—dimensional grid search

method is proposed to determine the values of the independent

29

30

design variablesf- N, Lc’ h and bf —-which would minimize
the total cost of the bridge. The feasible discrete values
of N and LC formulate the two—dimensional design space, and
those<lflland bf formulate the other two—dimensional design
Space.

The first step in setting up the procedure is to
establish the Space of the search, i.e., the limits and
increments of each of the independent variable. A typical
cycle of the grid search consists of (a) the determination
of the costs of designs corresponding to the, say, n points
of the grid which is centered on a given point, (b) compari—
son of these costs to locate the best (the lowest cost)
design, and (c) the use of the best design from the previous
cycle as the grid center for the next cycle, unless it coin—
cides with the center of the grid, in which case, the search
method is terminated. The design corresponding to the lowest
cost thus determined is taken to be the design desired. This
approach implies the assumption that the design space is

convex, and a local minimum is also the global minimum.

3.2. Problem Solution

 

For the application of the grid search method to obtain
pa near minimum-cost design of the bridge it is necessary to
determine the limits of the design space, i.e., the lower
and upper bounds of the feasible values of the design
variables. The size of the grid should be such that the
design space covers the practical possibilities of the

design.

31

3.2.1 Design Space and the Size of Grid. The criteria used

 

to determine the limits of the design space and grid size

are as follows. The lower and upper bounds of the number

of girders are such that resulting spacing between successive
girders should be within the range of 6.5' to 12.0'. In any
case, the number of girders shall not be less than three.

The increment of N is equal to one. The lower bound and
upper bound of length of cantilever LC are (Ll/10-2) and
(Ll/10+2), respectively. The increment of LC is set equal

to one foot.

The lower bound of the depth of the web plate h is set
by Section 1.7.11 of the Specifications which states that
the ratio of depth of steel girder alone to length of span
shall not be less than 1/30. Prevailing practice usually
considers 42" to be its minimum value. The upper bound of h
is equal to its lower bound plus 24" unless it is determined
by the grade of the roadway. The increment of h is taken to
be 6". The lower and upper bound of the width of the flange
plate bf are 14" and 22", respectively. The increment of bf

is taken to be 2".

3.2.2 Details of the Procedure. The procedure to obtain a

 

minimum—cost design of the bridge using the grid search method
is depicted in Figure 3—2, and described in the following.

1. Refer to Figure 3—1. A starting point in the design
space of N and LC is selected with N corresponding to minimum
girder spacing and LC about one tenth of the length of anchor

Span Ll. A 3x3 nine point grid (refered to as Grid I) is

32
centered on this starting point, and the values of N and
LC corresponding to the other points of Grid I are determined.

2. For a given set of N and LC at each of the nine
points of Grid I, determine the values of h and bf such that
the total cost of the bridge corresponding to the point of
Grid I is minimized. This is achieved by searching the design
space of h and bf. The following steps: 3, 4, and 5 des—
cribe the grid search in that design space.

3. A starting point in the design space of h and bf is
selected such that the values of both variables are minimum
allowable. A 3x3 nine point grid (refered as Grid II; see
Figure 3—1) is centered on this starting point. (It may be
noted that if the center of the grid is on the boundary, the
number of grid points will be less than nine.) The values of
h and bf corresponding to the points of Grid II are
determined.

4. For each of the nine points of Grid II, the design
of the bridge is completed by using the conventional elastic
design procedure, and the cost of the bridge is determined.
The costs of the nine designs are compared, and the design
with the least cost is found.

5. If the least—cost point of Grid II is the center
of the grid, the search in the design space of h, and bf is
terminated. Otherwise, the point corresponding to the
least cost design is used as the center of Grid II for the

next cycle, and the steps 3, 4, and 5 are repeated.

33

6. The cost of design corresponding to each of nine
points of Grid I is determined. These costs are compared
and the least cost design is located.

7. If the least cost point of Grid I is the center of
the grid. The search in the design space of N, and LC is
terminated. Otherwise, the point corresponding to the least
cost design is used as the center of Grid I for the next
cycle, and the steps 1 through 7 are repeated.

To reduce computational work, the information about the
designs corresponding to all grid points encountered in

the search process is stored in computer memory.

3.3 Practical Applications

 

The designs of three bridges are considered to illustrate
the application of the grid search method. The designs of
these bridges have been also obtained by linear programming

in Chapter II.

3.3.1 Example 1 The summary of the results of this example

 

is given in Table 3—1. The lower bounds of N, Lc’ h, and bf
are 5,9', 48", and 14", respectively. The upper bounds of
N, Lc’ h, and bf are 7, 132 54", and 22", respectively. The
number of designs required to be investigated to determine
the optimal design by exhaustive search is 150.

For the first cycle, N = 7, LC = 11, and h = 48, bf = 14
are the co-ordinates of the center of Grid I and Grid II,
respectively. The number of designs that have been considered

when the minimum—cost design by the grid search method is

obtained is 36. The values of N, LC, h, and bf which minimize

34
Z are 7,10', 54", and 14", respectively, and the corresponding
cost of the bridge is $184,840 or $15.66/sq. ft. of the

bridge.

3.3.2 Example 2. The summary of the results of this example

 

is given in Table 3-2. The lower bound of N, Lé, h, and bf
are 5,9', 48", and 14", respectively. The upper bound of N,

L h and bf are 7,13', 54", and 22", respectively. The

C,

number of designs for an exhaustive search is 150.

11, and h = 48,

For the first cycle, N = 7, LC
bf = 14 are the co-ordinates of the centers of Grid I, and
Grid II respectively. The number of designs covered when
a minimum—cost design by the grid search method is obtained
is 48. The values of N, LC, h, and bf which minimize Z are

6,13', 54", and 14", respectively. The corresponding cost

of the bridge is $160,130, or $15.33/sq. ft. of the bridge.

3.3.3 Example 3. The summary of the results of this example

 

is given in Table 3-3. The lower bounds of N, Lc’ h, and
bf are 3,5', 42", and 14", respectively. The upper bounds
of N, LC, h, and bf are 4,9', 66", and 22", respectively.
The number of designs required to be investigated for an

exhaustive search is 250.

For the first cycle, N = 4, LC = 7, and h = 42,
bf = 14 are the co-ordinates of the centers of Grid 1, and
Grid II, respectively. The number of designs investigated

when the minimum-cost design by the grid search method is

obtained is 32. The values of N, Lc’ h, and bf which

35
minimize are 4,8', 42", and 14", respectively. The .
corresponding cost of the bridge is $45,960, or $11.86/
sq. ft. of the bridge.

It may be noted that each of the above examples the
grid search method involved a substantially smaller number
of designs than required by an exhaustive search. Comparison
of procedures to obtain nearly optimal designs of the bridge
by the grid search and linear programming method will be
postponed after the next chapter. With a view to improving
the efficiency of the search method, the method of bounds

is considered and is described in the next chapter.

 

CHAPTER IV

DESIGN BY THE METHOD OF BOUNDS

4.1 Introduction

 

The computer—aided designs of two-span cantilever
highway bridges by using linear programming and grid search
method have been explained in Chapters II, and III, respec—
tively. The grid search method seems more efficient than
the linear programming method. The method of bounds, as
described in this chapter, is investigated as an improvement
upon the grid search method by reducing the amount of compu—
tational work. The key idea that makes the reduction possible
is that the search to determine a nearly optimal design may
be terminated if the cost of the current feasible design is
within a pre—specified "tolerance" of an "effective lower
bound".

Thus the first step of the method is the establishment
of an effective lower bound, Zf. It should be as close to
the largest lower bound as possible. The latter, of course,
is the Optimal solution and is unknown. A certain amount of
insight of the engineering problem is necessary in order to
establish an efficient Zf. Aside from its dependence on an
evaluation of Zf, the accuracy of the method also depends on
the tolerance, s. In fact, the method may net lead to a
solution if values of Zf and s are such that (Zf + s) is less

36

37
than the true optimal solution. This can be seen with the
help of Figure 4—1. The method is described in detail in

the following section.

4.2 The Method of Bounds

A nearly optimal design of the bridge by using the
method of bounds may be determined as follows. Refer to
Figure 4-1. (a) Determine an effective lower bound of the

cost of the bridge, Z (b) use a suitable search method

f.
to obtain a feasible design with cost Z, which is within
a pre-specified tolerance s of the effective lower bound,
and (c) terminate the search when first such design is
found.

A feasible design is said to be a nearly optimal if.

both of the following requirements are satisfied.

(4-1)

IA
N
:3

Z

Z < Z + 8 (4—2)

where Zu is an upper bound of Z (monotonically decreasing),
and s is an assumed tolerance.
An upper bound of Z is the cost of a feasible design.
During the search process, when a new feasible design is
found and if its cost satisfies the condition (4-1) that
cost will be taken as Zu for the next feasible design. In
this manner Zu would be monotonically decreased.
An effective lower bound Zf, is essentially a fully—
stressed design of the bridge. It is obtained by the following

procedure. The reinforced concrete slab is designed using

 

 

38
the conventional elastic design procedure. The welded
steel plate girder is designed such that the flange plates
satisfy the maximum bending stress criterion only, and the
web plate satisfies the maximum shear stress criterion, and
local buckling requirements. Other requirements such as
lateral buckling of the girder are not satisfied. Therefore,
the design is an infeasible one.

The depth of the web plate is set equal to the maximum
allowable, 1/20 of the span length L1' This results in less
material in the flange plates. The width of the flange plates
is set equal to the minimum allowable, i.e.,14". The values
of N and LC are such that Zf is minimized by using an
exhaustive search. As mentioned early, the main objective
of this technique is to determine a lower bound sufficiently
close to the least upper bound Z* without exceeding it, and
the procedure should not be overly timeeconsuming.

The value of the tolerance 8 may be assumed to be of
the order of 10% of the effective lower bound. If the value
of Zf and s are such that a nearly optimal design cannot be

located, then the values of Zf and/or 6 should be revised.

4.3 Practical Applications

The designs of the three bridges considered earlier in
Chapters II, and III are again undertaken to test the method
of bounds.

A design of the bridge is obtained using the computer
program "BRIDGE", which is modified for the method of bounds

by adding subroutines to determine Zf, and adding the

39
conditiOns (4—1), and (4—2) to check for termination of the
procedure. The program is available for use on the B 5500
computer system of the MDSH. Input data for the program is
the design data Of bridge and output consists Of the details
of a complete design. The User's Manual of the program is
given in Appendix I, the listing of the program is given in

Appendix II.

4.3.1 Example 1: A summary of the results of Example 1 is

 

given in Table 4-1. The values Of Z*, Zf + s, and Zf are
$181,900, $187,895, and $169,275, respectively. Three designs
were investigated to locate a nearly Optimal design. The

cost Of the nearly Optimal design is 2.5% more than that Of

the true Optimal design.

4.3.2 Example 2: A summary of the results of Example 2 is

 

given in Table 4-2. The values of Z*, Zf + s, and Zf are

$160,130, $162,616, and $146,501, respectively. Twenty three

4
Hmw‘ v

designs were investigated to locate a nearly Optimal design, I ,

and its cost is 0.24% more than that of the Optimal design.

4.3.3 Example 3: A summary of the results of Example 3 is

 

given in Table 4—3. The values Of Z*, Zf + s, and Zf are
$45,956, $46,344, and $42,751, respectively. Nine designs
were investigated to locate a nearly Optimal design. The
nearly optimal design of this example is the same as the
Optimal design determined by the exhaustive search.

The foregoing examples indicate that the method of bounds

requires less number of designs than that required by the

 

40
grid search method. Thus, the computer—aided near Optimal
designs Of three bridges have been completed by three
different methods of Optimization. A detailed comparison
Of the results Of these examples is presented in the next

chapter.

 

CHAPTER V
COMPARISON OF THE METHODS

OF DESIGN AND CONCLUSIONS

Computer-aided near optimal designs of two—span
cantilever highway bridges have been presented in the
previous chapters. Linear programming, the grid search,
and the method Of bounds have been used to Obtain designs
Of three bridges. Computer programs have been prepared to
implement the above methods Of Optimization. Comparison Of
these methods Of Optimization, and conclusions based upon
the results of the research are given in the following

sections.

5.1 General

Viewed as a mathematical programming problem, the cost-
Optimization Of the bridge design problem is a nonlinear
one. Only approximate solutions Of the problem are practic—
able. The approach considered consisted Of linearization
and the solutions Of a series Of linear programming problems.
However, such a procedure proved to be rather cumbersome and
with no guarantee Of good accuracy.

The search methods——the grid search method, and the
method Of bounds—-involve simple mathematical strategies

with results that are at least no less accurate than the

41

 

42

linear programming approach. Another factor worthy of

mentioning is the consideration of future changes in design

specifications. It would appear that to incorporate in

the computer program any change in the design requirements
would involve considerably less difficulty for the search
method than that for the linear programming method.

The major difference between the grid search method and
the method Of bounds lies in the criteria used to terminate

the search. By terminating the search in the method Of

bounds when the cost of a current design is within a pre—
specified percentage Of its effective lower bound, the

number Of designs which are investigated to locate a nearly

Optimal design can be reduced considerably from that required

by the grid search method. However, the formulation of the

method Of bounds requires considerable insight of the bridge

design problem in order to determine an effective lower

bound Zf and a tolerance s. It seems that only a trained
structural engineer would possess such insight.

the method without modification, would fail if improper

values Of Zf and e are used.

5.2 Comparison of the Results
The results of cost—Optimization of Examples 1, 2, and
3 Obtained by the various Optimization methods, are given

in Table 5—1. Computer times used to Obtain the solutions

of these examples are given in Table 5—2.

A comparison of the cost Of the bridge design Obtained

by using the traditional method with the costs Obtained by

Furthermore,

43
using the Optimization methods indicates that the latter
produce more economical designs. Similarly, a comparison
Of the costs Of the bridge Obtained by the methods of Optimi-
zation shows that the difference between the Optimal cost
determined by the exhaustive search and a nearly Optimal
cost determined by the grid search, or by the method Of
bounds, is less than 2.5% Of the Optimal cost.

The computer cost to Obtain a nearly Optimal solution
is about 0.01% to 0.02% of the total cost of the bridge,
whereas the average saving in the total cost Of the bridge
is about 4.0% Of the total cost of the bridge designed by
using the traditional method. Thus, the computer-aided

near Optimal design would result in a net saving of about

5.3 Conclusions

 

The following conclusions based upon the results Of
the present study.

(1) It is feasible to use computer—aided designs for
the bridges considered herein and that such practice would
result in engineering economy.

(2) The search method is more suitable than the linear
programming method for the type Of structural design problems
considered in which the Objective function and the constraints
are nonlinear, and the variables are largely discrete.

(3) The method Of bounds seems to be a more efficient

approach than the grid search method. However, this method

44
does require an insight of the bridge design problem
which only trained structural engineers are likely to
possess.

An extension of the present work may include the
application Of a similar approach to other types of bridges
and also to the use of the Load Factor design method allowed
by the AASHO specifications. However, a more basic problem
that needs study appears to be an automatic procedure for
setting the new values for Zf and e in the event that the
search procedure could not produce a design that satisfies

the bounds first set.

 

45

 

(a) General View

Figure l-l Photographs of A Two-Span Cantilever

Highway Bridge.

(To be continued)

v—r— Y 7"" “r *7 W'fir- wr w

 

(b) Cantilever and Hinge

Figure 1-1 (Continued)

47

 

_‘Ref. Line (Typ.)

 

BL = Length of Bridge

__‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:: \N; l \\\ Highways
1 L1 1 fl L2
Anchor Span '_ :=Suspended Span
Cantilever Length _k_ Brg. (TYP°)
Lc

Bridge Elevation

Bw 2 Width of Bridge \

 

 

 

 

 

 

 

A .
Lil: .

|___-

S
8

 

TL; T" T
l _ 1

[Girder
Spacing

Bridge Deck Section

 

 

 

 

Figure 1—2: Two-Span Cantilever Bridge Details

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hinge
[ tf1 [rm 1‘ tf3 {tf4 [th [th
f l l l 7 l i l
x/
h N
l l l A l l l
‘1 L 71 1: ‘L .
tfe L tf7 tf8 “tf9 L tf10 tf9
Bl KlLl B LC B2 K2L2 32
L1 .1 L2
1. Br‘so LB 1
B1 = 0.5 (1 - Kl) L1 B2 = 0.5 (1 — K2) L2
Elevation Of Two—Span Cantilever Girder
h = depth of the web plate (in.),
Kl = cut—Off ratio,
K2 = cut—Off ratio,
L1 = length Of anchor span (ft.),
L2 = length of suspended span (ft.),
LC = length Of cantilever (ft.),
tf = thickness of the flange plate (in.)
i = i=1, 2, ., 10.
Given: L1 and LB
Figure 1—3: Two—Span Cantilever Girder Details

(TO be continued)

49

 

 

 

 

 

 

 

AS//r———Shear Connector

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11
p
./'O _7; > T8
\ .IX X
\ 7‘—
N.A. -\
\
4
S .2
\
tor s
w2 \
b
bf f
Composite Section Non—Composite Section
AS 2 area of steel reinforcement in
slab (in.2/ft.),
bf = width of flange plate (in.),
T8 = thickness of slab (in.),
twl = thickness Of the web plate in the anchor
span (in.),
tW2 2 thickness Of the web plate in the
suspended span (in.),
N.A.= neutral axis.

 

 

 

Figure 1-3 (continued)

50

 

8,000 Lbs.* 32,000 Lbs.* 32,000 Lbs.*

14'—0" Varies, 14'—0" Min.

y

(iii if:) 30'-0" Maxu (3:)

Standard HS 20 Truck Loading

18,000 For Moment

 

Concentrated Load
in Lbs.

 

 

26,000 For Shear

 

Uniform load 640 lbs. per

linear foot of load lane

 

 

//"//\///////////////

HS 20 Lane Loading

 

* Weight on the axle.

 

 

Figure 1—4: Standard HS 20 Truck Loading

 

51

-———-\Read input datg/
l

q (index Of cycle Of iteration) = l

 

 

p (index of linear programming problem) = 1

1

Assume feasible values of N,Lc,h, and bf, and

 

 

determine X3, and 2% (q = 1) using conven—

 

 

tional elastic design method

1]
1 '1

Determine the elements Of matrices A,B, and C

 

 

to define the mixed linear programming problem

Minimize F = CX subject at A ng, and X>0

1

Obtain solutions X3 (p = 1,2,3,. . .) by

 

 

 

 

using the Land and Doig algorithm. Determine

(ZL)g, and ZS Of the solution vector X3.

..NQ__<<:Zq - Zq <:l% qu\\. Yes 11 Xq is a
o p‘\ O //r P

solution

 

 

 

 

 

 

 

 

 

 

 

 

 

q = q + l, and p = 1

l

 

 

 

 

 

 

 

Use N, LC (rounded off), h, Use N,Lc(rounded

and bf from the solution Off), h, and bf from
“r vector, and determine X3 X3, and complete

and 23. the design

 

 

 

 

 

 

i

write the details

of design

 

End

Figure 2-1: Flow Chart of the Iterative Method

Figure 3-1:

52

TO minimize Z = Z(N, Lc’ h, bf)

 

Feasible Design Space

4:4

 

‘

\ X ‘ ‘ V \ ‘
q
\ [

 

 

2T

 

~ ———Center of
‘ Grid I

 

 

 

 

 

 

 

 

 

N

,————Feasible Design Space
l 6" '
1 .

 

 

 

 

\ l--——Center of
~ ‘ Grid II

 

 

 

 

 

 

 

 

 

 

The Grid Search Method

 

 

53

(:ngrfj)——«J\Read input data/7

Determine the limits of design space, i.e.

 

 

 

 

 

lower and upper bounds Of N’Lc’ h, and bf

 

 

 

 

 

 

co—ordinate Of center Of Grid I = co—ordinates
Of the least cost design from the previous cycle.

For the first cycle use the specified co-

 

 

ordinates.

 

_-

 

1
Determine co-ordinates of a point of Grid I

 

 

 

at which corresponding design is to be minimized.

 

 

l
Yes Are N,LC corresponding to the grid NO

point within the design Space

 

 

 

 

 

Yes Are the designs corresponding to NO

nine points Of Grid I investigated?

 

 

Determine new co-ordinates of center of Grid I

 

 

 

 

corresponding to the least cost design.

 

 

 

Are the co-ordinates Of center Of Yes
NO
Grid I changed?

 

a

 

 

__ Write the details of a minimum—cost design

 

 

 

 

 

End (to be continued)

Figure 3-2: Flow Chart Of the Grid Search Method

 

54

 

w

 

 

CO—Ordinates Of center Of nine point Grid
II = co-ordinates of the least cost design
from the previous cycle. For the first cycle

use the specified co—ordinates.

 

 

 

 

 

l

 

Determine co-ordinates Of a point Of Grid II

at which corresponding design is to be com-

 

pleted.

 

 

l

f

grid point within the design space?

Are h, and b corresponding to the

Yes NO_

 

 

Complete the design and determine the cost

 

-*d Of the bridge Z, and store required design

information

 

 

 

 

 

 

 

Are the designs corresponding

Yes NO

 

 

to nine points of Grid II ::>

investigated?

 

 

Determine new co—ordinates of center of

 

P- Grid II corresponding to the least cost

 

 

design

 

i

No Are the co—ordinates Of center of Grid II Yes

changed?

Store Optimal values of h, bf, and Z (:>

corresponding to a given set Of N, and LC

 

 

 

 

 

 

 

Figure 3—2 (continued)

55

z: Z(N, LC,h,b)

f

it KI— Feasible Design Space

 

 

 

 

 

 

 

 

 

 

 

 

(N, Lc’ h, bf)

Z = cost Of a current feasible design,
2* = the least upper bound of Z,
ZL1 = upper bound Of Z, decreasing as the

search goes on,

Zf = an "effective lower bound,"
6 = tolerance which is some assumed percentage
Of the value Of Zf.

Figure 4-1: The Method Of Bounds

 

56
Table 2-1: Data for Design Of the Bridges

 

 

 

Dimensions and Loading Example 1 Example 2 Example 3
Bridge Length, BL .236.00 232.00 152.00
Bridge Width, Bw 50.00 45.00 25.50
Span Length, Ll 116.75 114.75 74.75
Span Length, LB 116.75 114.75 74.75
Truck Loading HS 20 HS 20 HS 20

 

 

 

Properties Of

 

Materials used for Construction Of the Bridges
Structural Concrete: minimum compressive strength
at 28 days is 3,000 psi.
Steel Reinforcement: Intermediate Grade Steel
deformed bars. Yield strength
Of steel is 40,000 psi., and
allowable tension is 20,000 psi.

Structural Steel: ASTM A—36

Cost of Material

 

The cost Of material includes costs of material and
labor.
Structural Concrete: $120.00/cu. yd.
Steel Reinforcement: $ 0.22/1b.

Structural Steel: $ 0.30/1b.

57

 

 

 

Table 2-2: Results Of Cost-Optimization Of Example 1
(Using Linear Programming)

Design 1 l l l 1

Variables X0 X1 X2 x3 X4
TS 7.50 7 50 8.00 8 00 8.00
A8 0 66 O 66 0.75 0 75 0.75
tfl 0 75 0.75 0.75 0.75 0.75
tf2 1.13 1.00 1.00 1.13 1.00
tf3 0.75 0 75 0.75 0.98 0.75
tfu 0 75 0 75 0.75 0.75 0.75
tf5 l 13 1 00 1.00 l 11 1.00
tf6 1.13 l 13 1.13 1.31 1.13
tf7 2 13 2.08 2.08 2.34 2 08
tf8 1.13 1.13 1.13 1.13 1.13
tf9 1 00 1 00 1.00 l 00 1.00
tflO 2.00 1.97 1.97 2.23 1 97
K1 0 64 0 64 0.64 0 64 0 64
K2 0 65 0 65 0.65 0.65 0.65
t 0 50 O 50 0.50 0 50 0.50
wl
t 0 50 0 50 0.50 0.50 0.50
w2
h 48. 50.16 53.10 48. 54.
bf 16. 16. 16. 16. 16.
LC 7.00 11.06 11.10 11.50 11.10
N 8. 7.36 7. 7. 7.
2 194,800 188,930 188,620
ZL 487,128 487,835 492,291 490,884

 

(To be continued)

 

 

58

Table 2-2 (Continued)

 

 

 

3::iggles X0 XI X3
T8 8 00 8.50 8 50
AS 0.75 0.75 0.75
tfl 0 75 0.75 0 75
tf2 1.00 1.18 1.25
tf3 0.75 1.02 1.00
tfu 0.75 0.75 0.75
tfs 1.00 1.21 1.13
tfe 1.13 1.44 1.25.
tf7 2.13 2.50 2.50
tf8 1 13 1.13 1.25
tfg 1 00 1.00 1 25
tflo 2 00 2.40 2.38
K1 0.62 0.65 0.65
K2 0 63 0.63 0 63
twl 0 56 0.56 0.56
th 0.50 0.50 0.56
h 54. 54. 54.
bf 16. 16. 16.
LC 11.50 11.50 11.50
N 7 6 '6
2 186,800 185,050 184,900
zL 459,478

 

 

 

59

 

 

 

Table 2-3: Results Of Cost-Optimization Of Example 2
(Using Linear Programming)

Design 1 1 1 1 1

Var1ables X0 X1 X2 X3 X0
TS 8.00 8.00 8.00 8.00 8.00
A8 0.66 0.66 0.66 0.66 0.75
tfl 0.75 0.75 0.75 0.75 0.75
tf2 1.13 1.13 1.19 1.25 1.00
tf3 0.75 0.75 0.75 0.78 0.75
tf4 0.75 0.75 0.75 0.75 0.75
tf5 1.13 1.13 1.18 1.22 0.97
tf6 1.13 1.19 1.31 1.35 1.13
tf7 2.25 2.25 2.47 2.53 2.19
tf8 1.13 1.13 1.13 1.13 1.13
tf9 1.13 1.13 1.13 1.13 0.88
tflO 2.13 2.15 2.36 2.41 2.10
Kl 0.65 0.65 0.65 0.65 0.65
K2 0.63 0.63 0.63 0.63 0.63
t 0.50 0.50 0.50 0.50 0.55
w1
t 0.50 0.50 0.50 0.50 0.56
w2
h 48. 51.56 49.08 48. 54.
bf 16. 16. 16. 16. 16.
LC 6. 10.48 10.03 10.03 10.03
N 7. 6.1 6. 6. 6.
2 173,500 166,500 161,300
ZL 436,471 437,756 438,822 438,362

 

 

(To be continued)

 

60

Table 2—3 (Continued)

 

 

 

3:31.521... X3 Xi X3 X3
T8 8 00 8 00 9 00 9.00
AS 0.75 0.75 0.81 0 81
tfl 0 75 0 75 0.75 0 75
tfz 1.00 1.03 1.34 1.25
tf3 0 75 0.78 1.03 0 88
tfu 0 75 0 75 0.75 0.75
tfs 1 00 1 00 1 32 1 35
tf6 1.13 1.24 1.56 1.50
tf7 2.13 2.29 2.77 2.88
tf8 1 13 1 13 1 14 1 50
tfg 1 00 1 00 1.00 1 38
tflo 2 00 2 17 2.64 2 63
K1 0 63 0.63 0 63 0.63
K2 0 62 0.63 0.62 0 62
twl 0.56 0.56 0.56 0.63
tw2 0 56 0.56 0 56 0 56
h 54. 54. 54. 54.
bf 16. 16. 16. 16.
LC 10.00 10.50 10.50 10.50
N 6. 5.65 5. 5.

2 161,200 167,000 167,800
zL 398,674 400,284

 

 

 

 

61

 

 

 

Table 2—4: Results Of Cost-Optimization Of Example 3
(Using Linear Programming)
D3333“ 1 1 1 1 2
Var1ab1es X0 X1 X2 X3 X0
8T 8.00 8.00 9.00 9.00 9.00
AS 0 66 0.66 0.81 0.81 0.81
tfl 0.63 0 63 0.63 0.63 0.75
tf2 0 63 0.63 0.63 0.63 0.75
tf3 0 63 0 63 1.01 0.85 0.75
tfu 0.63 0.63 0.63 0.63 0.63
tf5 0.63 0.63 0.63 0.63 0.63
tf6 0.75 0.86 1.32 1.13 0.88
tf7 1.34 1.45 2.15 1.88 1.63
tf8 0.75 0.75 1.02 1.00 0.88
tfg 0 63 0 63 0.63 0.63 0.88
tflO 1 25 1.35 2.00 1.75 1.63
K1 0 59 0.58 0.60 0.60 0.59
K2 0 60 0.60 0.60 0.60 0.60
t 0 44 0.44 0.44 0.44 0.56
wl
t 0 44 0 44 0.44 0.44 0.50
w2
h 42. 44.69 42 48 48
Df 14 14. 14 14 14
LC 7 50 7 50 7.50 7.50 7.50
N 4 3.63 3. 3. 3.
Z 46,440 46,320 46,260 47,860
2 105,671 108,075 107,962

 

 

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3-1: Summary Of the Results of Example 1
(Using the Grid Search Method)
Feasible Design Space
13 V x ‘ j ‘ ‘ ‘ ‘ W ‘ ‘ ‘1‘
12‘ . (54,14) 1 (54,14)
11 ‘ (54,14) (54,14)
* ‘l
10 : (54,14) : (54,14)
\ i
9 4 (54,14) (54,14)
4 5 6 7
N
13 v T1 W \ \ ‘ ‘ ‘ 1 V wi‘
12 : 185,860 ~ 192,010
. 186,460 1 186,240
11 N r
K
10 ~ 186.910 1 184.840
I \
9 14", . .. \ 189,200Li 186,470
4 5 6 7
N
Note: The Optimal values of h, and bf are Shown in the

first table, and corresponding values Of Z are
shown in the second table, for a given set Of N,
and Lc' O denotes the co-ordinates of the center

Of grid for the first cycle.

 

 

 

13

12

ll

10

13

12

11

10

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3-2: Summary Of the Results Of Example 2
(Using the Grid Search Method)
Feasible Design Space
(54,14) (54414) l (48,14)
E : e
8- (54,14) (54,14) . (48,14)
4 i J
“ l
\ (54,14) (54,14) (48,14)
. if
k
4 (54,14) ‘ (48,14)
K W
4 5 6 7
N
167,680 160,130 166,610
: 168,220 160,240 ‘168,450
} \
\ 168,080 160,520 )166,760
K N
4 162,980 x166,600
~ I
l \
4 5 6 7

 

 

64

Table 3-3: Summary Of the Results Of Example 3
(using the Grid Search Method)

Feasible Design Space

(48,14) (42,14)

 

\ ‘ \ X ‘ E
‘

(48,14). ‘ (42,14)

7

7

 

\ ‘4

‘ (48,14) ‘ (42,14)

 

\ N

‘ (48,14) ~ (42,14)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 4 5 6
N
47,620 46,320
. 47,870 4 45,960
4
~ 48,020 46,140
‘1
\ V
448,090 47,010
P
k
3 4 5 6

 

 

 

65

Table 4—1: Summary of the Results Of Example 1
(using The Method of Bounds)

2* = $181,900 2f = $169,275 zf + e = $187,895

Feasible Design Space;;;7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13 x \ x \ x V V x T x n
. 4(/
12 ‘
c \ \
ll .
k \
10 K \ (54,114)
9 \ \ x \ L x \ \ ¥A ‘W
4 5 6 7 8
N
13 (ff \ K \ \ \ VV V rt
12 \ ‘
\ \
K \
11 ‘ Q
\ V 1 4 840
10 K 8',
: \
K \
9 X \ \ \ \ L \ \\_1\ X L x \
4 5 6 7 8
N
Note: The Optimal values Of h, and bf are Shown in the

first table, and corresponding values Of Z are
shown in the second table, for a given set Of N,
and LC. 0 denotes the co—ordinates Of the center

of grid for the first cycle.

  

‘lllllllllllH

 

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4-2: Summary Of the Results Of Example 2
(using The Method Of Bounds)
2* = $160,130
zf = $146,501
zf + s = $162,616
Feasible Design Space
13 V ‘ v V ‘ \ \ \ \ ‘fi7
/.
12 ‘ 4 (48,14)
t i
\ W
11 (54,14) ( (48,14)
4
10 . (54,14) (48,14)
K \4
9 K\ A A \ L\ A _\ x x 1;
4 5 6 7
N
13 V x‘ x \ \ X \ \ \ K x
L w
12 ‘ ‘ 168,450
x 4
11 ~ 160,520 L166,760
L i
10 ~ 162,980 ‘ 166,600
“g 4
K \
g A L\ 3 x x x; L \\ x \
4 6 7

 

 

 

 

 

 

'Table 4-3:

67

Summary of the Result Of Example 3
(using The Method of Bounds)

2* = $45,956
2f - $41,751
2 + s - $46,344

§;—————Feasible Design Space

 

(42,14)

 

(42,14)

 

(42,14)

 

 

 

 

 

 

 

45,960

 

46,410

 

1

47,010

 

 

7 I I I

 

 

 

 

 

 

 

 

 

 

68

 

 

 

 

 

 

 

Table 5—1: Comparison of the Results
Method Independent Cost Number
Of Design $ per of
Optimization Variables sq. ft. Designs
N La h bf
Example 1
Traditional Method 6 7.0 48 16 17.05
Exhaustive Search 6 13.0 66 18 15.42 150
Linear Programming 6 11.5 54 16 15.68
Grid Search Method 7 10.0 54 14 15.66 36
Method Of Bounds 7 10.0 54 14 15.66 3
Example 2
Traditional Method 6 9.0 48 16 16.14
Exhaustive Search 6 13.0 54 14 15.33 150
Linear Programming 6 10.0 54 16 15.48
Grid Search Method 6 13.0 54 14 15.33 48
Method Of Bounds 6 11.0 54 14 15.38 23
Example 3
Traditional Method 4 7.5 42 14 11.87
Exhaustive Search 4 8.0 42 14 11.86 250
Linear Programming 3 7.5 48 14 11.93
Grid Search Method 4 8.0 42 14 11.86 32
Method Of Bounds 4 8.0 42 14 11.86 9

 

 

 

 

 

 

69

 

 

 

 

 

Table 5-2: Comparison Of Computer Time
Example Method Of Total Computer Computer
Optimization Time in Sec. System

Linear Programming 501 CDC 6500

1 Grid Search Method 321 B 5500
Method Of Bounds 199 B 5500

Linear Programming 502 CDC 6500

2 Grid Search Method 369 B 5500
Method Of Bounds 7 269 B 5500

Linear Programming 321 CDC 6500

3 Grid Search Method 305 B 5500

Method of Bounds 213 B 5500

 

 

 

 

 

 

LIST OF REFERENCES

 

 

LIST OF REFERENCES

Clark, J. W., and Hill, H. N., "Lateral Buckling Of
Beams," Journal of the Structural Division, ASCE,
Vol. 86, No. ST 7, July 1960.

Douty, R. T., "Optimization Of a Two-Span Cover—Plated
Steel Beam," Computers in Engineering Design
Education—Civil Engineering, VOl. III, University
Of Michigan, College of Engineering, April, 1966.

 

 

Gass, S. I., Linear Programming 3rd edition, McGraw-Hill
Book Company, Inc., New York, 1969, pp. 49—88.

 

Goble, G. G., and DeSantis, P.V., "Optimum Design Of Mixed
Steel Composite Girders," Journal of the Structural
Division, ASCE, Vol. 92, NO. ST 6, December,1966.

Hiller, F. S., and Lieberman, G. J., Introduction to
Operations Research, Holden—Day, Inc., San Francisco,
1968, pp. 565—5711

 

Kelley, J. E., "The Cutting—Plane Method for Solving
Convex Problems," Society of Industrial and Applied
Mechanics Journal, Vol. 8, NO. 4, December, 1960,
pp.’703-711.

 

Land, A. H., and Doig, A. G., "An Automatic Method of
Solving Discrete Programming Problems," Econometrica,
Vol. 28, 1960, pp. 497—520.

Moses, F., "Optimum Structural Design Using Linear
Programming," Journal of the Structural Division,
ASCE, Vol. 90, NO. ST 6, Proc. Paper 4163, December,

1964, pp. 89-104.

United State Steel (USS), "Composite: Welded Plate Girder,"
Highway Structures Design Handbook, United States
Steel Corporation, Pittsburgh, Pa., 1965, pp. 4.1 —
4.68.

70

 

 

 

APPENDIX I

A COMPUTER PROGRAM - BRIDGE

—USER'S MANUAL-

By
Kulkarni Sudhakar R.

Department Of Civil Engineering
Michigan State University

Ju1y, 1973

 

 

 

 

TABLE OF

1. Introduction . . . . .
2. Scope and Limitations
3. User's Card Deck . . .
4. Data Card Deck . . . .
5. Output . . . . . . . .

6. Example . . . . . . .

FIGURE

l-A Example Structure

71

CONTENTS

Fifi
.72
.. 72
72

72

1. Introduction

 

"BRIDGE" is a computer program for a nearly Optimal
design Of two-span cantilever bridges. The program is
written in FOTRAN IV, and it is available on the B 5500
computer system of the Michigan Department Of State Highways.
To use the program one needs the usual peripheral cards for
access to the computer and data cards for his program.

In the following paragraphs, the scope and limitation
Of the program, the makeup Of the user's card, the data
card deck, and the description Of the output are given. Next,
an example is given.

The program is written by Kulkarni, S. R. The subroutines-
which are required to design the welded plate girder are
provided by the computer programming section Of the design

division Of the Michigan Department of State Highways.

2. SOOpe and Limitation

 

The program is intended to determine a near minimum-cost
design Of the two-span cantilever bridge. The type of bridge
is reinforced concrete slab on welded steel plate girders with

sheer connectors.

3. User's Card Deck

 

A set of these cards can be Obtained from the computer
programming section. Usually, the first card of the set
shows the account number of the program. The second card
shows the computer time requested, and the remaining cards
are the system cards for the access to computer. For data

deck, see next section.

4. Data Card Deck

 

Card
NO.

 

Data

 

SQUAD, SQD
MONTH
IDAY

IYEAR

BRL
BRW
SPANLl
SPANLB

RDW

TUTIL
TSWLL
RAILWT

LOAD

STRESA
FC
RMOD

FR

C1
C2

C3

UBWEBD

73

 

 

 

Col. NO. Specification
ll 18 2A4
21 22 I2
24 25 I2
27 28 I2
11 20 F10.2
21 30 F10.2
31 40 F10.2
41 50 F10.2
51 60 F10.2
11 20 F10.2
21 30 F10.2
31 4O F10.2
41 43 I3
11 20 F10.2
21 30 F10.2
31 40 F10.2
41 50 F10.2
ll 20 F10.2
21 30 F10.2
31 40 F10.2
11 20 F10.2

Remarks

Identification

Date

Bridge

dimensions

Loading

Material

prOperties

Cost data

Bound Of variable

 

 

Program

M
BRL
BRW

Cl

02

03

FC

FR

LOAD

RAILWT

RMOD

SPANLl
SPANB

STRESA

RDW

TUTIL

TSWLL

UBWEBD

Problem

' Symbol

BL

Bw

74

' ExplanatiOn

Length Of bridge (ft.),

width of bridge (ft.),

cost of concrete ($/cu. yd.),

cost Of reinforcement ($/lb.),
cost of structural steel ($/1b.),
allowable compressive stress Of
concrete (psi.),

allowable tensile stress Of rein-
forcement (psi.),

truck loading, 1 = HS, and 2 =

H loading,

total weight of bridge railings (1b.),
modular ratio Of concrete, usually
10,

length Of anchor span (ft.),
length of span (ft.),

allowable stress in bending for
steel (psi.),

clear roadway width (ft.),

total utility (Oil or water pipes)
load (lbs./ft.),

total Sidewalk live load (lbs./ft.),
upper bound Of the depth Of web

plate (in.).

75

5. Output

On the output sheets the following information is
printed.

l. The input verification for the data of the problem
as shown on page 78.

2. The details of the minimum—cost design of the bridge
as shown on page 79, and

3. The details Of the design of the two—span cantilever
girder as shown on pages 80 through 83.

The input information as described in Section 4——Data
Card Deck--is printed on the first sheet Of the output.

The details of a near minimum-cost design Of the two—span
cantilever bridge as shown on page'fllconsists Of the following.
The bridge data, load data, cost data, the values Of the
independent design variables which minimize the cost Of the
bridge, Slab design, girder design and cost analysis. The
average weight of girder in both spans and corresponding
factor Of safety against lateral buckling are given under
girder design heading.

The details Of the design of the two-span cantilever
girder consists of the details of the design of girder in
suspended span L2, which are shown on pages 80 and 81, and
the details Of the design Of girder in anchor span Ll’ which
are shown on pages 82 and 83-

The explanation of the details Of the girder design is
as follows. First, the average weight Of girder is given.

Then, the section properties (moment Of inertia, section

76

modulus) are given for non-composite, and composite sections
Of the girder. Note that N denotes the modular ratio of
concrete. The length (ft.) and thickness (in.) Of flange
plates are listed, and the maximum bending stresses in tOp
and bottom flanges are listed next. The moments (ft.-kips)
due tO dead load, live load and sidewalk plus future wearing
surface loads are given. At the end Of the page, the maximum
reactions (kips) at supports are listed and the design Of the
bearing stiffeners is given.

The details of the girder design Of the suspended Span
are continued as shown on page 81. The deflections Of the
girder due to dead loads and ratio of live load deflection
to span length at mid span are given. Usually the girders
are designed without using the transverse intermediate
stiffeners, and this is indicated by the statement--"none
required between brgs." on the output sheet. The actual
factor of safety against lateral buckling is given. The
information — Input Verification — lists the data used by
the subroutine GIRDER to carry out the design Of the girder.
The thickness and depth Of web plate and width Of flange
plate are given here.

The details Of the design of girder in the anchor span

are described similarly and are shown on pages 82 and 83.

6. Example

Figure l—A shows the two—span cantilever bridge. The

data for the design is given as follows.

77

Bridge Dimensions: BL = 236.00 ft., Bw = 50.00 ft.,

L1 = 116.75 ft., LB = 116.75 ft., and RD = 46.50 ft.

Loading: Total utility load = 0., total sidewalk live
load = 0., weight of railings = 954 lbs./ft., live load = HS
20 truck loading.

Material Properties: Fb (A 36 structural steel) = 20,000
psi., fC = 3,000 psi., n = 10, f8 (reinforcement) = 20,000 psi.

Cost Data: The unit costs Of concrete, reinforcement and
structural steel are $120/cu. yd., $0.22/1b., and $0.30/lb.,
respectively. The upper bound Of the depth of web plate is

54 inches.

REINF.STEEL=

0.22
SIR. STEEL = 0.30
UPPER 8ND, 4E8 DEPTH a

'78

INPUT VERIFICATION
SQUAD :KULKARNI DATE =27- 3-73
BRIDGE L = 236.00 BRIDGE 4: 50.00
SPANLI = 116.75 SPANR : 116.75
CL ROAD 4: 46.50 TOTAL UTIL= 0.00
TOTAL SwLL = 0.00 RAILING NT: 954.00
TRUCK LOAD: I STRESS = 20000.00
TO IN 0040 = 1200.00 400. RATIO: 10.00
rs IN REINF: 20000.00

COST DATA
CONCRETE a 120.00- DOLLARS PER CU.Y0.

.DOLLARS PER L8.
DOLLARS PER L8.

54.00

 

IND-SPAN CANTILEVER BRIDGE
BRIDGE DIMENSIONS
BRIDGE LENGTH = 236.00
BRIDGE WIDTH = 50.00
SPAN 1 LENGTH a 116.75
SPAN 2 LENGTH = 116.75
ROADWAY WIDTH = 46.50
LOAD DATA
LOADCI=HS 2=H) = 1
RAILING WT. = 954.00
SDWALK L. LOAD = 0.00
FUT. WEARING = 25
TOTAL UTILITY = 0.00
COST DATA
COST OF CDNC. = 120.00
COST OF REINF. = 0.22
COST OF STEEL = 0.30
DESIGN VARIABLES
GIRDER SPLCING = 90.00
NO. OF GIRDFRS = 7.
DEPT” OF HER = 54.00
UPPER 8ND. HEB = 54.00
CANT. LENGTH = 10.00
FLANGE PL WIDTH: 14.00
SLAB DESIGN
SLAB THICK. = 8.00
TRANS. REINF. = 0.75
GIRDER DESIGN
AVEWT. SPAN 1 = 231.
AVEWT. SPAN 2 a 222.
FSAFTY SPAN 1 = 1.30
SAFTY SPAN 7 = 1.61
COST OF SLAB : 35982.72
COST OF STEEL = 131189.58
MIN.BRIDGE COST: 184839.52
15.66

BRIDGE COST 2

79

DESIGN
MATERIAL

FT. FSCREINF.)
FT. FC(CONC.)
FT. FS(STEEL)
FT. MOD. RATIO
FT. -
IRSO/FT.
L8$./FT.
LRs./59.FT.
LRS./FT.

s/LB.

S/LB.

TN.

TN.

IN.

FT.

IN.

IN.

SQ.IN./FT.

'LB$./FT.
LBS./FT.

tfléflfifi‘w

/SQ.FT.

II II II II

SQUAD

PROPERTIES

20000.
1200.
20000.
10.

KULKARNI

PSI.
PSI.
PSI.

 

 

 

80

P L A T E G I R D E R SQUAD KULKARNI
PRUB. N0. 1.
AVG. WT./FT. = 222.
G I R 0 E R COMP. SECT. N=8
I SCTOP) SCBOTT) I
24317. 774. 999. 69707.
31796. 854. 1670. 103273.
CENTER “0710. 1190. 1757. 104170.
31796. 854. 1620. 103273.
24317. 774. 999. 69707.
C O M P O S I T E S E C T I O N
N=10 N=30
I 8(T0P) stnan) I 5(109) SCROTT)
65904. 6840. 1430. 47164. 2030- 1298.
96055. 651/10 22°30 649090 2526. 90800
CENTER 97761. 6651. 2291. 69140. 2826. OIOIL
9.455. 651a. 2293. 64949. 2526. 9084.
LENGTH THICK,
27.6 0.625
O P P L A T E ’ 51.0 1.125
78 2 0.625
1806 10125
0 T T 0 M P L A T E 70.0 2.25C
' 1802 1012“
LT BRG 0.50L RT ORG
T R E S S E 8 TOP 0. 19422. 0.
ROTT 0. 19841. ‘ 0.
D M E N T 8 LIVE LOAD O. 1361. 0.
SN 9' FNS 0. 431. 00
nEAD LOAD 70.5 76.5
E A C T10 N S Lo Lo W/IMP 5405 571.5
(MAX) L. L. wO/IMP 44.8 4a.8

RRG STIFF 0.625 X 6.50 0.625 5 6.50"

81

P L A T E G I R D E R SQUAO
PRDR. NO. 1.

LT PIN CL
REAM 0.00 0.71
SLAR ADJ. SPANS 0.00 0.00
D E F L E C T . SLAR THIS SPAN 0.00 1.92
SIDEWALK 0.00 0.23
DELTA LL/L 1/1504
5 T I F E . S P A . NONE DEQD BETWEEN BRGS

FACTOR OF SAFETY AGAINST LATERAL BUCKLING 3 1.6
I N P U T V E R I F I C A T I D N

HEB THICK. 0.5000 5.”. LIVE LDAD
HEB DEPTH _ 54.00 5.". * E.W.S.
FLANGE WIDTH 14.00 CANT. LENGTH A_
SLAB THICK. 8.00 P (BEAM)
GIRDER SPA. 90.00 P (SLAB)
SPAN LENGTH 106.75 P (LIVE LOAD)
LIVE LDAD HS 20 P (SW + ENS)
DIST, FACTDR 1.10 CANT. LENGTH B
HAUNCH DEPTH 1.00 P (BEAM)
HAUNCH WIDTH 20.00 P (SLAB)
DESIGN STRESS 20000. P (LIVE LOAD)
UTILITIES O. P (5" + ENS)

ow
0 O
ONO
00‘.

DO
00000 0000

I O O.

o. o o
OOOOOOOOO

KULKARNI

RT PIN

0.00
0.00
0,00
0.00

82

P L A T E 0 I R D E R 50040 KULKARNI
PROBO N00 10
AVG. ng/FT. = 231.
0 I R 0 E R 0042. SECT. ~=8
I 51702) 5(80111 I
26212. 821. 1095.‘ 75762.
33607. 902. 1704. 108229.
CENTER 42594. 1236. 1849. 109241.
33607. 902. 1704. 108229.
26212. 821. 1095. 75762.
COMposITE sECTIUN
N=10 N=30
I 3(1021 5(BUTT) I S(TOP) SCBOTT)
71435. 6666. 1582. 50428. 2456. 1427.
100843. 6463. 2436. 67050. 2561. 7200.
CENTER 102292. 6607. 2434. 71661. 2859. 9220.
100843. 6463. 2436. 67450. 2561. 9200.
71435. 6666. 1582. 50428. 2456. .427.
LENGTH THICK.
2899 0.625
0 P P L A T E 50.0 1.125
47.8 0.625
20.9 1.250
0 T T 0 M P L A T E 69.0 2.375
36.8 1.250
LT BRG 0.48L RT 690
T R E s s E 5 TOP 0. 19443. 19876.
0077 0. 19692. 14394.
DEAD L040 0. 1536. -6no.
0 M E N T 6 LIVE LOAD 0. 1502. ~542.
sw + rws o. 430. ~177.
DEAD LOAD .1 166.0
E A C T I 0 N S L. L. H/IMP .5 91.4
(MAX) L. L. w0/1HP .2 '74.6
RRG STIEE 0.625 0.25 0.875 Y 6.25

83

P L A T E G I R D E R SQUAD KULKAQNI
PRUB. N0. 1.

LT PIN CL RT PIN

BEAM 0.00 0.63 “0.18

SLAR ADJ. SPANS 0.00 '0057 0.34

D E F L E C T . SLAR THIS SPAN 0.00 2.60 '0.77
'SIDEHALK 0.00 0.26 ’0.05

DELTA LL/L 1/1267 1/ 045

S T I r E . S P A . NONE RE00 BETNEEN BRGS

FACTOR OF SAFETY AGAINST LATERAL BUCKLING = 1.3
I N P U T V E R I r I c A T I 0 N
HEB THICK. 0.5625 S.w. LIVE LnAo 0.
HEB DEPTH 54.00 s.w. + F.N.S. 302.
FLANGE NIDTH 14.00 - CANT. LENGTH A 0.00
SLAB THICK. 8.00 P (BEAM) 0.0
GIRDER SPA. 90.00 P (SLAB) 0.0
SPAN LENGTH 116.75 P (LIVE LOAD) 0.0
LIVE LnAD HS 20 P (58 . FWS) 0.0
DIST. FACTOR 1.10 CANT. LENGTH B 10.00
HAUNCH DEPTH 1.00 P (BEAM) 14.0
HAUNCH wIDTH 20.00 P (SLAB) 41.2
DESIgN STRESS 20000. P (LIVE LOAD) 44.8
UTILITIES o. P (SW + FHS) 16.1

 

 

8 1!-

 

Ref. Line (Typ.)
B = Length of Bridge (ft.) _J

 

L

 

 

 

 

 

 

 

 

 

 

: .Highways

 

 

 

 

 

 

 

 

4 L1 _ ‘ L2
Anchor Span Suspended Span
Cantilever Length _*_ Brg. (Typ.)

Lc

'Bridge Elevation

 

Bridge Railing

Bw = Width of Bridge (ft.).

“mm if My

 

 

Ti. T

\
L. ...>J
T

 

 

 

S 2'—6" (Typ.) =l
Girder '
Spacing
Bridge Deck Section
Given: BL’ and Bw'

 

Figure l—A: Example Structure

 

 

 

APPENDIX II

COMPUTER PROGRAM

A.2.l Description of Routines

 

The computer program BRIDGE consists of the main rou—
tine BRIDGE, and the following subroutines: GRIDl, GRIDQ,
SLAB, PRELIM, DESGIR, GIRDER, PROPTY, MOMENT, STRESS, DELTA,
SHEAR, FINDM, DETAIL, DETGIR, FSDES, MINBM, and MINC. The
subroutines GRIDl, and GRID2 use the grid search method to
determine the values of number of girders, the length of
cantilever, the depth of web plate, and the width of flange
plates such that the corresponding cost of the bridge is
minimized.

The subroutine SLAB determines slab thickness, and
transverse reinforcement in slab. The subroutine PRELIM
determines utility and sidewalk loads carried by a girder.
The subroutine DESGIR determines the data for calling the
subroutine GIRDER, and completes a design of the two—span
cantilever girder. The subroutine GIRDER designs the welded
steel plate girder. The subroutine PROPTY determines the
section properties such as moment of inertia and section
moduli of a composite and non—composite section. The sub—
routine MOMENT determines moments due to dead load and live
load (HS 20 truck loading) at a section along the length of

girder. The subroutine STRESS determines_the bending

85

 

 

 

86
stresses due to the moments. The subroutine DELTA deter—
mines the deflections due to dead load and live load at the
mid—span and at the end of cantilever. The SHEAR determines
the shear due to dead load and live load at a section along
the length of girder. The subroutine FINDM selects the sec—
tion properties to be used to determine the spacing shear
connectors along the length of girder. The subroutine
DETAIL is used to write the details of a bridge design.
The subroutine DETGIR is used to write the details of a
girder design.

The subroutine FSDES, MINBM, and MINC are used to
determine an "effective lower bound" of the cost of the
bridge. The subroutine ESDES determines the value of num—
ber of girder such that an "effective lower bound" is mini—
mized. The subroutine MINBM determines the length of canti—
lever which minimizes an absolute sum of area under a bend—
moment diagram of the two—span cantilever girder. The sub-
routine MINC determines the cost of a "fully—stressed"

design of the bridge.

A.2.2 List of Identifiers

 

The important identifiers, used in the program BRIDGE,

are identified below in alphabetical order.

ALLOWC = Allowable live load deflection ratio at midfspan

ALLOWE = Allowable live load deflection ratio at the end of
cantilever

AVEWT = Average weight of girder (lb./ft.)

BCOST = Total cost of the bridge ($)

BMDL

BMSW

BMSWLL

BOTPL (M)
BOTTPT(M)=
BRL =

BRW =

CANTB

CG (M,K)

C1 =

C2 =

C3 =

DELT(J,K)
DISTLL =
FATIGC =
FATIGT =

PC =

FS(M) =

FSAFTY

FR =
GRNO =
GSPA =

HDEPTH =

87
Bending moment due to dead load at a section X
along the length of girder (ft.-kips)
Bending moment due to sidewalk dead load
(ft.-kips)
Bending moment due to sidewalk live load
(ft.—kips)
Length of bottom flange plate (ft.), M=location
Thickness of bottom flange plate (in.)
The length of the bridge (ft.)
The width of the bridge (ft.)
The length of cantilever (ft.)
Center of gravity of section, leocation,
K=Modular ratio
Cost of concrete ($/cu.yd.)
Cost of reinforcement ($/lb.)
Cost of structural steel ($/lb.)
Dead load deflections (in.)
Live load distribution factor
Allowable fatigue stress for butt weld (psi.)
Allowable fatigue stress for butt weld (psi.)
Allowable compressive stress in extreme fiber of
concrete (psi.)
Actual bending stress (psi.)
Factor of safty against lateral buckling
Allowable tensile stress in reinforcement (psi.)
Number of girders
Girder spacing (in.)

Haunch depth (in.)

 

 

HWIDTH
I(M,K)
LOAD
NEGLL
PBMB
PSLB
PSWB
PLLB
PLATEW
POSLL
RAILWT

RDW

REINF(2)

RMOD
SB(M,K)
SLABT
SLABW
SPANLl
SPANL2
SPANB
ST(M,K)
STRESA
STUD
STUDS
SWFWS

SWLL

TOPPLCM)

TOPPT(M)

88
Haunch width (in.)
Moment of inertia (in.u)
l for HS loading, and 2 for H loading
Negative live load moment at point X (ft.-kips)
Cantilever load (kips)
Cantilever load (kips)
Cantilever load (kips)
Cantilever load (kips)
Width of flange plate (in.)
Positive live load moment at point X (ft.-kips)
Total railing weight (lbs./ft.)
The width of roadway (ft.)
Transverse reinforcement (in.2/ft.)
Modular ratio of concrete
Section modulus (in.3)
Slab thickness (in.)
Effective width of slab (in.)
Length of anchor span.(ft.)
Length of span 2 - suspended span (ft.)
Length of span B = SPANL2 + CANTB
Section modulus of tOp flange (in.3)
Allowable stress in bending for steel (psi.)
Stud spacing (in.)
Number of studs/row
Sidewalk + future wearing course (lb./ft.)
Sidewalk live load (lb./ft.)
Length of top flange plate (ft.)

Thickness of top flange plate (in.)

TUTIL
UBWEBD
UTIL
VDL
VSW
VLL
WEBD
WEBT
WTFTSL
X

ZMINC

89
Total utility load (lbs./ft.)
Upper bound of the depth of web plate (in.)
Utility load (lbs./ft. of girder)
Shear due to dead load at point X (kips)
Shear due to sidewalk dead load (kips)
Shear due to live load at point X (kips)
The depth of web plate (in.)
The thickness of web plate (in.)
Slab weight (lbs./ft. of girder)
Distance from left support (ft.)
of the cost

Value of an "effective lower bound"

of the bridge ($).

000000

90

A.2.3 LISTIVL 01 FRUGHAM

 

PHUUHAM RHIUGL

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COMMON/UET/[UPVL12THPPL22TUVFLjpfiflTVLlrhU[PL2.UUIPL32XLLNT.

A LIOPLptsrUpprUPRptthTILpfdeTlDEBUT IRIUMULAthULC)D'MULfiflb‘ML-LA)

8 UMLLL.8MLL828NSNA2HMSWC2MNSNU;wSTIFF2SIUUISF1FA2P120|llthlw2

C STIFH.ST1F825T1FL

COMMON/uRIU/ UnthU.GNNULU!NGpCAkILfipCANTUd2WLdUUH!anuLDINU.

'A PLdepPLNUH.leplYCpiXGCtlYCL;IXHIIYDp1AHC;IYUCIZULLDIS)I

B LSLAU.CHLIN+.CSTLLL18CUSI:1H8(5;5);SAFIY(5,b)pPLw(5:b)p

C 8104(pr)M’H’T'erv‘))HJPIB(52‘))2(]PTG(‘J25)2UPIC(325)'5AP 11(5’5)

COMMON/SHCH/UCCST28bKNU;bCANIpdwEBUABPLATLIUA[UMRUHACP
CUMMUN/LHC/LANI./MINC

“MCOCLD

l FURMAICIOX2/A422X22(12'1X)!12)

2 FURMAI(IUX»5+10.?)

3 FORMAT(10123FlQo/IIJ)

4 FURMAI(IOX»PIU.2)

6 FURMAT(1H1;///////
A 40X, IVHINPUI VLHIFICAT[fiNp///ZUX;12H58uAu :1
B 2A424X211HUAIP 322(1221H')212p/)

7 FORMAI(£UA!12HHNIUGL L :2TlOoZ22X211HhHIduE 8:2110.2;//
A ZCXI17HSPANL1 =Ith.z.2X.11hSPAnb =2f1U822/)

8 FURMAI(ZOX212HLL HUAU w=3T1U.Z22X211HIUIAL UT1L=ttlu.d:// 20X:
A IZPYUTAL SALL :2}10.8.2X211HNA1LILL. HI=IFIUod2//)
9 FORMAI(ZOK212HIHULK lUAD=21329anlHSIRtbb =.F10.d2//
A 20X.1?HFL [A CuNC :1110.242A.11FMcu. HAIIU=2T10o22/)
10 FURNAI(ZOX21?HFS 1N HLLNF=1+1U.Zn//QOAIIUHCdST UATA://)
11 FURMAI<ZOAAI?HCUNLRETL 3’T5.3'7X918HUULLAN5 PER Lbquo2//
A 20x,12nRL1N1,STLLL:.1A.2.zx.15HnULLANS PEN Lu.://
8 2CK217HSIH. STEEL :2F8.222X215HUULLAHS PER Lu.://)
12 FUHMAI(“0X'15HIWIIIAL uLSIuhp/l/20Xn13nblflutfi SP =211U.2;8x.

0000

O

91

A . 10HNU.UF GIHUERS=,+4.0,8A:11HNLH ULPIH=2F10.2:
8 IAHCANT. LLNUTH =,14.0.//)

14 FURMAT(20XIIJHFLANGE WIUTH=2F10.222X26HWLUI=2F1Uo22//)

15 F0RMAT<ZOX2J2HCUST UP THE INITIAL ULSIGN =2ilOo227HUULLAH52)

16 FURMAI(ZOX226HCPT BCUSI AT IKIIIAL “SPAzpflquDIHUULLAd 2//)

20 FURMAT(ZOX211HNU.UF IILH=21&22X25HGRN032FQ.URZXASHLSFA=!F“oU1//)

21 FORMAI(ZOX;32HCUST UT THL CUHRLNI DESIGN =2r10.2:/fiUULLApr//)

22 FORMAI(&OXIISHCPTIMUM UESIuNp//?OX25HGSPA=II501.2X2DHNLDU=2F541.
A 2X27HPLAILN=r5.1.2K20HCANIb22F5o12//)

23 FURNAT(£OX;14HHNIUGL CUST =.I6.222X.18HUULLAHS PLH Su.rl.)

24 FURHAICdox:17HR1N. udIUuL CuST=,F10.2:?x.ZHUULLAHS://)

25 FURNATIZOX222HLPPLH HNU. WLU ULPTH =.IIUoZ’//)

30 FURMAI(ZOX24HH01N22X15HuCUDI22X22F10.2)

31 FORMAICZOAolaHCUSI UF SLAb 32511.2.2X27HDULLAR52/
A ZOXIIQHCUSI UF RLINF32F11.222X27HDULLAHS)

33 FURMAI(£OA.14HCUsT OF SILLLGf11.222Xp/HDULLARS)

40 FURMAI(ZOX22HUNU GHNUI ZFIU.ZIIQ)

“I FURNAT(ZOX210H5NU CANIU. JIIUoZ)

“2 FORMAI(20X!9HDKU WLdD! 2FIU.22 I“)

43 FURMAIIZOX2IIHEND PLAILW. ZF1067)

45 FURMAICZOX.$2HCU‘UHUINAILS UP “1“. COST ULDIGN2/ZUX22(14'2X))
46 FORMAIIZOXAJbHDLSIhN VAHIAdLLb FUR MIN-CUbI ULSIGNI/ZUA’DIIU.Z)
58 FORMAT(G3X2QHIIME22X.2F/.2)

TU HLAU DLSIGN uAIA SUCH AS BRIUGL UINLNSIUN$,MAILHIAL
PHUPLRTIL32HIGHWAY LUADINC AND CUST UAIA

170 REAU(1212LNU = 180) SQUADybu02MUNTHDIUAY2IYEAR

REAU(1;Z) BRL;8RN.SPANL1pSPANd.NUN
REAU(1:J) TuIIL,Tb~LL:NAILw|:LUAC
REAU(1:2) SIHLsAp1C;HMUu;FR
REAU(122) C1.C2,C3

REAU(124) Ubwtnu

INPUI VLHIFICAIIUN

HRIIE(J26) SQUAUASQU.MUNIHAIUAY2IYEAR
HRIIE(JI7) dRL9dHW2SPANL125PANU
HRIIECJAB) RUw:IUIIL:ISwLL2HA1LAT
HRIIE(3;9) LUADpSIRLSApFCpHMUU
NRIIE(3:10) FR
NHITL(3.11) C1:C2.C3
WRIIE(3225) UUALBU
TU ULILHMINL 18E LIMITS OF lHL UESIGN SPACL
NUMdtR UP hHlULH CHUJCES AND CANIILLvLR LENGIHS

100
110

120
125

130

135

140

145

150

92

IU£I= l

GRNU = 20.

DO 100 A = 121/

GRNU = URNG ' 1.0

GSPA = ( BRA - a.*2.b)*12./(GHNU ' 1.)

IF(uSPA .GL.78.) 00 TU 110

CONTINUE

UdGRNU = GHNO

DU 120 A = 124

GRNU = \JRN‘J "1.

GSPA = ( 88A ‘2.*2.5)*1&./CUHNU - 1.)

IFCUHNU .EA.3.) GU 10 125 '

IFCuSPA .GT. 144.) 80 IU 130

CONTINUE

GRNUL8 = GRNU

NG = Ile1086RNU'URNULd)+1

GO ID 155

GRNULa = GNNU + 1.

N8 = IFIX(U80RN0-0RNUL8)+1

CANIB = FLUAT(IFIA(SPANL1/1U.))

CANTUd= FLUAI(IFIX(SPANL1/ld.)) + 2.0

CANTL8= FLUAI(IPIX(SPANL1/IU.)) -2.0

T0 ULTLKMINL UPPER AND LUNLN LIMITS OF ULquN SPACL
rOH 8E8 PLATE DEPTH AND rLANUE PLATL WIUIH

WEBIJ : 56.0

00 1&0 K = IfiIU

NEBD = NLBU + 6.0

11'((SRANL1*12./AL60).LL. 30.1011 10 145

CDNIINUL

WEBULU 3 WEdD

HEBUUB = wEdDLH + 24.0

N0 = 4 t I

IF(UBWLUU .LQ.lbO.) 60 [U 130

ND 3 IFIX((UHWEHU ' WEdULB)/6.U) + 1
PLNLB 14.0

PLWUd 22-0

HRIIE(3240)GRNUL82UHGRNU.NG

HRIIE(3241) CANILB:CANIU:CANIUH

NRIIE(3;42) WEBULUpHLUUUHpNU

NRITL(3:43) PLWLfipPLNUd

NOW THE LIMITS UF IHE ULSIUN SPACE ARL ULILKMINLD
STAR! THL SEARCH FUN THL MINIMUM'CUSI ULSIGN BY.
USINu THE GRID SEARCH TELHNIUUL

CALL FSUES

CALL GHIHI

CO'UHUHINAILS Ur IHL MINIMUM‘CUST DtSIGN ANU GUS!

HRIIE(3245) IXG021YCC

 

50

180

A

A
['1

WMCOGZ)

8
C

D

93

GRNU = UBUHNU ' (TLUATLIXGC ' 1))81.

NR11L(J:SO) CRNU

FURMAI(ZOX24HURNU2 +6.0)

GSPA = (BRA - 2.*2.b)*12./(UHNU ‘1.)

CANIB = FLUAT(IFIX(SPANL1/1Uo)) '2.U +(FLUAT(IYCC'I))*1.

WLBD = UPTH(IXGC:IYCC)

PLAan= 0P18(lqu.IYCC)

WRIIE(5246) CNNU.LSPA.CANT8.wL80.PLATLA

T0 UBIAIN [ME FINAL DESIGN

WRIIE(3'22) GSPADWLHU1PLATLNICANTG

CALL SLAB

CALL PHLLIM

IUET = 1

CALL ULJGIH

CSLAB = C1*(88L*bhw*SLAdT/(12.*Z7.) + (URL*HWIUIH*HDLP|n/(144.*
2,.)'GRI\U))

CREINF = C2*A?C.*bNL*bHN/54.*HEINF(Z)

CSTLEL = C3*(nG(I)*(SPANL1 + CANTJ) + WU(2)‘SPAHL2)*GNNU*IOIS

BCOSI = C1*(BRL*HRW*SLAbI/(12.*27.) + (HHL‘HNIU[H*HULPln/(lua.*
27.)*GHNU)) + C26490.AUKL*BHw/ja.AHLINF(2)
+C3*(wu(1)*(5PANL1 + CAAIH) + w8(2)tsPANL2)*uRNU*I.10

CALL DLIAIL

HRIIE(3:31) CSLABACHLINT

NRIIE(3:35) CSILLL

HHITL(3224) BCOST

UUCOSI = BCUSI/(hHLthN)

HRITE(3223) UdCUSI

PTIML = TI~L(2)/6U. - PIIML

OIIME = TIML(3)/60. - UIIML

WRITE(JDS8) PIIHF2011ML

60 TU 1/0

STOP

END

SUBHUUIINE bRIUI

COMMON dMDLpBMLL.beWthMSWLLfidMIRK25CTIFTCO)2CANTAJCANId!CL1(614)p
DELILAiULLILd2UELTLC2UELI(324)2UISTHb2LMPACI2LXIHA2rAIIGCZ)!
I5(4).GSPA2HULPIH2HWIUTH!I(02A).LCADAMtNLGLL9PBMA2PDM62PLLAI
PLL52PLAIEI(Z)JPLATLH.PUDLLpPSLA2PSLH.bLABW!bU(6.4)23LA5I25PANL
151(0’4))$IAIM(6))SIRLSAISIUUS)SWFWS)SWLLDIUPPI(6))UIIL’VDLD
VLL!VLLMIN2VSW;VIUIALwauUpthIwaFIb(b)2N[Iiprx.PbNA2PbWb

2XL.XR2LL2ISATTYpAVbWT20CUELL

CUMMUN/URUAIA/HHL28HN25PANL11SPAABISPANL22RUN.ILDRMUU27N2CIICZ.
C32hHN02hCUSI2bCUbII;WG(2)2UPTHLH(10).UPISPA(lU)2UFIVL(IU)2
UPTCANCIU).PCA)2TUIIL2TSWLL2HAIIWT;RLINL(Z)ILILH9FSATIItnLDUIp
UHWLdH2TVI(I)erI(1)2IULI

0

9H

COMMON/URIU/ UbQHNU)UHNULBIH62CANIL62CANIUU.NLUUUBDWLUULb’NU.
A VLHLUJPLWUB!IXU’IYC’IXGC!IYCCDIXH!IthIXHC2IYBC!lUC(3’D)J
8 CSLAd,CdLINT.CSTLEL28CUST:1NUC525):SAFTY(Sp5).PLH(b2b)p
C wH(b,b)2UPTH(525)RUPTB(325)2GPTG(525)2UPTC(325)2SAFTI(525)
COMMUN/SRCH/UCUST!dbRNUIBCANIJHNEBUDBPLATL!UAIUM’UHDUP
COMMON/LBC/CANT.ZMINC

10 FORMATCZOX217HOUTPUI UNIDI uHNOn// 5(TIU.Z1ZX))
ll FORMAT(ZOA;17HCUTPUT GRIUI CANI2// 5(T10.2.2A))
12 FORMATCZOA217HCUTPUT GHIUI wLbU2// 5(F109222AI)
14 FORMATIZOX219HUUTPUT GHIUI PLAILNpl/ 5(T12.222X))
15 FORMAICZOX217HUUTPUT GHIDI CUSI2// 5(I10.2’2X))

16 FURHATCZOXAZTHCHECH FUN CUNVEHULNCfitl/ 20X22T106224(1422A))
17 FORMAT<1HI://// 20X11SHUUTVUT SUMMAHYDZX!3HGRIU1)
IR = 1
IS = 3
IF THE STARTING PUINI IS THE UPTIMUM POINT
IXGT = l
IYCT = J
DU 60 K
DU 75 J
ZGC(K.J) = U
75 CONTINUL
80 CUNTINUL
ISPLCI = I
IF<ISPLC1 .LQ.1) CU TU 90
IR = 3
IS = 3
CO'URDINATLS UT CLNTLN U1 GNIUtbPECIFT UNL UT IHL IULLanNU
CLNTLR 0F ULSICN SPACE UK
SPLCIFIEL BY THE DESIGNER
90 IXGC = 1H
IYCC 2 IS
TO UEIEHMINL GRID CU'UHUINATES AND CCHHLSPUNUINU ULSIUN CUSTS
DU 120 A = 1.3
00 110 J = 1.3
IXG = [AGC + K21 - g
IYC = ITCC + J*1 - 2
TO CHECK IF GRID POINT IS IN ULSIGN SPACE
IF(IXC .LT. 1 .UR. TAG .CT. NC) GU TC 110
IF(IYC oLT. I .UH. IYC .GT. 5) 00 TU 110
TO CHLCH IF CRIU POINT NAS A PART UT THL PREVIUUS URIU SLAHCH
TU AVOID REPLATING TnL SAVL UESIGA
IF(LCC(1xG.11C) .NE. 0.) CU T0 110
T0 UETLHMINL ULSICN Cfibl AT THE POINT (IXbIIYC)
GRNU = UBCHNU ' (FLUATCIXG ' l))*1.
GSPA = (BRA - 2.*2.b)*12./(hHNU '1.)
NRITE(3:20) GNNUpCSPA
20 FORMAT(20X:5HGH1DI: 2FIU.2)

H II
o—ar—o

 

 

105

106
110
120

121

130
135

95

CANTO = FLUAT(ITIA(SPANLI/1U.)) '2.U *(FLUAT(ITC ‘1))*1.

CALL GRID?
HEBU = NEUULB + (TLUAT(IXHC ‘ I))*6.C

PLATEN = 14.0 + (TLOAT(IYBC ‘ 1))*?.C
TU STUHL OPTIMUM PLATLW AND WLdU AT POINT (IXUfilYC)

0PTG<IXHAIYC) = GNNU
0PTC(IXU;IYL) = CANTB
OPTH(IXU;IYC) = NEBU
0PTU(IXUpIYC) = PLATLN

CALL SLAB

CALL PRELIM

CALL UEDGIH

CSLAU = C1*(BRL*BKN*SLAHT/(12.*27.) + (8HL*HHIUIH*HULHTH/(Iaa.*
27.)*GRNU))

CREINF C2*A90.*HHL*BHN/54.‘HLIAF(2)

CSTLEL C3*(WG(I)*(SPANL1 + CANTB) + WU(2)*SPANL2)*GKNU‘I.16

BCOST = CI*(BHL*BHW*$LAUT/(12.*27.) + (URL*HNIUTH*HULPIH/(104.*
27.)*GNNU)) + C2AA9U.*BRL*UNw/Sa.*HEINT(a)
+c3*(wu(1)*(SPANL1 + CANTO) * wG(2)*SPANL2)*uRNO*I.Id

ZGC(IXG2IYC)= BCOST/1000.

IF((ZGC(IAC:IYC)*IOUU.) .LT. AMINO) GU TU 105

GO TO 106

IXGC IXG

IYCC IYC

GO TO 1/0

CONTINUE

CONTINUE

CONTINUE

HRITE(J)15) ((ZUCCJIK)2N = 125)IJ= 125)

T0 LOCATE THL CENTER 0? THE GHIU

SMALLI=£GC(IXGC;IYCC)

WRITET3216) ZGC(IAGC:IYCC)»SMALLI:TXGC2IYLC2[HATS

DU 135 A = 123

00 130 J = 1,3

IXG 3 IAGC + K*1 - 2

IYC = ITCC + J*I ' 2

TO CHECK IF THE OHIO POINT IS IN THE ULSIUN SPACE

IFClXOoLT.1 .UH. IX0.0T.NG) GO TO 130

IF(IYC.LT.1 .UH.IYC.OT.5) GU TU I30

IFCLGC<IXU:IYC) .LT. SMALLI) GO TO 121

GO TO 150

SMALLI = ZGC(IXG’IYC)

II II

’
-
-
-

IXGT = 1xu

IYCT = IYC

CONTINUE

CONTINUE

SET C0 URUINATES UF MIN. CUST ULSIGN AS CU UHUINATLS UT NLN CLNTLn
IXGC IXHT -
IYCC IYCT

no

170

10
11
12
1A
15
16
17
18

96

TU CHECK THE CONVERGENCE IT THE MINIMUM-LUST UESIUN
IS THE CENTER UF GRIU CUNVERUENCE IS OUTAINEU

HRITE(3216) ZGC(IXGC.TYCC)2SNALL121XGC2IYCCRIRDIS

IFCIXGCoEUolR .ANU. IYCC.EU.IS) GU TC 170

IR 3 IXUC

IS 3 IYCC

SMALL1=£GCCIXGC2IYCC)

GU TU 90

CONTINUE

HRITEC3217)

WRITEC 3.10) ((UPTG(J.K)2K=I’5)2J 3125)

WRITE(3’11) ((CPTC(J.K)2K=125).J = 1.5)

HRITEC3’12) ((CPJH(J.N).K=1.5)2J=I.S)

NRITE(3214) ((UPTBCJ.K)2R= 1:5);J = 1,5)

NRITE(3215) ((ZGC(JpK)2K : 125)!J= 1.5)

RETURN

END

SUBRCUTINE URTCZ
COMMON UMUL.BMLL»BMSN,UMSNLL20MTRNpUCTTPT(o)pCANTA;CANTd:CO(o:A)p

UELTLA2UELTLBJULLTLCtUELT(324)2DISTH62LMPAC11LXTRA2FAT16(2):
T5(412CSPApHUEPTH2HNIUTH2I(6!“)OLUAU.M2NLGLLdeMA2PUMprLLA.
PLLflipLAIET(2)2PLATEW.PUbLLnPSLALPSLH.5LA8WI§B(O.4)JbLABprPANL
’ST(O.A)RSTATM(6))STRESA2STUUS2SWFN82SNLL:TUPPT(6)2UTIL!VUL2
VLL'VLLMINAVSW2 VTUTAL2WE602NLBT2WTTTU(5).WTF[SLIXIPDWAIPSNE
.XLIXR2LL2FSATTYpAVGWTfiCUUELL

COMMON/GRUATA/HRL BRW25PANL1! SPANB. SP ANL22RUW.PL2RMUU.TH2C12C2.

8 C3.GKNUpUCUST2UCU5TI2WGCZ)1UPTWEB(10)IUPTSPA(10).CP|VL(1U)I

C UPTCAN(IU)2P(A)2TUTILtTSWLLDRAILWTpHLINF(2)11TLR2F5AITIINEOUIp

D UBWEUD2TPT(I)25PIC1)JIDLI

COMMON/URIU/ UBURNU:GRNULBRN62CANTLH2CANTUd.NLUUU52WLbULUINU.

A PLWLdpPLNUUDIXUAIYC2IXGC11YCCpIXleYUAIAHC2ITBC.£GCIDIS).

B CSLAd.CHEINI2C5TLEL2HCUSI21Hb(5.5)2SAFTY(b.D))PLw(5!D).

C WHI525)2UPTH(525)2UPT81525)20PTG(525)9UPTC(325)2SATTI(5.5)

COMMUN/ORCH/UCCSTJBURNUIBCANI.BWEbofiaPLATL’UATUMDU“!LT

CUMMUN/LBC/CANTolMINC

“MCOQD

FORMAT(ZOX212HUUTPUT GRIU22AFGRN022X2F5.UprCAN1622X2TDoU)
FORMAT(ZOX.6HPLATEW.// b(fo.l:2x1)

FORMAT(ZOX2AHNEUD:// 5(P6.T!2X))

F0RNAT<20x.5HSATTV.// 5(16.2.2x>1

FURNAT<20x.480061.// 5(E10.2.2x))

FORMATTdOA;l/HCUNVEROENCE CHECK;//20X,2(FIU.2,2A):A(14:28))
FURMATCIH12//// 2UX215HUUTPUT SLMNARY22XIDHGRIUZ)
"FORMATCZOXADHSAFTI:// 5(F6.2,2x))

NRITE(3:17)

WRITE(J:IO) GHNUICANTB

 

 

75
80

90

95

100

97

DO 60 K
00 75 J
ZRB(K;J) = U.
WH(KDJ) = 0.
PLH(K2J) = d
SAFTY(KRJ) =
CONTINUE
CONTINUE

IP = 1

IO = 1 ,
IF THE STARTING PUINT IS THE OPTIMUM POINT

IXHT = I ‘

IYBT = 1

ISPECZ = I

IF(ISPLL2 .EO.1) CU TU 90

CU-URDINATES UF CENTER UT NINE POINT ORIU IS ASSUMEU TO

CENTER Ur UESICN SPACE

IP = 3

IO = 3

IXHC = IP

IYBC = IQ

GIRDER UESION AT GRID CENTER MUST SATISTY THE REQUIREMENTS

UF LATERAL BUCNLING

NEBU = dEdULB + (FLUAT(IXHC ' 1))86.C

PLATE" = 14.0 * (FLUAT(IYBC - 1))82.0

CALL SLAB

CALL PRELIM

CALL DEoGIR

CSLAB = C1*(HRL*8RNASLAUT/(12.*27.) + (nRL*HNLUTH*HULPTH/(104.‘
A 27.)*GRNU))

CREINF = C2~A90.fibRLtBRn/54.*REINF(2)

CSTLEL = C3*(NC(1)*(SPANL1 + CANTO) T wu(2)*SPANL2)tUNNU*1.18
BCOST = C1*(BRL*BHW*SLABT/(12.*27.) + (BRL*HNIUTH*HUEPTH/(144.8

II II
.—
V
UT

O
040

A Z7.)'GRNU)) + C2*A9U.*HRL*8RN/ba.iREINT(2)
8 *C3*(WG(1)*(5PANLI * CANTH) + NG(2)'SPANL2)‘URNU*I.IU

lHB(IXHCpIYUC) = bCUST/IOOO-

F.s. IN SAPN I

SAFTY(IAHC,IYNCI = TSATTY

F.S. IN SPAN 2

SAFTI(IAHC.IYBC) = FSATTI

PLw(IXHC.IYdC) = PLATEN

HHCIXHCDIYBC) = HEUU
'IF(FSATTI.GE.1.25.ANU. TSAFTY.OE.I.25) U0 TU IOU
IYBC = IYBC + I

GO TO 9)

TO DETERMINE ORID CU'URUTNATLS AND CURRESPUNUINU ULSILN COSTS
DU 120 K = 123

DU 110 J = 1:3

IF((ZHB(IXHCDITUC)*IOOO.) .LT. ZNTNC) CU TU 1/0

 

('5

107

108
110
120

'98

IXH IAHC + Ail - 2

IYB IYBC + J*1 ' 2

T0 CHECA IT GRIU PUINT IS IN ULSTGN SPACE

IF(IXH .LT. 1 .UR. IXH .GTo NU) GU T0 110

IF(IYB.LT. I .UN. IYB.UI. 5) CU TU 110

TO CHECK IT GNIU POINT «AS A PART OF THE PNLVIOUS uxlu DLAHLH
TO AVOID REPEATING THE SANE DESIGN

IF(LHB(IXH»IYB) .NL. 0.) 00 TU 110

TU UETLHMINL DLSICN CUST AT THL PUINT(IxH:IYB)

NLBU = NEBDLB + (FLUAT(IXH ' 1))*6.0

PLATEw = 14.0 + (TLUAT(IYd ' 1))*2.0

CALL SLAB

CALL PRLLIM

CALL DLSGIN

F0 80 IN SPAN 1

SAFTYCIXH»IYB)

F.So IN SPAN 2

SAFTI(IAH:IYB) = FSAFTI

PLH<IXH»IYB) = PLATEN

NH(IXH!IYB) = WEBU

CSLAU = CI*(BRL*BHN«SLAuT/(12.*27.) + (bRLiHWIUThflHDLPTH/(laa.i
27.)*GHNU))

CREINF = C2*a90.*HRL*wa/54.*HEINF(2)

CSTLEL = C3*(NG(1)*(SPANL1 + CANTd) + NG(2)*SPANLZ)*GHNU*1018

BCOST = C1*(BRL*BHN*SLABT/(12.*27.) + (bRL*HNIUTH*HULPTH/(144.*
27o)'GHNU)) 4‘ C2*490¢*BRL*UHw/5a0*HL1NT'(Z)
+C3*(wu(1)*(SPANLI + CANIB) + NG(2)*SPANL2)*uRNU*I.Id

ZHB(IXH:IYH) = BCOST/1000. .

MLTHUU UF BOUNDS CRIDZ

IF(SAFTY(IXHlea).LT.1.z5 oUR.SAFTI(IXH,IYd).LT.I.25) CU TU 110

TO CHECK IT GRIU POINT SATISFIES BCUST LT (MING

IF((ZHB(IXH;IYB)*IOUU.) .LT. (MING) GU TU 10!

60 TU Ida

IXHC IXH

IYBC 1Y3

GO TO I/O

CONTINUE

CONTINUL ‘

CONTINUL

T0 LOCATE ThL CENTER UT THL CHIU

SMALLZ = AHd(IXHC:IYdC)

DO 135 N = 1:3

00 130 J = 1:3

IXH = IAHC + Ktl . 2

IYB = ITHC + J*I ' 2

TO CHECK IF GRID POINT IS IN ULSIGN SPACE

IF(IXH.LT. 1 .OR. IXH.CT. NU) CU T0 130

IF(IYB.LT.1 .UH. IYB.GT.S) uU TU 130

T0 AVUIU THL UNSTABLE GIRDER uESIGN FROM CUMPARISUN

IF(SATTT(IXHpTYd) .LT. 1.25) GU TU 130

TSAFTT

O

121

130
135

170

A
B
C
D
E
F

B
C
U

99

15(bATTI(IAn;IYd) .LT. 1.25) UU TU 130

IF(£hB(IXH:IYH).LT.SMALLZ) UC TU I?!

GU TC 150

SMALLZ = lHDCIXHpITd)

IXHT = IXH

IYBT = 1Y5

CONTINUE

CUNTINUE

SET CU URUINATES UF MIN. CUbT DESIGN AS CU UNUINATES UT NLN CENTER

IKHC : IXHT

IYBC : IYHT

TU CHECK THE CCNVLNNENCE II THE MINIMUM CUST DESIGN
IS THE CENTER UF UHIJ CUNVENGENCE IS UUTAINEU

HRITE(3!16) lHH(IXHL;1YUC):SNALL?»IXNCpleC:IP:IU

IF(IXHC.EN.IP .AND. IYdC.EU.IC) GU TC 1/0

IP = IXnC

IQ = ITUC

SMALL2=LHd(TXHC;TYbC)

GU TU 9U

CONTINUE

HRITEC3DIIJ ((PLW(JtK)9N= 1:3)1J3 1:5)

NR1TE(5!IZ) ((WHCJnh)pN = 1:5)IJ3 1,5)

WNITEIJ:15) ((SAFTI(J-N);K = 1:5),U = 1:5)

WRITEI3!IA) ((SAFTY(J;N):K=1Ib)!d=1!5)

WRITE(3!15) ((ZHH(J1K))A =115)’J=1n5)

RETURN

END

SUBNUUTINE SLAB ‘

CUMMUN dMUL,HMLL,bMSw,dMSWLL,dMTRnpuflTTkT(6);CA~TA:CANTd:Cu(o,A),
ULLTLAyULLTLrSIULLTLC9UELT(3,4))DISTHDILT49ACT)T.XTRA)T~AT16(2)!
T5(4)!U5PA’HUEPTHpHHIDTHI[(bta)9LUAUpMidtULLIPdMAfiPdMUIPLLA!
PLLfltPLATET(Z)yPLATanPUbLL:PSLA'PSLhnbLABN!bB(b!4)!SLABT:bPANL
’ST(ép4)ISTATMC6)pSTHLSADSTUUS!SWFWS:SNLL!TUPPT(b):UTIL!VDLD
VLL’VLLMIN)V5W)VTUTALDNEDIJ‘IWLRTfiWTF TL2(D)!WTFTSL)X:T"DWA1P5WD

)XL:XH’LLpTSATTYpAVUWTnLUULLL

CUMMUN/dRDATA/UHLpHHNpSPANLlnSPANB-SPANL29HUNpFC3HMUUprNIC1)CZ!
CJ)GRNU:UCUST9UCUSTIIWG(2)9UPTWEU(IU)pUPTSPA(10))UPTPL(TU)!
UPTCAN(1U)DP(4)'TUTIL:TSWLL!HAILNT9HLINT(2)’ITLH:F5HTTIDWLDDI:
UBWEdUpTPT(I)pbPT(1)!IUET

DATA CUVERP/Z.73/:CUVENN/2of5/

ASSUMPTITJNI)

._... .L ,

 

 

0000000

on

100

MAIN REINT PERPENUICULAN TU TPE TRAFFIC

g—o

2 TOP ONE INCH OT THE CUNCntTE TS HE CONSIOEREO AS A HEARING

COUNJE;ANO «ILL NUT dE USEO TON O DISTANCE IN DESIbNo

3 WHEN CHECKING SLAB THICKNESS FUR NEGATIVE MUMENTtTHL HAUNCH

HILL NUT BE USED TO CALCULATE THE LFTECTIVE O.
A SLAO CUNTINUUUb UVEH THREE OR MORE SUPPORTS.

IF(LOAO.EO.1) SLMIN :
IF(LUADOEQ02) SLMIN :
SK=1./(l.+(TH/(RMUD*F
Sdglo'SA/3o
RF=005*IC*SN*SU

/.5
/.U
C)))

TO DETERMINE TPE MOMENTS IN SLAU DUE TO UEAU LUAU ANU LIVE LOAD

ESPAN=OSPA-PLATEN/2.

SLUT: GSPA/IZ.

K=O

SUMOLP=(12.3*SLOT +25.)*ESPAN*ESPAN/ZUTS.

SBMULN=(12.5*SLOT +23.)*ESPAN*ESPAN/IAAO.

IF(LOAO.EO.I) GO TO A

SOMLL=CESPAN/12. +2.)/5a.*laCOO.*O.8*1.J

GO TO 7

SUMLL=<ESPAN/12. +2.)/32.*IOOOO.*0.8*1.5

SBMTLP=bBMDLP +SHMLL

SUMTLN=SBMULN +SHMLL

SLAUDP=JQHT(SUMTLP/HF)

SLAUDN=UQNT(SHNTLN/HF)

SLAOTP=SLAOOP+CUVERP

SLAdTN=SLAHON+COVLRN

SLAUTH=AMAX1(SLAHTP15LAOTN:SLMIN)

SLBT = aLAdTH

K=K+1

IF(K’2)ID212

S : SLUT

SLABT=AMAXI((FLOAT(IFIAIS*2.+.99)))/2.:SLMIN)

REINFP=SHMTLP*12./(20OOO.*O.9*(SLAHT'COVEHP))

REINTN=SBMTLNt12./(200UO.*U.9*(SLART'COVEHN))

REINF(2)=AMAXI(NI[NTPpNEINTNJ

SLAUT= 10.

IF(GSPA.LE.IAA.) SLAOT= 9

IF(OSPA.LE.129.) SLAHT= 9

IF(USPAOLLOIIQO) bLABT= 8
8
/

IFCUSPAoLEo 99o) SLABT:
IF(USPA¢LEo 78.) SLAUT=
REINF(2)= 100

IFCGSPA-LEoIAA.) NEINFI2)= 0.88

IF(USPAoLEo123o) HEINF(2)= Uodl
IF(G$PA¢LE0108¢) NEINF(Z)= 00/5

U‘CU’CU’

‘nmoomb

B
C
0

TMUOGD

101

IF(USPAoLE. 87.) HEINFI2)= U.bb
RETURN
END

SUBHUUTINE PRELIM

COMMON dMDLrBMLL:UM5NpUMSWLL'UMTRNpUCTTPT(O):CANTAICANTU:CU(6:A):
UELTLA:UELTLU:UELTLC)DELT(3'4))UISTHDDEMPACTyEXTHA’TATIG(2)!
TS(4)pUSPA:HUFPTHpHNIDTH)I(6)4)pLUAU:MANEULLpPBMApPUMbtPLLA!
PLLBJPLATET(2)pPLATEN;PODLL!PSLA!PSLB}bLABNtSB(6:4)ISLAUT35PANL
pST(Op4)nSTATM(6)’STRESArSTUUS:SNFNS.SNLL:TUPPTCO)!UT1L1VDL9
VLL!VLLMIN9VSN:VTUTAL)NEUU;NLHT»NTTTGC5)0NTITSL!X’FJNA9P5ND

pXL;XR»LLoFSAFTYpAVuNT1CUUELL

CUMMON/uRDATA/UHLdeN'bPANLltSPANB)SPANL2tNUN9TC:HMUU)TH!CIDCZD
C3:GNNU:GCUST;UCUSTI;NG(Z):UPTNE6(10)’UPTSPA(10):UPTPL(10)!
UPTCAN(IU):P(4))TUTILtTSWLL'RAILWT;HEINT(2)91TLRIFSAFTltflEBDI;
UdHEdD;TPT(1)pBPT(I)1[ULT

TU DETERMINE UTILyswLL! ANU SNFNS TN LBS. PER FUUT UT dEAM

UTIL = TUTIL/GRNU

SWLL z TSNLL/GRNO

SNFNS = I RAILWT + 25.U*ROA)/CNNO

HDEPTH = 1.0

HNIUTH = PLATEN + 6.0

TU UETENMINE CANTILEVEH REACTIONS

CANTA = 0.
PBNA = O.
PSLA = U.
PSWA =09
PLLA = U.
RETURN

END

SUBHUUTINL ULDGIR

COMMON UMUL'BMLL!UMSWDUHSWLLIUMTRKDHC‘T TFT(0)9CANTA)CANIrjrCUIépI-J);
UELTLA;UELTLH!ULLTLCIDELTC394))UISTRD’EMPABTIEXTRAnFHTIUIZ)!
I5(4)’U5PAIHUEPTHpHWIUTH)I(0!“)nLCAUrM,NEULLIPBMA9PGM69PLLA’
PLLUIPLATETCZ)pPLATENpPUOLT-pPSLA9PSL595LAHNIDUCOIQ)IbLABTybPANL
:ST(O)4)ISTATM(6),STRESA:STUUS)SNFWS;SWLL!TUPPT(6)DUTIL!VUL!
VLL,VLLMIN,waDVTOTALJWtUU'WLHTONTT’.T(I(5)PWTI ISL)XIPDNADP5W5

:XL;XR:LL,TSAFTY.AVCNT:COOELL
COMMUN/dRUATA/BHLIHHN;bPANLI!SPANU)SPANLZIHUNpTL9RMUU1PK!C11C21

B LJIUNNUfibCUbT»GCHSTI}NG(Z):UPTNEB(IU):UPTSPAIIU)9UPTVL(IU)!
C UPTCANCIU))p(0)fiTUTILDTSWLL)HAILWTDRLINT(2)DITLH)F5ATTI’HEUUID
U UBNEdD;TPT(1)1UPT(I)pIDET

on

on

0000

102

TU UESION GIRDER IN SPAN 2

CLB = CANTO

SPANL = SPANB - CANTO
TUPFT(I) = U.
BOTTPTCI) = O.

NEBT = O.

IF(LUADIEQOZ) CUDELL : 400
DISTLL = 1.1

HUEVTH = IoU

HHIUTH = PLATEW + 6.0
STUUS = 4.0

CANTB 3 0.

PBMd = O.

PSLU = O.

PSNU = O.

PLLU = O.

CALL GIRDER
FSATTI = TSAFTY
IF(IOET.EO.I) GO TO 90
CALL DETGIN

90 NOIZ) = AVGNT

T0 ORIGINAL VALUES UF CANTTLEVER
CANTB = CLU

T0 UESIJN GIRDER IN SPAN 1

TO UETERMINE CANTILEVER REACTIONS FOR 5

SPANLZ = SPANd - CANTO

NTFTSL = I.OaZ*(GSPA*SLABT * HWIUTH*HUEPTH)
IF(NG(2).E0.0.) PUMB = O.2>*SPANL2/2.
IF(NG(2).NE.0.) PbMd = NG(2)*SPANL2/2000o*1015
PSLB = «TITSL*SPANL2/ZOOO.

PSNU = SNFNS*SPANL2/20UU.

IFCLOAU.EU.1) GO TO 110

PLLU = AMAXICIAO.'T12./SPANL2)*(GSPA/(120.*1.1)):

A (26. + O.32*SPANL2)*(OSPA/(IZO.*I.I)))
GO TO 111
110 PLLd = AMAxI((I2. - 672./SPANL2)*(G$PA/I12Uo*1-1))p
A (26. + O.32*SPANL2)*(GSPA/(IZO.*I.I)))
III SPANL = SPANLI
NEBT = v.0

TOPPTII) = O.
BOTTPTII) = 0.
CALL GIRDER

TPTII) TOPPT(I)
BPTCI) BJTTPT(T)

II N

 

 

A

A
B
c _
0
E
F

A

A

d

C

103

IF(IUET.EO.I) GO TO 11:

CALL DETGIR

NG(I) = AVGNT

GCOST = (NU(I)*(SPANL1 + CANTO ) + NG(2)*SPANLZ)*1oId*CJ
RETURN

ENO

SUBRDOTINL OIRCLR
P L A T E b I R D E R O E S I G N
MARCH 29; 19/2.
REAL IpITOPpINEOpIBOTT:N,LENCalTOEI:NECLL:IY:LLR:LLNAAR,LLRMAX,
LLNWUI9IX
COMMON OMDL;BMLLpOMSNpOMSNLLpHMTRKnUDTTPT(o):CANTA:CANTO:CC(6;A),
OELTLApDELTLd:OELTLC:DELT(3:4);DISTRO,ENPACT:EXTRA;FAT16(2):
TS(AIpUSPApHuEPTH;Hw[DTH:[(6pa)1L0A0pM;NEGLLpPBHAdeMdpPLLA;
PLL8:PLATET(a)pPLATEN,PDSLL;PSLA,PSprSLABw:SB(b'A)oSLABT;SPANL
)$T(Op4))STATM(6):STHESA:5TUUSJSWFW515NLL)TUPPT(6)’UTIL)VDL1
VLL’VLLMIN'VSWIVTUTALpWEbUpWLBTnWTFTUCb)!WTFTSL!X'VbNAIPSWU
pr;XR:LL.FSATTTpAVONTpCUUELL '
COMMON/DESI/ALLONCIA);ALLDAE(A);NOT(12):PROdLL(A)’TRAT10(2)
COMMON/OESZ/ADJIA):OLRT(2):LLR(l):LLRMAX(2):PSTOD(9):STIFT(9):
TSTITF(2);TNLH(9);LLRNUII2) ‘
COMMON/OET/TUPPLI;TOPPLZ,TOPPL3;HOTPLI:uOTPLz,BOTPL3:xCLNT:
ITOPL,FSTDPpFTOPRpTOOTTLyFSBOTTnFHOTTR:dMOLAyuMOLC,OMULB:BMLLA:
UMLLC;UMLLO:OMSNApHMSNCpuMSNb»wSTITF:$TUU:STIFAbPT:bTIT:STIN:
STIFR,STIFa-STIFL
DATA E/Z9OUUOOO./pEMPMAA/IoJ/pPLMIN/O.3/5/fiPSTUUM/24o/

XL = 0.0
XH =10U
LOAD = I
IFCCUULLLoLN.QU.) LOAD = 2
IFCCUUELL.EU.85.) LUAU = 3
IF<CUUELLoEO.BO.) LUAU : a

UISTLL=1.I

IF(LUAU.EU.I)LL=2U

IF(LOAU.EU¢Z)LL=20

IT(LCAU¢EN.J)LL=85

TOTALL ‘ CANTA+SPANL+CANTB

EMPACT AMINIILMPMAXOIoO*5Uo/(bPANL*125o))
IF(LUAU.LQ.3) EMPACT = 1.0

RRIMP :-100+(6O.'JO*SPANL**2/16UCO)/TOOO
IF(3PANL.UTo80.U) RHIMP = IoC+C36o+600o/($PANL'JO))/IUU.
DISTRB = EMPACT*GSPA*LL/II2U.*PHUBLL(LUAU)*UISTLL)

IFCLUAUoEOoJ)DISTRB = LL*GRSP/12./(OISTLLYPRUBLLILUAU))

123

126

130

134

152

154

10%

IF(LUAD.FU.4) BISTRO = RRTMPfiuSHA/(120.*UISTLL)
EXTRA = 1.16

IF<WEUTOLNQU0)TEXTHA :1012

ADJUST 3 ‘1.0

PLG 7- UOU

IFITUPFT(1)rNE.U.) DU TU 126

IF(PLATLH.NE.U.U) 00 TO 123

PLG = 100

PLATEw AMAX1(3.U+IFIX(SPANL/ZU.0)*2.Op]TIAINEdU/Ibo0*4060))

PLATEN = PLATEN + 2.0
HWIDTH = PLATEN+6.U
ADJUST 3 1.

TOPPTII) = 0.3/3

BUTTPT(1) = 0.3/5

NTFTSL = IoUA2*(GSPA*SLAHT+HHIDTH*PUEPTH)

IF (NEUT.GT.O.C) DO TO I30

NEBT = AMAKI((FLUAT([FIK(NEUU/150.*T6.+U.99)))/16.UpPLMIN)

HTFTCIII = :00.

CALL SHEAR (0.0)

VL = VTOTAL

CALL SHEAR (8.0)

T = AMAAIIWLBD/ISO.»(WEUD*ANAXI(VL»VTUTAL)*IUOU./7500.**Z)**0.353)

TREOD = AMAA1(ITTA(T*10.+U.99)/16.09PLMIN)

NLBT = TREND

IF(LCAD.EO.A)NEOT= AMAAI(ITIX(WEBD/lAS.*16.+D.99)/10.0;PLMIN)

IFCLOAUTEO.A.ANO. STRESA.LI.23500.)Ng§igANAxIIITIx(wEuu/IIO.t16.
+0oi9)/16.0;PLMIN) ‘ " '”‘

P L A T E T H I C N N E S S E S + L E N u T H S
X = SPANL/Z. + 1.0
IF ((PBMA+PSLA)*CANTA.LE.(PUNB*PSLH)*CANTB) X = SPANL/Jo
X 3 X'QOU
FSBDTT = "99999.
CALL PRUPTY (1)
CALL MOMENT
CALL STRESSIIJ
IF(TS(2J.LE.FSUDTT) 00 T0 134
X a X + 1.0
FSBUTT = FSCZ)
GO TO IJZ
IF(AUJUbT.LE.0.0) CU TO 162

CODEX: 0.0
XTDP = AMAXI(PLMINpTUPPT(1)*F5(1)/(STRESA+IUU.)pPLATEN*

A SQRT(FS(1))/3250.pPLATEN/24.)

XBOTT = AMAxIIPLMIN:BOTTPT(1)*FS(2)/(STRESA+IOU.)yPLATEn/ZA.)
IF(AUS(TOPPT(1)-XIOP).UE.O.UOI) CODE = 1.0
IF(ABS(TOPPT(1)-XT0P).CE.O.OU/) TOPPT(I) = XTOP
IF(ABS(ouTTPI(I)~XdOTT).CE.O.OO/) COOL = 1.0
IFCABS(d0TTPT(1)‘XdOTT).GE.0.00/) RUTTPT(I) = xoUTT
IFICODEoEQ. 1.0) DO TO 134

 

 

162

164

172

116

224

228

232

234

238

 

105

TUPPT(I) : 1F1X(TuPPT(I)~N.U+U.99)/D.U
BOTTPTII) = IFIX(RDTTPII1)*6.0+O.99)/H.O
XCENT = X

00 I64 M=Iibp1

TOPPT(M) = TOPPT(I)

BOTTPT(M) = BUTTPT(1)

CALL PRUPTY (M)

NTFTDL = NTTTG(1)+WTFTSL+OTIL

CALL MOAENT

CALL STRESS(1)

FSBUTT = FS(2)

X = -0.299

X = X‘IOO

CALL MOMENT

CALL STRESSII)

IF (BNUL+PUSLL*UMSW+UMSWLL-LT.U.O) GO TO 1/2
XL = X/SPANL

X = SPANL+O.299

X = X'IOO

CALL MDMFNT

CALL STRESS(1)

IF (BMDL+POSLL+OMSN+OMSNLL.LT.0.0) GO TO 116
XR : X/SPANL

F L A N D E P L A T E C U T D T P 5
TUPFT(2) = IFIX(AMAAI(TUPPTII)*4.+00518.D*(PLATEW/(32500/
A SDRT(STRESA))*0.12Q)))/Oc0
IF(TOPPT(2)+00145.GL.TUPPT(I)) DU TU 26b
00 22“ N=31551
TOPVT(K) = TUPPT(2)
X = XCENT
CALL PKUPTY (2)
CALL MDMENT
CALL STRESSIZ)
IF(TS(1).LE.STHLSA uANU. FATIGCI) .EC. 0.0) CU T0 231
X = X'Ioo
GU TD 2Z8
TOPPLI = A+CANTA

X = XCENT
CALL PHUPTT (3)

CALL MOMENT

CALL STRESSI3)

IF(TS(I).LE.STRESA .ANO. FATIUIIT.E0.0.0) DO TO 236
x = x + 1.0

GO T0 254

TOPPL3 = SPANL-x+CANTH

ToPPLZ = TOIALL-TOPPLI‘TOPPL3

 

 

 

 

 

240

250

254

258

260

262

266

270
272
274

BUTTPTI
BOTTPT(
IFCcDTT
IFIUCTT
X = XCE
CALL PR
CALL NO
CALL ST
IF(FS(2
X = x-I
GO TO 2
BUTPLI

X = XCE
CALL PR
CALL NO
CALL ST
IF(F$(&
X = x +
GO T0 2
BOTPL3

BOTPLZ

GO TO 2
BOTPLI

BOTPLZ

BOTPL3

BOTTPTI
BUTTPT(
GO TO 2
TOPPLI

TDPPTCZ
TUPPLZ

TOPPL3

GO TO 2

DO 2!?
ADJ(K)
BMOLA
BNDLB
BMLLA
BMLLB
TFILOAD
IFILOAO
BMSWA =
BMSWD =
FTOPL =
FOOTTL
FTUPR =
FBOTTR
IF(ADJU

N T! IT IT

106

4) = IFTXCAMAXIIROTTPTII)iA.+.b;b.*(PLATEn/ZA.+.IZU)))/6.0
)) = BuTTPT(A)
PTIA)+D.125.CE.6OTTP ( )) GO TO 262

T 1
VT(A)+U.250.UE.UUTTPTIT).ANC. TUTALL.LE.OU.0) CU T0 264
NT
UPTY (A)

AENT

NESS(A)

).LE.STRESA .ANO. FATIG(&).EU. 0.0) 00 TO 254
.0

JO

3 X+CANTA

.v‘f

OPTY (5)

MENT

NESS(5)

).LE.STRESA .ANU. FATIU(2).EQ. 0.0) CD TO 260
1.0 -

38

= SPANL'X+CANTN

= TOTALL'HUTPLI'UOTPLS

IO

= 0.0

= TOTALL

= 0.0

4) = BUTTPT(1)

J) = UDTTPT(1)

(0

= 00K)

) = TUPPT(I)

= TOTALL

; 000

40

C A N T I L E V E R S T R E S S E a
A=1un1
= 0.0
~(NTFTU(4)+NTTTSL+UTIL)*CANTA**2/ZUUU.'(PUMA+PSLA)*CANTA
'(NTFTG(S)+NTFTSL*UTIL)*CANTH**2/ZUDU.'(PUNU*PSLU)‘CANTb
-CANTA*PLLA*I.3 ,
-CANTB*PLLH*I.J
.EU.J) dMLLA = dMLLA/1.3
oEN.J) BMLLD = HMLLU/I.3
“SWFNS*CANTA**2/ZOUU.‘CANTAiPSNA
'SWFNS*CANTd**2/20UU.'CANTH*PSNU
-(HMOLA+HMLLA+UMSWA)*12UOU./ST(a!1)
= -(OMDLA+OMLLA+OMSNA)tIZOOO./SB(A;I)
'(NMDLH+HMLLU+UMSNU)*TZUOU./ST(511)

= -(uMuLO+HMLLB+szwo)*12000./Sd(5:1I

5T.LE. 0.0) GU TU 3/6

 

 

277

278

279

280
281

284

107

ALCUMP = AMINICSTRESA+1UO.I(JZDU.*HUTTPTCA)/PLATEN)**¢)
IF(ITUPL.LE.STHESA+IOD. .ANU. FUUTTLoLE.ALCUHP) GU TU dl/

AUJII) = 1.0
IFITTUPL.OE.STRLSA+IUU. .ANO. TUPPT(A).LE.UUTTPT(4))

ATOPPT(A) = TOPPT(A) + 0.125

IF(FUUTTL.6E.ALCUMP) HUTTPTIO) = UOTTPTIA) *U.145
IT(DUTTPTIQ).LE.TUPPT(4)) DUTTPT(A) = TUPPT(4)

CALL PROPTY (A)

GO TO Z/A

IF(AOO(I).LO. 0.0) U0 IO 260

ADJII) = 0.0

IFITOPPT(A).EA.TOPPTI1)) GO TO 279

X =3 XCEIVT

IFITOPPTCA).LE.TOPPT(1)) GO TO 27a
BOTTPT(£) = dDTTPTIA)

CALL PROPTY (2)

X = 1.0

IF(TOPPT(A).CL.TOPPTII)) CALL MUNENT
IFITOPPT(A).CL.TOPPT(I)) CALL STRESS(2)
IT(TOPPI(A).LE.TOPPT(T)) CALL MOMENT
IF(TOPPT(A).LE.TDPPT(I)) CALL STRESSCA)
IF(IS(I).LE.STRLSA .ANO. TATICII) .EC. 0.0) CD TO 26C

IF (TOPPT(A).LT.TOPPT(I)) X = K‘T.O
IF (TDPVTIA).CT.TOPPTII)) x = x+1.0
GO TO 218

TOPPLI = 0.0

60 TU 2d1

TOPPLI = X+CANTA

TOPPLZ = TOTALL-TOPPLT‘TDPPLS
IF(UOTTPT(A).EG.DOTTPTII)) DO TO 285

X = XCLNT

IF (BOTTPTIA).CT.OOTTPTII)) X = 1.0
IF(UOTTPT(A).GE.HUTTPT(I)) CALL POPENT
IF(BOTTPT(A).CE.ROTTPT(1T) CALL STRESS(2)
IFCUOTTPTIA).LE.ROTTPT(1)) CALL NONENT
IFIUOTTPT(4).LE.BOTTPT(I)) CALL STRESSIA)
IF(TS(2).LE.STRESA .ANO. FATIO(2) .EG. 0.0) DO TO 266
IF (BUTIPT(A).LT.DOTTPT(1)) X = X-I.0

IF (bDTTPTIA).GT.bOTTPI(1)) x = x+1.0

GO TO 20A

BOTPLI = 0.0

60 TO 267

BOTPLI x+CANTA

BOTPLZ TOTALL-ROTPLI'ODTPLJ

ALCONF AMINI(STRESA+IO0.:I3250.*BOTTPT(SJ/PLATEN)**2)
IFIFTOPR.LE.STRLSA+IOO. .ANU. FUGTTR.LE.ALCUMP) CD TD (09
AOJ(3) = 1.0

IF(TTOPR.GE.STRESA+IOO. .ANU. TOPPT(5).LE.bOTTPT(5))

H II N

289

290

291

292

293

296

297

298
300
302

ATUPPT(5) =

108

TOPPT(5) + 0.125
IF(FBOTTR.GE.ALCDMP) OUITPTIS)

BCTTPTIS) + 0.125

IF(60TTPT(S).L£.TOPPT(S)> UOTTPT(5) = TOPPT(5)
CALL PROPTY (5)

GO TO 27a

IF(AOJ(J).EO. 0.0) DO TO 30:

ADJ(3) = 0.0

IF<TOPPT<ST .EC. TOPPT(1)) at TO 291

X 3 XCENT

IF(TUPPT(5).LE.TDPPT(I)) GO TO 290
BOTTPTIJ) = BOTTPT(5)

CALL PRUPTY (3)

x 3 SPAIVL'I .0

IF(TOPPT(5).GE.TOPPT(1)J CALL MOPENT
IFITOPPTIS).GE.IOPPT(I)) CALL STRESSI3)
IF(TOPPT(S).LE.TOPPT(1)) CALL MONENT
IF(TOPPT(5).LE.TOPPT(I)) CALL STRESS(5)
IF(TS(1).LE.STRESA .ANU. FAIIO(I) .59. 0.0) GO TO 294
IF (TOPPT(5).LT.TUPPT(1)) x = x+I.O

IF (TOPPT(5).OT.TOPPT(I)) x = x-I.o

GO TO 290

TOPPL3 = u.u

GO TO 293

TOPPL3 = SPANL-x+CANTH

TOPPLZ = TOTALL-TURPLI'TOPPLJ
IF(UOTTPT(5).EC.BOTTPT(1)) uc TO 29!
IF(ADJ<4).LO. 0.0) 00 TO 302

AOJ(A) = 0.0

X = XCENT

IF (BOTTPTI)).GT.OOTTPT(I)) x = SPANL'1.0
IT(BOTTPT(5).CE.HOTTPT(1)) CALL NONENT
IF(dOTTPT(b).GE.ROTTPT(I)) CALL STRESS(3)
IF(bOTTPT(5).LE.HOTTPT(I)) CALL NUNENT
IFIUOTTPT(5).LE.BOTTPT(1)) CALL STRESSCb)
IF(TS(2).LE.STRESA .ANO. FATICIZ).EO. 0.0) CO TO 290
IF (OUTTPIIS).LT.ODTTPT(I)) x = x+I.0

IF (BOTTPT(5).CT.OOTTPT(I)) x = x-1.o

GO TO 296
BOTPL3 =
GO TO 300
BOTPL3 =

0.0

SPANL‘X+CANTB

BOTPL2 = TUTALL-RUTPLT'UUTPL3
TOPPT(2) = TUPPT(A)
IF (TUPPL1.LT.BDTPLI)
BUTTPTCZ) = BOTTPTII)
IF (TUPVLIoLT.BUTPL1) UUTTPT(2)
CALL PRUPTT (2)

TOPPT(3) = TOPPTCB)

TUPPTIZ)

TOPPT(I)

= BOTTPT(A)

      
  
 
   
   
  
   
  
  
  
 

'l = I'lij‘a:‘
.-.-l-Ju‘)T!
" ' I! 41 '38313
:;nq JJ‘:
. K J! 00
i’uUA117
.x : “Jun
L'L'quu!":
'. 1]“: g "
NIH)?!
. :I‘E'f-l
-".'-‘- JJI.)
.-'s-: a. I

'}""!':l')11g.
."EIYU

'Hi

1"!

:3]

mi“: '.'I II

3;

 

on

on

320
322

324

345

352

354

356

358

360

109

IF (IuPHL3.LI.Hu1PL3) IUPPI(J) = TCPPI(1)
aortptcs) = BUTIPI(1)

IF (IUPPL3.LT.BUTPL3) UUTTPI(3) = BUTIP[(5)
CALL PRUPTY (3)

C A N I l L L V E R D L F L L C T l U N 5
CALL DELTA (BUTPLInbUTPL2;dUTPLJpIUPPL1;IUPPLZ:IUPPLJ)
IF(UELTLA.GL.ALLUHL(LUAU)-Q.C.UH. CAAIA.EC. 0.0) GU lb 322
IOPPTCA) = TOPP](4)+U.125
AOJ(l) = 1.u
IF (TOPPTCA).LE.BUTIPT(4)) C0 10 320
BUTIPICA) = 80TIPIC4)+U.125

’AUJ(2) = 1.0

CALL PRUPIY (a)

GO to 2/A

IFCUELILB.GL.ALanL(LnAu)-Q.O.UR. CANTB.EH. 0.0) GU IU 548
TOPPI(5) = lUPPI(b)+U.lab

AUJ(3) '1 IOU

IF (IUPPT(S).LE.5UTIPI(D)) DC IU 324

BOTIPT(b) = BUTIP1(5)+0.1?S

AUJ(4) = 1.0

CALL PHUPTY (5)

GD 10 2/A

C E N T L H L I N E U E r L E C I 1 U N 8
IF (ULLILC.CT.ALLUWC(LUAU)) GU ID 3/6
IF(SIUDb.EQ.0.0.ANU.TUPPT(1).LL.BUIIPI(1)) IUPP|(1)=IUPPICI)+U.125
80TTPI(1) = BUTIPI<1)+U.125
CALL PRUPIY (1)
IF (TOPPLlouT.HUTPL1) GU T0 354
IF (BOTTPT(1).LL.bUfTPI(2)*2.0) GU TU 3:6
BUTIPT(2) = BUTIPT(2)+0.125
CALL PRUPTY (2)
BOTTPT(A) = BUTIPT(4)+U.125
CALL PHUPIY (A)
GU T0 330
BUTIPI(Z) = BUTTPI(1)
CALL PRUPTY (2)
IF (BOIIPT(1).GI.HUTTPI(4)*¢.U) GU TU 352
IF (IUPPL3.GT.HUTPL3) GU TU 360
IF (BOTrPT(1).LL.BUIrPT(3)*2.U) GU TC 312

BUIIPI(J) = BUTTPTC3)*U.I25

CALL PHUPTY (3)

BUTIPI()) = BHIIPI(5)+U.125
CALL PRUPTY (5)

GO ID 3/2

BUTIPI(3) = BUIIPI(1)

110

CALL PNUPTY (3)
IF (do!IPT(1).Gl.bOIIPI(S)‘z.0) GU TC 356
372 CALL DELTA (BOTPLI,uUTHL2,bUTPL3.TUPPL1,TUPPLz:{UPPL3)
60 T0 214
376 x : xctwr
IF(ICPPI(1) + 0.125.NL.IUPPI(A) .ANU.
A IOPPI(1) + 0.125.NL. IUPPI(S)) GO TO 3/8
TOPPILI) = {OPPI(1)+O.1aS
GD 10 162
378 IF(uUlfPT(1) + 0.125.NL. BOIIPI<A> .ANU.
ABUTTPT(I) + .12: .NL. HUTIPI(5)) CC TU 360
BOTIPT(1) = BUIIPI(1)+U.125
GO T0 10?
360 CALL MUMENT
CALL STdESS(l)
400 FSTUP = LSLI)
FSBUII = FS(2)

BMULC = BMDL
BMLLC = PUSLL
BMSNC = HMSA

CALL DELTA (BUTPLIIUUIVL296UIPL3:IUPPLI;TUPPLZ!IUPPL3)

412 AVGAI = (CIUPPL1*IUPPT(Q)+IUPPL3*IGPPT(5)+dUIPL1*BOIIPILH)*UUIPL3*
A dUlTPT(5)+IUPPL2*IUPPI(I)‘UUIPL2*BUTIPI(1))*PLAILW/IUIALL*
B wEthwLBI)*3.Q

s H L A H C U N N E C 1 U H s + b T 1 F r L N L H‘s
SSPAMN = 99999.
IF(LXIRA.EU.1.15) GU TU 416
CALL SHLAR (0.0)
VL = VIUTAL
CALL SHLAR (8.0)
T = AMAA1LWLdD/ISU.;(WLUDtAMAx1(VL;VTUTAL)*1000./7500.'*2)**0.333)
IF(LUAUoEH.4) T = WEHD/bl. _
IF(LUAU.£9.A .AND. SIRLSA.LL.23500.) I = deU/buo
IREUD = AMAXICIFIX(I*IO.+O.99)/l6.0pPLM1N)
IFCIRLUU.LE.WLBI) 00 TU “16
WEB] = IRLuu
60 10 123
“16 DU “20 J=IJ991'
VN = J'I
CALL SHLAH (VN)
X = VN*¢PANL/do
CALL FINDM (IUPPLl!IUPPLJAUUIPLIABOIPL3)
PSIUD(J) = b.96*SIUdb*l(M;J)/((VLL+VLLM1N)*STAIM(M)+C.UUI)
IF(LOAD.L9.J) PbTUU(J) = PSIUU(J)*7.32/5.9b
ANEB = NEHT*(WLHD+IUPPI(M)+UCIIPT(N))
THEU(J) = WLBU*SQHT(IOUU.*VIUIAL/AWLB)//SOU.
IF(LUAU.FC.4) IwEb(J) = WLHu/blo
IF(LUADQE_Qoq .AND. STRLbA.LL0235000) IWLdLJ) = WLBU/OUC
SIIFF<JJ = AMIN1(IIUUO.*del/SNHT(1000.*VIUIAL/AHL6);ALUU)

 

lll

IF(LOAD.LU.4) SIILF(J) = AN1A1(10500.*WLUI/bNHI(IOUUo*ViuIAL/
A AWEB)37290)

Q20 SSPANN = AMIN1(:SPAMN;SIIFF(J))
IF(LUAU.EC.1) PCTV = AU./72.
IF(LOAD.EN.2) PCfv = 8./AO.
PLLL = VLLA*(1.U+PCIV*CANTA/(PSLAtIOCO./wlFI5L+U.001))
VL = IOUO.*(PHNA+FSLA+PLLL*1o3*PSNA)+(WIFIdL4)+nIFT5L)'CANIA*UoUI
STIFA = AMIN1(1IOUU.*WLbT/SufiI(VL/(WLBI*(WLUU+IUPPT(4)+5UTIPT(aJ)

A )):NLdU)
THEUL = deU*5UHT(VL/(WLHT*LAEBU+IUPPT(A)+dUITP[(Q))J)//500.
PLLR = HLLH*(1.U+PCTV*CAN1b/(PSLH*IUOO./WIPISL+U.001))
VR = 1090.*(P5Md+P5L6+PLLR*1.J+PSWH)+(WIFIC(S)+WIF15L)*CANTu+u.01
STIFB = AMIN1(11000.~wLbT/Sun1(VR/(HEBthwLBD+TUPPT(b)+dUTIPTCS))
A ))!HEHU)
TWEUR = NLCU*SCHI(VR/(wLBT*(WLdu+TUPPT(5)+UUTTPI(5))))//DOO.
IF (AEUI.GT.HEUU/150.) 00 IU 420
422 STIFL = 0.50
STIFH = 0.50
GU IU 4A0
426 00 428 A=1’5!1
IF(IHLB(K).LE.WEHI) GU 10 “$0
428 CUNIINUL
G0 10 422
430 IFCK.EU. 1) SIIFL = 0.0
IFCK.GEo2)SIIFL= (K'l)/d.-(NLUI'TWEH(K))/d./(IWLH(K'1)'IWLB(K))
DU 440 K=59991
IP(IWLB(14'K).LL.WLHI) CU IU «42
440 CUNIINUL
GO [0 022

442 IFCK.LQob) STIFN = l 0
IFCKoGLo6)bIIFR: (13” K)/6. +(WLHT TWLB(IA'K))/6. /(IWLd(ID' K)’

A IHLU(1A-K))
4A6 IFCSIUUb.L9.0.U) CU TU 46)
PSTUDI = 'MINICPSIUU(1), PSTCO(2) PSTCD(5):PbIUD(d):PSI00(9))

H1 = ((IUPPT(1)+BUIIPI(I))*PLAILN+WLHUAAEUI)*bIHLSA/C. )5

H2 = 0. dS*JUOO.*SLAbI*SLAdW

PSIUDZ = SPANL*(XH'AL)*12.*5IuUS/(2o*AM1N1(HlpHZ)/(U.OD*ZU7UO.))

SIUU = IFIX(AMINI(PSIUUIDPSIUUZIPDTUUM))

IF<SIIFL.LN.O.C .ANU. SIIFR.LC.1.U .AND.(CAATA*12o0oLLo5IIPA .ANU.
A wEBI.GE.INEBL).ANJ.(CANIot12.0.LL.$TIFb .ANU. NLdI.uL.TwLBn))
H 60 In 461

STII 0.375

SIIW AMAKI(2.0+WLHU/JU.O:PLAILW/A.O)

IF<AMUD(ST[W1005)OEQ0 U00) DIIW = SIIW'Uob

SIIN oIIw-AMUUCSTprU.5) '

450 Sle oIIW+O.b
STIJ ZG.O*wLBu**2/AMINI(SsPAMNASTIFApsrle)**z'20.C

II II II

 

   

CO

“52

054

461

A
B

A
B

A

A
B

A

B

A

112

SIIIR AMI~I(SSPAMNASIIFA:bIIFB)*wLBT**3*STIJ/10.92
STIIA STII*$I1W**3/3.U

IF(5IIIA.G£05TIIR) GU TU “61
IF($IIWCGT0(PLAIEN'WEBI)/ZOU'UOZS) GU TU “54
IFC$T[I*16oUoGE.SIIW) GU IU 450

SIII = bTII’OoIZS

IF($TINoGT¢(PLAIEW'NE8I)/2.U'0025) GU TU “52

GO TO 490

L A I E H A L B U C K L I N G

AMINT ‘ (TUPPT(A)+TUPPT(5))/2.0

PLTTOP = TUPPI(1)tAMIN1(AMINT/IUPPT(l)+U.Z’I.U)

AMINB ' (HUTTPI(A)+BUTIPT(S))/2.0

PLTUUT = HUITPI<1)*AMINI(AMINU/UOTTPT(1)+U.291.U)

H8 3 (PLAILW*PLITUP*(PL[TOP/2.+NEHU*PLIHUI)+WEBU*NEBI*LWLUU/2o*
PLIHUT)+PLATLW*PLIHUT**2/Zo)/(PLAILW*(VLIIUP+PLIdUI)+WLBU*
NLHI)'PLTHUT/2.U

HI = PLI[UP/2.0+WLBU+PL[BUT/2.9-HH

Ix = (PLATLw*(PLIIUPAAJ+PLTauI**3)+NEBT*NLdU*t3)/12.+PLAILwr
PLTTUP*HI**2+WLHT*NERU*(HI-PLTTUP/2.'wEdu/d.)**2+HLATLWt
PLTBOTiHBttZ

[Y = (PLATEN**3*(PLTIUP+PLTUUI)+NEBU*HEHI**J)/lz.

TURC = (PLATLN*(PLTTUP**3+PLTHUI**3)+wLbU*wEUI**J)/J.-d.&1*(

PLIIUP**A+PLTBOT**A) .

(PLITOP+WLUD+PLTBUI)**2*PLTIOP*PLIUUIAPLATEwtto/(144.*IY)
(HIwPLTIUP-HdtPLTBUI)tPLATEu:.3/(IZ.*IY)

HARPC
ECCLN

CUNSTJ = ECCLN+(HU*(PLIUOT*PLAILW**3/12.+PLTBUI*PLAILW*HB*#Z*
HBi*3*~EBT/4.I'HT*(PLIICP*PLAILN**J/12.+PLIIUV‘PLAILW*
HT**2+HT**3*wEur/a.)J/(2.*IX)

FCR = Iol3*3.14**2*E*IY/(IK/H[*bPANL**2*144.)*((UoAS*LCCLN*Uo52*
CUNSTJ)+SUHI((0.45*ECCLN+U.52*CUN5IJ)t*2+wAHPC/IY*(1.+L/2.6*
IURC*3PAAL**2*144./(J.IQ**2*E*HARPC))))

FSBLAM 3.*WTFIG(1)/LXTRA*(SPANL**2/2.‘CANTA**2'CANTU**Z)/bT(1!1)

FSAFIY = FCH/FSUFAM

R E A C T I U N S + B E A R I N d S T I r I .
DU 47? K=19291
CANTX = CANIA
SPANLX = 2.*PSLA/HIFTSL*IOUU.
II(K.LQ.2) CANIX = CHNId
IF(K.EUo2) SPANLX= 2.*P5LB/WILTSL*IOOO.
EMPA = (EMPACT+1.3)/2.
LLR(1) = (.64*(SPANL+CANIX)**2/2.+(.6A*5PANLX/2.+ZO.)*(5PANL+

CANTX))*LMPA/SPANL

IFCCANTX.LU.O.O)LLR(1)=(U.6A*5PANL/2.+ 26.)*£MPACT
IF(LOAD.EU.2) GU T0 465

IF(CANTX.£U.0.0) CU T0 463
LLR(2) ? (l2.*SPANL+72.*CANTX-6l2.)*EMPA/SPANL

113

LLRIJ) = (AU.*SHANL'IIZ.+32.*(bPANLA+LAnTx-IA.)t(bPAhL+LaNTA)/
A SPANLA)*LMPA/SPANL
LLR(A) 4 (3¢.*SPANL+((SPANLA+CANTX'IA.)*32.+(5PAALX+LANIX'20.)*o.I
A *(bPAAL+CANTX)/SPAALX)tEMPA/SPANL
LLR(S) = (52.*(5PANL+CAAIA'IA.)+32.*(SPANL+CANIA)+5.*(5PAALA-IA.)*
A (SPANL+CANTX)/5PANLX)*LNPA/SPANL
LLRLO) = ((/2.*5PANLA'C/2.)‘(5PANL+CANIX)/bPANLA)*I.J/DVANL
“63 LLRL7) = (7a.*SPANL'672.)*LMPACI/SPAAL
GU IL “/0

465 IF(CAAIK.LC.O.U) b0 TU Ab?

LLR(2) ~ (d.*(SPANLX'14o)/SVANLA+J?o)*(5PAAL*CANIX)*1.J/bPAwL
LLR(3) = (5.*SPANL*D.*CANIK’1A.)*b.*EMPA/SFANL
LLRIA) = (3Zo*(bPANLX+LANIX‘14o)/SPANLX*(SPANL+LANTA)+U.*5VANL)*
A EMHA/SVANL
LLR(5) = (d.*(SHAwLX+CANIX'IA.I/SPANLX*(bPANL+CANIX)*3¢o*bPANL)*
A LMPA/SPANL !

467 LLR(6) (5o*SPANL'14o)*H.*LVFACI/SPANL

M II

070 LLMAXR AMQXI(LLH(I))LLR(2)ILLH(3)!LLH(4)'LLK(D)ILL“(°)ILLK(i))

A *UISTHn/LMPACi

RSWLL = SNLL/ICUU.*TUTALL*(ILIALL/?.-CAmlx)/SHA~L

LLHMAACA) = AMAXI(LLMAXH;(LLNAXH+RSNLL)ifloo)

LLHHUI(A) = AMAA1(LLR(I)/LMPA;LLR(?)/LMFA9LLH(J)/LMPA;LLH(4)/LMPAn
A LLHCb)/LMVApLLH(6)/1.J:LLH(l)/LMPACI)
IF(CANIA.LQoU.0)LLHNUI(A): ANAA1(LLH(1)/LMPALI)LLHWHI(A))
IF(LUAU.IU.Z)LLHWUI(K)= AMAx1(LLH(1)/LMLA,LLH(2)/1.J;LLH(3)/LMPA,
A LLHCA)/LMPA;LLH(S)/LMPA»LLH(6)/LNPACI)
IF(LUAUotu-2.AAU.CA~[x.Lu.0.)LLHAUI(K):AMAA1(LLA(1)/LMHALI,
ALLRWOI(K))

LLRNUI(A) = LLHHUI(K)*UISTHn/LMPACT

LLRWUI(A) = AMAAI(LLRNUICA):(LL«AUI(K)+H5wLL)*U.8)

IF(LUAU.£U.J) LLRMAXCK) = UISIHd*(0.085*(bPANL+LANTA)**2/2.+(
A O.Cdj*SPANLx/z.)*(bPAwL+LAAIA))/5VANL
IFCLUAU.EU.J) LLRwUICK) LLHMAX(K)

IF(LOAU-LU-4) LLRMAACA) b.1336AtLL*UIbIRu*(5PANL+dI.IZJI/IOU.

II {I

LLRMAX(n)/HHIMP

IF(LOAU.EU.A) LLRwUI(KJ
DU Q72 L=lvlvl
472 LLR(L} = 0.0

DLAVC = (WITISL+AvaT*LxrAA+LIIL+5w+h5)/luuu.

DLRI(2) : (ULAVu *IUIALL*(IUIALL/Zo'CAh[A)+(PbMu+PbLu+anB)t
A (SPQNL+CANTA)‘(PHMA+PSLA+P§NA)ififlNTA)/5PANL

DLRICI) = ULAVUiTUIALL+PHMA+P5LA+PSNA+Pth+PSLd+PSAd'ULNIL2)
WSTIFI = (PLATEw-WEHTI/d.-AMCU((PLATLw-wtbi)/2.yU.2§)-J.Zb
TSTIFICA) = wSlIIF/12.*5CRTLbU./33.)

IF(SIHLOA.LL.23bUUo}ISIIFF(h)= WSIIFF/lZ.*5URI(JO./3J.)

TSTIFF€A> = AMAAI<IFIX([STIfF(<)*d.+C.999)/d.pPLMIN)-u.lab
47A ISTIFF(A) = TSIIFF(K)+U.123

BRGSTH 3 (LLHHAA(K)+ULH[(K))/(2.*ISIIFF(K)*(H5IiiF‘UobuI)

 

A

A
477
478

”Mcomb

114

1; (uRuoTR.uT.Qu.) CU TU Ala _

IF(uHGSIR.Cl.29. .ANU. bTRLbA.LL. 23500.) 6U TU #74

RUYR = (2.*ASIIIF*T5T1FI(K)*((WLHT*W3TIFF)/a.)**2+2.*IslIF¥(K)
thlIFFw*3/12.)/(2.*NST1FF*TSTIFF(K)+1d.tw£u]**g)

FALLCw = IGUOU.-O.3*NEHU**2/HGYH

If (STHLSA.UT.?3500.) VALLUW = 22000.'0.SO*WLBU**2/RCYK

IF ((LLKMAK(K)*ULRT(K)I/(2.*W5T1FF*TSTIIF(K)+18.*HLUI+ta).bl.

FALLUfl/IOOC.) GU TU 474
CUNIINUL
CONTINUL
RETURN
ENU

 

BLOCK UATA

DESIGN CUNSTANTS
CUMMUN/UEbl/ALLUWL(4))ALLUWL(A)INUT(12)!PHUdLL(A)ITHATIULZ)
DATA ALLUWC/IUUUoIHUUa’UotbAUo/

DATA ALLUWL/350.950U.AU.!00/

DATA HUI/0,1116Ut91!121)I53!162,213AZAAAZ7QA3U51335/

DATA PHUBLL/ZUO!?U0IUDC!800/

DATA IHATIU/O.:24./

END

SUBHUUTINL PRUPIY (M1)
P H U P E R T I L S
REAL IIN '
UIMLNSIUN ~(d)
CUMMQN UMUL16MLLAUM5NpUMSWLL9uMTRKAHCTTFT(U)rCANTA’CANIU'CG(6’“)'

ULLILA9ULLILUAUELTLCAUELI(J:4):UISTHUALMPACTALXTHAIPATIGLZI!
I5(AJAUSPAAPULPTH9HWIUTHA[(Opa)oLCAU:MANEbLLAPUMAIPDMUAPLLA:
PLLUIPLATLT(a);PLATLA)PUbLL!PSLA:PSLUASLABWfibU(O)A)IbLABIpbPANL
’SI(O)Q))SIAIM(6)ISIHLSA’STUUSISWFWS'SWLL,IUPPI(6)IUTIL’VDLD
VLL!JLLMIN’V5W)VIUIALDWEUU)NLBTDWTFTU(5)IWTFTSL)X!P°"ADPSND
:XL9XR’LL1FSAPTYpAVbWT'CCULLL

DATA N/9999990’9999990'999999.!999999g,9999990’50’IUODJUO/

M = M1

NTFTG(M) = ((TUPPTCM)+dUTlPI(M))*PLAILW+WLUU*HEBI)*JoA*L*THA

SLABH SLAUI+HULPTH+TUPPT(1)‘TUPPT(N)

SLAUW AMIN1(GSPA15LADI*12ot12o*SPAAL/Ao)

DU 8 K3194)I

J = K+“*IFIK(5IUU5/SU.+U.99)

,,-t_ (,2. _.

 

CO

A
B
C
D

r'czom»

*WOOmp

A

A

A

115

(2.*N(d))+(TUPPT(M)*PLATLW)*(SLABH+TUPPT(N)/20)*(NLBT*WEb0)t
(SLAbH+IUPPT(M)+~LnD/2.)+(dnIIPI(M)tPLAILW)*(SLAUH+IUPPT(H)+
nLdD+dUTIPI(M)/2¢})/((bLAUW‘HWIUTH)*5LAdT/N(J)+5LAUH*HWIDTH
/N(J)+I0PPI(M)*PLATEA*ALUI*NLBU+BOTTPT(MIfiPLATEw)

IchK) = (SLABHuHNIUTH)*SLAUT/N(d)i((LG(M:K)'$LAbT/2.)**2+SLABT**2
lid.)*SLAHH*HWIUTH/N(J)*((ABS(CG(MAK)'SLABH/2¢))**2+SLABH**2
/12.)+IUPPI(M)*PLAIEN*((AUS(CG(N.K)'SLABH'TUPPT(M)/2.))**2+
TUPPTCN)**2/12. )+HLBT*WEUD*((AUSCCUCM!K)' SLABH' TUPPT(M)-nEb0/
2.))**2+w£50**2/12.)+BuTIPI(M)*PLATLn*((SLABH+TOPPI(H)+NLUU+
an:rPr(M)/2.-CC(M»K))**2+BUTTPI(M)**2/12- )

ST(M:K) 1(H9K)/(Cb(M)K) 'SLABH)

88(M9K) I(Mph)/(SLAUH+TUPPT(M)+NEBD+BUTTPT(M)'CG(M:K))

SIAIM(M) = (CG(M)1)‘CG(M’3))*WTFTG(M)/(3oA*EXTRA)

RETURN
END

catnynl ' ((SLAuw-HAIUlfi)*5LAuI**2/(2.*A(J))+HWIUTHt:LAdH**£/

 

SUBHOUTINL MUMENI

M U M L N I S

DIMENSIUN FATIUC(2)'FATIGT(Z):R(6)

CUMMUN UMULDHMLLPUMSHDUMSWLLfiuMIRKDHUIIVT(°))CANTAICANIU’CGLbIQ)I
DELTLAIDLLTLUDUELILCIUELI(3!“)IDISTRHflLMPACIUEXIKAIrAIIG(2)’
fS(A):GSPA9HUFPTH9HWIDTHA1(694)ALUAU;MANLGLLAPUMA'PdetPLLAr
PLLUtPLATLICZ):PLATLNpPUbLLIPSLAvFSthSLAbw’bH(6:A):bLAH]ASPANL
'ST(°D4)!SIAIML6))SIRESADSIUUSDSNFW598WLL!TOPPI(6)IUIILDVDLD
VLL’VLLMthVprVTUTALANEUUANLBTANTFTb(5)!NTFTSL:X:PaWA9Pwa

AXLAXHALLAFSAITYpAVbHTtCUULLL

X1 = (CANTA+SPA~L+CANTU)*(CAAIA+SPANL-CANTU)

X2 = CANTA+X

NTFIUL = wIFISL+w1FIG(1)+UTIL

BMDL = (wTFIUL*x1/(ZOUU.*SPAAL)+(PBMA+P5LA)tccANTA+SHAwL)/8PANL-

(PUM5+HSLH)ACANTH/5PANL)Ax-AIFrnL*x2*x2/2000.-(P5MA+PSLA)fixz
8M8" : (SWIwswxl/(ZOUO.*SPANL)+PSWA*(CANIA+SPANLI/SPANL
-Powu*CANTu/5PANL)*x-SwIwS*x2*x?/2UOU.'PSNA*XZ

BMswLL = SWLL*X*(5PANL'X)/2UUU.

NLGLL = ’1.3*(PLLA*CANIA*(SPANL'X)+PLLU*CANIB*X}/SPANL

IFCLUAU.IU.3) Cd 10 b

BMLI = X*(SPANL- X)*(.32+18./SPANL)

IF(LUAU EUol) GU T0 2

BMIHK = AMAX1((sPANL' X)*(AU.*A'112. ), X*(AUo*($PANL' XI-IIZ.))/5PANL

GU TO A ‘

XMAA': AMINI(K»3PANL-x)

BMTRK = AMAX1(XMAA*(72.*(SPANL'KNAX)‘674.)/SPANLAXMAK*L72o*($PANL'

XMAX)-336.)/5PANL-l12.AAMAA*(6A.A(5PANL-XMAX)'AAU.I/SPANL)

A PUSLL = AMAAI(HMTHK:HML1)*UISTRH

A

A
B
C
U
E
F

A

A

116

BMIRK = HMINKAUISTHH
GU ID /
6 BMTHK = 0.0

Z = AMIN1(X/SPA~Ly1.-X/SPANL)
PUSLL = Io.*LL*uI3THA/65.*(J.099n71*SPANL+U.UIJUAZOASPANL**2+

3.6/10**9*SHANL**S'ZU.3021)*£*(A.16-4.34*l)
IF(LUAU.LU.5) PUSLL = U.085*UISTRB*X/2o*(SPANL'A)
IF(LOAU.EU.3) NLGLL = NLGLL/1.3
CONTINUL
RLTURA
END

SUBHOUTINE 3THL55(M1)
DIMENSIUN FATIGUC2))FATIGT(Z)9H(6)
COMMUN HMUL9HMLL.nMsw,UMswLL;HNIRApHOITPTLO)ACANIA9CANId9CCI6AA);

UELTLAAULLTLMAULLTLCpULLI(J90)pDISIRbrtMPACTAEXTRAAPAIIGLZ):
.I5(a))USPA1HULP[H’HWIDTHD1(014)fiLOAD)MINLGLLIPBMAlPUMBIPLLA)
PLLH!PLAlLT(Z):PLATENAPUSLL9PSLA’PSLb:SLAUNISB(6:4)!bLABT:5PANL
’ST(O:Q)ASTATM(6);STRESAISTUUS:SNFWSASNLL»TUPPT(6):UTIL1VDL:
VLLCVLLMINpVbW,VTUTAL;WLUU;WLHToNTFTb(5),WTF[SLIX9P3NA1P5HB
)XL!XR!LL)FDAIIYyAVhHI'CUULLL

M = M1

N1 = 3

N2 = A

IFCA/SPANL.LE.AL .UH. A/SPANL.GL.KR) N1 =‘I

IF(A/SPANL.LE.XL .UR. K/SPANL.UL.KR) N2 = 1

TOPPUS (nMnL/5T(M;I)+bMTRn/5T(V:Nl)+UMSW/3T(M:N2))*IZUUO.

TUPNEG ('UMDL/ST(M)1)‘NLULL/SI(M9N1)‘bMSN/ST(MIN2))*IZUOU.

R(I) = 1.U*IUPPUb/TUPNLG

BUTPCS (dMUL/bb(M,1)+UMIHh/Sfi(N;Nl)+HMbN/§B(MAN2))*IZUUU.

ButmEG ('dMUL/Sd(M;1)‘NEGLL/SU(M¢NI)‘bMSN/SU(M:N2))*IZUOUo

R<2I = '1.U*HUTPUS/UUTNLU

DU 5 J=I)2!1

FATIG(J) = 0.0

IF (AUS(R(J)).GI.1.U) H(J) = 1.U/R(J)

IF (STHLSA.UT.ZJSUU.) IATICC(J) 2/OUO./Adb(1.'1.43*H(d))

IF (SIRLSA.LL.235uu.) LATIUL(U) 20000./(I.-o.d9*R(q)I

IF (SIHLSA.UT.2J500.) FATICICJ) IHOOU./(1.”U.02*H(g))

IF (STRLSA.LL.ZJSUU.) FATICIIJ) 17200./(I.'U.02*H(C))

CONTINUL

IF (IUPPUS.CT.FATICC(1) .UH. TUPNLG.GI.+ATICT(1)) FATIuCII

1F (BUTPUS.CT.FATIGT(2) .UH. dUTNEG.GT.IATICC(2)) FAIIuCZ)

Fscl) = AMAKI(TUPNLUITUPP05+(PUSLL'HNTRA)/5T(M!N1)*IZUUU.9
(IUPPUS+(PUSLL'HMIHA+BM5ALL)/ST(N,N1)«12000.)Ao,a)

F$(2) = AMAAI<BUTNLUrBUIPUS*(PUDLL‘HNTRKI/SUCMINI)*12000o!
(BUTPUS+IPUbLL-bMTHn+BM5ALL)/SB(N,N1)*12000.1*0.8)

RETURN

END

III“

H H II II

H.»
O.
C‘CL

 

10

12

14

16

TWMCOCID

117

SUBHGUTINE DELTA (BOTPLIABUIPLZAUUIPL3:IUPPLIAIUPPLZAIUVPLJ)

U L F L E C T I U N 5

REAL IUtFfiIIIIZII

DIMLNSIUN HMD(21)pIUEF(42)'P(21)9PL(A)!PR(Q)!V(ZI)tWT(4)

COMMON dMDLrBMLLthSWdeSHLL’UMlRKAHUTTPT(O)DCANIAICANIb!CG(6)A)p
UELILA)UELILU:DELTLC;DELl(3:4I’DISTRUALMPACT)EXTRAAPATIG(2):
I5(a)!GbPA1HULPIHDflNIUIH’I(6)4I’LUAUIM)“tGLL)PBMA!PdMU!PLLA!
PLLUAPLAIEIC2)pPLATLWpPUbLL)PSLAJPSLB:bLA8H:bH(6;A)!bLABIerANL
’SICODQ)ISIAIM(6)fi5INLSA)SIUUS)SNFWS!SWLL!IUPPI(6))UI1L’VDLD
VLL'VLLMIN:V5W1VTUIALAWLU09NLBTANIITb(5)nWTFISLJXJPaflAprWo

:XLAXRALLoFbAFTY9AVUNTALUULLL

SWFACT = (SwaS-QU.¥GSPA/12oI/CSHFWS+O.UI)

IF (SWFACToLI.0.0) SWIACT = 0.0

DU 40 Kl:1!d:l

DU 1 K=IAZIAI

PCK) = U00

GU IU ((96:10:14))K1

00 A K3192191

X 8 (K'I)*SPANL/20o

CALL FINDM (TUPPLIATOPPL3ABUIPLIIBUTPL3)

IUEF(K) : I(M:1)

IDEF(K+21) = 1(Mpa)

P(K) = (WTPIG(N) +UTIL)*SPANL/ZU.

BMD(1) = 'PUMA*CANTA*IOUO.'AIFTU(A)*CANTA**2/2o

BMO(21) = PPHMUACANTB*IUOO.'WTFTG(5)*CANTU**2/2o

PLCI) = PUMA
PR(1) = PBMU
WT(I) = WIFTG(1)
GU TU 10

BMU(1) = 'PSLA*CANTA*1UUO.
BMDLZI) = 'PSLEACANTB*IUOO.

PL(2) = PbLA

PR(2) = PSLd

GO IU 10

BMD(1) = -wIFTSL*CANTA**2/2.

8MD(21) = 'nTFTSL*CANTd**2/2.

DU 12 K=lt2111

P‘K) = "IfTSL*IQOo/ISOO*SPANL/ZUO
NT(3) = WTFTSL*IAU./150o

GU TO 1d

BMD(1) = -PbHA*DWFACT*CANTA*1UOU.
BMD(21) = -PSw8*SwFACT*CANIo*IOOO.
DU 16 K=132191

P(K) = OWFflb*SAFACI*SPANL/ZU.
ID£I(K) = IUEFCK+ZI)

IUEFCI) = 1(4911

IUEI(21) = [(5:1)

PL(A) = PSWAASWIACT

 

 

 

 

 

118 \

PR(A) = PSwdtswaCT

18 V(1) = (~8MU(I)+8MU(21I)/SFANL
DU 20 K31 1911

20 V(II = V(II+P(K+II*(2U' KI/ZUo
DU 24 K3111991

24 V(K*1I = V(KI'P(K+1I

DD

26 3MD(K+1I = UMD(NI+V(K)*6PANL/20o

RL
RR
00
RL
30 RR
XM
00

30 XM

: RL+(41-2*K)*(BMU(K)/IULFLKI+BNU(K+1IIIUEF(K+I))*U. U9*SPANL

26 K=lfil9tl

 

3 000
= 000
30 K3132011

= RR+(K*2- I)*(bMD(K)/IDEF(n)+BMD(K+I)/IUEF(K+I))*0. 09*SPANL
= 0. U

 

SA K=1 10:1
= xM+(21- ZAK)ACBMU(K)/1ULF(K)+BND(K+1I/IUEI(fi*1))*00 09*SPANL > i
DELI(I:KI) = “CANIA*12o/29UUOOUU. *(KL'WT(K1I*18.*CANIA**3/IUEF(1) t

A

-PL(K1)*CANTA**2*46000./IULF(III

DELT(2’K1I = (HL/Zo'XMI*SPANL*12o/29000000o
- 60 DELT(3!K1I = 'CANIB*12o/29000000.*(HR'WT(K1)*18.*CANTU**J/IUEF(21)

A

-PH(K1)*CANTU**2*AUOOU./IDLF(2III

TOPPL = (TOPPL1+TUPPL3I/2o
BOTPL = (BUIPL1+BUTPL3I/2o

II
12

A

= (1(2:2I+I(392II/2o
= (I(ApZI+I(b:2II/2.

EQVIB = lo/(lo/I(1:2I+14.0/SPANL**A*(TUPPL**3*(SPANL'TUPPLI/I(1:2)

t(1(1:2)/11'l.)+BUIPL**3A(5PANL'BOIPLI/l1*(1I/IZ-IoIII

DELTLI 3 o000496*(SPANL+AS I*SPANL**3

IF
IF

(LUAU.EUo1I UELTLZ = .0894*(SPANL**3' 555o*SPANL*AIUU I
(LUAU.LU.2) UELTLZ = .0496*(SPANL**3' 200.*$PANL*161U0 I

DELTLC :AMINI(999901EUV18*5PANL*120/(AMAXI(DELILI’DELIL2I*DISIRbII

H1
CL
A

= (WIFTG(A)*NTFTSL+UIIL+5HFWS)/1000. _
= SPANL+CANTA-((w1*(sPANL**2'CANTA**2)-Z.*(PUMA+PSLA*PSNA)*
CANTA)/(H1*SPANL))**3/SPANL**2*(1.-I(4:1)/LQV18)

DELILA =AMIN1(9999.9CANTA*6UA.*I(AAII/(PLLA*1.3'CANTA*'2*UL*.UOIII

HI

= (WIFIU(5)*NTFISL+UIIL‘bhFWbIIIOOUo

DR 3 SPANL+CANTH'((W1*(bPANL**2'CANTB**2I'Zo*(PBMH*P5Ld*PSNDI*

A

CANTBI/(WI*SPANL)Ir*3/$PANL**?*(I.’I(ba1I/LQV18I

DELILB =AMIN1(9999.;CAN[8*6UN.*I(501)/(PLLU*1.3*CANTB**Z*DR+.001)II

IFCLUAUoEUoJ) DLLTLC = AMINIC12oALUVIBAjUA-t29000000./(bo*1]26.t

A

IF<L0A0.CA.5> DLLILA
IFCLUAU.EU.3) ULLTLU

SPANL**3*LL*UISTRB)!9999I
ULLTLA*1.3
ULLILU*IC3

IF(LUAD-EQ.AJDELTLC= SHANL*12.*LQv18/((101.6'9.u9*SPANL+U.23a*

A

SPANL**Z+6.43L'5*SPANL**4+1.67E'7ASPANL**5I*LL*UIb[HUI

RETURN
END

119

SUBROUTINE SHEAR (VN)
S H E A H 5
COMMON dMUL:BMLLABMbW;dMSWLLIUMTRthflIIPILOI)CANTA!CANId:CG(6;A);
UELTLAAUELTLUAULLTLC90ELI(3:4):DISTRUALMPACI’LXTHA:FATIG(2I:
I8(4))GSPA1HULPTH1HHIUTHI1(61A)yLUAU;M)NEGLL)PBMAJPUMD:PLLAp
PLLB’PLAILI(2ItpLAILWJPUbLL)PSLA’PSLHDSLABH’DB(6I4)fi5LAUIISPANL
’$I(O)Q)DSIAIM(6)!SIHLSAISIUUS’SWFWSISWLL'IUPPI(OIIUIIL’VUL!
VLL,VLLMIN'VSWtVTUTAprfibUAWLHTnWTFTGIfiIpWTFTSLAX9PbNA993nD
, :XLAXRILLAFBAFTYpAVUWTILUULLL '
VX = AMINI(VN;8-VNI
IF {LUAU.LU.2I VLL = AMAXI(D.U*(8.'VX-22.A/$PANLI:(U.'VXI**2*
A UoOUD*SPANL+26.0'3.ZS*VX)‘UIDTHB
1F (LUAU.LU.1) VLL = ANAX1(9oU*(8.'Vx-?A.l/SPANLI,(6.'VXI**2*
A 0.003*SPANL+26.0' 3. 2)*VXI*UIbTRB
IF(LOAU.EU.J) VLL = 0. UdSiUISIHb*(A. 'VX)*SPANL/b.
IF (LOAU.LU.4) VLL = 6.15384*LL*UISTRB*((bc'VXI*SPANL/d.+
A 5 21.125)/100.
XI = (CANTA+SPANL+CANIU)*(CANIA+SPANL- CANTd)/ZUUO.
VLUL = ((HTFTSL+WTFTG(1)+UTIL)*XI+(PUMA+P6LA)*(CANTA+5PANL)-
A (PdMB+PSLB)*CANTBI/SPANL
VLSH = (SWFHS *X1+PSWA*(CANIA+SPANL)‘PSwB*CANTUI/SPANL
VLSWLL = SWLLtSPANL/ZOUU.
VOL : AUS(VLDL‘PHMA-PSLA-(wTFTSL+WTFTC(I)+UTIL)*(CANIA*VN/8o*
A SPANLI/lUOU-I
sz = Ads(VLSW'PSHA- swrws*(CANIA+VN/8.*SPANL)/1000.)
VSNLL = ABSCVLSNLL'VN/d.*SWLL*SPANL/IUUU0I
VTUTAL = AMAXI(VDL+VLL+VSW:(VUL+VLL+VSWLL+V3W)*U.8I
VLL = AMAK1(VLLp(VLL+V5NLLI*0.8I
VLLMIN = AMAXI(a.*Vx;b.*VX'l12./SPANL;3.25*vx+.UOS*VXAt2*SPANLI
IF(L0AU.EU.1)VLLM1N= AMAX1(VLLMINoBo*VX-AAd./$PANL99o*VA'6/Z./
A SPANL)
IF(LOA0.LE.A)VLLMIN= AMAX1(VLLMIN*DISTRU:PLLB*I.3*CAth/SPANL)
IF(LOAUoGL.D) VLLMIN = AMAXI(VLLNIN*DISTRB!PLLA*1.3*CANTA/SPANL)
RETURN
END

“WCOCDD

SUBRUUIINE FINUM (IUPPLIDIUPPLJ)dUIPL1)dUIHL3)

FIND M (UHULH LEFT IU RIGHT. “32,1!3’5)

COMMON UMUL:BMLL:UM5N;dMSWLL:BMTRKpHGTIPT(6)pCANTAtCANIdtCG(6:A):
UELILA:ULLTLUIDLLTLC;UELT(3AA)pDISTRUILMPACTpEXIRA:FAT16(2):
ISCQIDGUPAIHUEPIHDHWIDTH!ILO’QIOLOAU’MDNEGLL3PUMA1PUMUIPLLA!
PLLBFPLAILI(2I1PLATLNpPUéLL)PSLA)PSLDDSLABNIbBCbta)fibLABIASPANL
:ST(OpA)»SIATM(6)ySIHESAASTUUS;SWFWS;SWLL:TUPPT(6)pUTILAVDL:
VLL’VLLMIN)VbWIVTUIALIWEUUIWLBIIWTFIUISIIWIFISL!X’P5WA)PSNU

:XL)XR)LL

M = I

VF'CJC‘ACID

 

 

O

C

120

IF(X+CANTA.LE.TUPPL1 .AND. 2+CAN1A.LE;BUTPL1) H = A
IF(X+CANTA.LE.TUPPL1 .AND. X+CANTA. GE. BOTPL!) M = 2
IF(X+CANTA.GF.TUPPLI .AND. X+CANTA. LE.BUTPL1) M = 2

IF($PANL+CANTB'X.LE.TOPPL3 .ANo. SPANL+CANTB~ x. LE. BUTPL3) H
IF<SPANL+CANTR'X.LE.TOPPL3 .ANo. SPANL+CANT8'X. GE. BUTPL3) M
1F(SPANL+CANTB*X.GEITCPPL3 .AND. SPANL+CANTB-X.LE.BUTPL3) H
RETURN

END

II "II
UWU‘

 

SUBRDUTINF DETAIL

COMMON BMDL!BMLLDBHSH)RMSWLLIBMTRKIBUTTPTCé)!CANTA}CANTRICG(6IAI)
DELTLApDELTLB)DFLTLC!DELT(3!“)!DISTR81EMPACTIEXTRAIFATIG(2)I
FSC4)DGSPAIHDEPTH’HWIDTHDI(6)4)PLOAofiufiNEGLLIPBMA}PBHBIPLLA’
PLLB’PLATFTCZI!PLATEHrPUSLL)P$LArPSLRpSLABN9SBC6IAIysLABTISPANL
’STC6FQIJSTATM(6)iSTRESA!STUDS’SHFNSISNLL!TUPPT(6)!UTILfiVUL!
VLL,VLLMIN¢VSH9VTUTALAHEBDpHEUT:HTFTG(5))HTFTSL1XpPSNA:PSWB

’XL’XRILL’FSAFTYIAVGNTICODELL

COMMON/DE.SI/ALLUHC(“);ALIOHE(A):MUT(12): PRORLLCA);TRATIO(2I

COMMON/0ES7/ADJCA) DLRT(7) LLR(7)9 LLRHAXCZ)IPSTUDC9I)STIFF(9I)

A TSTIFF(7) TMIEBC9I LLRWOIC7I

CUMMDN/BRDATA/BRL;BRN,SPANL1'5PANBcSPANL?,RDW;FC;RMUD:FH:CI,CZ;

B CBIGRNO,GCOSTAGCOSTI9HG(2)pOPTUEB(103,0PTSPA(IOIIOPTPL(10)p

C OpTcAN(IO)’P(a))TUTILDTSWLLDRAILNT9REINF(2)fiITERIFSAFTI!WEBDI)

0 UANEBD,W9T(1);BPT(I),InET

COMMON/DET/TOPPLI’TOPPL7ITUPPL3IRQTPLI’BnTPL2!BOTPL3DXCENTI

A ‘FTUPL!F3T0P}FT0PRpFR0TTL)FSROTT!FRUITR,BMDLA!BMDLCIRHDLBQBMLLA,

B BHLLCARMLLBvBHSWAABHSWCIBHSHB;NSTIFFISTUD;STIFAAPT;STIT98TIV9

C STIFRISTIFBJSTIFL

 

WNCOG’J)

MILL-HIT

OUTPUT
125 FORMAT(IHIp///////'
A 20X133HTWU'SPAN CANTILEVER BRIDGE DESIGN;
8 ISXISHSQUAU:3X18HKULKARNIt/I _\
101 FORMAT(20Y,IRHBRIUGE DIMENSIONS)IéX'ZOHMAIERIAL PROPERTIES’II

- 102 FORMAT(20Y;16HBRIDGE LENGTH =9F10.2y2X!3HFT.:3X112HFSCREINF.) =:

A F890'9YyaHPSI.)/2OX1I6HBRTDGE WIDTH =1FIO.2)2XI3HFT.)3XI
B _ IZHFC(CONCoI SIFB. O!ZXI4HPSI0I

103 FORMAT(20x,16HSPAN I LENGTH =.FIO. 2,2X93HFT.;3X IZHFSCSTEEL) .=.
AF8.0,2XAAHP$I.:/20x'16HSPAN 7 LENGTH =:F10 2)2X)3HFT0’3XI
'8 IZHMOD. RATIO =IF8.O) .

104 FORMAT(20X.16HRUADWAY WIDTH =:FIO.2;2Y:3HFT.;//20X:
A AIOHLOAn DATAo/I

IOS FORMAT(20X,I6HLOAD(1=HS 23H) =CI7fi/2OX!16HRAILING NT. =9FIO.2:
A2x’8HLRS./FT.:/70X:16HSDHALK L. LOAD :pFIn.2t2X’8HLBSo/FT-I

.106 FORMAT(ZOXII6HFUT0 WEARING =pSXpIBH25 LBSo/SQoFT.’/

A 20X316HTUIAL UTILITY =;FIO.212X;8HLBS./FToI//
a 20x.10chSI DATA,/)
107 FORMATCZOXDI6HCUST UF CONCQ =,F10.292X!8H$/CU.Y093/
A 20XDI6HCOST UF REINF. :1FIO.2I?Y}SH$/an’/
8 20X916HCUST 0F STFEL =1FIO.2;2X!5H$/LR.I//2OX’
C I7HDESIGN VARIAHLES;/)

 

 

121

108 FORMATc20x,16HCIRDER SPACING =,FIO.2.2x,3HIN.p/
A 20X916HNO. 0F GIRDERS =1FR.O)

109 FORMAT(?OY116HUEPTH OF HER =:FIO.2)2X:3HIN.;/
A . 20X516HUPPER 8ND. HEB =;FIO.292X:3HIN,;/
B 20X)I6HCANT. LENGTH =1FIO.2)2XA3HFT.)
110 F0RMAT<2OX.16HFLAMCE PL wIOTH=.FIo.2;2x,3HIN.,//
A 20x.12HSLAR DESIGN,/)
115 FORMATc2O¥516HSLAB THICK. =»F10.2;2XI3HIN.9/
A 20x,16HTPANS. HEINF. =;FIO.2;2X»IOHSQ.IN./FT.1/)
120 FORMAT(20X;IOHGIRDFR DESIGN./) .
121 FORMAT(?0y;16HAVEHTo SPAN I =pF8.0rAX!BHLBS./FT.)
122 FORMAT(20¥;16HAVENT. SPAN 2 =nFfl.0:AXr8HLBS./FT.)
123 FORMATCZOYIIéHFSAFTY SPAN 1 =,FIo.2./
A 20X:16HSAFTY SPAN 2 =9F10.2»/)
WRITEC3,I75)
WRITE(3!IOI)
WRITECBAIOZ) BRL,FR,BRM,FC
WRITE(3AIO3) SPANLIySTRESAISPANBpRHOD
HRITE(3110A) Rnw
HRITE<3,105) LOADpRAILHTpTSHLL

 

HRITEC3AIO6) UTIL
HRITEC3IIO7) CI,02,C3
WRITEC3)I08I GSPAIGRNO
HRITE(3;109) WEUDAUBNEBDICANTB
HRITE(3;IIO) PLATEW
WRITEC3IIIS) SLABT.REINF(2)
HRITE(3JI7OI

WRITF(3r1?1) we(1)
HRITE(3pI??) waIZ)
WRITE(3'I?3I FSAFTYAFSAFTI
RETURN

(END

SUBRUUTINE DETGIR
COMMON BMDL,RMLL,BMSN,BMSWLLCBMTRK;BUTTPT(6);CANTA;CANT99CG(6,A),
A DELILAIDELTLRIDFLTLCAUELTC3!a)ADISTRRpEHPACTIEXTRA’FATIG(2)I
B FSCA)JGSPADHDEPTHtHWIDTH)I(6)0)ILUAD,MINEGLLIPBMADPRngpLLAD
C PLLB’PLATFT(2))PLATEHAPOSLLDPSLA9PSLB)SLARHI88(6)“)!SLART!SPANL
D 'WSTC6'A)!STATH(6)ASTRESAASTUUSISHFHSASHLL:TUPPT(6)IUTIL:VDL:
E VLL,VLLMINyvsprTUTAL,I‘IERIIAVIEBTtI‘ITF-TG(5)yNTFTSL)X)PSNA,PSHB
F )XL’XRILL)FSAFTYJAVGNTICODELL
COMMON/DESI/ALLUWC(“I:ALLUWE(AI'MUT(17)rPRUBLL(AIpTRATIO(2)
COMMON/DES2/ADJ(A),DLRT(2):LLR(7):LLRHAX(QJIPSTUD(9),STIFF(9),

 

122

A TSTIPF(2)»TKLH(9)1LLHWU1(£}
CUMMON/dRUAIA/th:dRN:SPANL1,8PANdpSPANL2;RUW;fO’RMUU:rH1C1;Cd:
U C3AQHNU:UCUST:UCUSTlpWG(ZJpOPTwEfi(10):UPTSPA(10)t0P0PL(IU):
C UPTCANKIU):P(A)»TUTILyTSNLLtHAILNTARLINf(2)111LR:FSAFTIpWLbUl:
U UBNEUD:TPT(1):BPT(1)pIDLT
COMMUN/UET/TUPPLI!TUPPLZ!TOPPL32HUTPL19bUTPL2)BUTPL3IXULNT!
A FTOPL,FSTOP:TTUPRdeUTTLpFSSUTTpFBUTIRydMULApBMULC;dMULB;8MLLA;
B HULLbndMLLBtHMbWAIBMSWC'UFSWbpHSTIFF151U023TlFA'PIIOTITtsTIN:
C STIFH)STIFB:§T1FL
l FORMATCOX31ASIOK’Ffioo)
2 FURMAT(6X:7F10.A)
3 FORMAT(OX;F10.Q:F/o1,13,5?10.4)
4 FORMAT (1H4:19x,?3HP L A T t u I R U L HDIOXIDHSQUAU’OX’
ASHKULKAHNI,l/Zux;9HPRUG. Nb.pF5.0)
16 FORMAT (///17X;AAHFACTUH 0F SAFtTY AGAINST LATLRAL ULCALINU =r

 

A +5.1,////22x.3/H1 N r u l v E. H 15 It A l 1U N://12}U

8 10H~Lb THICK.;F13.4;12K’14H5.W. LIVt LUAL),H.0) ‘
11 FURMAT <12x,oHALu DEPTH,F1A.2.12X;13H5.A. + F.w.S.:Fluou;

A /12X:12HFLANGE W10TH;F11.2)12XAIAHCANI. LENGTh A9i9o2)

19 FORMAT (12X211HSLAB THICK.:+12.2»12Xp8HP (dLAM):f15.1/12X;
A 11HGIRDER SPA.»F12.2:12K28HP (SLAB)prlbol/12A111H§PAN LLNGTH:
B F120Z912X)13HP (LIVL LOAU)IF1001)

23 FORMAT (IZKAQHLIVL LUAUABX2f5.0:IJ'12Xp12HP (5w + wa):F11o1:
A /12X:1?FUISTo fACTUH2F11o2:12X:luHLAwT. LtNUIH 02t9o2)

25 FURMAT (1?X:12FHAUNCH ULPTH;F11.7»12X:6HP (dEAM)AF15o1'
A /12Xn12HHAUNCH n10Th:F11.2:12X;8HP (bLAb):F15o1)

2? FURMAT (12Xp13FUFSIUN STRESbIFlUOU’lZXDleP (LIVL LUAU)1F10.1;
A /12X;9HUIILITIEb;f1a.0,lzX;12HP (5W * FWb))f1101)

29 FURMAT (36X:Fb.2;1HL:FlU.0)

31 FURMAT (12X221hb I I f r o 5 P A .pAXpruobULpFlocU)

32 FORMAT (15X;1H(,F5.3,3H" XAF4.1!2H");7X15HU.62LAF10.0)

3a FORMAT ( //u7x:18HLLNuIH IHICK.:19xp1H1p/)

36 FURMAT (ISXpIBFT U P P L A T E:F19.1pF12o3)

38 FURMAT (AZX;F10.1pF12.J)
40 FORMAT (15A923Hu U T T U M P L A T L;rla.1’+12.3)
42 FORMAT (//52X96HLI dHU»4X;Fu.291FLn5XprHT dRG:/15X:9Ha T R E S;

A 0H 5 E 5:5X33HTUPyFZUoO)ZFlC009/35Xp4HbUITl}19.0IZFan0)
as FORMAT (//35Xp9HDLAU LUAupr14.0,?F1U.n;/15A;13HM U M L N I b:/x;
A YHLIVE LUAU1F1400p2F10.U:/35‘n6H5W * PWSIFl5obdeIUoU)
AB FORMAT (//3bX;9HDtAU LUADpf14.1;F20ol/15x’1/HR L A C T I U N 525x;
A 12HL. L. W/IMP9F11.11f20o1/91X95HCMAA)I9AIIZHLo L. HU/IMPt
B FII-IIFZO.1/15A:9HURU SIIFAtFloo3:2H x:+b.2:F13.3;éH X:r5.2>

52 FORMAT (///bOA'6HLT PIN;6X!ZFCLp6Xn6hRT PIN//32A!4HutAM1fl9o2:2r
A 10.2/32X115H5LAU ADJ. SPAN52F8.2)?F10.2/IZX!15HU L f L L C T .1
B 5X!14HSLAB THIS SPANyF9o2’2F10.2/32X18HSIULWALK2F150222}10.2)

57 FORMAT (32XAIUHUFLTA LL/Lp/Ap2(2h1/pIQ:AX):2H1/IA///)
56 FURNAT(J2X:10HU£LIA LL/LIIIAI2H1/fiIQIQXI2Hl/Il“///)

 

 

 

123

59 FURNAT(J2X110HULLTA LL/LAIXpZFI/AIAJAADZHl/pIQ///)

60 FURMAT k32A»thuLLTA LL/L;1/X»2H1/yIA///)

61 FORMAT (//jax,l/HC U M P n S 1 T EpHXpleS L L l I U N/fi5bX)
A 4HN=10226X)4HN=30/,?1Xn2(7Xn1H196Xr16H5(TUP) SCOUTT)))

64 FORMAT ( 16X)6FLENTEH9FIO.UIZF9.0!F12.C;2f9.0)

65 FURMAT (72XpF1C.O;2F9.U;F12oC)Zf9.0)

66 FURMAT (12At7HS T U UIJK913hS P A C I N Gplb9OH ATpIA’lh",
A SXIQHSIARTb AT)F6.291HL/)5/X)9HENU5 AT)?O¢211hL///)

7O FORMAT (/211p1UHAVh. WT./FT. =pf7.09////35Ay11Hu I R O L R»18x.
A 16HCUMP. SECT. N=8»/ZBX:1HIp6xn6H5(TOF):3x:/HS(GOII);19xer1)
72 FORMAT (37%»5HCANT.;FIU.O)
74 FORMAT t 16A;6FCENTERpF10.0,2i9.C:14X;F10.U)
75 FORMAT (22x,F10.0,2T9.0;1ax,F10.0)
80 FORMAT (/12X)21HS T I T L 0 S.P.A..:IUXI
A 22hNUNE KLHU dLTwELN ORGb)
81 FORMAT (03x91AMLErT CANT. ARN,F9.O:2HIN)
82 FORMAT (ijtleHIhHT CANT. ARM»+8.0.2HIN)
83 FORMAT (47X:2H(,Fb.3:JH1NX:T4.1,?HIN)
1 S T U U T P U T b H t t I

 

PROUzl o

WRITL(3JA) PMUH

NRITL(3:7U) AVGwT

IFC5TUUD.LN.O.) GU TU 400
IF(BUTPL1.NL.U.)WRIT£(3:75)1(421)JST(Q:1)ADDCAII)!I(4t2)
IF([UPPLloNtouo)WHITECJI75)1(2)1)15T(2p1):bu(2)1)!1(214)
WRITE (5:74) 1(191):ST(1p1);Sb(l;l)vI(1'2)
IFCTUPPL3.Ah.0.)HHIIE(Jp75)1(3)1)pST(3p1)péDCJIL):l(3;ZJ
IF(UUTPL3oNt.Oo)WRITE(J;75)1(bpl)pST(btl):50(521)!I(5,2)
GU TU 45?

“80 IFCULTPLI.Ntouo)leTL(J:75)1(Qnl)pST(AOI)JOd(A!1)
IF([UPPLloNL.Oo)WRITEC3;/5)l(2,l)tSl(2r1)250(2!1)
WRITE(JI74) 1(191)!$T(I)I)ISH(1!1)
IF([UPPL3.NhoOo)WRITE(3!75)I(3!1)ASTC391)9bd(Jfil)
IF(bUTPL3.NtoOo)WRITL(3;75)1(521)95TC5:1)!5d(b!1)

GU T0 490

“82 WHITEk3261)
IF(DUTPL1.NL.O.)WHITE(J:65)([(ApK)n5T(A:K)ISb(4;K):K=5'~)
IF(IUPPL1.Nt.U.)WRI]L(J;6b)(I(22K)p5T(2pK)!58(2:K)!A=JI4)
WRITL(3I64) [C1rJ))ST(193))Sd(193)nI(194j'éT(194))bU(1!“)
IFCTUPPL3.Nh.Oo)WHITL(3)65)([(32K)25T(JAK)!DU(3IR)!K=3’R)
IF(uUiPL3.Nt.0.)wnlIL(J,65)(I(5:K).ST(5.x);Sd(b,A),K=3:u)

490 WNITE(J!34)

IF(TUPPL1.NL.O.)WRITE(5p38) TUPPthlCPPICAJ
NRITECJIJO) TUPPL2;TUPPT(1)
IF(IOPPL3oNtoUoJWhITE(J;36) TUPPLJ;IUPPI(5)
WHITE(3:38)
IF(bCTPL1.wL.U.)WRITL(J;38)HCTPLT:HUTTPT(4)

     

12H

wRITE(3:AU) bOTPLGdUTIPT(1)
IF(OUTPL3.Nt.U.)WRIT£(3;35)uGTPL3pHOTTPT(b)
XCENT = XCEwT/SPANL
HHITE(3:A2) ACLNTATTOPL»TSICP:+TUPNpTBuTTLpFSMOTTpfuulIn
HRITE(3:A5) dNULA;uMULL’BNULUdeLLApHMLLC!dMLLutflMSnAIUMSMLnONSND
NRIT£(5»Ab) ULHT(I)AULHI(2))LLHWAX(I)!LLHMAX(2)ILLHWUI(I)D
A LLHWUI(2),TSTI?F(1)ywSTIFFpTSTI+F(2):WSTIFI

2 N U U U I V U I b H L E I
WRII£(3’A) PRUd
HRIIE(J152) ((UELI(A;L)1K=1!3))L=1:4)
IF(CANIR.NEoO. .ANUo CANTBONEoUo)WRIIt(J!b/)UELILA’ULLILC’UtLILD
IFCUANIA.NE.O. .AND. CANTB oAEo 0.)WRIIL(J!§9) UELILApULLILC
IFCCANTA.E0.0. .ANU. CANTB.NE.Oo)NRITL(315b) ULLTLC:ULLILB
IF(LAATA.LN. 0. .ANU. CANTH.LO. O.)WRITL(J!OO) ULLTLL
IF(5TOUJ.Lw.O.)wHITt(3:06) bIUOSoSIUflnyIXN
IF(bIIIL.ENo 0.0 .AND. JTIFHoLQo 1.0) GU TU 495
IF (CANIA*12.0.UT.SIIFA .ANUo WLBI.LI.IWEUL)HRITE(3!/2) SIIFA
DU 492 J:1;491 '
PI = (J'II/d.+.U01
IF (PIooIoL3III'l )WNIIIV.(J)?9) Pl

IF (PToLLobTIFL .ANO. bTIFL.Ol.U.OOb)leIht5:29) PT'SIer(J)
“92 IF (STITL.OI.PT .ANU. bllFL.LT.PT+O.123 .ANU. SIIFL.NEoU.5)
ANRIIE(3;29) STITL;5TIFI(5)
If (STIFL.LI.O.5)wRITE(5931)
IF (STITL.LN.U.5) ~N1TE(3p31) STIPFCS)
IF (STIrF?.ul.O.b)vMLITE(J;32) tsflTpSTIw
If (STIrR.L0.0.S)wRITE(J:32)SIIT'STIW;STITF(6)
DU “96 J26r911
PT = (J'1)/d.
IF (STIFR.OL.PT .AND. J.NE.0)WHITt(Jp29) PT
IF (STIFR.OT.PT .ANU. blIFR.LT.PT+.125 .ANU.STIFH.LT.U.995)
AWRITL(3'29) STITRySTIIF(6)
496 IF (STIFR.LT.PT .ANU. J.NE.O)WRITE(3;29) PIpSTIfF(J)
IF (CANIH*12.O.OT.STIFU .ANO. wLBT.LT.Tw£bH)wRITL(3:/2) bTIIB
GO TO 4’9

“QB'WRIIE($:80)
IF (CANIA*12.0.0T.SIIFA .ANU. WLHI.LToTwLbL) wHITL(3!61) STIFA

IF (CANIH*12.O.OT.STIVd .ANO. thToLT.TAEbn) WRITE(Jt62) STIFU
IF ((CANTA*12.0.GT.bTIFA .Awu. NEOT.LT.IHLUL) .uH. (CANIU*12.U
A .uT.STIFB .ANO. wtuT.LI.thBR)) wRITL(3:dJ) STITrbTIH '
499 WHI|L(5:16) PSAFTY:MEd[;SWLL

WRIIL(3'17) MfdlhbwfWprLAltw'LANTA

WRIIL(3!19) bLAHTfiPUMApOSPAthLA;SPANL;PLLA

WHIT£(3'?3) LOULLL:LLpHSNAyCISTLLACANTO

WRITL(3;25) HULPTHpFHMUprLUTHpPSLH

WRIT£(3:21) STHEbApPLLUpUIILypth

RETURN

END

 

 

 

 

00000

60
61

90

125

SURKCUIINL T3UL5 .

CUNMUN DNULIBNLLprSW)UMSWLL!dNIRKDUFTIFI(O)9CANIA)CANId!Lb(O)Q)p
UELILAAULLILUAUELILCIULLIC394))”ISIHUALMPACIALXIRAtfAIIUKZ)!
IS(“);USVA)FUEVIH;HWIDTEDI(6)0).LCAU;MANLGLLAPBMAthMDJPLLA:
VLLUIPLAILII2))PLAILN:PDDLLIPSLADPSLHAbLAUw:bb(OJQ)IDLAC'II5PANL
351(0p4).SIATMIO)’SIKLSA)5IUUSISNFH515WLL'IUPPI<6I£UIIL1VULt
VLL.VLLMIN,VSW.VTOTAL.NLUU.WL8T.NTFTb(5).HT+ISL.x.PaAA.PSw5

)XLDXHILLtFSAI’IYpAVUWT9CUUtLL

COMMON/DESI/ALLUWC(A).ALLUNL(A).MOT(12).HRO5LL(4).THAIIUCZ)

CUMMON/UE52/ADJ(A)AULHI(2)9LLH(/)1LLRMAA(2)AP5IUU(9)25|IIF(9)’

A ISIIIF(Z);IWLH(9)1LLNWUI(2)
CUMMUN/URUAIA/bKLABthbPANLI'SPAthbPANL2:KUH,PU1RMUU1PHIC1)C2!

U CJIUNNUIGCUSIJDCUSII!Wb(Z)!UPIflLH(10)DUPISPA(1U))CPIPL(IU))

C UPICAN(IU))F(4)AILIIL'ISALLAHAILWTpfithT(2)21TEH;f5HfIIINLDUI;

U UUNLDDyIPT(1)ybPICI)IIDL| ’
CUHMUN/ULI/IUPPLI:IUPPLZIIUPPLjyfiflTPLIobUIPL2:UUIPLJ)XCLNT;

A fIUPL.FblUP.FTUPRpFUUTTLpFdeTI.FBUTIH.6MULA96MULC.oMULfi:BMLLA.

B oNLLc,dMLLb.UMSWA.UMch.uwab.w5TIFr.STUU.ST1FA.PI.SI1T.STIW:

C bIlffitSIlFflthlIL
CUMMUN/URlb/ UHhHNUpGHAULUDNGpCANTLdeANIUU)WLUUUU!WLUULbINU;

A PLWLU;PLAUBIIXQIIYC:IXGCfiIYCCOIXHIIYb)lAHCIITHCAZOL(5)5)I

b CSLAUpCHLINF9C5IELL!UCUSI’ZHU(525)ISAFIY(5r§)!PLW(b!§3!

C NH()pb)pUPTH(btb)rUPTB(b):))0PTG(S;5))UPTC(D!§)1SAPII(5!5)
COMMUN/bRCh/UCfibT)UGHNUIBCANI)dWEdflrBPLAILAUAIUMAUHfitP
CUMMUN/LBC/CANIpZMINC
FORMAT(20X’AHMA1N»AT10.2.2X:2+10.2)

FORMATCZOX)I3HA LUWLN UUUNU:2X:2FIO.C)

T0 UEIEKMINL A LUWLR UUUND PCH SLVERSIHUCIURL GUST
SLAB DESIGN AS PER AASHU 1.5.2 ANU
bIHULR DESIGN ' NEd PLAIL SAIISFILS bHLAH ANU LUCAL dUCKLING
FLANUE FLAIR SAIISFILS ULAUING SIHLSS UNLY

EXHAUSTle bLAHLH TU OLTLHMLAL A LOWER bUUNU

GHNU = UBIJHNU

VTMCOGZD

OATUM = 1000000000.
STUUS = 4.0
EXTRA = 1.10

PLATLw = 14.

GSPA = (BRA - 2.*2.b)*12./(uNNU '1.)
CALL SLAB

CALL PRLLIM

DU 120 10 = 1,5

CALL MINC

IF(dCOST.LT. OATUM) GO IO 9U
GO TO 93

OATUM = BCUDT

BURNU = GHNU

BCANI = CANIU

UNEUO = WLUU

BPLATE = PLAILA

 

 

95

120
125

“FCC‘TIID

A
U
C
U

A

H
C

126

OHNU = nRND - 1.

GSPA = (BK/J " 20*205)*120/(URI\U -10)
IF(UHNO.LT.URNELB) bu TU 12)

CALL SLAB

CALL PHELIM

CONTINUL
HHITE(3:60) BORNO:OCANT.OwtbflpfiPLAThovATUMIUCUbT

THE VALUE UT LPSILUN IS 11 PLRCLNT OF A LUnER dUUNU
ZMINC = UATUN*1.II

WRITE(3.61) ZNINC:DAIUM

RETURN

END

 

SUBHUUIINL MINbM

CUMMUN DHUL)BMLL!UMbT‘.pUMSWLLIBMIRKDUCIIPT(O)1CANIA!CANIU!CU(6)4))
ULLILAAULLILdtUELILCpUELI(J)“)nDISTRUALMPACI:LXInA:FAI16(2):
Ib(4);b$PA.FUEPIH:HNIUTH1I(0’4)nLUAU;MJNCGLLIPBMAIPOMdpPLLA:
PLLd'PLAILT(2)pPLAILWpPUbLL'PSLAnFSthbLABHIbB(624)ibLABI;bPANL
:SICO;A):STATM(6))SIRESA:SIUUSpSNFWSySWLLAIUPPIIOIpUIILJVDL:
VLL,VLLMIN.wa.VTUTAL:WthD.NL8T.NTFIb(5)pNTfTSL)X:PJ~A:P5Wb
IXL)XHJLLfiFOAFIYAAVUWI!MCUELL

CUMMUN/UEbI/ALLUWC(N))ALLUNLCA)fiNUI(I2):PKUGLL(4):THAIIUIZI

CUMMbN/UFSZ/AUU(A)'ULHI(2)’LLH({)9LLHMAX(2)DPSIUU(9)!SIIIF(9)I
TSTTrT(2).TALH(9).LLnNUI(2)

CUMMUN/de)AIA/HKL)UHW)5PANLII§VATVBobPANLZIHUWIT'CIHMUUprJCIICZ)
L31UHNIJ2L5CUST)IJCUSII»W(1(Z)9I3PIWEH(IU)»UPISPA(IUIAUPIPLIIU))
UFICAN(1U);P(A))IbIILtTSWLLpflAILNTpRLINf(2))IILHpFSAFIIDNLDUI)
UBWEUUIIPTII);UPI(1)!IDEI

CUMMCN/URIU/ UdaHNU.CRNuLu.NG.CAATLn.CAHTUU.NLUuUU.wLnULU:Nu.
PLWLdpPLwUBAIXb,IYC’IXGC;IYCCtIXH)IprIKHCpIYBCA£QCI3I5)!
CSLAU.CHLINF:CbTLLLIdCUSI2LHUC5n5).SAFIY(595)'PLW(5!3)9
HHI).5).UPTH(5:5)2UPIUID:5):CPTU(5A5)JUPIC(5!5);SAPII(5:5)

CUMMUN/OHCH/UCIJSI.0BITRNUtflc/ANIDBWEHDJBPLAILIUAIUM2UH30T’

COMMON/LHC/CANIpZMINC

IO FURMAT(20A15HMINHM92XprUHNU =92X1FA.U:hHGbPA =p

A

2X.FO.210HnEdl) =:FO.2:2X18HPLAIEW =2T-6.2)

11 FORMAT(2OX.3HMINBM.ZX.IHCANTU =.F6.2.2X.7HTSUMA :,

40

A

TIU.2:IHFT.KIPS)
TU ULTENMINL CANTO 5U IHAI bUM OF AREA UNULN u.M. UlAunAN 15 MIN.
TU ULTLRMINL SUM OF U.M. IN SPAN IWU
CANlu = CANTLB
CLB = CANTH
SPANLZ = SHANO - CANTO
SPANL = ST’ANL?

 

50

60

65
A

127

CANTO = O.

PBMO : U.

PSLb = U.

PLLb = U.

PSWU 3,0

SUMA = U.

X = O.

WTFTG(1) = ZOO.

DISTLL = 1.1

LMPMAX 3 103

EMPACT = AMTN1<LNPMAA.1.0+bU./(5PANL+125.)J
DISTRB = EMPACTthPA/(120.*UISTLL)

wTFTSL = 1.U02*(GSPA*SLA8T + HWIDTH*HOEPTH)

CALL MOMENT

BMTl = AMAX1((HMDL + BMSw + BMSNLL + POSLL)*U.5.(BMUL+uMaw+POSLL))
BMTZ = AMAXl((HMOL+5MSA+HMSNLL+NEOLL)t0.8.(dMoL+UMSw+NLuLL))
SUMA = DUMA + AUS(AMAX1(BMI1,UMI2)) '

X = X + 1.
IF(A.LE.SPANL2) GO TU 50
SUMAZ = SUMA

CANIILLVEH HLACTIUNS

PBMU = NTTTU(1)*SPANL/ZUOO.

PSLU = ~TFTbL*SPANL/200U.

PSHU = awFwaSPANL/ZOOO.

PLLb = AMAx1((/2. ' b?2./SPANL)*(OSPA/(120.*l.1)1.

(26. + C.32*SPANL)*(uSPA/(120.*1.1)))
TO ULTLNMINL SUM OF u.M. IN SPAN UNL
SPANL = SPANLI
EMPMAx 1.5
EMPACT AMINICLMPMAX;1.0+5U./(bPANL+125.))
BISTRO LMPACTACSPA/(120.*1.1)
CANTO = CLU
SUMA = u.
X=Oo
CALL MUAENT
BMTl AMAK1((BMDL + HMSW + UMSHLL + POSLL)*U.d;(5MOL+oMaN+VUbLL))

BMT2 AMAX1((BMDL+OMSN+BMSNLL+NEGLL)*0.6.(dMuL+eMSN+NLuLL))

SUMA SUMA + AuS(AMAX1(BM11.dMT2))

X = x + 1.

IF(X.LL.SPANL1) GO TO 60

TU ULTENMINL SUM UF 6 M OVLN'CANTILLVER

X = CANTO

RMTZ = A*(PUMM + PSLB * PSWd + PLL8*1.3) +
WTTTu(1)*X*X/?UOO. + wTFTSL*X*X/ZOUU.

SUMA = DUMA + AHS(HMT2)

X = X “ 1.0 .

IT (x.G|. 0.) GO TO 05

SUMAI = SUMA

TbUMA = SUMAI + SUMAZ

IF(TSUMA.LT.ISLMA) UO TO IU

II II II

 

CO

105 WRITL(3.11) CANT .TbUMA

128

GO TC 90

70 ISUMA = T5UNA
CANTd = CANTO + l.
IFICANTU.GT.CAATUU) GO TO IUU
GO TO AU

90 CANT = CANTO - 1.
ISUMA = ISUMA
GO TO ldS

100 CANT = CANTU "1.0

TSUMA = IbUMA

RETURN
ENU

 

SUUHOUTINL MINC

COMMON dMULpBMLL.HMSN.UMSwLL.uMTHA.NOTTPT(0).CANTA.LANTU.CC(6;AJ.
UELTLnfiutLILU’UELTLUIUELT(J!4)!DISTHUILMPACI’LXIRADTATI“(2)!
TS(“):b5PA.HULPTH)HwIUTHI1(6ta)ALUAUpM;NEULL;PBMA!POMD:PLLA:
PLLU’PLATLT(2)!PLAILW’PUDLL'PSLA'PSLH'bLABW’Dfi(6IQ)fiDLAdlfispANL
’bT(0.4)ASTATM(6)ASTHLSA’5TUU89SNFWSASWLL’TUPPTIOJAUTILfivoL:
VLL,VLLMINpVSWIVTUTAprtubpwtfil.WTTIu(b)!WTTISLfixtPDWA)PSWb
:XLAXHALLJTSATTY.AVUWT;LLULLL

CUMMDN/Ufibl/ALLUWL(“)fiALLUNL(Q)’~UI(I?)IPNUHLL(4)!TRAT10(2)

CUMMUN/UEbZ/ADU(A):ULRT(2):LLN(I):LLRMAX(2);P5TUU(9)16TITT(V):

A TSTITF(2).TnLB(9).LLnWOl(2) ’
CUMMUN/URTTATA/bfiL)UHN1 bf'ANLIDSPATNUDSPANLZIHUAJT‘L9RMUUAT’H1C1iCk’I

b CJACKNUIOCUSTAOCUSTI;NG(2);UHTNEH(10);UPTSPA(IU):UPTPL(IU):

C UPTCANC IU)0P(‘T)!TUTIL.TDT‘TLLDNAILNTpHLINT' (2)21TLN:T5AT’ T IAWLOUI:

U UUNLDUDTPTCT)AUPT(1))IULT
COMMON/URIU/ UBORNU:GHNUL8)NO:CANTLHACANTUU9WLBUUUIthJLupNu.

A PLdepPLwUHtbefiIYC’IXGCDIYCC:1XH'IYD!I‘HCpIYHCylbLIDIB);

H CSLAM.CHLINF.CSTELL:UCUST:LHbTbnb):SAFTY(5p5)JPLWTDID)!

C WH(5;5)9UPTH(515)AUPTBIb’b)ICPTG(595)IUPIC(D!S)9SATTIIBAS)
COMMON/DRCH/UCUbI)BURNU’HCANTIdflfiaolfiPLATL’UATUM!UH!UP
CUMMUN/LBC/CANTpZMINC

I0 FURMAT(ZOXPAHMTNC:2X!4HCUST22XI3F10.2)
11 FORMATIZUXAQHMINC:23HCANT. TCH MIN WEU thLLAZA:

A TIOQIIZTIITojfiIIOOU)

12 TURNAI<ZOX1I3HFLANGL CUbT =)TIU.Up
A IOHWLH COST =.F10.UJ
I“ FURNAT<ZOXIQHMITNCAZX1JHSTFIZXpTDHTPLThpgT10.2)
15 FURMATIZUXIUHMINC)2A;/HTLANuLSA2Xp2T10.4)
16 FURMATIZOXAQHMINC.ZHXLI2XIZHXH!ZXJ7F10.2I
17 FURMATIZOX!5HTPLTH12X.T10.2)

TmC.("{II>

TU UETLNMINL CANTILLVLH LLNUTH SUCH THAT bTRUCTURAL

 

 

00

no

00

129

bTLLL 1N WLH Ib MTNITV‘zIJM

ALLOHALULL aHLAH 1N OINULR NLU - CROSS SECTION 18 12 mol.
ICANT = 3

T0 DLTLMINL NEG THICKNLSS 1A SPAN 2
SHLUI = loUUOUUUOU.
CANTO = CANTLH

NTFTC(1) = 200.

HTFTSL = 1.U42*(GbPA*SLAHT + leDTHihULPTH)

OISTLL = 1.1

 

CLB = CANTri

SPANLZ = SPANU - LANTH

CANTO = O.

PBMb = U.

PSLd = U.

PLLd = U.

PSWU = .

SPANL = SPANLZ

EMPMAA = 1.5

EMPACT = AM1N1(LMPMAA.1.U+SU./(5PANL+125.)1
DISTHb = EMHACTtGSPA/(120.*UT§TLL)

HEBU = oPANL1*12./ZU.

CALL SHLARCU.)

VL = VTUTAL

CALL SHLAR(N.U)

PLMIN = 0.1/5 '
T = AMAA11ALUU/15u..(NLdOtAMAXI(VL.VTUTAL)*IUUU.//bOU.'*2)**O.333)
TREUO = AMAAICTTIx(T*10.+O.99)/16.0;PLM1N)
WEBT = TREUO

T2 = deT

TO ULTLNMINL NLU THICKNLSS 1N SPAN 1

CANTO = CLN

SHANLz = SPANN - LANTd

PdMu = NTFTUC1)*SPANL/ZUOUO

PSLU = NTFTbL*SVANL/ZOUO.

Pswd = oNTNb*SPANL/ZOOU. .

PLLU = AMAX1((72. ' 672./SPANL)*(USPA/(12U.*1.1)):

(26. + 0.32*5PANL)*(USPA/(120.*I.1)))
SPANL = SPANLT

EMPMAX = 1.5
EMPACT = AM1N1(LMPMAX.1.0+DU./(5PAAL+125.))
DISTRB = EMPACT*OSPA/(120.*l.1) '

NLBU = JPANL1*12./ZU.
CALL SHLARCU.)

VL = VTUTAL

CALL SHLAHCN.O)

PLMIN = 0.3/5 ‘
T = AMAKTCNLUU/150.p(WEdU*AMAAI(VL.VTUTAL)*IUOU./7SOC.'*2)**0.333)

THEUU 3 AMAXI(ITIXT1*16.+0.99)/16.0.PLM1N)

 

AS

50

SS

60

100

102

130

HLBT = TTHuiO
T1 = NLUT

STELLA = NEUO*((SPANL1 + CANTU)*T1 + SPANL2*TZ)*3.4*URNU
IF(1CANT.LU. 1) GO TO GO '
TF(5TEELH.LT.SWLU1) GO TO 4b

GO TO 5U

CANTS = CANTH

SNEUI = STELLw

CANId = CAANTH * I.
IFCCANTd.UT. CANTUU) GU TU 55
GO 10 AU

CANTd = CANTS

ICANT = 1

GO TO AU

WHITE(3'II) CANTHIleTZISTLLLW
USE CANTILLVER NHICH MINIMIILS AREA UNDLR U M DIAGRAM
CALL MINBM

TU ULTLRMINL STRUCTURAL STLLL IN FLANCLS SPAN a
CANTB = CANT
CLU = CHNTU
SPANL? = SPANM ' CANTd

CANTU = O.

PUMU 3 LT,

PbLb = U.

PLLO = U.

PbNU : U.

SPANL = SPANLZ

EMPT‘TIAX : 10.3

EMPACT = AMINICLMPMAX:1.0+SU./(bPANL+125.))
DISTHU = LMPACTAGSPA/(120.*l.l)
X = 1.

TO ULFINF xL AND AH

XL = U.

XK = 101.)

TPLTH = O.

TUPPT(1) = .0625

BUITPTTI) = 0.0625

WLBT = 12

CALL PRUPTY(1)

CALL MOMENT

CALL STRESS(1)

IF(TS(2).GT.200UO.) HOTTPT(1) = BOTTPT(1) + U.002b
IFTTS(1) .UT. ZUOUU.) IUPPT(1) = TOPPTTT) + 0.0025
IT(Tb(21.GT.ZUOUO. .OR. 18(1) .uT.20000.) UU TO 100
TPLTM = TPLTH + TUPPTCI) + bCTTPT<1>

IF(A.LU. 55.) CH TO 102

GO TO 1U3

WHITL<J:15) TOPPT(1).BUTTPT(1)

 

 

131

103 X = X + 1.

TDPPT(1) = 0.0625

BOTTPT(1) = 0.0025

IF(XOGL.3PANL) 110 TU 10‘)

GO TO 100

T0 ULTEKNINL STHULTUHAL STELL IN FLAAuLb SPAN 1
105 STFZ = TPLTH*14.*1.*3.4*CRNO*1.51

WRITE(3:IA) STF2.1PLTH

CANTO = CANT

SPANL2 = SPANO - CANTO

 

 

PBMb = NTFTO(1)*SPANL/ZUUO.

PSLU = ~TFTSL*SHANL/ZOUU.

PSWB = awaS*5PANL/ZUOU.

PLLB = AMAA1((72. - b/Z./5PAAL)*(GSPA/(120.*1.1)).
A ‘ (26. + O.32*SPANL)*(USPA/(12O.*l.1)1)

SPANL = SPANLT

LMPMAX 10.5

EMPACT = AMINITLMPMAX.1.0+bU./(bPANL+IZb.)1
BISTRO = LMPACI.GSPA/(120.*1.1)
X : SPANLl
106 x 3 X .10
CALL MOMENT
IF((BMUL + HUSLL + DMSN) .LL. 0.) GO TO 100
XR = X/DPANLI
X = 1.
TPLTH = O.
TOPPT(1) = 0.0625
BOTTPTCI) = 0.0025
WLBT = 11
110 CALL PHUPTY(1)
CALL MOMENT
CALL STRESS<11
IF(TS(1) .CT. 20000.) IUPPT(1) = TOPPT(1) + 0.0025
IF(T5(2).GT.?0000.) HOTTPT(1) = BUTTPTCI) + 0.0025
IFCTS(2).GT.ZOCUO. .UH. FSCI) .UT.20000.) uU IU 110
TPLTH = TPLTH + TOPPT(1) + DUTTPT(1)
IF(XIEQO350)GIJTU112
GO TO 113
WRITL(3’15) TOPPT(1).BOTTPT(1)
X = X + 1.
IF(XOGLOSPANL) UT] IU 120
TOPPT(1) = 0.0625
BOTTPTCI) = 0.0625
GO TO 110
FLANGE PLATL THICK. OVEN CANTILLVLR
120 X = CANTO
WHITE(3:1/) TPLTH
125 TOPPT(1) = 0.0625
BOTTPT(1) = 0.0675

pap
Hp
CAIN

 

126

A

132

CALL P4111311“)
BMDLB: '(WTFTGII) * WTTTSL +CTIL)*X**2/ZOUU. ' (PfiMb + P5LU)*X

BMLLb = -x*PLLH*1.3
BMsne = -SWTHS*A**2/ZUUU. - XinWU
FTOPR = -(UMDLU + bMSNo + bMLLd).12000./5111.1)

FBOTTH =-(OMULB + HMSwU + UMLLU)*12000./Su(1.11
FSCI) = FTUPR '

FSCZ) = FHOTTH ‘

IF(Tb(1) .01. 20000.) TuPP1(1) = TCPPT(1) + 0.0025
IF(TS(2).CT.20000.) UOITPT(1) = BOTTPT(1) + 0.0025
IF(TS(2).GT.20000. .ON. FS(1).GT. 20000.) CU TO 126
TPLTM = TPLTH + TuPPT(1) + bOTTPTCI)

X = X'lo

IF(X.CT.0.) GO TO 125

STFl = TPLTH*1A.*1.*3.A*GRNU*1.31
HHITLIJITA) SIFInTPLTH
TO ULTLRMINL ORIOCL C051

CbLAB = C1*(ONLAHNW*5LAUT/(12.*27.) + (5RL*HNIDTH*HULP1H/(1aa.*
2/.>*GRNU)) -

CRLIAT = C2*490.*URL*ORw/SA.*R£1NF(2)

CSTLLL = CJ*(8111 *STFZTSTELLN)*I.IH

BCUbT = CSLAH + CHETNT + CSTELL

NRITL(3:10) CbLAU.CR£INT.CSTELL

CSTT = C3*(aTF1 + STF2)*1.ld

CWEU = CS*STEELw*1.18

wRITE(3.12) CbTT.CNLd

RLTURN

FND

 

 

 

 

 

 

 

         
     

 

 

 

 

 

 

 

        

   
   

           

 

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