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A DISCRETE-ELEMENT METHOD FOR TRANSVERSE VIBRATIONS OF BEAM-COLUMNS RESTING ON LINEARLY ELASTIC OR INELASTIC SUPPORTS
by
Jack Hsiao-Chieh Chan Hudson Matlock
Research Report Number 56-24
Development of Methods for Computer Simulation of Beam-Columns and Grid-Beam and Slab Systems
Research Project 3-5-63-56
conducted for
The Texas Highway Department
in cooperation with the U. S. Department of Transportation
Federal Highway Administration
by the
CENTER FOR HIGHWAY RESEARCH THE UNIVERSITY OF TEXAS AT AUSTIN
June 1972
The contents of this report reflect the views of the authors, who are responsible for the facts and thE! accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the Federal Highway Administration. 'Ji1lis report does not constitute a standard, specificatjLon, or regulation.
ii
PREFACE
This study describes a method developed to analyze beam-columns subjected
to either static fixed loads or dynamic loads. The supports of the beam
column may be either linearly elastic or nonlinear and inelastic. The method
incorporates previously developed discrete-element beam-column techniques.
This work was done under Research Project 3-5-63-56, "Development of
Methods for Computer Simulation of Beam-Columns and Grid-Beam and Slab Systems,"
conducted at the Center for Highway Research, The University of Texas at Austin,
as part of the Cooperative Highway Research Program sponsored by the Texas
Highway Department and the Department of Transportation Federal Highway Admin
istration.
The authors wish to acknowledge Mr. John J. Panak, Research Engineer, for
his valuable suggestions in programming, writing, and other aspects of this
study. Appreciation goes to Miss Darlene Neva for typing this report. Other
personnel of the Center for Highway Research are gratefully acknowledged for
their assistance. The excellent facilities of the Computation Center of The
University of Texas and its cooperative staff made a significant contribution
to this work.
June 1972
iii
Jack Hsiao-Chieh Chan
Hudson Matlock
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LIST OF REPORTS
Report No. 56-1, "A Finite-Element Method of Solution for Linearly Elastic Beam-Columns" by Hudson Matlock and T. Allan Haliburton, presents a solution for beam-columns that is a basic tool in subsequent reports. September 1966.
Report No. 56-2, "A Computer Program to Analyze Bending of Bent Caps" by Hudson Matlock and Wayne B. Ingram, describes the application of the beamcolumn solution to the particular problem of bridge bent caps. October 1966.
Report No. 56-3, "A Finite-Element Method of Solution for Structural Frames" by Hudson Matlock and Berry Ray Grubbs, describes a solution for frames with no sway. May 1967.
Report No. 56-4, "A Computer Program to Analyze Beam-Columns under Movable Loads" by Hudson Matlcok and Thomas P. Taylor, describes the app lication of the beam-column solution to problems with any configuration of movable nondynamic loads. June 1968.
Report No. 56-5, "A Finite-Element Method for Bending Analysis of Layered Structura 1 Systems" by Wayne B. Ingram and Hudson Matlock, describes an alternating-direction iteration method for solving two-dimensional systems of layered grids-over-beams and plates-over-beams. June 1967.
Report No. 56-6, ''Discontinuous Orthotropic Plates and Pavement Slabs" by W. Ronald Hudson and Hudson Matlock, describes an alternating-direction iteration method for solving complex two-dimensional plate and slab problems with emphasis on pavement slabs. May 1966.
Report No. 56-7, itA Finite-Element Analysis of Structural Frames" by T. Allan Haliburton and Hudson Matlock, describes a method of analysis for rectangular plane frames with three degrees of freedom at each joint. July 1967.
Report No. 56-8, "A Finite-Element Method for Transverse Vibrations of Beams and Plates" by Harold Salani and Hudson Matlock, describes an implicit procedure for determining the transient and steady-state vibrations of beams and plates, including pavement slabs. June 1968.
Report No. 56-9, "A Direct Computer Solution for Plates and Pavement Slabs" by C. Fred Stelzer, Jr., and W. Ronald Hudson, describes a direct method for solving complex two-dimensional plate and slab problems. October 1967.
Report No. 56-10, "A Finite-Element Method of Analysis for Composite Beams" by Thomas P. Taylor and Hudson Matlock, describes a method of analysis for composite beams with any degree of horizontal shear interaction. January 1968.
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Report No. 56-11, I~ Discrete-Element Solution of Plates and Pavement Slabs Using a Variable-Increment-Length Model" by Charles M. Pearn, III, and W. Ronald Hudson, presents a method for solving freely d iscor:tinuous plates and pavement slabs subjected to a variety of loads. April 15'69.
Report No. 56-12, "A Discrete-Element Method of Analysis for Combined Bending and Shear Deformations of a Beam" by David F. Tankersley and William P. Dawkins, presents a method of analysis for the combined effects of bending and shear deformations. December 1969.
Report No. 56-13, "A Discrete-Element Method of Multiple-Loacing Analysis for Two-Way Bridge Floor Slabs" by John J. Panak and Hudson Matlock, includes a procedure for analysis of two-way bridge floor slabs continuous over many supports. January 1970.
Report No. 56-14, "A Direct Computer solution for Plane Frames" by William P. Dawkins and John R. Ruser, Jr., presents a direct method of ~jolution for the computer analysis of plane frame structures. May 1969.
Report No. 56-15, ''Experimental Verification of Discrete-Elenent Solutions for Plates and Slabs" by Sohan L. Agarwal and W. Ronald Hudson, presents a comparison of discrete-element solutions with small-dimension test results for plates and slabs, including some cyclic data. April 1970.
Report No. 56-16, "Experimental Evaluation of Subgrade Modulus and Its Application in Model Slab Studies" by Qaiser S. Siddiqi and W. Ronald Hudson, describes a series of experiments to evaluate layered foundation coeff:i.cients of subgrade reaction for use in the discrete-element method. January 1970.
Report No. 56-17, 'Dynamic Analysis of Discrete-Element Plates on Nonlinear Foundations" by Allen E. Kelly and Hudson Matlock, presents a numerical method for the dynamic analysis of plates on nonlinear foundations. July 1970.
Report No. 56-18, "A Discrete-Element Analysis for Anisotropic Skew Plates and Grids" by Mahendrakumar R. Vora and Hudson Matlock, describes a tridirectional model and a computer program for the analysis of anisotropic skew plates or slabs with grid-beams. August 1970.
Report No. 56-19, "An Algebraic Equation Solution Process Formulated in Anticipation of Banded Linear Equations" by Frank L. Endres and Hudson Matlock, describes a system of equation-solving routines that may be applied to a wide variety of problems by using them within appropriate programs. January 1971.
Report No. 56-20, "Finite-Element Method of Analysis for Pla,ne Curved Girders" by William P. Dawkins, presents a method of analysis that mB,y be applied to plane-curved highway bridge girders and other structural melIlbers composed of straight and curved sections. June 1971.
Report No. 56-21, "Linearly Elastic Analysis of plane Frames; Subjected to Complex Loading Conditions" by Clifford O. Hays and Hudson ~1atlock, presents a design-oriented computer solution for plane frames structures and trusses that can analyze with a large number of loading conditions.. June 1971.
vii
Report No. 56-22, I~nalysis of Bending Stiffness Variation at Cracks in Continuous Pavements," by Adnan Abou-Ayyash and W. Ronald Hudson, describes an evaluation of the effect of transverse cracks on the longitudinal bending rigidity of continuously reinforced concrete pavements. April 1972.
Report No. 56-23, "A Nonlinear Analysis of Statically Loaded Plane Frames Using a Discrete Element Model" by Clifford O. Hays and Hudson Matlock, describes a method of analysis which considers support, material, and geometric nonlinearities for plane frames subjected to complex loads and restraints. May 1972.
Report No. 56-24, "A Discrete-Element Method for Transverse Vibrations of BeamColumns Resting on Linearly Elastic or Inelastic Supports" by Jack Hsiao-Chich Chan and Hudson Matlock, presents a new approach to predict the hysteretic behavior of inelastic suppo=ts in dynamic problems. June 1972.
(p) indicates Preliminary Report.
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ABSTRACT
A discrete-element method of analysis for transverse vibration of beam
columns resting on linearly or nonlinearly-elastic, or nonlinearly-inelastic,
supports is presented. The applied forces include static fixed loads and time
dependent dynamic loads. The program is an extension of programs BMCOL 43
(Research Report No. 56-4) and DBCI (Research Report No. 56-8) to cover in
elastic supports and time-dependent axial thrusts. Two multi-element models
are used to simulate the inelastic characteristics of supports, one which
allows the beam to lift off the support when it deflects, upward or downward,
and the other which considers the resistance to either upward or downward
deflection. An internal damping factor, which is related to the first deriva
tive with respect to time of the curvature of the beam, has been included, in
addition to the conventional external viscous damping factor.
The method is based on an implicit difference formulation of the Crank
Nicolson type. A computer program has been written to check the validities of
the proposed multi-element models of tracing the loading paths of the nonlinearly
inelastic supports and of the implicit formulation of the Crank-Nicolson type.
The results compare well with the theoretical results and with experimental
data.
KEY WORDS: dynamic, static, nonlinearly-inelastic supports, multi-element
model, beam column, implicit formulation, discrete-element method, computers,
piles, bridges, earthquake, wave forces.
ix
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SUMMARY
A computer program, DBCS, is presented which can efficiently analyze a
beam-column resting on linearly or nonlinearly-elastic, or nonlinearly-inelastic
supports and subjected to either static fixed loads or dynamic loads.
The path-dependent history of loading of the nonlinearly-inelastic
resistance-deflection curve of the support is considered in this study. Two
multi-element models, which are used to simulate the nonlinear characteristics
of the inelastic resistance-deflection curves, have been introduced. For prob
lems with nonlinearly-elastic or nonlinearly-inelastic supports, the iteration
process compares successively computed deflections until a specified tolerance
is satisfied. An option available in the program allows a switch from a
spring-load-iteration process (adjusting both the stiffness and the load from
one iteration to the next, which is known as tangent modulus method) to a
load-iteration technique (adjusting only the load) when the supports yield or
disconnect from the beam-column. An internal damping factor, which is related
to the first time derivative of the curvature of the beam, has been considered
in addition to the external viscous damping factor which is normally encountered
in a dynamic problem.
The results of an analysis can include
(1) solutions for the member under static fixed loads,
(2) solutions for the member under dynamic and static loads at each time station, and
(3) plots of computed deflections or moments along either the time or the beam axis as required.
Seven example problems typical of those encountered by highway and founda
tion designers illustrate the uses of the program and the options available.
Included are a three-span beam loaded with an AASHO standard 2-D truck moving
at a uniform speed of 60 mph, a simply supported steel rod loaded by axial
pulses, a partially embedded steel pipe pile loaded by idealized wave forces,
and a partially embedded steel pipe pile excited by sinusoidal, earthquake
induced forces.
xi
xii
A guide for data input is presented which allows routine application of
tho method of analysis with little necessary reference to the body of the main
report. Any number of analyses may be run at the same time.
IMPLEMENTATION STATEMENT
The utilization of numerical methods to describe computer models of
problems in structural dynamics has been an interesting subject to which many
structural engineers have devoted their efforts in recent years.
In this study, an efficient computer program, DBC5, is developed for the
analysis of beam-column problems, static or dynamic, which have either linearly
elastic or nonlinearly-inelastic supports. The path-dependent history of
loading of the nonlinearly-inelastic resistance-deflection curve of the support
is considered. Potential applications include study of the dynamic response
of actual truck-loaded bridges, prediction of the response of offshore piles
tc wave forces, analysis of railroad loadings on continuous spans which are
supported on soil foundations, analysis of transverse response of partially
embedded piles to earthquake-induced forces, and prediction of the hysteresis
effect of inelastic supports under pavement slabs.
Recommendations are made for further research in developing other better
multi-element computer models for nonlinear supports so that buckling, fracture,
softening, and relaxation (creep) of the support could also be considered in
the program.
It is further recommended that this program be put into test use by
designers of the Texas Highway Department to further evaluate its uses, and
to investigate needed extensions or modifications to make it more usable for
the practicing design engineer.
xiii
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TABLE OF CONTENTS
PREFACE
LIST OF REPORTS
ABSTRACT AND KEY WORDS
SUMMARY
IMPLEMENTATION STATEMENT
NOr-1ENCLA TURE . . . . . .
CHAPTER 1. INTRODUCTION
Purpose and Scope of Program DBCS • Application . • • . • . • . •
CHAPTER 2. DEVELOPMENT OF THE IMPLICIT OPERATOR
Discrete-Element Model for Dynamic Beam-Columns Equation of Motion of Beam-Columns ••.... Internal and External Damping . • • • • • . . . • • Linearly Elastic and Nonlinearly-Inelastic Supports . Stability of the Implicit Operator •••••..•
CHAPTER 3. RESISTANCE-DEFLECTION CURVES FOR THE LATERAL SUPPORTS
Multi-Element Models .•. . . . . . . . . . . . . Baushinger Effect of Nonlinear Resistance-Deflection
Cu rv e ...•• ... . . • . • • Systematic Approach for Following Loading Paths . . .
CHAPTER 4. DESCRIPTION OF PROGRAM DBCS
General . Procedure for Data Input Tables of Data Input Error Messages •.•. •• Description of Problem Results Plots of Results ••. . . . .
xv
iii
v
ix
xi
xiii
xvii
1 2
5 5
18 21 24
27
29 36
41 41 43 46 47 48
xvi
CHAPTER 5. EXAMPLE PROBLEMS
Example Problem 1. Inelastically Supported Mass Under Free Vibra t ion . . . . • . . • . . . . . • . .
Example Problem lA. Free Vibration of Mass by Specifying Initial Displacements .•......••..•
Example Problem 2-1. Lateral Vibration of a Simply Supported Beam with Axial Compression Force
Example Problem 2-2. Lateral Vibration of a Beam on Elastic Foundation . . . . . . . . . . . . . . • .
Example Problem 3. Three-Span Beam Loaded by an AASHO Standard 2-D Truck Moving at a Speed of 60 mph . . .
Example Problem 4-1. Simply Supported Steel Rod - Axial Pulse Base Time = 0.10 Second ...••......•
Example Problem 4-2. Simply Supported Steel Rod - Axial Pulse Base Time = 0.026 Second •••..•.•....
Example Problem 4-3. Simply Supported Steel Rod - Axial Pulse Base Time = 0.006 Second .••.•...••••
Example Problem 5. Partially Embedded Steel Pipe Pile Loaded by Wave Forces . . . • • . • . • • • • . • •
Example Problem 6. Three-Span Beam with One-Way and Symmetric Nonlinear Supports Loaded with a Transient Pulse ..•..•...••..•..•..••••.•
Example Problem 7. Partially Embedded Steel Pipe Pile Excited by Sinusoidal Earthquake-Induced Forces
CHAPTER 6. SUMMARY AND CONCLUSIONS
.
.
53
55
. . . 55
61
61
72
76
76
76
90
. . . 95
107
REFERENCES • • • . • • . . . . . • . . • • • • • . . . • • • • • • • .• 111
APPENDICES
Appendix A. Derivation of the Dynamic Implicit Operator Based on the Crank-Nicolson Implicit Formula ..•.•
Appendix B. Derivation for Recursive Solution of Equati~ls (Extracted from Appendix 2 of Ref 4) ..•••••.••
Appendix C. Stability Analysis for the Dynamic Implicit Operator with Internal Damping Coefficient
Appendix D. Guide for Data Input for DBC5 . . . . • Appendix E. Glossary of Notation for DBC5 . • • • • Appendix F. Flow Charts and Listing of Deck of Program DBC5 Appendix G. Listing of Data for Example Problems ••• Appendix H. Partial Sample Computer Output for Example
Problems . . . . . . . . . . . . . . . . . . . . . .
THE AUTHORS
115
127
133 141 169 175 209
215
229
Symbol
'\+1 ' '\-1
bk+l , bk _l
C
ck+l ' ck ' ck _l
D
(De)
(Di
)
dk+l , dk
_l
e
E
ek+l , ek
_l
f. k J,
F
h
ht
I
j
k
M. J
M. k J ,
NOMENCLATURE
Typical Units
in-lb
lb-sec/in
lb-in2-sec
in
sec
. 4 1n
in-lb
in-lb
xvii
Definition
Coefficients in stiffness matrix
Coefficients in stiffness matrix
Concentrated applied couple
Coefficient in stiffness matrix
Lumped internal damping factor divided by time increment length
Lumped external viscous damping factor
Lumped internal damping factor
Coefficient in stiffness matrix
Base of natural logarithms
Modulus of elasticity
Coefficient in stiffness matrix
Coefficient in stiffness matrix
Flexural stiffness, E1
Beam increment length
Time increment length
Moment of inertia of cross section
Beam station number
Time station number
Static bending moment at beam station j
Dynamic bending moment at beam station j and time station k
R
Rl
, R2
, . . .
S S R
l, R2
RA
, R B
S~ J
N S. k J,
T
TT
V. J
V. k J ,
W. J
in-lb/rad
lb
lb
lb
lb/in
lb/in
lb
lb
lb
lb
in
Definition
Concentrated applied static transverse load
Concentrated applied dynamic transverse load
Static support reaction at beam station j
Dynamic support reac~ion at beam station j and time station k
Retained projected values of resistance of segments 1, 2, on support curve
Concentrated rotational restraint
Resistances in the sub-springs 1,2, ... of the nonlinear support
Residual resistances in subsprings 1, 2
Resistances at points A and B on support curve
Concentrated transve:cse linear spring restraint at beam station j
Concentrated transve,::se nonlinear spring restraint at beam station j and time station k
Retained values of slopes of segments 1, 2, ... 011 support curve
Static axial thrust
Dynamic axial thrust
Static shear of bar j
Dynamic shear of bar j at time station k
Static transverse de:Election at beam station j
CHAPTER 1. INTRODUCTION
In recent years, a great deal of interest has been focused on the utili
zation of numerical methods to describe computer models of problems in struc
tural dynamics. Many computer programs (for instance, Refs 3 and 10) are
available for solving for the dynamic response of beams or slabs which are
either linearly or nonlinearly supported; no program has been found, however,
which considers the path-dependent history of loading of nonlinearly-inelastic
supports. As a result of the necessity to consider the important effects of
energy lost due to the hysteretic behavior of the supports, much of the present
work has been devoted to the development of multi-element models to simulate
the nonlinearly-inelastic supports. With these models, general rules in FOR
TRAN logic can be observed and used to predict the loading paths of resistance
deflection curves of the supports. The models for the nonlinearly-inelastic
supports described in this work are not time-dependent; therefore, relaxation
(creep) of resistance of the support is not included in the model. The retar
dation, or delayed elasticity, of the support can be considered by installing
an external viscous dashpot in parallel with the support model. The softening
of the nonlinearly-inelastic support is also not included but strain-hardening
may be considered.
Purpose and Scope of Program DBC5
The primary purpose of this investigation is to develop an efficient com
puter program for solving for the transverse dynamic response of a beam-column
resting on linearly or nonlinearly-elastic or nonlinearly-inelastic supports
which are simulated by the proposed multi-element models described in Chapter
3. When the hysteretic behavior of the supports is considered, the program
is able to predict more accurately the dynamic response of beams, piles, slabs,
or even bridges which are supported by soil foundations.
A marching method of solution is used which is based on an implicit
formula introduced by Crank and Nicolson (Ref 2) to solve second-order heat
flow problems. Salani (Ref 10) is credited with applying this implicit
1
2
formula in determining the transverse time-dependent linear deflections of a
beam or plate. Essentially, the beam is replaced by an arbitrary number of
rigid bars and deformable joints, and time is divided into discrete, equal
intervals. The representation readily permits the flexural stiffness, the
elastic restraints, the mass densities, and the applied external loadings to be
discontinuous and lumped at the deformable joints which connect the rigid bars.
The governing partial differential equation at each joint is approximated by
a difference equation that includes several unknown deflections of the joint,
which occur at specified time intervals. All difference equations are based
on the assumptions of linear elasticity and the elementary beam theory. The
effects of transverse shear and rotatory inertia are neglected.
The nonlinear characteristics of the supports are considered by using
either a spring-load iteration process (adjusting both the stiffness and the
load from one iteration to the next, which is known as tangent modulus method)
or a load-iteration technique (adjusting only the load). The iteration pro
cess compares successively computed deflections until a specified tolerance is
satisfied. Only three nonlinear characteristics of support curves are con
sidered: the first is exhibiting the same resistance to either upward to down
ward deflection, hereafter referred to as the symmetric resistance-deflection
curve; the second is allowing the beam to lift off the support when it de
flects upward, hereafter referred to as the negative one-way resistance
deflection curve; the third is allowing the beam to lift off the support when it
deflects downward, hereafter referred to as positive one-way resistance-deflec
tion curve. Two multi-element models to simulate the three types of support
are introduced in this work.
Application
Computer Program DBC5 is versatile and efficient for
(1) solving for the transverse response of beam-columns with linear and nonlinear supports under free or forced vibration;
(2) computing slopes and shears of the bars, bending moments, and support reactions of the deformable joints, statically or dynamically; a~
(3) plotting computed deflections or moments along either the time or beam axis for the requested monitor stations.
Applied forces include static fixed loads, time variant axial thrusts,
and time variant lateral loads. Static solutions of beam-columns under fixed
loads may also be obtained.
The computer program is intended to provide an efficient tool for anal
yzing many problems encountered by highway and foundation designers which are
complicated and unsolvable using classical methods. A variety of highway
structures, such as bridge girders, guard rails, or even a whole bridge floor
under moving loads or suddenly applied impact, can be simulated by the computer
model of Program DBC5 and solved efficiently. The capability to treat supports
that behave nonlinearly and inelastically provides for direct solutions of
transverse deflections of railroad rails under moving loads and the prediction
of the responses of offshore piles to wave forces, as well as the study of
transverse responses of piles to earthquake-induced forces, since the parts
of rails and piles supported by the soil can be reasonably represented by the
proposed multi-element models for considering the path-dependent history of
loading of the soil supports.
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CHAPTER 2. DEVElDPMENT OF THE IMPLICIT OPERATOR
Program DBC5 is developed for the dynamic analysis of beam-columns rest
ing on linearly-elastic or nonlinear supports under time variant axial thrusts
and lateral loads. The program uses a discrete-element model for developing
the equation of motion of beams. An implicit formula of the Crank-Nicolson
(Ref 2) type is then utilized to form a marching operator. At each point in
time, a set of simultaneous equations for the unknown deflections at the de
formable joints can be obtained by systematically applying the marching opera
tor at all joints. A recursive procedure is then used to solve the simultane
ous equations. A brief discussion of the method of the recursive solution
pTocedure is included in Appendix B.
Discrete-Element Model for Dynamic Beam-Columns
Matlock and Taylor have developed a static model composed of rigid bars
and springs (Fig 1) which can be used to simulate a beam-column. In Ref 5
the efficiency of this model has been proved by solving a variety of structures
which can be simulated as beam-columns. By the addition of masses and damping
factors lumped at the deformable joints, a dynamic model (Fig 2) is formed
that can closely simulate the dynamic response of real beam-columns.
Equation of Motion of Beam-Columns
The equation of motion for transverse vibration of beams can be obtained
by summing all the forces, internal and external, at a particular joint j
and a particular time station k (see Fig 3). The concept is based on D'A1em
bert's principle. Thus an equation can be written in terms of the unknown
deflections w, moments M, internal damping factors i D • •• ,
5
6
1 ..... 11------ h -----.. 1
~E h
(a)
h
Deformable
(b)
1------ h -----
\
I Wj_1
Joint NlITIber ... j-I
Bor Number ... .. j+ I
(c)
Fig 1. Mechanical model of beam-column.
w
Beam STA.
, .... j -I
7
Internal Dashpot
Beam Axis -----II ......
Deformable Joint
--..."......-- Lumped Mass
Linear + Nonlinear Spring ----
External Dashpot
Detail of Deformable JOint
Fig 2. Dynamic model of beam-column.
Fig 3. Free-body d' lagram of a porti on of th e dynamic beam-col umn model.
j+1
Tj+2+ TT+2,k
Vj+l,k
N hS. kW ' k J, J,
(2.1)
By the USe of Crank-Nico1son's implicit formula, Eq 2.1 can be further
represented as
where
O.2ShR. 1 J-
(2.2)
h2 .T. + 0 5 TT +
L J • \. j ,k-l
9
f. k J, ==
Detailed derivations of Eqs 2.1 and 2.2 are included in Appendix A. In
Fig 4, an implicit operator of the Crank-Nicolson (Ref 2) type is shown for
Eq 2.2.
11
To start the dynamic solutions, a static model as described in Ref 2 is
used twice, at time stations -2 and -1, for solving the initial deflected
shape of the beam-column due to static loads. Since the implicit operator re
quires the deflected shape at only the two previous ttme stations, there are
no considerations of the initial velocities and initial accelerations of the
deformable joints of the beam. This is consistent with normal practice, since
deformable joints of a beam normally have no initial velocities or initial
accelerations.
All deflections at time station k+1 are unknown. To solve for the un
known deflections at time station k+1 , the operator, which requires the de
flected shapes of the beam at time stations k and k-1 to be known, is ap
plied systematically at beam joints j = -1, 1, 2 ••• M+1. This procedure
establishes a set of simultaneous equations wherein each equation includes
five unknown deflections. These equations are solved by a two-pass, recursive
technique described in Ref 4 (see Appendix B) for the unknown deflection of
every joint. Once the deflected shape of a member at time station k+1 is
known, the slopes, bending moments, shears, and support reactions for the mem
ber at the previous time station k can be determined by using the procedures
described below.
Slope. Equation 2.3 is the simple-difference expression for slope of the
individual bar j in terms of the deflections of joints j-l and j and the
beam increment length h (see Fig l(c».
e . J
== (2.3 )
The concept applied for both the static model and the dynamic model. It
will be seen in results printed from the program that the slope is printed be
tween the beam stations; this slope is that of the bar between the indicated
beam stations.
12
c 0 -0 -CJ)
Q)
E ~
00 I
I I I I I I
K+2
K+I
K
K-I
K-2 I
I I I I I o
4 ht
•
0- - - - - - j-3
ak+1 bk+1 Cktl dk+1 ek+1 I
Ck + f- k j,
+
ak-I bk-I Ck-I dk-I ek-I
~h"'"
j-2 j -I j+1 j+3-- -- --M
Beam Station
Fig 4. Implicit operator of Crank-Nicolson type used in Program DBCS.
13
Bending Moment. In conventional beams, the bending moment is equal to
the product of the flexural stiffness and the curvature. In the finite-element
model the flexibility of the beam and the curvature are lumped at the beam sta
tion points. The corresponding relation for bending moment M in the static
model is
(2.4)
In the dynamic model, the internal damp ling factor lumped at the joint con
tributes its effect in addition to the conventional bending moment. Thus,
M. k J, =
+ 2w j ,k_l - wj+l,k-l + wj-l,k+l
\
- 2wj ,k+l + wj+l,k+l)
Shear. Shear V. in the static model is found fram the equation of J
moment equilibrium of bar j (Fig l(b». Thus,
V. J
= -M. 1 + M.
J- J h (
-w.' l+w, ) .. J - J - T j -----~h -----::.-
(2.5 )
(2.6 )
For the dynamic model, in addition to the effects of conventional bending
moments and static axial thrusts, the moments contributed by the internal
factors (Di )j-l and ~Di)j , and the time dependent axial thrusts
are also found in the equation of moment equilibrium of bar j (Fig 3).
damping T
T. k J, Thus,
V. k J, =
14
~ ') d (W'_l k - 2w, k+ wj + 1 k) + D~ - J 2 Ja z j dt h3
(2.7)
Substituting
and
d (W, 1 k - 2w, k + w'+1 k) J- I h 1 a =
dt h3 1 3 (-W, 1 k 1 + 2w, k 1 2h h J- , - J, -
t
\
-wj+1,k-l + wj-1,k+l - 2wj ,k+l + wj+1,k+l)
into Eq 2.7, thus,
-M, 1 k + M, k ( \ l (-W + W ) V'k = J-, J, -IT,+O.ST: +T: ) .• j-l,k j,k J, h L J J,k-1 J,k+l"';l h
15
\ + W j _ 1 ,k+ 1 - 2w j ,k+ 1 + W j+ 1 ,k+ 1 ) (2.8)
As seen in Fig l(b) (or Fig 3), V. (or V. k ) is the shear throughout J J,
the length of bar j. Therefore Program DBC5 is written such that the shear
computed in each bar is printed between the adjacent beam stations. In the
program, the rotational restraints and the applied couples, which convention
ally are considered to be concentrated at a point, are acting on the beam as
equal and opposite loads separated by two increments, as shown in Fig 5.
Therefore, the shear for only the bars adjacent to beam stations with applied
couples or rotational restraints is affected by these loads and is not the
same as conventional shear.
Support Reaction. As described in Ref 5, the support reaction for the
static model can be obtained by Eq 2.9 or Eq 2.10.
If joint j is supported on a linear spring Ss. h . QS , t e react~on . J J
is
Q~ = J
s S .w.
J J (2.9)
If joint j is supported on a nonyielding support (deflection W. equal J
to zero), the support reaction is
Q~ J
=
Rj _1Wj _2 - Rj_1W j - Rj+1W j + Rj+ 1Wj+2
4h2
TjW j _ l - TjW j - Tj+1Wj + Tj+lwj+ l h (2.10)
16
~d Transverse Couple)
Rj (Rotational Restraint)
(a) Mechanical model.
]-1 ]+1
(b) Equivalent forces.
Fig 5. Rotational resistance R and applied couple C acting on the mechanical model.
17
For the dynamic model, the support reaction can be obtained by Eq 2.11 or
Eq 2.12.
If joint J LS supported on a linear \s~) or
( s N) J S or both
or
S, + S, k ,the support reaction Q, k J J, J,
s S ,w, k
J J,
nonlinear \S~ k) J, spring
is
(2.11)
S where Q, k is the iterative correction reaction of the nonlinear support at
L, time station k •
is
If joint j is supported on a nonyie1ding support, the support reaction
R Q, k J,
T T (Q, k-1 + QJ',k+1\ 1
- l. J, 2) \ \ - 4h2
\
+ Rj+1wj +2 ,k)
( -WJ'-2,k-1 + 2wJ'_1,k_1 - w + w j,k-1 j-2,k+l
\ - 2wj _ l ,k+1 + Wj ,k+1)
18
- wj+1 ,k_1 + wj - 1,k+1 - 2w j ,k+1 + wj+ 1 ,k+1J
+
\
I\w j , k- 1 - 2w j , k + W j , k+ 1 j +
(2.12)
Internal and External Damping
ii', le\ In Eq 2.1, two symbo 1s \p) j and (D) j are introduced to represent
the internal damping coefficients and the external viscous damping coefficients,
which are both lumped at joint j. The rheological models of the two damping ! i\
constants are shown in Fig 6. The typical units of IP)j
1b-in2-sec/sta and 1b-sec/in/sta. The internal damping is
, , i e "
and \D)j are
due to the internal
dynamic viscosity of materials. Boltzmann first proposed the hereditary
theory which attributed the loss of energy, due to internal dmnping, to the
elastic delay by which the deformation lagged behind the applied force.
Coulomb also proposed a viscous theory which assumed that the viscosity
effects are proportional to the first time derivative of strain. The coeffi
cient of proportionality (constant for each material at constant temperature)
is called the coefficient of viscosity. In Ref 13, Volterra has shown a
mathematical relationship between the viscous and hereditary damping theories.
The relationship of stress 0 to strain E: for the material based on the
hereditary theory can be assumed as
Deformable Joint j-I Deformable Joint j+ I
Bending ....... ·.,. ... '0 ...... Lumped Mass
(a)
Deformable Joint
s n 5'+5',1< j j
(b)
Fig 6. Rheological models of internal and external damping coefficients.
19
20
o =
where
Pn =
co E E: + I: Pn
n=l
n-l d E:
n-l dt
(_l)n-l J To 1 - - 'T
n - ¢('T)d'T (n-l) !
o
(2.13 )
'T is an instant of time between 0 and To, To is the period of heredity
(¢('T) = 0 for 'T > To), and ¢('T) is the memory function which can be found
from the experimental data.
to
If the coefficients Pn with n > 1 can be neglected, Eq 2.13 reduces
o EE: + dE: z.,-
dt (2.14 )
where z., = Pl. Equation 2.14 is the stress-strain relationship of the mater
ial based on viscous theory. Therefore, viscous theory is included in here
ditary theory.
In the rheological model shown in Fig 6(a), the internal damping coef-. \ ficients i Dl. i are re lated to the first time derivative of curvature. Thus
\ /j
M. J
= (2.15)
gives the relationship between the moment M. and the curvature (jJ. • J J
Equation 2.15 is used in Program DBC5 for calculating the bending moments
contributed by the flexural stiffness and the internal damping coefficient at
the deformable joint j.
It will be shown in problem 4 described in Chapter 5 that the internal
damping factors have little effect on the solutions of lower frequencies of a
vibrating steel beam. The vibrations at higher frequencies, however, are
damped out with time due to the effects of internal damping and rapid changes
of curvatures.
21
The external viscous damping force is defined as few. k) = _(De) .w. k J, J J,
The constant De (in tons per unit velocity, lb-sec/inch, or kip-sec/ft) is
called the coefficient of viscous damping. This type of damping occurs for
small velocities in lubricated sliding surfaces, dashpots, and hydraulic shock
absorbers. The so-called viscous resistances are produced by the slow motion
of immersed bodies in fluid, either liquid or gas. The value of the coeffi
cient (De). depends essentially on the nature of the fluid, as well as on J
the form and the dimensions of the immersed body.
With these two types of damping factors, the Program DBC5 can solve the
problems characterized by the so-called visco-elastic nonlinearity to a great
extent.
L~nearly-Elastic and Nonlinearly-Inelastic Supports
In the discrete-element model, lateral supports of various types can be
represented by either an equivalent linearly-elastic spring or a nonlinearly
inelastic spring at each of the beam stations. For example, if the beam is
supported by columns or piers, the axial stiffnesses of the columns or piers
can be approximately estimated and included as equivalent linear spring con
stants. For more reality, the true behavior of many lateral supports can be
better represented by the nonlinear characteristics of the axial resistance
deflection curves of columns, or piers, and soil supports.
If the beam is laterally supported by a soil foundation, better results
are obtained by using the nonlinear characteristics of the resistance-deflec
tion curves of the soil than by using the equivalent linear spring constants
of the soil resistances, since these characteristics represent the true be
havior of the soil supports. In Program DBC5, the nonlinear characteristics
of each lateral support are described by a curve consisting of straight line
segments. Only three types of the nonlinear characteristics of the lateral
supports, symmetric, negative one-way, and positive one-way, are considered
in this work.
The symmetrical nonlinear resistance-deflection curve is shown in Fig 7.
The force developed against the beam-column model by the supports is plotted
on the vertical axis and the model deflection is plotted on the horizontal
axis. For both load and deflection, the positive sense is upward. The posi
tive and negative one-way resistance-deflection curves are shown in Fig 8.
22
Resistance
I~
0,
Deflection
-04 -03 -02
-0 ,
9 ~1 W2 .. I .J WI
F 7. Symmetric resistance-deflection curve.
2
Resistance to Downward Deflection _-P\
Resistance
No Resistance to Lift-off 6 Deflection
(a) Negative one-way resistance-deflection curve.
No Resistance to Lift-off
Resistance
"'1 Deflection
I
,---- Resistance to I Upward I
Deflection I I I
5 16
(b) Positive one-way resistance-deflection curve.
F 8. One-way resistance-deflection curves.
23
24
For the negative one-way support, resistance to deflection is developed only
when deformable joints deflect in the negative or downward direction; for the
positive one-way support, resistance to deflection is developed only when de
formable joints deflect in the positive or upward direction. In order to be
consistent with the multi-element model (see Chapter 3) used to simulate the
nonlinear resistance-deflection curve of the supports, several limitations
are placed on the nonlinear characterization:
(1) The curve must pass through the origin.
(2) The curve must be continuously concave as viewed fr~n the horizontal axis, except that horizontal lines of zero stiffness can be input for representing ideally plastic behavior.
(3) No softening of supports is permitted; that is, no reversal of slope sign of the segments on the support curve is p,ermitted.
Stability of the Implicit Operator
Reference 10 has shown that when a uniform beam with well-defined boundary
conditions under free vibration is analyzed using the implicit operator of the
Crank-Nicolson form, the solution is stable for all positive v,a1ues of EI ,
p, h, and ht. , . \
If the same beam, with internal damping (Dl.)j lumped at joints, is
under free vibration, the quadratic equation for evaluation of the stability
criterion becomes
where
F ;;;
D :::::
q2 :::::
if; r 2 2 , 9 e , 2 2 J l_ 4(F+D)(cos ~ -1) + q
r 4(F-D) (cos +1
L 4(F+D)(cos
EI
(D i ) ./ht J
2 Ph3 /h2 t
n
o (2.16)
The derivation of Eq 2.16 is included in Appendix C.
) ~ '- .)
From Eq 2.16, the condition for bounded solutions as time approaches in
finity becomes
2 2 2 2 l6(F -D )(cos ~ -1) + 8Fq > 0
n (2.17)
The above inequality is true, since D is usually less than F for a reasona~'
ble time increment length ht
, and the positive value of the term 8Fq2 is
1 th h . 1 f h l6(F2 - D2) (cos p, _1)2 a ways greater an t e negatlve va ue 0 t e term ~ n
when ht
is extremely small.
For more complicated cases, such as a beam with internal damping, non
linear spring supports, and rotational restraints under forced vibration, the
analytical proofs for the stability of the implicit operator are not feasible.
However, stable numerical solutions have been obtained for most complex prac
tical problems that are physically stable. For beams with nonlinear supports
under forced vibration, cautious selection of a reasonably small time increment
length (for instance, less than 1/10 of the fundamental period) must be made,
since the basic assumption of the Crank-Nicolson implicit formula is that the
time-dependent forcing function and the time variant nonlinear springs are
smoothly varied with time. Extremely small time increment length is not
possible for most practical problems due to the present limitation of a maxi
mum of 1000 time stations provided in Program DBC5.
!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!
44!5"6!7$1*'*0!8$($.$9'.$/-!")':!
CHAPTER 3. RESISTANCE-DEFLECTION CURVES FOR THE lATERAL SUPPORTS
Three types of lateral supports are considered in this work: linearly
elastic, nonlinearly-elastic, and nonlinearly-inelastic. Chapter 2 discusses
how the nonlinear support characteristics can be reasonably described by non
linear resistance-deflection curves which are approximated by a series of
straight-line segments. This chapter introduces two multi-element models each
of which is made up of several sub-elements that can be used to simulate the
nonlinearly-inelastic characteristics of either a symmetric or a one-way re
sistance-deflection curve. A short discussion of the Baushinger effect (Refs 9
and 11) as it relates to the formation of the loading paths of nonlinearly
inelastic resistance-deflection curves is included. Finally, a systematic
approach to tracing the loading paths of nonlinearly-inelastic resistance
deflection curves in dynamic problems is explained.
Multi-Element Models
A multi-element model consisting of several parallel sub-elements for
simulating the symmetric resistance-deflection curve is shown in Fig 9(a).
Each sub-element has a linear spring connected in series with a Coulomb fric
tion block. The sub-elements are assumed to behave as perfectly elastic
plastic and their resistance-deflection curves are shown in Fig 9(c). We see
that it is possible to represent a symmetric resistance-deflection curve with
a number of perfectly elastic-plastic simple elements in parallel. Buckling
and fracture phenomena are not considered in this work. Now let us study the
behavior of the sub-elements. If the multi-element model is starting to de
form downward, all of the four sub-springs will deform elastically and the
resistance will be built up to point A, as shown in Fig 9(b).
At this point the resistance force in the sub-spring Sl is just equal to
the friction force created at the sliding surface of the Coulomb friction
block. Beyond the point A the entire sub-element 1 will slide and no resist
ance, therefore, will be taken by this sub-element. If the model continues to
deform downward the resistance will be taken by sub-springs S Sand S 2' 3' 4
27
28
Multi - Element Model K'!!r-w <1>><
IDe::(
Force . SI ~ Sub-springs
Gravity
Fixed Datum Plane - Joint of
Beam - Column
Fric tion Ellocks E D S 1 Resistance Sliding Surfaces I I I I I I I
~c I S3+S4 I B I S2+ S3+S4
I I A I I I I I I SI +S2+S3 +S4
I I Deflection
D' E'
Procedure of Constructing the Above Curve From (c)
1. Use slope SI + S2 + S) + S4 and deflection We to construct pOints A and A';
2. Use slope S2 + S) + S4 and deflection Wb to construct points Band B';
3. Use slope S) + S4 and deflection We to construct pOints C and C';
4. Use slope S4 and deflection Wd to construct points D and D';
5. Extend from pOint D to point E and from point D' to point E' by using zero slope and deflection We'
(b)
(a)
E I
Lwe I I
+
+
+
Resistance
A'
Deflection •
E'
Deflection •
,'-----B'
Deflection
~"'----C'
RE!sistance
Deflection ~
~ D'
(c)
Fig 9. Multi-element model of symmetric nonlinear resistance-deflection curve of the support.
29
up to point B. At point B the sub-element 2 will start to slide. Beyond the
point B the resistance will be taken by sub-springs 53 and 54 up to point C.
At point C the sub-element 3 will also start to slide. Beyond point C the
resistance will be taken only by sub-spring 54 up to point D. Finally at
point D the entire support will start to yield plastically and offer no more
resistance to the deflection. Based on the above assumption, a continuously
straight-line-segment curve which represents the nonlinear characteristics of
the support can be easily obtained by the summation of all ideally elastic
plastic resistance-deflection curves of the sub-elements. The procedure of
constructing such a curve is shown in Fig 9(b). Theoretically, the more seg
ments there are in a curve, the smoother the representation of the nonlinear
characteristics will be. Practically, however, a symmetric curve that consists
of a maximum of 19 straight-line segments (9 sub-elements) is suitable to rep
resent the nonlinear characteristics.
A multi-element model, as shown in Fig 10(a), is used to simulate the
negative one-way resistance-deflection curve. The model is similar to the one
used in the symmetric case except that there is no connection between the
springs and the friction blocks. The spring-blocks work the same way as the
ones used in the symmetric case when the deflection is downward (negative).
While the deflection is upward (positive), the springs lift off the friction
blocks and therefore no resistances can be built up in the sub-springs. Once
the sub-springs are contacting the friction blocks again, the resistances in
the sub-springs will again begin to build up and the spring-blocks act exactly
as if the model is deflecting downward. The procedure of constructing a nega
tive one-way curve from the individual sub-elements is shown in Fig 10(b).
The same multi-element model as shown in Fig 10(a) can be used to simu
late the positive one-way resistance-deflection curve. In this case, the
support model is to the right of (or above) the joint of beam-column; there
fore, the sub-springs disconnect from the friction blocks and exhibit no re
sistances while the deflection is downward (negative).
Baushinger Effect of Nonlinear Resistance-Deflection Curve
If a specimen of mild steel is subjected to compression after a previous
loading in tension, the applied compression stress, combined with the residual
stress induced by the preceding tensile test, will produce yielding in the
most unfavorably oriented crystals before the average compressive stress
')0
E
~'~r---W Multi - Elemenf Model ~~ Gravity Force .
S, ",r;;:- Sub- spnngs
Fixed Datum Plane ~~~
Sliding Surfaces
(a)
Deflection
Procedure of Constructing the Above Curve From (c2
1. Use slope 8 1 + 82 + 83 + 8 4 and deflection Wa to construct pOint A;
2. Use slope + 83 + 84 and deflection Wb to construct pOint B;
3. Use slope + 84 and deflection We to construct point C;
4. Use slope 84 and deflection Wd to construc t point D;
5. Extend from point D to point E and from point 0 to point E' by using zero slope and deflection We'
(b)
E
C
r- Joinf of I Beam - Column
II--~//Il_-
C(lntact Knobs
Friction Blocks
+
+
+
Resistance
Deflection ... E(
I I
I I
Deflection :
'" E' , I
I Resistance I
I I
f . I
De lectlonl If'
E' I I I I I
Resistance I I
(c)
I I
Deflectio{
E'
Fig 10. }iulti-element model of one-way resistance-deflection curve of the support.
31
reaches the value at which slip bands would be produced in a specimen in its
original state. Thus, the tensile test cycle raises the elastic limit in ten
sion, but lowers the elastic limit in compression. This phenomen is called
the Baushinger effect (Refs 9 and 11).
From the multi-element models shown in Fig 9(a) and Fig 10(a), we can
see that each individual sub-element is allowed to slide during the loading
unloading process. As a result of sliding or plastic deformation produced on
loading, a system of residual resistances is introduced on unloading. Obvious
ly, these residual resistances will influence the deformation produced by sub
sequent loads. Now let us study the so-called Baushinger effect based on the
multi-element models shown in Fig 9(a) and Fig 10(a). For simplicity, con
sider a single spring support which has a nonlinear symmetric resistance
deflection curve with only three segments of resistances as shown in Fig ll(b).
The nonlinear spring can be constructed by a multi-element model consisting
of only two sub-elements each with an ideally elastic-plastic resistance
deflection curve as shown in Fig ll(c) and Fig ll(d). If a vertical load Ql is applied downward to the spring, it will produce resistances Rl and R2
in the sub-springs Sl and S2 ,respectively. Thus,
Rl = QlSl
(Sl+S2) (3.1)
R2 = QlS2
(S 1+S2) (3.2 )
By gradually increasing the load Ql we reach the condition at which the
friction block in the sub-element 1 starts to slide while the sub-spring S2
continues to deform elastically. This condition corresponds to point A in
Fig lIb. If at point A the deflection is equal to a value wI' then the
corresponding resistance RA is found from the equation
= (3.3 )
32
(<l )
(b)
(c)
(d)
Sub-element I
S,
Resistance
I I I I I I I
_L Rs
-r
I I I
ResIstance I ' I I
Sliding Surface
Normal Forces
Coulomb Fricti()n Block
I i Deflection --~~~--~~~~--~+-~
Sub-element 2
i I l I
:+ I I
R. I
eSlstancel
Rs 2 Deflection
Fig 11. Baushinger effect of symmetric non linear resistancedeflection curve of the support.
33
If we continue to increase the load, the sub-element 1 will slide and exhibit
no resistance, and the additional resistance RB will be taken by the sub
spring S2 • The additional resistance ~ corresponding to the deflection
will be
(3.4)
The relation between ~ and is shown in Fig 11(b) by the inclined line
AB. If we begin unloading the model after reaching point B on the diagram,
both sub-springs Sl and S2 will behave elastically and the relation between
the removed resistance and the upward displacement will be given by Eq 3.3.
In the diagram we therefore obtain the line BC parallel to OA, and the
total vertical displacement upward during unloading is
(3.5 )
Substituting Eq 3.3 into Eq 3.5 yields
(3.6)
From Eq 3.4 we obtain
(3.7)
Comparing Eqs 3.6 and 3.7, we find that w2 is greater than w3 • We see
that because of the sliding of sub-element 1, the model does not return to
its initial state and the permanent set OC is produced. The magnitude of
the permanent set is found from the equation
OC (3.8)
34
Because of this permanent set there will be positive residual resistance
will be balanced by the negative
8 1 • The magni tudE~s 0 f R~ and
and Fig 11 (d), respeGtive1y.
in the sub-spring 82 ' and this resistance
residual resistance R~ in the sub-spring
R~ can be shown graphically in Fig 11(c)
To show the Baushinger effect let us consider a second cyG1e of loading.
At small values of load both sub-springs deform elastically, and in Fig ll(b)
the second loading process will begin at point C and proceed along the straight
line CB. When we reach point D the sub-spring 81
will be relieved of the
residual negative resistance R~ and during further loading will have positive
resistance. At point B the positive resistance in the sub-sprtng 81
will
equal the friction force created by the friction block, and sliding of the
sub-element 1 will begin. It is seen that the sliding resistance of the sub
element 1 is raised because of the residual resistances. The Bliding resist
ance was at point A with resistance RA in the first cycle and at point B
with resistance RA + ~ in the second cycle. The process of unloading in
the second cycle is perfectly elastic and follows the line BC •
Let uS now reverse the direction of the force and apply an upward load
Q1. The deformation will then proceed along the straight lin(~ CE, which is
a continuation of line BC • At point E the total negative resistance is
RA - RB ' and the negative resistance in the sub-spring 81
is equal to the
friction force created by the friction block. As the load is increased, the
deformation proceeds along the line EF. If the model is unloaded after
reaching point F. it will return to the initial state represented by point
O. It is seen that by loading the model downward to point B, the positive
sliding resistance in the sub-spring 81 was increased from liA to RA + ~ •
At the same time the negative sliding resistance in the sub-sp:ring Sl was
diminished from to R -A ~. This illustrates the Baushinger effect.
The area of the parallelogram OABCEFO gives the amount of me l:;hanica1 energy
lost per cycle. The same concept discussed above can be expanded to a model
consisting of any number of sub-elements.
For a negative one-way resistance-deflection curve as sh01. ... n in Fig 12b,
the Baushinger effect for this model can be illustrated by the loading paths
indicated by the parallelogram OABCO and the area defined by OAB'DEFO •
Notice that unless the model reaches the condition at which both sub-elements
are sliding during the loading process (point D) there will be no permanent
set after unloading from point B in the diagram. This is because the sub-spring
(a)
(b)
Sub -element I
(c)
Sub-element 2
(d)
Fig 12.
SN -Contact Knobs
.... 1ff4Ai
Normal Forces
Coulomb Friction Block Sliding Surface
t Resistance
5 1+52
0 Deflection I !
II
Resistance
-~~-......
Deflection
, t Resistance
521 I Def lection
Baushinger effect of negative one-way resistancedeflection curve of the support.
35
36
Sl and the friction block are disconnected after reaching the point C in the
diagram, and hence no residual resistance can be built up in the sub-spring
Sl. hTith further unloading the deformation proceeds along the line CO and the
model will return to the initial state represented by point O. At this point,
if we reverse the direction of load, the deformation will proceed along the
new curve line OCBB'D. This is because the sub-spring S 1 lifts off the
friction block before reaching the point C.
It is seen that because the sub-springs are able to lift off the friction
blocks, no negative resistances can be built up in the sub-springs and there
fore the loading paths can generate only on the negative side of deflection.
The above concept also can be expanded to the model which has more than two
sub-elements. The Baushinger effect of a positive one-way resistance-deflection
curve is similar to a negative one-way resistance-deflection curve, except
that the loading paths can generate only on the positive side of the deflection.
A general procedure for tracing the loading paths of either a synnnetric
nonlinear or a one-way resistance-deflection curve of any support which has
more than two sub-elements will be discussed below.
Systematic Approach for Following Loading Paths
In Figs 9(c) and 10(c), we see that the multi-element models for simu
lating the nonlinearly-inelastic supports can be represented by a number of
ideally elastic-plastic sub-elements. Although it is possible to retain the
ideally e lastic-p lastic res istance-deflection curves of each i::l.dividua 1 sub
element in the program, a more efficient and easier way is to :retain the over
all curve which is obtained by the summation of all ideally elastic-plastic
resistance-deflection curves of the sub-elements as shown in Figs 9(b) and
10 (b) •
In Program DBCS, nonlinear supports can be described for .'lny beam station.
The nonlinear resistance-deflection curve of the support is described by a
simple tabular input which generates a curve of straight line segments with a
series of point numbers. After defining the resistance-deflection curve by a
series of points, which are numbered in order, the resistance and deflection
of each point must be retained sequentially in the variables of Q and W ,
respectively. The retained values of the deflection w must be increasing
positively, while that of the resistance Q must be decreasing or equal for
the consecutive points.
A typical symmetric nonlinear resistance-deflection curve is shown in
Fig l3a.
37
In order to be able to systematically generate the loading paths of the
curve during the loading-unloading process, the slope of each segment, the
projected horizontal (deflection) value and the projected vertical (resistance)
value of each segment must be retained sequentially in the variables of S ,
6w , and 6Q ,correspondingly. Notice that the retained values of the varia
ble S must be in descending order.
Now let us study the procedure of constructing the new path of the origi
nal resistance-deflection curve shown in Fig l3(a), when the deformation of
the support starts to reverse its direction at point A as shown in Fig l3(b).'
The systematic procedure consists of (1) drawing line AB parallel to segment
4-5 so that the projected value of line AB on the horizontal axis is equal T
to 6W l and the projected value on the vertical axis is equal to T
6Q l (Z) drawing line BC parallel to segment 3-4 so that the projected value
line BC the horizontal axis is equal T and the projected value on to 6wZ T the vertical axis is equal to 6QZ , (3) connecting point C to point 7 on
of
on
the original curve, and (4) renumbering the new path of the resistance-deflec-
tion curve as 1, Z, 7 as shown in Fig l3(b).
We see that the number of points on the new curve has been reduced to 7
from 8, which is the previous number of points on the original curve. If the
deformation of the support again starts to reverse its direction at point A',
shown in Fig l3(b), another new path of the resistance-deflection curve, num
bered 1, 2, ••• 6, will be generated by a procedure similar to that described
above, except that three lines, A'B' , B'C' ,and C'D' , are drawn before
the new curve connects to the previous curve.
For a typical negative one-way resistance-deflection curve, as shown in
Fig l4(a) , the logic of tracing two new curves can be shown graphically in
Fig l4(b). The procedure is similar to the one used in the symmetric case.
It is seen that loading paths are generated only by downward (negative) de
flections and positive resistances. We also see that the second adjusted
curve is generated when the deformation of the support, after going back to
the initial state at the end of the first loop, continues to proceed along the
line 5CBA2A' and starts to reverse its direction at point A'. The deforma
tion at point A' is greater than the deformation at point A, where the first
Jh
-10 I
(a)
(b)
.~ Slope No.
I 2 3 4
6wT
S (Total)
3000 0.2 2000 0.2
16~7 0.6 0.6
Fig 13. Logic of tracing hysteresis loops for symmetric nonlinear resistance-deflection curve.
6QT I I
(Total) Segment
600 4-5 400 3-4,5-6 400
I
2-3,6-7 0 1-2,7-8
2
10 as
Resi stance (Q)
IOxl02 Slope S b.wT
No. notal)
I 3000 01 2 1500 0.2 3 667 03 4 0 1.0
r Detlection (W) I I I
39
b.OT Segment (Totol)
300 4-5 :DO 3-4 200 2-3
0 i 1-2,5-6
101--- l:J. W4 = 0.7 ----.I
(a)
Resistance ( Q )
tJ.. 2
- 1.0 1-05 I I I . I I I I I I I I
::!~ I
(b)
0---0--0 Ori gi na I Cu rve (1,2,----6)
• • • First Adjusted Curve (OJ, f2l ,--- [Z] )
~ Second Adjust~ Curve (&, £ ,---~)
Deflection ( W)
Fig 14. Logic of tracing hysteresis loops a negative one-way resistance-deflection curve.
40
adjusted curve started. The logic of tracing the loading paths of a positive
one-way resistance-deflection curve is similar to the one used in a negative
one-way resistance-deflection curve, except that the loading p,aths are on the
positive side of the deflection.
A detailed flow chart re lating to the FORTRAN logic, whidl is used in
the program, of tracing the loading paths of the nonlinearly-inelastic resist
ance-deflection curves of the support will be found in Appendix F.
CHAPTER 4. DESCRIPTION OF PROGRAM DBC5
General
Program DBC5 is written in FORTRAN language for Control Data Corporation
(CDC) 6600 computers. With minor changes, the program would be compatible
with IBM 7090 computers and with other systems. Four available FORTRAN sub-
routines are included in the program:
(1) INTERP3 (Ref 5) for interpolating the input data,
(2) TICTOC (Ref 5) for evaluating the computation time,
(3) SPLOT9 for plotting the results by using the printer plotting method, and
(4) ZOTI for plotting the results by using the microfilm or ball-point paper plotting method.
The program uses in-core storage which requires 58,500 words. Although
the use of auxiliary tapes will reduce the storage, it raises the computation
time to a great extent due to tape operations; therefore, the in-core solution
method has been chosen rather than using many auxiliary tapes. A summary flow
chart of the program is presented in Fig 15. The definitions of symbols used
in the program and a listing of the program with several general flow charts
are included in Appendix E and Appendix F. Subroutines SPLOT9 and ZOTI were
developed particularly for the highway research projects at The University of
Texas at Austin.
Procedure for Data Input
A guide for data input is included in Appendix D. The guide is designed
so that additional copies may be furnished as separately bound extracts for
routine use. A parallel study of the guide will help the reader to understand
the following discussion.
Any number of problems may be stacked and run together. The sequence of
problems is preceded by two cards which describe the run. The first card of
each problem contains the problem number and a brief description of the problem
41
42
I START I
I IREAD & PRINT Input Data I
I Advance time stepl
I Dynamic load for time stepl
\.
Iterate with spring and load I Form y
coefficient matrixl (or load only) for closure I at each time step lcomput~ deflection~
recurs~on procedure
N~ Closure?
Yes
Support curve Yes adjustment necessary?
I No I Compute slopes, moments, shears, reactions for previous time step
support ]
I I Reta in informa tion for plots I
I I PRINT iterational monitor deflections I
and computed results
I Return for new time step'
I I Plot results if requested'
I ISTOP I
Fig 15. Summary flow chart of DBC5 program.
The program continues working problems until a blank problem number or an
error is encountered; then the run is terminated.
Tables of Input Data
43
Table 1 is the data-control table. It consists of a single card which
must be input for each problem. The number of cards in the remaining tables
and the hold options of these tables are specified in this table. The hold
options for the various tables are independent of each other, but care must
be exercised to insure that the data in the various tables are compatible.
For example, if a previous 40-increment problem had a distributed load from
station 0 to station 40, then the user should not change the total number of
increments in the new problem to some number less than 40 unless he erases
the loads pas t the end of the new beam.
Table 2 contains
(1) number of increments into which the beam-column is divided,
(2) beam increment length,
(3) number of increments into which the time axis is divided,
(4) time increment length,
(5) printing option for the computed results,
(6) number of monitor beam stations which are requested for printing the computed results,
(7) iteration switch to show whether or not the problem has nonlinear supports,
(8) number of monitor beam stations which are requested for printing the iteration data,
(9) option to choose the method of plotting the results, and
(10) option of using lines or points for the plots.
If the number of time increments is 0, the problem is considered to be a
static case and the complete results of the static solution will be printed
at every beam station. In this case, two kinds of plots, which have either
deflection or moments for the vertical axis and beam stations for the hori
zontal axis, may be requested by inputing "0" for the time station in Table 9.
For any station in the beam, a deflection, slope, or both, in either the
initial or the permanent condition, may be specified in Table 3. If an initial
condition of deflection, slope, or both is specified, it will be disregarded
44
after the dynamic solutions are started. A specified deflection is equivalent
to a single lateral support that is very stiff and is unable to deform freely.
A fixed-end support can be simulated in the computer by the specifica
tion of a permanent zero deflection and a permanent zero slope at the same
beam station. The ability to specify a permanent deflection other than zero
provides the user with a simple method of studying problems in which the sup
ports settle. The ability to specify an initial deflection other than zero,
on the other hand, provides the user with capability to study dynamic problems
with initial displacements. Due to the method of simulation (see Fig 5), the
specified slopes and deflections must conform to the following minimum spacing
requirements:
(1) A slope may not be specified closer than three increments from another specified slope.
(2) A deflection may not be specified closer than two increments from a specified slope, except that both a slope and deflection may be specified at the same beam station.
Slope and deflection conditions may be specified at no more than 20 beam
stations. Each specification requires a separate card. The c.ards may be
stacked in any order within the table.
Fixed loads and restraints are described in Table 4. The input values
are beam stiffness, fixed static loads, linear support springs, static axial
thrusts, rotational restraints, and applied couples. Couples and rotational
restraints appear only as concentrated effects, while axial trrusts are usually
distributed. The remaining values may be either concentrated or distributed.
The method used for the description of distributed data is illustrated in
Appendix D. All of the input values of Table 4 are accumulated algebraically
in storage. Therefore, there are no restrictions on the order of the cards,
except that within a distribution sequence, the beam stations must be in
ascending order. Axial thrusts must be described in the same manner as other
values in this table because there is no provision in the program for auto
matically distributing the internal effects of an externally applied axial
thrust. For example, an axial thrust applied to the ends of the beam-column
must be specified as either an axial tension or an axial compression at each
interior beam station. The number of cards accumulated in Table 4 cannot
exceed 100.
45
Table 5 is used to describe the lumped mass densities, the lumped internal
damping [actors, and the lumped external damping factors ~t beam stations.
Data in this table are input the same as in Table 4. The maximum allowable
number of cards accumulated in this table is 100.
Time-dependent axial thrusts are input in Table 6 as values applying to
designated bars. The values of time-dependent axial thrusts specified in this
table may vary linearly both in the beam axis direction and in the time axis
direction. There are no restrictions on the order of cards, except that beam
stations which are stated as the starting station and the ending station in
the same card must be in ascending order. A concentrated effect of time
dependent axial thrust at a beam station is not permitted to be stated in this
table since it would have no physical meaning.
The method used for the description of distributed data is also illus
trated in Appendix D. The number of cards accumulated in this table cannot
exceed 100.
Table 7 is used to describe time-dependent lateral loadings applied at
the station points. The variation and the distribution of the input data are
the same as in Table 6, except that the concentrated effect of the lateral
load at a single station is allowed in this table. The number of cards accu
mulated in this table cannot exceed 100.
Nonlinear resistance-deflection curves of the supporting beam stations
are input in Table 8. Every particular nonlinear support curve is described
by three consecutive cards. The values input on the first card are resistance
multiplier, deflection-multiplier, number of pOints, symmetry option, deflection
tolerance, and resistance-tolerance. The values input on the second card are
resistance values which are multiplied by the resistance-multiplier to obtain
the final resistances of each point on the curve. The values input on the
third card are deflection values which are multiplied by the deflection
multiplier to obtain the final deflections of each point on the curve.
There are no restrictions on the order of support curves, except that
within any distribution sequence, the beam stations must be in regular order.
More than one curve may be placed at a beam station. The maximum number of
support curves cannot exceed 20 (60 cards).
Four kinds of plots may be requested by inputing the name of the hori
zontal axis, the name of the vertical axis, the particular beam or time sta
tion, and the multiple plot switch in Table 9. As many as five of the same
46
kinds of plots, each of which is described by one separate card, may be
superimposed by using the multiple plot switch. There are no restrictions
on the order of cards, except that beam or time stations must be in regular
order within a sequence of the same kind of plots. Superimposed plots must
all be of the same kind. The maximum number of cards input in this table can
not exceed 10.
Three kinds of plotting methods, namely printer plots, mierofilm plots,
and ball-point paper plots, are built into the program. The s,.itch which
tells the program to choose one of the above three methods for plots is input
in Table 2. Various plot options are illustrated in example problems described
in Chapter 5.
Error Messages
Program DBC5 provides many checks for common types of data errors as well
as for particular errors which occur due to either violations of FORTRAN logic
or solution errors considered by the program.
Conditions which are considered to be of the type of data errors are
(1) the allowable number of cards for an input table is exceeded;
(2) a slope or deflection is specified too close to another slope or deflection;
(3) data are input at stations beyond the end of the bear~-column;
(4) the station numbers in a distribution sequence are out of order;
(5) the symmetry option is not equal to 1 when only one point is input on a support curve;
(6) the final values of deflection and resistance, which are input for defining the points on a support curve, are improperly specified;
(7) the allowable number of nonlinearly supported beam s1:ations is exceeded;
(8) the number of cards input in Table 8 is not zero when the problem is a linear case; and
(9) repeated data are input in Table 9.
The conditions which are considered solution errors are
(1) calculated displacements exceed maximum allowable deflection specified by the user,
(2) calculated displacements exceed limits of support curves during the iteration process, and
(3) solution does not close within the specified number of iterations.
An error message which defines the error will be printed if any of the
above conditions is encountered.
47
In addition to the specific error messages, a general purpose error mes
sage is provided for a number of unlikely errors. If the message lUNDESIGNATED
ERROR STOP I1 is printed, the user must investigate the program to determine
what caused the error. Any error detected by the program will cause the en
tire run to be abandoned.
Description of Problem Results
The output of results is arranged so that the input quantities of Tables 1
through 9 are available for all problems and are printed with explanatory head
ings.
Table 10 presents the calculated results which are tabulated according to
the following order:
(1) monitor deflections during the iteration process of the static solutions (none if the problem is a linear case or if the number of monitor beam stations for printing the iteration data is zero),
(2) complete results of the static solutions,
(3) monitor deflections during the iteration process of the dynamic solutions at time station 0 (see explanation in the parentheses of procedure 1,
(4) complete results of the dynamic solutions at time station 0,
(5) monitor deflections during the iteration process of the dynamic solutions at this time station (see explanation in the parentheses of procedure 1),
(6) monitor or complete results of the dynamic solutions at this time station, and
(7) repeat outputs of procedures 5 and 6 until the last time station specified in this problem is encountered.
The proper headings are printed at the top of the outputs of each pro
cedure described above.
If printout of the calculated results are not requested, the message
I1PRINTING RESULTS IS NOT REQUESTED I1 will be printed under the headings of
Table 10.
48
Plots of Results
Options are available which allow the user to obtain plot~; of the
deflections or moments along either beam or time axis. Three kinds of plot
ting methods, namely printer plots, mircofilm plots, and ball-point plots,
are available in the program. Both microfilm and ball-point p~ots fit in an
8-inch by 9-inch space. The optimum scales for the plots, both horizontal and
vertical, are automatically selected by the program. Two fictitious points,
one with the minimum horizontal and vertical values of all the points to be
plotted on a single frame and the other with the maximum horizontal and verti
cal values, are plotted on the microfilm or ball-point plots for the purpose
of choosing the proper vertical scale. These two fictitious pJints are not in
the stored values of the points to be plotted on this frame and, therefore,
should be disregarded. An example plot of the deflections alo'~g the time axis
for inelastically supported mass under free vibration is shown in Fig 16. The
plot in Fig 16 is made using the ball-point plotting method; t~e problem num
ber and the identification of the plot variables are printed beneath the plot.
The plot variables which identify either a microfilm plot or a ball-point plot
include three sets of characters. The first set to the right of the problem
number is one of the following four symbols: DVT or MVT (deflection or moment
along time axis for a particular beam station), DVB or MVB (deflection or
moment along beam axis for a particular time station). In the center is either
BEAM STA = or TIME STA On the right are the station numbers for which the
deflections or moments are plotted. The plot in Fig 16 is from Example Prob
lem 1, described in Chapter 5. Another example plot, which is plotted using
the microfilm plotting method, is shown in Fig 17. In this example, the
moments along the time axis for the mid-point of a three-span beam on which
an AASHO standard 2-D truck is moving with a speed of 60 mph is plotted on
the microfilm, which then can be developed on an enlarged magnetic print. The
plot is from Example Problem 3, described in Chapter 5. Many more ball-point
plots which have more than one curve plotted on one frame can be found in
Example Problem 5, described in Chapter 5.
A typical example plot which is plotted by using the printer plotting
method is shown in Fig 18. The printer plotting method plots the horizontal
axis vertically on the sheet. The plot is from Example Problem 6, described
in Chapter 5. Different symbols (as many as five) are used on each frame of
CI .. t .!..
+ 1 ~j tV ..... I ::
;~J ~ Deflection
proble~l n', T BE"RI'-, ('Tr' -:: number'" UJ CJ v - . , -' j
versus Time
0- Station number
Fig 16. A typical ball-point paper plot.
Problem number
.. C -
o . o •
•
:1 ~ 20.00
~~Moment versus Ti~ Station number
~ BEA" STA = ~ Fig 17. A typical microfilm plot.
o
VI o
51
PROr;RA~, ntlC"> - MASIER - ,)t\CK (:H,.,I\I - MI\TLOCK _ nF"CK1-REVIStf'l!\j I'IIITE = 20 .1 11"1 71 t:.Xt>Mt'Lf' t'~Ofll.F""'S FOR ppnr:oA'" f)I1Cc, RY JACK C'"4AI\! ,/lINf lq7\ UYn/HIC Kt:AM-rnlll"1N PR()~OI\M ( 5-1-C, YMPLliIT ()PF:~I\TnR 1
PRO~ (CO"ITlJl
b TtiPEf-'3P>I"I 8EAM ·>lIT'" ONF"_WA'I' ANO C:;VMMFTRTC NONl r'-JF"Ao C:;lIP lOAUED ;:Iy T.p -- .... - PlOr OF I)EFlECTIO"1 VC; T!~F" HlP RI='AM <::ThTTI'PIiS OFt ...... _
CURVF ,.) = I{
2 I"'UkVF" (. ) ::;; 11
PT.NO o"Allli;-"
-1 -1.3Ur-lf, • 0 -b.!'>OlF"-17 • 1 -6. bb~F"-O:l • -r ? -1.b13/:-U2 • .or 3 _2.~91F-O? • ·r 4 -4.!:>76F"-IJ? • • T 5 -J.47,+F"-J2 • 0 T 6 -1.71':lC"-v? • -r 7 _e.odbl='_,)4 • 0
g -3.0.31 F" -,) 3 *T • 9 -1.4t1!:>>'-O? or • 11) -1. 73i'lF"-')2 • oy
11 -".0 30"'_)2 • O! 12 -3.5boF"-12 • " T 13 -3.C1'10F"-O? • .. T 14 -3 ... 2"/:-,)2 I> T 1 ') -1.0'1:'/:-02 • or 16 7. b12 F - ,13 ... 17 -3. 9'17F-(; 1 or • lA - 3. 4!:.dF-')2 .. t 19 _3.b 44 ,_)2 • ° Y 20 -2. 72'-fF"-·12 • o! 21 -1.94~E'-1)2 • *T 22 -5. Ut;4F-')3 or • 2 3 i:I.:'O)Fw,)3
° 24 3.0'1":1f.'-03 * • 25 -2. '1jU-p or co - 3. 0 b 'I '" - ,) ;: • or n 1.bbcF-02 • * 28 4.137F"-J? • Til. 29 -1. !:Sd2F-'!2 *r • 30 -H.?'+lF"-O? ° r • 31 -1. 3bl:\F"-I!? °T. 32 6.972F"-02 • T I>
33 4.04<1F"-<J? • T* 34 -h.4!:>UF-v2 • .. T 35 -B.3itt.>F"_)2 (t T • 36 9.59'-1I:-J< * • 37 7.1211'"-q? T " • 38 4.241lF-02 • !* 39 -b. 1 !:>YE-I) 2 • It r 40 -7. St;4F-'"I;:> • * 1
Fig 18. A typical printer plot.
52
the printer plots for identifying different curves superimposed on this frame.
The identifications of the plot variables (such as description of program and
run, problem number and problem description, description of the type of the
plots, and beam stations or time stations) are printed on the cop of each
frame of plots. The numerical values printed vertically to the left of the
plots are those of the first plot on this frame. The scale of the vertical
(deflection or moment) axis is not plotted on the plots, and the horizontal
(beam or time station) axis is scaled equally for each point on the plots and
plotted by the symbols I. A series of numbers corresponding to the beam or
time stations is printed vertically on the left end of the plots for the pur
pose of indexing. The printer plots fit on 8-l/2-inch by ll-i~ch standard
letter-size paper.
CHAPTER 5. EXAMPLE PROBLEMS
In this chapter seven example problems are solved by Program DBCS.
Problems 1 and 2 each have two different cases and Problem 4 has three differ-
ent caSes. Problems 1 and 1A demonstrate the validity of the program in pre
dicting the formation of hysteresis loops on the nonlinear resistance
deflection curve of the spring which supports a freely vibrating mass.
Problems 2-1 and 2-2 present a comparison between the numerical solutions
solved by DBC5 and the theoretical solutions of vibrating beams illustrated
in Ref 12, pages 375 and 378. Problem 3 demonstrates the capability to use
the computed dynamic tire forces of a moving truck from another available com
puter program as input data in Program DBC5 to predict the dynamic responses
of a highway structure. Problem 4 compares the experimental data with the
computed results of a simply supported steel rod under axial pulses. Problem 5
demonstrates the capability to predict the response of off-shore piles to wave
forces. The solution of a three-span beam (30 beam increments) which is
simply supported at the ends and loaded by a transient pulse at station 17 is
illustrated in Problem 6. Finally, Problem 7 demonstrates the capability to
predict the response of a partially embedded pile to earthquake-induced forces.
A listing of input data is included in Appendix G, and the sample computer
output listing is included in Appendix H.
Example Problem 1. Inelastically Supported Mass Under Free Vibration
Figure 19 shows a mass of density 1.0 Ib-sec2/in. supported by a nonlinear
ly inelastic spring; the mass is loaded by a static load of 100 pounds for the
initial deflection and then released by applying a constant dynamic load of
100 pounds. The weight of the mass is neglected. The nonlinear characteris
tics of the resistance-deflection curve of the spring are also shown in Fig 19.
Time increment length is equal to 6.283 X 10-3
second, which is approximately
1/100 of the natural period. The vibration of the mass is solved at 800 time
stations. This first problem is intended to provide confidence in the proposed
multi-element models to predict the loading paths of the nonlinear-inelastic
53
54
Problem 1
density
100 lb
2 1.0 lb-sec lin
-100 lh/each time sta
Number of beam increments 1
SN Beam increment length = 1 inch
Number of time increments = 800
Time increment length ~
-3 6.283 X 10 seconds
Frob lem
Nonlinearly-Inelastic Resistance-Deflection Curve of SN
2 Resistance (Q)
LB ------ 120
60
36
Deflection (W) 10
----~--J\~~----------+--4-4~~~--+---------_+----~~ I .. .8 2
10 I I I I I I , -36 I I
-60 ----- 8
I
I I I I I I I I I I I I I I I I I I I I I I
I I I I I 1 I 1
19 110 -120 --------------~--""'____J,
19. Example Problems 1 and lAo Inelastically s..lpported mass under free vibration.
support. From the computer output, a plot of deflections versus support
reactions can be obtained, as shown in Fig 20.
55
It is seen that due to partial yielding of the support at the beginning,
a hysteresis loop is formed in each cycle of vibration and mechanical energy
is lost in each loop. The mechanical energy lost from point A to point J in
Fig 20 is equal to the area defined by JKEFGHIJ, which is equivalent to the
difference between the summation of the areas defined by ALDCBA and DEFPD and
the summation of the areas defined by FPNHGF and JMNIJ. Therefore, the de
flections are damping out with time until the response finally becomes a free
elastic vibration within the range of greatest stiffness. This can be proved
from the tabular output of the program when large time increment lengths -2 (6.283 X 10 second) are used. The computer plot of deflections versus time
is shown in Fig 21.
Example Problem lAo Free Vibration of Mass by Specifying Initial Displacements
An additional run on the preceding problem was made by specifying an
initial displacement of 3 inches instead of applying a static load for the
initial deflection. From the plot of support reactions versus deflections
(Fig 22), it is seen that a permanent set is obtained because of initial
yielding of the support. The mass is finally freely vibrated at a position
about a mean permanent upward deflection of approximately 1 inch. This can be
shown in the computer plot of deflections versus time in Fig 23.
Example Problem 2-1. Lateral Vibration of a Simply Supported Beam with Axial Compression Force
Figure 24 shows a simply supported beam which is compressed with an axial
force of 3.70 X 105 pounds and placed under free vibration by specifying an
initially deflected sine curve for which the maximum deflection at the center
is 3.952 inches. The beam is 120 inches long and has a uniform flexural stiff-9 2
ness of 1.08 X 10 1b-in The beam is divided into 10 elements, each is 12
inches long. -1 2
The mass density of the beam is 1.085 X 10 1b-sec /in./sta.
The number of time increments is 200, and the time increment length is
3.752 X 10- 4 seconds. The theoretical natural period of the vibration is
3.752 X 10-2
seconds (Ref 9, p 375).
Inelosfically Supported Moss
~ Under Free Vibration
I ~ I ~ I ~ I I I ! , I
I ! I
I~
Resistance (Q) LB
100
0-- -0 Original Q W Curve of SN
)(-X-il 1st Hysteresis Loop
~ 2nd Hysteresis Loop
+--t-+ 3rd Hysteresis Loop
Mechanical Energy Lost from A to J
50 = Area Defined by J KEFGHIJ
!~~~O _______________ ~_+ __ . _________ ~~~~~~~ ____ -+~~t·rO ________ ~L ______ ~~ -10·0
Fig 20.
p
dr:f= 100 LB
M
at
E== TIME
-50
100
Deflection ( I INCH
) I I I I I I I I I I
I I
J'- I "--,,-- I
~I
"- ! "~
teres is loops of computed resistance-deflection curve of Example Problem 1.
- --~I
\ r-"\ ,- r· ) ' ... ' -J.-i
1 CVT Blh'l ':'; h "::
21. Example Problem 1. Response of the nonlinearly supported mass.
58
Free Vibration of Mass By Specifying Initial Displacements
~-~
: "\ i "\
i \ I \ I
-110 -30 -2.0 -\.O
Resista nce (Q) LB
150 0--0 Original Q-W Curve of SN
~ x )( First Hysteresis Loop
~ Second Hysteresis Loop
100 [}--{}-{] Third Hysteresis Loop
t-t-t Fourth Hysteresis Loop
Mechanical Energy Lost from
A to I = Area defined by IJDEFGHI
Deflection (W) INCH
2.0 3.0 10 ~--4-------~-------4------~~--~~~~~--~-------+I-----~
D= 3 in. -50
-100
-150
\ \
"\ '\
"\ 'b-----
Fig 22. Hysteresis loops of computed resistance-deflection curves of Example Problem lAo
l I I I I I I I I I ~
\ it '.l
0\-, I +
) ,
C !
t ~ J,
...
+ .I-I •.
--r---.i::; _ '0:'
Fig 23. Example Problem lAo Response of the nonlinearly supported mass.
60
w Uniform EI t tp
T T Beam Axis
• I nv Uh:: 12 I'n,
-3.952 sin T I! l-- i :: 120 in'-~I..-.l"1
~ III Beam I Sta. 0 9 10
Beam dl
d9
Sta. 0 1 2 3 4 567 8910 d2 d
S \ ~I 4 , f I , , J d91 d
3 d7 ~ d2 de ,-
d4 d6
E1 1.OS x 109 lb-
Mass density 1.OS5 X 10- 1
2 lb-sec /in/sta
T -3.70 X 105 Ib
Number of beam increments 10
Beam increment length 12 inches
Number of time increments 200
Time increment lE~ngth
-4 3.752 X 10 seconds
-3.952 x sin ISo -1.221 inches
-3.952 x sin 36° -2.323 inches
-3.952 X sin - 3 .197 inches
-3.952 x sin 72° -3.759 inches '( d3d d d d~/ '* 14 5 ~ -J-.lJ d5
-3.952 X sin 90° -3.952 inches
24. Example Problem 2-1. Lateral vibration of a simple supported beam with axial compression force.
61
The theoretical responses (Eq d, p 375, Ref 12) and the computed responses
based on a time increment length of 3.752 X 10- 3 seconds (1/10 of the natural
period) are plotted for the center of the beam on the computer plot of deflec
tions versus time for beam station 5, as shown in Fig 25. The responses of
the small time increment length (1/100 of the natural period) are almost per
fectly matched with the theoretical responses. The responses of the large
time increment length (1/10 of the natural period) show more variation when
compared to the theoretical responses in the first period, and after the first
period are shifted with a phase angle. The more increments, however, the beam
is divided into, the better the results. The computed moments, based on both
small and large time increment lengths along the time axis for the center of
the beam, are shown in Fig 26.
Example Problem 2-2. Lateral Vibration of a Beam on Elastic Foundation
Figure 27 shows the beam in Problem 2-1 resting on an elastic foundation
which has a spring constant of 1.2 X 104
lb/in./sta. The beam is under free
vibration with an initially deflected shape of sine curve, in which the maxi
mum deflection at the center is 6.672 inches. The number of time increments -4
is 200, and the time increment length is 1.540 X 10 seconds. The theoreti-
cal natural period is 1.540 X 10-2
seconds (Ref 12, p 378).
The comparisons between the computed responses, based on both small and
large time increment lengths, and the theoretical responses (Eq c, p 378,
Ref 12) for the center of the beam are shown in Fig 28. The difference be
tween the computed responses of the small time increment length (1/100 of the
natural period) and the theoretical responses is slight. The computed moments,
based on both small and large time increment lengths, along the time axis are
shown in Fig 29.
Example Problem 3. Three-Span Beam Loaded with an AASHO Standard 2-D Truck Moving at a Speed of 60 MPH
This problem demonstrates the capability to input the dynamic tire forces
interpreted from the computer output of the program described in Ref 1 into
Table 7 of Program DBC5 to predict the responses of a three-span beam; the beam
is loaded with an AASHO standard 2-D truck which is moving at a uniform speed
of 60 mph, as shown in Fig 30.
62
I I I I
g I I I
_ Theoretical ____ 1/\0 of Noturol
Period """"" \/100 of Natural
Period
1 I I , I I I I I I I I I j
fig 25. E~.mple y<oblem 2-1. Deflection vs time fo<
the center of beam-
o
o Cl
C)
~J
COl I I
l*r r ,> ·-<:']1-..1
c.:? I
C)
u Cl
U rD.
!
u (J
C) ,
I..C
Cl! c' .; . i
c; <l i CJ \
I
CJ .,-,
21 nv T f3E~ CjTR -= S
63
1/10 of Natural Period ~ 1/100 of Natural Period
Fig 26. Example Problem 2-1. Moment vs time for the· center of beam.
64
lw Uniform E I ,<.p Beam
Axis
lDlIDluJ _~. -6.672 sin ~x ~h-12 In.
I I 1 I-- £. = 120 In. 1 ... 1
~ I I Beam 1 1 I
S~. 0 9 10
Initially Deflected Sine Curve
d9
dS d
7 d
6
EI 9 2
1.08 X 10 lb-in
Mass density 1.085 X 10- 1
2 lb-sec /in/sta
Elastic spring constant
1.2 X 104
lb/in/sta
Number of beam increments c 10
Beam increment length ~ 12 inches
Number of time increments = 200
Time increment length ~
-4 1.540 X 10 seconds
- 6.672 X sin 18° -2.061
-6.672 X sin 360
-3.922
- 6.672 X sin 54° -5.398
-6.672 X sin 720
- 6.345
inches
inches
inches
inches
- 6.672 X sin 900
- 6.672 inches
Fig 27. Example Problem 2-2. Lateral vibration of a beam on elastic foundation.
. I ;
\.-: 1
C" :
?-2 DVT BEH'! 5TH
J
1 J
t: C!
3.
I I I I I I I I I I I
J
"\ \ \ \ \ \ \ , \ \ \ \ \ \ . r-\ \ \ \ \ \ \ \ \ \ \
Theoretical 1/10 of Natural Period
*** 1/100 of Natural Period
Fig 28. Example Problem 2-2. Deflection vs time for the center of beam.
i Dl II
Cl
f 2 nVT BEi>¥1 STP
1/10 of Natural Period ~ 1/100 of Natural Period
Fig 29. Example Problem 2-2. Moment vs time for the center of beam.
WI = 74 in
W2=70 in
Beam Station 0 10
Static Weights of Wheels
Axle 1 right 3139 lb Axle 1 left 3012 lb Axle 2 right 7780 Ib Axle 2 left 7103 lb
S ta t ic Loads of Axles
Front 3075 lb Rear 7441 Ib
Dynamic Forces of the Front and Rear Axles
XI2 2-D Truck
XI X l 2 153 inches SIR SIL' 535 lb/in S 2R 0 S2\. T
SIR T SIL
ST2R ~ ST21. '"
DIR~DI\"c
D2R = D2L ~
DljR=DljL
DT2R ~ DT2 L =
20 30
Beam Properties
FI 4.5 X 10 in. Ib-in F2 9.0)( 10 in Ib-in S 2.0 X 10 Ib/in
3750 Ib/in
4500 Ib/in
9000 Ib/in
7.3 Ib-sec/in 6.7 Ib-sec/in
2.8 lb-sec/in
5.6 Ib-sec/in
40
PI 0.5964 Ib-sec /in/sta P2 0.2982 Ib-sec /in/sta Beam increment length 60 inches -2 Time increment length 5.666 X 10 sec
QHLB)
3 - 2xl0
3 -I X/O
o 5 10 15 20 25 35 40 Beam Sta
30. Example Problem 3. Three-span beam loaded by an AASHO standard 2-D truck moving at a speed of 60 mph.
67
68
The beam is simply supported at the ends and has a length of 200 feet.
The beam is divided into 40 increments, each of which is 60 inches long. The
two linear springs are located at beam stations 10 and 30 and have a value of
2 X 105 lb/in. The first and third spans of the beam have a uniform flexural 11 2 2
stiffness of 4.5 X 10 lb-in. and a uniform mass density of 0.5964 lb-sec /
in./sta. The second span of the beam has a uniform flexural stiffness of
9.0 X lOll lb-in.2
and a uniform mass density of 0.2982 lb-sec2/in./sta. The
computer model used in Ref 1 for simulating the AASHO standard 2-D truck is
shown in Fig 30. The input data for the springs and dashpots \>lhich simulate
the tires of the truck and the static weights of the wheels are also shown in
Fig 30. The computed dynamic forces of the four tires are averaged to give
the two axle forces which then are input into Table 7 of Progrc~ DBC5. The
distribution of these two axle forces on both the beam and time axes is shown
in Fig 30. The time axis is divided into 80 increments, each of which has a -2
length of 5.666 X 10 seconds so that the front axle will move one beam
increment length per time station.
The computed deflections and moments along the time axis Eor the center
of the beam are plotted by the program as shown in Figs 31 and 32, respectively.
The plots are produced on microfilm which then can be enlarged.
A comparison of the dynamic response to the static respon.se of the deflec
tions along the beam axis is shown in Fig 33. This static res"ponse of the
plot in Fig 33 is computed by specifying the static load of th= front axle at
beam station 21 and the static load of the rear axle at the ce":lter of the bar,
between beam stations 18 and 10; the dynamic response of the plot, therefore,
is the computed deflection along the beam axis for time station 21. The
dynamic effect on the responses of the beam produced by a moving 2-D truck
is fairly small compared to the static responses produced by the same truck.
This is because the dynamic forces generated by the 2-D truck, which is moving
on a smooth pavement, are smoothly varied with time, and therefore there are
small dynamic effects. Any other AASHO standard truck or semi-trailer can be
used in the program described in Ref 1 to generate the corresponding dynamic
tire forces, which then can be input with a limited amount of interpretation
into Table 7 of Program DBC5 to predict the dynamic responses of highway
structures that can be simulated by beam-columns.
-- -------.----4.00
-------.-- - -------,---- ----,-- ---------.-------/--...-, --14.% '9.00 24.:JO ]4.00
Static ,. A " Dynamic
3 DVB TIME 5TR ~ 21
Fig 33. Example Problem 3. Comparison of dynamic to static response (deflection along the beam axis).
71
72
Example Problem 4-1. Simply Supported Steel Rod - Axial Pulse Base Time 0.10 Second
In the following three problems, a simply supported steel rod is loaded
by three axial pulses with different peak loads and base time, as shown in
Fig 34. The problem is a real experiment which was done by Matlock and Vora
(Ref 7). The steel rod is 153.75 inches long and 11/16 of an inch in diameter.
The steel rod is divided into 41 increments, each of which has a length of -3 2
3.75 inches. The mass density of the rod is 1.023 X 10 1b-sec /in./sta.
The static weight of the rod is 3.95 X 10-1
1b/sta. An internal damping fac
tor of 20 1b-in.2-sec is input at each station.
The first axial pulse input for this problem has a peak value of 175
pounds and a base time of 0.10 seconds. The axial pulse was rl~corded on the
oscilloscope in the experiment by Matlock and Vora described i'l Ref 7. The
true axial pulse is described at 20 points by scaling from the recorded output
of strain gages, which then can be input in Table 6 of Program DBC5. The time -3 increment length is set equal to 5 X 10 seconds and 40 time stations are
solved for this problem.'
Figure 35 shows the comparison of the computed bending monents along the
time axis to the measured bending moments in the experiment for the center
station of the steel rod. It is seen that good agreement exists between the
predicted results and the measured results before 0.9 second, while the mea
sured bending moments are smaller than the predicted bending moments after
0.9 second. This deviation is mainly due to the assumption that the beam
properties are fully elastic in the dynamic model of Program DBC5. Other
factors, such as the errors obtained by scaling from the recorded output, the
difficulty in establishing the ideal hinged supports, and failure to consider
the axial deformation of the rod, also affect the accuracy of the predicted
results.
An additional curve of the bending moment along the time axis for the
center of the rod, which is computed without inputting the internal damping
factor, is also plotted on Fig 35. Little is changed by ignoring the effect
of the internal damping factor, since the change of the curvature at this
point is very smooth along the time axis. The computed deflections along the
time axis for the center station of the steel rod are plotted by the program,
as shown in Fig 36.
Hangers
900 r r Problem 4 - 3 Peak Value:: 850 Ib
800
[Hammer I I
Aprox.3.75 in.--j
700 ,Problem 4 2 ~ Peak Value:: 635 Ib
Beam Station'" 4140 .
II 116 In. Steel Rod
Axial Pulse Bridge
II Bending Moment I' . I I Bridge ILRubber
r:: 150 In. ~mmmm~_ ~I Spring
--1 t-h = 3.75 in. 20190
600 Beam -3 2
Hass density 1.023 X 10 Ib-sec /in/sta Properties
-·500 (/) ::;) ...
.t:!
t- 400 o )(
<l:
300
200
100
E
I
EI
30 x 106 psi
0.01094 in4
3.28 'X 105
Weigh t
Ib-in2
-1 3.95 X 10 Ib/sta
Problem Number of time increments
Time increment length
Internal damping 20 lb-in2
sec/sta Beam increments 41 Beam increment length 3.75 inches
4-1 40
5 y 10- 3 sec
4-2 300
1 X 10- 3 sec
4-3 400
-4 5 X 10 sec
----- Problem 4 - I ------~~ Peak Value:: 175 I b
O~~~~----~-~--~----~----~------L-----~==~~~==------~ 01 02 03 04 05
Time, seconds
06 07 .08 .09 .10
Fig 34. Example Problems 4-1. 4-2, and ~-3. Simply supported steel rod loaded by axial pulses. ....... w
0
C! 0 r.£:! If)
0 0
a co '<l'-
0 0
0 0 ...-
0 0
0 (" en
0 0
0 '<l'-N
0 0
a r.£:!
o a a co
4-1 MVT BERM STR; 20
I I I f I l I I I Experimental
I - -- Without internal damping "I ........ With internal damping
I I I I I
Fig 35. Example Problem 4-1. Comparison of moments vs time for the center of beam.
/
I 01 r" '
~H
0: ttl ;
';J , I
~J 4-1 OVT BER~ STR = 20
Fig 36. Example Problem 4-1. Deflection vs time for the center of beam.
76
Example Problem 4-2. Simply Supported Steel Rod - Axial Pulse Base Time 0.026 Second
In this problem, the steel rod in the preceding problem i~ loaded by
another pulse with a shorter base time but larger peak value. The base time
of the pulse is 0.026 second and the peak value of the pulse i~: increased to
approximately 635 pounds. The axial pulse is described at 26 points by scaling
from the recorded output of the experiment.
The computed bending moments along the time axis, with and without inter
nal damping effect, and the measured bending moments are plotted for compari
son, as shown in Fig 37. Results similar to those for the preceding problem
are observed, but the effect of the internal damping factor is a little
greater than in the preceding problem. Figure 38 shows the canputed deflec
tions along the time axis for the center station of the steel :cod.
Example Problem 4-3. Simply Supported Steel Rod - Axial Pulse Base Time 0.006 Second
In this problem the loaded axial pulse has a base time of 0.006 seconds
and a peak value of 850 pounds. Again the computed bending mo~ents along the
time axis, with and without internal damping factor, and the measured bending
moments are plotted as shown in Fig 39. Because of the higher frequency of
vibrations, the mode shapes of the beam vary rapidly with time and cause large
variation of curvature; therefore, the effect of the internal damping factor
is much greater than in the preceding two cases. The estimated value of 2
20 Ib-in. sec/sta input for the internal damping factor is higher than the
real value. Figure 40 shows the computed deflections along the time axis for
the center station of the steel rod.
Example Problem 5. Partially Embedded Steel Pipe Pile Loaded by Wave Forces
This problem shows how DBC5 can be used to solve the dyna.mic response of
laterally loaded piles to wave forces. Figure 41 presents a ~teel pipe of
30-inch outside diameter and 28-inch inside diameter which is partially
embedded in the soil to a depth of 80 feet and extends slightly above the sur
face of the water which is 40 feet deep. The steel pipe is di.vided into 31
increments, ?ach of which has a length of 48 inches. The flexural stiffness
of the pipe is 2.877 X 10 inch lb-in.2
. The mass density of i:he pipe is 3.21
,'---4C.:'C;
, c c:1 ~1
! o u
o
---- Experimental --- Without internal damping *** With internal damping
, - .. ~ --.. - --_., ---.-- ---,----- -- ---··--.--------~I --- ---r--.\~--~
40~OC 3~·SG ~2C.GS :bC.~~C 2GD.~3 :40·GS ~28.
4-2 MVT BER~ STR 20
Fig 37. Example Problem 4-2. Comparison of moments vs time for the center of beam.
i 1
c:. I
~1
:f'\ I '\. ! •
i
S21
~1 I
:J I
4-20VT BERM STR c:: 20
I ------------.-" ----,-1--
lGO.~S 2~U ~S 2~C.~S
Fig 38. Example Problem 4-2. Deflection vs time for the center of beam.
I
320 ~ :;~J
, 0 1
4-3 MVT BERM STR = 20
--- Experimental
I \
---- Without internal damping *** With interna 1 damping
j -,--~-----r--------~-------.-------'
15C. GG ZOO" r: ,> ""V
Fig 39. Example Problem 4-3. Comparison of moments vs time for the center of beam.
sr-.-S-G ~~~I GG--~,);; .. -~;o;;-c-;s-, --!-"s-c .-JO ---2SG.SG--------zsC~GS-'--'3C=_; . S-,~-" -·-~~-s--7c~ . CJC
I
i,:;1 ~1
I
4-3 DVT BlRM STR = 20
Fig 40. Example Problem 4-3. Deflection vs time for the center of beam.
co o
Beam Sta. 7------
§·!!?rw
Q))( CIl<[
Mass= 32.1 S Ib-se~in
81
EI =,2.877 X 1011 lb-in2
Time Ma ss dens i ty
3.21 1b-sec2/in/sta
Force Internal damping factor
oJ=-2x 1041b Current
Drag Force
Qb=-IO Ib II d~~~~~ ~~~~~
Q
--10~""""W
1.918 X 106
lb-in2-sec/sta
S = 3 X 107
Ib/in
13- -- - -- :Of (Soft ClaYJ'----"k----i~: R ~ 2.1 X 1010 Ib-in/r~d QT == -2 x 10
4 Ib 16---- -
A
d
Qb -1 X 104 lb d
60 ft. (Stiff Clay) I
I
Beam increments = 31
Beam increment length ~ 48 in.
Time increments = 400
31- ---- 1 +w Time increment length = -2
1.0 X 10 seconds -, r-30 in
A28in Section A-A
Idealized Wave Force at Beam Station 1
o
4 -1.6 x 10
Fig 41.
6.91 X 103 Ib (Sta. 160)
1.91 X 103 Ib (Sta. 280)
400 Time Station
-4.89 x 1021b (Sta. 310)
-4.122 x 103 Ib (Sta 230)
-1.309 x 104 1b (Sta. 60)
Example Problem 5. Partially embedded steel pipe loaded by idealized wave forces.
82
2 . 1b-sec I~n. from station 1 to station 31. A large mass, which is arbitrarily
2 set at 32.1 1b-sec lin., is lumped at station 0 to represent the weights of
the structure, equipment, etc. The static weights of the pipe and the large
mass lumped at station 0 are considered to be the axial compreEsion forces,
which are distributed from 1.220 X 104
pound at station 1 to 4.924 X 104
pound at station 31. An estimated value of 1.918 X 106
1b-in.L:-sec/sta is
input for the internal damping factor.
The pipe is assumed to be simply supported at station 31 and assumed to
have a linear spring support of 3 X 107 1b/in. as well as a rotational re
straint of 2.1 X 1010
1b-in./rad at station 1. From station 1J. to station 31,
the pipe is laterally supported by soil. The soil from station 11 to station
16 is assumed to be a soft clay which has a shear strength that varies from
200 1b/ft2
at the top to 333.4 1b/ft2
at the bottom. The soil below station 16
is assumed to be a stiff clay which has a shear strength of approximately
4751b/ft2
. Generation of the resistance-deflection curves fOl~ the soil sup
ports at stations 11 and 13 is based on a technique presented by Matlock
(Ref 6). The resistances and deflections for both support curves are described
at 20 points as shown in Fig 42. The remaining support curves between stations
11 and 16 are linearly distributed.
The nonlinear characteristics of the resistance-def1ectio'1 curves for the
soil supports from station 16 to station 31 are obtained from the data for the
experiment done by Matlock and Vora in Ref 8. The nonlinear resistance
deflection curves are also described at 20 points, as shown in Fig 43. 4
Current drag forces are assumed to vary from -2 X 10 pound at station 1 4
to -1 X 10 pound at station 11. Idealized wave forces are shown at the
bottom of Fig 41. The wave forces are assumed to vary linearly from station 1 -2
to station 11. The time increment length is equal to 1.0 X 10 seconds.
Four hundred time stations are solved for this problem.
The computed deflections and moments versus time for stations 10 and 21
are plotted by the program, as shown in Figs 44 and 45, respectively. Figure
46 shows the computed deflections along the beam axis for time stations 60 and
160. Figure 47 shows the computed deflection along the beam axis for time
stations 230 and 280. Figure 48 shows the computed moments along the beam
axis for time stations 60 and 320. All plots in this problem are automatically
plotted using the ball-point plotting method.
2
Support
B~am Station 13
II 2 h~\" ~ : --------__ ~~3 I I I
I I I
Support Curve at Beam Station II
Resistance (Q) LB
1.5 x 104
Support Curve at Beam Station 13
Point Q W Point Q W
1 14380 -20.000 11 - 2650 0.034 2 14380 - 5.426 12 - 4700 0.202 3 11750 - 2.963 13 - 6100 0.418 4 10500 - 2.119 14 - 7490 0.769 5 9240 - 1.444 15 - 8365 1.072 6 8365 - 1.072 16 - 9240 1.444 7 7490 - 0.769 17 -10500 2.119 8 6100 - 0.4181 18 -11750 2.963 9 4700 - 0.202 19 -14380 5.426
10 2650 - 0.034 20 -14380 20.000
Deflection (W)
2 5 INCH 20 3 4 I
~~--~-----1------~----~------4------4~----~------~----~------~----~----~~~~ -20 -5 -4 -3 -2
Support Curve at Beam Station 11
Point Q W Point Q W
1 5800 -20.000 11 -1070 0.034 2 5800 - 5.426 12 -1932 0.202 3 4750 - 2.963 13 -2465 0.418 4 4240 - 2.119 14 -3022 0.769 5 3735 - 1.444 15 -3380 1.072 6 3380 - 1.072 16 -3735 1.444 7 3022 - 0.769 17 -4240 2.119 8 2465 - 0.418 18 -4750 2.963 9 1932 - 0.202 19 - 5800 5.426
10 1070 - 0.034 20 -5800 20.000
-I I 1 I 1 I 1 ,
~------____ ~19~~'0
19
I 1 I I 1 1 1
1
210
Fig 42. Symmetric resistance-deflection curves of soft clay at beam stations 11 and 13.
• Resistance (Q) I LB
Support Curve at Beam Stations 16 to 31
I I I I I 1
10
+3X 104
~+------+-----~------~------~-----+------4----
-20 -5 -4 -3 -2 -I
II
104
-2 x 104t
-3 x 1041
Point
1 2 3 4 5 6 7 8 9
10
2 -+---
Q H
28235 -20.000 28235 - 5.426 26305 - 2.963 25340 - 2.119 23652 - 1.444 21722 - 1.072 19552 - 0.769 15696 - 0.418 11840 - 0.202
7020 - 0.034
3 4 ·······1 I
18
Point Q W
11 - 7020 0.034 12 -11840 0.202 13 -15696 0.418 14 -19552 0.769 15 -21722 1.072 16 -23652 1.444 17 -25340 2.119 18 -26305 2.963 19 -28235 5.426 20 -28235 20.000
Deflection (W)
5 INCH 20 I ~ ...
I I I 1 I 1 I
19 10
Fig 43. Symmetric resistance-deflection curves of stiff clay at beam stations 16 to 31.
c; :
;;J I ,
C) i
~1 01 t<?1
C:1 ,
5 DVT BERM, 5TH =
Beam STA 21
10 21
Fig 44. Example Problem 5. Deflection vs time for beam stations 10 and 21.
, 'I I :~ -j
I
C")
, I r, I c\jl
I
I
~'" I, r_._ ~'" r. ,0 ',_, ----,----_" '~" ,'J' , r,,' C -', ',r-J C-.-~_ --'J --'-------.- - - --, '--,----'-------,----- --j _ ' ',' "'.' , r:; G L Gr.') ::;; C 0 ~; C
5 MVT SEW., STR = 10 '" Ll
1 ---,
3~C,C~ ACO~~
Fig 45. Example Problem 5. Moment vs time for beam stations 10 and 21.
\ \
\ " '-'.~
C'
c::: ,~
I
:. - -1
L~ , \ C·
I
.- j
i n I r~
A
C)
'" -J
!
- --.------ -------,--- ---- -- -~------. ~ ~:~ ·(?~CC 1r:?c:~
l I 1 t
// I
/ / ;j
II Time , I
\ STA 160 //1 I \ / I
\.. // / \", ----
\
\ / Time STA 60
5 ovB TJME' STR = 60 160
~-,~,,-~;,~~~===ri ~----~. //~,~,~~ ~<r.c'G 2~'J~ j2.~C.
Fig 46. Example Problem 5. Deflection along the beam axis for time stations 60 and 160.
~ \'
"':J
, \
\ \ \
\ \ \
\ Time \' STA 280
\
\ \
Time STA 230
I I
I I
5 ova TIME 5TR = 230 280
i I'
I' I; ,;
.GC
Fig 47. Example Problem 5. Deflection along the beam axis for time stations 230 and 280.
co co
u:
/ 1/
'-4 ~-~'J"*. ~'.;C ~';. --~r(""" - • -J -....J _, -.,..J _J ~ , _!J
O! CJ , I
~~ \ ) \ ) I
~~ i "\ ;' / ~j .\\. /1 ., \ / I , t::> I ,!
C
c::;,~ r" '-,
I 01 O!
~J
Time STA 60
;--'
I~.\ I / \ ..
! / \;'. ! /
Time STA 320 \ 1\
~J NVB T f tjj:' 5TR = 60
±=::[== ... ~ ....
Fig 48. Example Problem 5. Moment along the beam axis for time stations 60 and 320.
90
Example Problem 6. Three-Span Beam with One-Way and Symmetric Nonlinear Supports Loaded with a Transient Pulse
This problem is intended to demonstrate that the option of switching from
a spring-load (tangent modulus method) iteration to a load iteration technique
which is built into Program DBC5 is useful when the tangent modulus method
fails because of lift-off of the one-way support or yielding of the nonlinear
supports. Figure 49 shows a three-span beam (30 beam stations) which is
simply supported at the ends, one-way supported at station 8 anj nonlinearly
supported at stations 16 and 17 and is loaded with a transient pulse at
station 17. The beam has a uniform flexural stiffness of 2.4 X 1010
lb-in.2
and a uniform mass density of 5.184 X 10- 1 lb-sec2!in.!sta. A rotational
restraint of 3.0 X 1010
is applied at beam station 0 and a constant axial 5 compression force of 10 pounds is assumed. The negative one-way resistance-
deflection curve of the support at beam station 8 is shown in Fig 50. The
symmetric resistance-deflection curve of the support at beam station 17 is
also shown in Fig 50. The nonlinear characteristic of the support at beam
station 16 is linearly distributed between beam station 15, which has no
resistance to deflection, and beam station 17. The applied transient pulse
at beam station 17 is shown at the bottom of Fig 49. The time increment length
is equal to 4 X 10-2
seconds and 40 time stations are solved for this problem.
The computed results and the monitor deflections at beam Etations 8, 15,
16, and 17 during the iteration process are printed on the comI~ter output.
It is seen that the tangent modulus method failed at time stations 5, 9, 13,
16, and 24 due to abrupt changes of time-dependent loads or lift-off of the
support at beam station 8 at these time stations; therefore the program
automatically switches to the load iteration method and succeeds in obtaining
the closed solutions. If the time increment length is reduced to 4 X 10-3
seconds and the same problem is rerun for 400 time stations, the tangent
modulus method works nicely at each time station except three at which the
beam starts to lift off the support at beam station 8. This is because
smoother changes of time-dependent loads are obtained due to tne small time
increment length and, therefore, the tangent modulus method only fails at
some of the critical time stations at which lift-off of the support occurred.
The computed deflections and moments versus time for beam stations 8 and 17
are plotted using the printer plotting method, as shown in Figs 51 and 52,
Beam Station
0-----
8----
V R~E.!!! W
QlM met
El, 'f
E1
T
2.4 x 1010
lb-in2
0.5184 lb-sec2/in/sta
10Slb 10
R - 3 X 10 lb-in/rad
SB and SN see Fig 50
Beam increments ~ 30
Beam increment length
Time increments ~ 40
48 inches
91
16 - OT ----17 Time increment length
-2 4 x 10 seconds
30
-10 4
Transient Pulse 0 T
Time Station
Fig 49. Example Problem 6. Three-span beam loaded by transient pulse.
92
Negative One - way Support Curve at Beam Station 8
2 3
-4 -3 -2
Symmetric Support Curve at Beam Station 17
Point Q W
1 13,600 -4.0 2 13 , 600 - 2 .0 3 8,600 -1. 0 4 3,000 -0.3 5 - 3,000 0.3 6 - 8,600 1. 0 7 -13,600 2.0 8 -13,600 4.0
-I
Resistance (Q) LB
NElgative One-Way Support Curve at
Beam Station 8
Point Q W
2 40,000 -3.0 3 39,000 -2.0 4 35,000 -1.0 5 20,000 -0.5 6 0 0.0 7 0 4.0
Deflection (W) 2 3 4 INCH
t7 .. I I I I I I I I I
7 18
Symmetric Support cu~ at Beam Station 17-~
Fig 50. Resistance-deflection curves of the supports at beam stations 8 and 17.
93
PHO"RA~' f)rlC., - ~ASIEf< • ,JilCK CH""I _ M"TLOCIo( _ nFCt<)-REvlc;ynN 11r.Tf = 26 .111"1 71 t.)(~''''''''LF "'14011(I""'S FOR ppnr.PA!-I [)RCc; .. V JACK Cl-lfII\1 clUNE 1Q71 IH"IA'~IC )~~A'-1·r""LJI'~N PRnr;OA'-1 I 5-1.C; T'-1Pl,H'IT ()PF~ATl'ln )
pJ.(O~ (CO'"TO) 6 THPI'f.-SPA"J HEI\'" I~!TH ()~F-i.//\v ANI) <;VM .... FTRr( NON( r'JFAQ <iUP LOAOEI) ;:tV T.p
0000* PLnT OF I)EFLECTln~ V<; TrMF FI')D R~AM ~TATTn~S OF. 00000
1 rURVF (" ) ::; ~
2 rUI'n/1" (+ 1 ::; 17
PT.'I)O °VALd!:!>
-1 -1.31H:-lf:l +
0 -b.601F-17 + 1 -6.00";1" -In + or ? -1.613I"-iJ2 + or 3 _2.9'tlF-Ol' + "'r 4 -4.:.7bF-l)? + * r 5 -J.4(4E-J? + ;; r I:> -1.71'fF-J? + or 7 _2.6l;;61"_04 + 0
f'l -3.031F-03 *r • 9 -1.4t:1!:>I".,)? °T • 10 -1.73111"-')2 + or 11 _2. u JoF_')? • or 12 -3."bbF_l? + 0 T 13 -3. '7\-1bF-,)? • 0 T 14 -3 .... .:cl"-,J? + 0 T
1 '" -1.0 __ 5F-I)'? • *r
\1:> 7.bl2E-d) .. • 17 -3.977F- rn or • If< -3.4t:i3f-')2 • .. T 19 _3.1j44F_.J? • 0 T 20 -2.12'lF_'12 + *1 21 -1.'1 ... SI"-J? + °T ?2 -5. Ul:l4F-,J3 "T + 2 3 /:I. StIlF-,) 3 * • 24 3.U'1!:>!="-03 0 • 25 -2. t;3ll"-,P • *r 26 -3.00"lF-I)2 • '>1 ?7 1.60cl"-,)2 + 0
28 4.1371"-02 • ro 29 -1.I:H~2f-')2 °T • 10 -tJ.'?/tll"-OZ 0 T • 31 -1.Jt;I;F-02 or. 3? b.972F-J2 • T 0
33 4.04~F-()? + TO 34 -6.4:;UF-v2 + * T 35 _!:I.3ott.lF_i)': 0 T • 36 '1.599F-01 * • 37 7.1211"-'12 T 0 + 38 4. 2411f-1)? + T* 39 -b. 1 !:>'fE-(lZ + .. T 40 -'.51:l4F-()? + * T
Fig 51. Example Problem 6. Deflection vs time for beam stations 8 and 17.
94
PROGRA~ n!:lCS - MASTER - JACK CHA~J - MATLOCK' - nECKl·REVISI~N nATE ~ 26 JU"I 71 EX-AMPLE ?RORLFMS FOR F'Rnr.RAM DHCIi RY JACI< CHAN JUNE 1971 DYt,jAMIC ~EAM-COLIJ"'N PR("It;PIIM ( 5-\-C; H4PUCIT OPERAT("IR ,
PROq (CO'lITO) 6 THREE-SPAN BEAM WITH ONF-WAY ANt) 5Vlo1MFTRTC IIIONLINFAI:! sup LOADEn RY T.p ... _. PLoT OF nENU • MOM. vs Ttr.lF F'nR RF'AM CiTATIONS OF. •••••
I CUHVF (. ) 1;1 8 2 CURVE (+ ) • 11
PT.t,jO *VALUE·
-1 -3.358F-I0 + 0 -3.89lE'-10 + 1 -3.518E'+04 • T + 2 -9. 53 81=:+0 4 • y + 3 -1.812f:+05 • T + 4 -2,122F+05 • T + 5 -2,1l5E+05 • T + 6 -1.409E'+05 • T + 1 .4.657f+04 • T + a 1.139E+04 t· + 9 -5.331E+U3 • +
10 -6.111F+04 • T + 11 -l,aZ8E+05 • T + 12 -2.815E+v5 • T + 13 -2.433E'+05 • T + 14 -1.a28F+OS • 1 + lIS -S.922E+04 • T + 16 3.1 18F+04 J - + 11 1.2Z2F+04 T * + 18 -9.1 51E+04 • T + 19 -2,51;91'.:+05 .. T + 20 -2.96SE+I)S * T + 21 -1.227E:.OS • + ! Z2 7olb6E+04 ,. + • 23 1.01I;E+O<; T • + 24 -1.266E+U4 *' • T 25 -8.17SE+V4 eo !+ 26 _3.985E+04 • • ! 21 _1.949F+04 + .Y 28 -4.J40E+04 • T + 29 .3.05ZF.'+04 • ! ~.
30 2.709E+04 T • • 31 1.262r::.04 T + 32 .. S,033E+04 + • T 33 -1./)2bE+v4 + * 'T 34 _S,11 40 F:+040 • + 3S -1.049E+04 • + ]6 2.866f"+U4 T * + 31 4.1b3F+04 T • + 38 _4.1/)401='+1)4 +* ! 39 .. 9,1931':+04 + * T 40 -1.125E+~4 * + T
Fig 52. Example Problem 6. Moment vs time for beam stations 8 and 17.
respectively. Figure 53 shows the computed deflections along the beam axis
for time stations 4, 8, and 20. Figure 54 shows the computed moments along
the beam axis for time stations 12, 16, and 30.
Example Problem 7. Partially Embedded Steel Pipe Pile Excited by Sinusoidal Earthquake-Induced Forces
95
The capability of Program DBC5 to analyze the response of a partially
embedded steel pipe pile which is excited by sinusoidal earthquake-induced
forces is illustrated with problem 7. Figure 55(a) shows a 12.75-inch outside
diameter X 0.5-inch wall steel pipe pile embedded in the soil to a depth of
40 feet and extending 10 feet above the ground surface. The pile has a
flexure stiffness of 10.9 X 109
1b-in.2
and is divided into 50 beam increments.
Each beam increment has a length of 12 inches. An axial compression force of
10,000 pounds is assumed in the pile. From beam station 0 to beam station 40,
the lumped mass density of soil and pile is assumed to be 5.184 1b-sec2/in./sta.
From beam station 40 to beam station 50, the lumped mass density of soil and
lateral load is assumed to be 2.592 1b-sec2/in./sta. The negative and positive
one-way resistance-deflection curves, which describe the lateral soil supports
from beam station 0 to beam station 40, are shown in Fig 55(b) and (c), res
pectively. The time increment length is 5 X 10-3
seconds and 160 time stations
are solved for this problem.
The pile is excited by a sinusoidal earthquake-induced force, as shown in
Fig 55(d). The sinusoidal earthquake induced force has a base time of 0.1
second (equivalent to 20 time stations) and is induced from an assumed sinu
soidal acceleration with a maximum value of 1/10 G (G = 386 in./sec2
) and a
base time of 0.1 second.
applied transverse load
Based on the discrete-element beam-column mOdel, the
QT is equal to the product of deflection Wand
spring force SN. By double integration of the equation of sinusoidal acce1-
eration and evaluation of the constants of integration, the relation between
the maximum deflection (W ) and the maximum acceleration (W ) is found to max max
be
W max
where t is the base time and TI is equal to 3.1416. If the nonlinear spring c
96
f.lRO"HA~ n,:!C':l * "'lASTEH - ,IIII";K CHl\N - MI\TI.()CI( - r)FCKI-REVJ5rli~1 t)l\Tf :& 2b ,III~I 7' I::XI\'1,.)LE ~ROtlLF'-\5 FOR PPIir:QAM f)HC" HY JACI( r:~At,1 IIINE 1971 UY!\jfl~TC IlEAM-t(lL'It~N P'?Iir,0I\M ( ~-1-C:; TMPI TCfT IiPF~ATnR "
PROq ICO",TI,)) (, THQEt:-SPAIIl HE A'~ '~l T", (lNF-\OIAY ANI) <:;'1Mo.,IFTQTC NONL I'~F'IIP C:;JP U)AOF.n ~v
PL,,'T **'*** OF 11FfLt:CTrON<; ~LONr; ~I=' III~ A'{15 liT TTIo1F c:;T AT 101\15 1')1", *It***
rUHlIF (*l ::: 4 ? CURVF ( + I ::: Ii 3 CURVF ( ... ) ::: 2n
jolT. ~IO <>V"UII::*
-1 S. 3iH~F-:)3 l(
0 o. l( 1 5.7~"'F'-')3 l(
2 1.71;)7[-.)? +( '3 3.1"'''E-');;> +l( ... .... 173F_')2 +*l( 5 ....37'/E-J?' +*')( 6 3 .. :.'''i3F-J2 +*X 7 .... 5 ... 91"-.)) +x ~ - .... S7bF-');> It'+ 9 _1.2221"_'11 It' +
10 -c.2 U3f- 1J l l( + 11 -3.35JF-iJl l( + 12 - .... 625F'- ill )( T+ 13 -5.9701='-U1 'I( T+ 14 -7.33'1F-IH 'to T + 15 -1::l.6~"'F-i)1 't* T + 16 -9.9S1F-'Jl )t *' T + 17 ':'1.1111"+<)0 J( * T + 18 -1.1~2F'+'IO X. * r + 19 -1.23 1-1".')0 X * T + 20 -1.25uF+1)ll X *' T 21 -1.22'~e::+JO A * f
22 -l.17HF+110 l( * T 23 ·1.0~~F.+I)O )( .. T ~4 -9.9,,7F-01 )( .. T 25 -8.6"u F -uJ l( ..
I 26 -7.15~r-Ol l( .. T + 27 -5.5101"-01 )( * T • 28 -3.7 ... 31"-\11 J( * T + 29 - 1 • 8 '12 F - 'Jl X * I + 30 o. l( 31 1.d'12E-'l1 + T .. )C
Fig 53. Example Problem 6. Deflection along the beam axis for time stations 4, 8, and 20.
+
f.P
+ + + +
+
PROr,RAM nBC':) - ~AS1E~ - J'\CK CHttt,j - Mil nocl< - nEC~I-REVJ~I~N nATE :0: 20 J11"l
p~o~
0 ~~ ........
PT.t>lO
-1 0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 1b 17 18 19 20 21 22 23 24 25 2b 27 28 29 30 31
EXIIMPLE P~OHLF'MS FOR p~nr.RAM OBCe; ~V JACI' C~AM ,IUNE 1 eni DYt,jA'lIC I:'IF:AM-rnLIlMN PRnt;I"III"I ! 5-1-e; TMPUC IT OPFi(ATI"IR )
( CO·iTl.ll TrlREE-SPAI\J BEtH IOIiTl-l nNF:-wAV ANI) c:.v"IMFTRIC NONLII\)FA~ ~lJP LOAOE'I)
PLnT OF REIltD. MOM. 1\ LON(; RF"M AX I S liT TT>.4F ~TATIONS r'lF, ~ ... ~~ ...
1 CURVE ( 1. ! ;: 12 2 r.UHVF (+ I = 1& 3 CURVE (X) = 30
·VAUJE.
O. l(
&.:;031'"+04 1(+ ... 7.6731"+'14 l( .. .. 2.1':)UE"+04 It +.
_3.3d5E"+04 11 + -8.86SF+\)4 .. xr+ -1.4~2[+O5 • Kr+ -1.937F"+OS • l( ..
-2.424,,=+OS .. T )( + -2.875F+J"i .. T l( -2.bOO,:+0.::, ... T + x -2.2d3f+05 .. T + )(
-1.930E+O'i .. T + )C
-1.54~E+0'i .. T + X -1.153F.+OS ... T + ~ _1, 64 OE+04 • T + X -4.0bB[+04 .. T + .(
-1.0 77F"+U4 .. + '(
2.11)0[+05 T + X .. 3.5851'"+05 + T ~ 4.323F+0'i X+ r .. 4.89H+Ot; X + T 5.271F:+OS l( + T 5. 446f+J'5 )( + 5.4071':+05 X + 5.1491="+115 /.. + T 4.679[+05 X .. T 4.01 2[+o':i X + T • 3.173E+,15 X + T • 2.1'171::+0<; l( + T 1.123E+OS J( + T ..
-9.2 52F-09 l(
O. y.
Fig 54. Example Problem 6. Moment along the beam axis for time stations 12, 16, and 30.
>IV
..
.. ..
97
71
T.p
.. .. ..
98
50 ,---10 ft
40 1 30
20 40ft.
10
o Beam Station
J ~.~ Q.! )(
W £I.J <[
10,000 Ibs
Loads
~;;g;;;;--12.75 in. 00. x 0.5 in. Wall Pipe
10,000 Ibs
(a)
F o. E1 30 x 106
x ~4 [(12.75)4
10.9 x 109
lb-in2
Mass Density:
Sta 0 to 40: Weight of soil + pile ~ 200 lbs/ft
:. p 200/386.4~. 5.184 lb-sec2/in/sta
Sta 40 to 50: Weight of pile + lateral load
~ 10,000 Ibs. Distributing on 10' length.
p = 10,000/(10 x 386.4) " 2.592 lb-sec2/in/sta
Number of beam increments = 50 Beam increment length = 12 inches Number of time increments 160
-3 Time increment length = 5 X 10 seconds
One-Way Resistance-Deflection Curves
2
-10 -0.05
Resistance (Ib)
100
S~ Deflection (in) 3 4
10 -10
SN - 100 - 2000 Ib/. - 0.05 - lin
(b)
Resistance (lb)
Deflection (in) 2 0.05 10
S~ -100 -
I I • I I I I I I
~ 1 3 4 (c)
Earthquake-Induced Forces Let X - 1/10 G ~ 38.6 in/sec
2 max
100 78.2 Ibs
0 5 10 15 20 Time Station
~ tc=O.l sec -1 (d)
Fig 55. Example Problem 7. Partia lly sinusoidal earthquake-induced
2 2000 x 0.1 X 38.6
2 n
~ 78.2 lbs
embedded pile excited forces.
by
force SN can be assumed to be constant, then the maximum sinusoidal
earthquake-induced force can be obtained by the equation
T Qmax
N 2 ., 2 S tWin c max
99
The earthquake-induced forces are applied at beam stations 0 to 40 to approxi
mately describe the ground movements.
The computed resistance-deflection curves of the negative and positive
one-way supports at beam station 20 are shown in Fig 56. The first loading
path is seen to form a loop which starts from point A and follows the path
of ABCDAEFGA. The mechanical energy lost in the first loop is equal to the
area defined by the loading path. The second loading path follows the path of
ADHDAGIGA; no mechanical energy is lost in this path because the energy is
absorbed by or transformed to adjacent supports. Between points D and G in the
second loading path, the supports at beam station 20 exhibit no resistance and
hence a hole is produced around the pile at this station. The third loading
path is similar to the second loading path except that in the third the maxi
mum positive deflection increases from point H to point J, and the maximum
negative deflection decrease from pOint I to point K. The succeeding paths
are similar but have progressively smaller maximum positive and maximum nega
tive deflections until finally no more energy is absorbed by or transformed
to adjacent supports and the entire pile exists in a state of free vibration
since no damping factor has been included. Figure 57 shows the plot of deflec
tion versus time for beam station 20. Here the response is tapering down to
the two limits at which the entire pile is in a state of free vibration.
A similar computed resistance-deflection curve can be constructed for any
other support, with a similar result expected unless the support (for instance,
at beam station 40) has not yet reached the yielding condition; then the com
puted resistance-deflection curve will be a straight line. Eigure 58 shows
the computer plot of deflection versus time for beam stations 10 and 30.
Figure 59 shows the computer plot of deflection versus time for beam stations
40 and 50. Figure 60 shows the computer plot of deflection along the beam
axis at time stations 10 and 20. Figure 61 shows the computer plot of deflec
tion along the beam axis at time stations 50 and 150. Figure 62 shows the
computer plot of moment along the beam axis at time stations 30 and 100.
FE
Negative ResistanceDeflection Curve
-0.05 I
-0.025
Resistance (LB)
100
50
-50
-"- First Loop of Loading Path
------- Second Loading Path
--- Third Loading Path
The Mechanical Energy Lost in the
First Loop = Area defined by ABCDAEFGA
0.05 Deflection (INCH)
Positive ResistanceD~flection curvi
"""'--___II'__... J_~ "'----J>.----'I_ B C
Fig 56. Computed resistance-deflection curves of the one-way supports at beam station 20 of Example Problem 7.
>-' o o
(f) Q)
..c u c
c 0 +-U <l> -Q)
0
Q) (f) ~ Q)
> (f)
c: 0 .... l-
.08
. 06
.04
.02
0
t I
-02
-.04 I
50 100
Time Increment Length
= 5 x 10-3 sec .
Fig 57. Example Problem 7. Deflection vs time for beam station 20.
Time Sta.
r---,..,,.., r,.., - '- ,~ .• .J.....,
.; ! c:
'_. -I
7 OVT BER~ STR - 10 3C
A j i \
f \ i
f
I I
V
. Fig 58. Example Problem 7. Deflection vs time for beam stations 10 and 30.
r-' o N
I ~. I
01 c; l
I I
'" I ell
~~ I
! ~'1 ,
L" I I
LY~
en C; I
0--, I
7 OVT BER~ STR =
\ ~ Beam STA 40
.. -~! *---.----~(f' '-is· ~
\ ,
\ 1 r-"i r." \ ~ •. " .JJ
\
I
\LBeam STA 50
40 sc
Fig 59. Example Problem 7. Deflection vs time for beam stations 40 and 50. ~
o w
Time STA 20
~~ j ,-------::::---11----6 r.-. C)"'.I r''-'
- ·Jv 'l,",0 --,-------~I --- -,-------
8.CJO ~6.00 ;:/i.OG I
3Z.!J:J
I
~I
~;l STA 10
" t J ;
°1 O.J ,
7 OVR TIME STR - 1 0 20
Fig 60. Example Problem 7. Deflection along the beam axis for time stations 10 and 20.
~-1 x
~.\\ -1 c
\
\ ~Time STA 150
gl .--- " - ----~ .. -~~-.-, ----", -·8·0::; (~~-C,::; 5.,(,J 15,OJ
7 OVr3 T1ME 5TH 50 150
Fig 61. Example Problem 7. Deflection along the beam axis for time stations 50 and 150.
,J Ll i
~ ~~_ji Jr. «1
:-, I ~~J ~ I
7 MVB TIME STR ~
r\ Time STA
STA 100
i \ I i
30 100
Fig 62. Example Problem 7. Moment along the beam axis for time stations 30 and 100.
CHAPI'ER 6. SUMMARY AND CONCLUSIONS
In this study, the dynamic solution of the discrete-element beam-columns
of Ref 10 (DBC1) and the static solution of the discrete-element beam-columns
of Ref 5 (BMCOL 43) have been modified and extended to cover nonlinear supports
and loads with time-dependent axial thrusts; the program has the added capabil
ity of producing plots of the deflections or moments along either beam or time
axis of any requested monitor stations.
The path-dependent history of loading of the nonlinearly-inelastic resist
ance-deflection curve of the support is considered in the study. Two multi
element models, which are used to simulate the nonlinear characteristics of
symmetric and one-way resistance-deflection curves, have been introduced. An
internal damping factor, which is related to the first time derivative of the
curvature of the beam, has been considered in addition to the external viscous
damping factor which is normally encountered in a dynamic problem. A computer
program which has been presented, DBCS, allows the user to obtain a variety of
results and plots. All of the uses and capabilities of DBC1 and BMCOL 43,
except construction of the envelopes of maximums provided in BMCOL 43, have
been incorporated in the DBCS program.
This method is based on an implicit difference formula of the Crank
Nicolson type. Problems in which only linear spring supports are specified
are not subject to the instability of dynamic solutions. Problems in which
nonlinearly-inelastic supports are specified. however. rAquire cautious selec
tion of a time increment length, to pick one reasonably but not extremely small
(for instance, less than 1/10 of the fundamental period), since the basic
assumption of the Crank-Nicolson implicit formula is that the time dependent
forcing function and the time variant nonlinear springs are smoothly varied
with time. For problems with nonlinear supports, fue iteration process compares
the successively computed deflections until a specified tolerance is satisfied.
The option of switching from the spring-load iteration technique (tangent
modulus method) to the load iteration technique for problems with nonlinear
107
108
supports is useful when the support yields or disconnects from the beam-column
since the spring-load iteration method fails in these cases.
This report is intended to encourage use of DBC5 as a tool in computer
aided design. Highway structural and foundation designers can use it to solve
many problems, static or dynamic, which they encounter and which presently are
solved by rough approximations or by tedious hand calculations. A very impor
tant use would be for a study of the dynamic response of actual. truck loaded
beam-columns, as illustrated in Example Problem 3 described in Chapter 5. A
highly sophisticated computer program (Ref 1) is available to compute the time
variant dynamic forces of the tires of a truck moving at a uniform speed on
highway pavements. With limited interpretation, the computed dynamic tire
forces could be input in the table provided for the time dependent lateral
loads in Program DBC5.
Further application to design problems that are difficult to solve by
conventional means is possible. The possible variations include analyses of a
bridge structure as foundation supports settle, railroad loadings on continu
ous spans supported on soil foundations, the effect of axial l,)ads induced
from tractive forces or temperature changes, approach slabs connected integral
ly with the bridge structure and supported on soil foundations, offshore piles
loaded with wave forces and partially embedded in the soil, ani transverse
response to earthquake-induced forces.
Future extensions to the model and the program might include, for highway
pavement analysis, the coupling of a vehicle model and pavement roughness
characteristics to the beam-column model for generation of dynamic loads (simi
lar to the model described in Ref 1) and the inelastic analysis of beam proper
ties.
The nonlinear solution capabilities should be extended to the beam bending
stiffness variable. Nonlinear moment-curvature relations could be incorporated
in the iterative procedure, thereby permitting analysis of the beam material
for stresses in the nonlinear range. The hysteresis effect of the nonlinear
moment-curvature relations should also be considered.
Studies of the nonlinear closure procedure should be continued. Methods
for accelerating closure in the load iteration technique should be developed.
In Program DBC5, the iteration process is performed at each time step. It is
possible, however, to iterate only at the critical time stations; at the rest
of the time stations, the support stiffness can be assumed to be unchanged
109
since a small time increment length must be used in these problems. The
existing discrete-element model requires the user to know and specify the dis
tribution of axial thrust throughout the beam. A valuable extension of this
work would be modification of the model to include axial deformations and
development of the force-deformation equations for axial thrust.
Finally, experimental data are required for further evaluation of the
method, especially in predicting the path-dependent behavior of the supports.
Research of this nature will furnish additional data on the nonlinear behavior
of the supports, which could be applied to modification of the existing multi
element models for nonlinear supports so that buckling, fracture, softening,
and relaxation (creep) of the support could also be considered in the program.
!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!
44!5"6!7$1*'*0!8$($.$9'.$/-!")':!
REFERENCES
1. A1-Rashid, Nasser I., "A Theoretical and Experimental Study of Dynamic Highway Loading," Ph.D. Dissertation, The University of Texas at Austin, May 1970.
2. Crank, J., and P. Nicolson, "A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of Heat Conduction Type," Proceedings, Cambridge Philosophical Society, Vol 43, 1947, pp 50-67.
3. Kelly, Allen E., and Hudson Matlock, "Dynamic Analysis of DiscreteElement Plates on Nonlinear Foundations," Research Report 56-17, Center for Highway Research, The University of Texas at Austin, January 1970.
4. Matlock, Hudson, and Allan T. Haliburton, "A Finite-Element Method of Solution for Linearly Elastic Beam-Columns," Research Report 56-1, Center for Highway Research, The University of Texas, Austin, September 1966.
5. Matlock, Hudson, and Thomas P. Taylor, "A Computer Program to Analyze Beam-Columns Under Movable Loads," Research Report 56-4, Center for Highway Research, The University of Texas at Austin, June 1968.
6. Mat lock, Hudson, "Correlations for Design of Laterally Loaded Piles in Soft Clay," Paper No. OTC 1204, Offshore Technology Conference, 6200 North Central Expressway, Dallas, Texas.
7. Matlock, Hudson, and Mahendra Vora, unpublished paper on axially loaded steel rod, The University of Texas at Austin, 1971.
8. Matlock, Hudson, and Mahendra Vora, unpublished paper on cyclic loaddeformation curves of stiff clay, The University of Texas at Austin, 1971.
9. Reiner, M., "Building Materials, Their Elasticity and Inelasticity," North-Holland Publishing Com~any, 1954.
10. Sa1ani, Harold, and Hudson Matlock, "A Finite-Element Method for Transverse Vibrations of Beams and Plates," Research Report 56-8, Center for Highway Research, The University of Texas at Austin, June 1967.
11. Timoshenko, S., "Strength of Materials," Part II, pp 411-416, D. Van Nostrand Company, Inc., 1956.
111
112
12. Timoshcnko, S., ''Vibration Problems in Engineering," pp 324-380, 'Q, Van Nostrand Company, Inc., 1955.
13. Volterra, Enrico, and E. C. Zachmanog1ou, "Dynamics of Vibrations," Charles E. Merrill Books, Inc., Columbus, Ohio, 1965.
!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!
44!5"6!7$1*'*0!8$($.$9'.$/-!")':!
APPENDIX A. DERIVATION OF THE DYNAMIC IMPLICIT OPERATOR BASf~ ON THE CRANK-NICOLSON IMPLICIT FORMULA
The dynamic model used in Program DBCS is discussed in Chapter 2 and the
free-body diagram of the model is shown in Fig 3.
Since the rigid bars connected between the deformable joints are only
responsible for transferring the bending moments, the shears, and the axial
thrusts from one joint to another, the equation of motion of the beam-column
can be obtained by summing up all the forces, internal or external, at any
deformable joint where the beam properties are lumped. For the convenience of
the reader, the free-body diagram of a portion of the dynamic beam-column
model is shown in Fig A1.
If we take equilibriums for the moments in Bar j and Bar j+1 and for the
vertical forces at joint J, then
for Bar j
i d (CPj-1,k)
i d ( cp j , k) M. 1 - M. + (D ). 1 - (D ).
J- J J- dt J dt
(T. + T
+ w. k) = 0 + V.h + T. k) (-w . -1 k J J J, J, J,
(A. 1)
for Bar j+l
(A.2 )
for Joint
T - R. 18 . 1 + R. 18 . 1 ]-]- ]+ 1+ Vj - Vj +1 + Qj,k + Qj + 2h
llS
Bar Number - - - - - - - - - - - - - -
Fig Al. Free-body diagram of a portion of the dynamic beam-column model.
117
(De). d ( ) J dt wj,k = 0 (A.3 )
Solving for the shears Vj
into Eq A.3 and multiplying by
and V. 1 J+
in Eqs A.l and A.2 and substituting
N hS. kW ' k J, J,
h yields
s - hS W j j ,k
= 0
Based on the Crank-Nicolson implicit formula, we can assume that
M. 1 = F. 1 J- J- 2
- 2wj _l ,k+l + wj,k+lJ
(A.4)
M F __ 1 __ fw - 2w + W + 2 j j 2h2 L j-l,k-l j,k-l j+l,k-l wj-l,k+l - wj,k+l
+ W j+ 1 , k+ 1 ~ ,
118
:t>l j +1
6 j+l
N S. k J,
2w j+ 1 ,k_l + wj+2 ,k-l + 'It.'j,k+l
,
- 2w·+l k+l + w·+2 k+lJ ' J, J,
=
1 == h22h L-wj-1,k-l + 2w j ,k_l - wj+1,k-l + wj-1,k+l
t
- 2w j ,k+l + W j+l,k+lJ '
=
- 2w j+ 1,k+l + wj+2 ,k+1J '
1 = 4h L-Wj,k-l - wj,k+l + wj +2 ,k-l + wj +2 ,k+lJ '
1 == h 2 LWj,k-l
t
2w j ,k + W j ,k+ 1 J '
119
d 1 I (w. k) = L -w. k-1 + w. k+ 1 : dt J, 2h
t J, J,-I
1 I w j-1,k
:: LW j - 1 ,k-1 + wj - 1 ,k+1J 2
1 r l I
w. k :: L w. k-1 + w. k+ 1 J J, 2 J, J,
1 r
\1·+1 k = L W ·+1 k - 1 + w. + 1 k+ 1 J J , 2 J, J,
T 1 r T T -: Q. k = LQj,k-1 + Qj,k+1J J, 2
T 1 ; T T-: T. k = LT. k-1 + T. k+ 1 J ' J, 2 J, J,
and
T 1 1- T T-(A.5 )
T j+1,k ::
LT j+1,k-1 + T j+1,k+1J 2
Substituting Eq A.S into Eq A.4 and rearranging the order, we obtain
120
i J 2r
T T T + (D )j+1 + h LT j + Tj+ 1 + 0.S(Tj ,k_1 + Tj ,k+1 + Tj+ 1 ,k-1
3 S N
2h p. . J
+ (S J. + S. k+ 1) J + J, h 2
t
3
[ 4h p.-
= _--,,"J J w. k + h 2 J,
t
1 [ iii ] 2··· T + h (D )j-1 + 4(D )j + (D )j+l - h LT j + Tj+1 + 0.S(T j ,k_1
t
121
(A.6)
Collecting the terms and rearranging the order in Eq A.6, we obtain a
dynamic implicit operator:
where
+ d w. + e w. + f k-l J+l,k-l k-l J+2,k-l j,k
= -2(F. 1 + F.) J- J
O.25hR. 1 J-
O 5( T T) . • Tj,k_l + Tj,k+l J '
Ck+ l = (F. 1 + 4F. + F.+l ) + hI L' (Di). 1 + 4(D i
). + i J- J J t J- J (D )j+lJ
+ h2 :T + T + 0 5(TT + TT T T L j j+l • j,k-l j,k+l + Tj+l,k-l + Tj+l,k+l) J
122
TTl + O.S(T'+l k-1 + T'+l k+1)J· , J, J,
O.25hR j + 1 '
2 (Di) -1 a
k_
1 = -ak+ 1 + h
t ,
r ' -1.
(D i) j J ' bk
_1 = -bk+ 1 - L (D ) j-1 + h
t
L (D i). 1 + 4 (D i) , + i ~
2h3 (De) c
k_
1 = -ck+ 1 + h (D ) j+1J + J- J h
t ,
t
r • i -d
k_
1 = -d - L (D 1.) j + (D ) j+1J k+l ht
2 (D i ) e
k_
1 = -ek+ 1 + h
t
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APPENDIX B. DERIVATION FOR RECURSIVE SOLUTION OF EQUATIONS (Extracted from Appendix 2 of Ref 4)
Any of the fourth-order difference equations for beams or beam-columns
can be written in the following form:
f. J
(B .1)
Definitions of the coefficients a. J
particular beam formulation considered.
through f. will vary according to the J
The general process of simultaneous
solution of a complete system of such equations will be the same in any case.
The matrix formed by writing coefficients a. through e. at all stations J J
has non-zero terms along only the five main diagonals, and the most efficient
method of solutions, therefore, is a direct process that amounts to simple
Gaussian elimination. In a forward pass, two unknowns are eliminated from
each equation, resulting in a triangularized coefficient matrix of only three
diagonals. On the reverse pass, the solution is completed by back substitu
tion. The necessary recursion equations for the process are derived below.
Assume, temporarily, that w. 2 and w. 1 can be eliminated so that w. J- J- J
can be written in terms of deflections at two stations to the right. In
general
w. = J
Writing equations of this form for w. 2 J-
= A. 2 + B. 2w . 1 + C. 2w . J- J- J- J- J
=
Substituting Eqs B.3 and B.4 into Eq B.l,
and w. 1 ' J-
a.[A. 2 + B. 2(A. 1 + B. lW' + c. lW' 1) + c. 2w'J· J J- J- J- J- J J- J+ J- J
127
(B.2 )
(B.3 )
(B .4)
128
f. J
(B.5)
Multiplying and collecting terms,
(a.B. 2B . 1 + a.C. 2 + b.B. 1 + c.)w. J J- J- J J- J J- J J
+ (a.B. 2C, 1 + b.C. 1 + d.)w '+1 J J- J- J J- J J
:= f. J
(a.A. 2 + a.B. 2A. 1 + b.A. 1) J J- J J- J- J J-(B.6)
Equation B.6 can be rewritten in the form assumed in Eq B .• 2:
(B.7)
where
A . = D. (E .A. 1 + a.A. 2 - f.) J J J J- J J- J
(B.7a)
B. = D.(E.C. 1 + d.) J J J J- J
(B.7b)
C. :::; D.(e.) J J J
(B.7c)
and where
D. J
1j(E.B. 1 + a.C. 2 + c.) J J- J J- J
(B. 7d)
E. :::; a.B. 2 + b. J J J- J
(B.7e)
For beam and beam-column problems, the coefficients A., B., and C. J J J
can be thought of as expressing the physical continuity of the system. In
129
these coefficients all of the known input data are digested and stored. The
coefficients at anyone station depend not only on the load and stiffness
data at that station but also on effects from all previous stations. These
coefficients have therefore been termed "Continuity Coefficients."
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44!5"6!7$1*'*0!8$($.$9'.$/-!")':!
APPENDIX C. STABILITY ANALYSIS FOR THE DYNAMIC IMPLICIT OPERATOR WITH INTERNAL DAMPING COEFFICIENT
In studying the stability criterion for the dynamic solutions of a beam
with hinged ends and M increments, a solution to the equation of motion
described in Chapter 2 (Eq 2.1 or 2.2) can be assumed to be
w. k ],
in which A is a constant, j '" 0, 1, 2, ••• M, and k
If we consider only the flexibility of the beam
(C.1)
1, 2, 3, ••.• co •
F. , the mass density of J
the beam o .• and the internal damping coefficient of the beam (Di ) .• then . J J
Eq 2.2, becomes
133
134
3 4h p., . J ' \ 2 )w. k
h J, t
r 1 [i i ' + 2 1. (F J' + F J' + 1 ) - h (D), + (D ). + 1 J j w, + 1 k-l
t J J J,
i
L~ (D ) j+l-- F j+ 1 - h J W j+ 2 , k-l
t
(C.2 )
Assuming that the beam has a uniform cross section, uniform elastic
properties, uniform mass density. and a uniform internal damping coefficient,
Eq C.2 can be simplified as
(F + D)Wj _2,k+l - 4(F + D)Wj_l,k+l + 6(F + D)Wj,k+l - 4(F + D)
Wj+l,k+l + (F + D)W j+2,k+l + (F - D)W j _2,k_l - 4(F - D)Wj_1,k_l
+ 6(F - D)Wj,k_l - 4(F - D)Wj+l,k_l + 6(F - D)Wj,k_l
- 4(F - D)Wj+1,k_l + (F - D)Wj+2,k_l
(C .3)
where
where
Let
Fall F terms (EI)
terms divided by h t
p ~ all p terms
Substituting Eq C.l into Eq C.3, we obtain
(F + D) f (~ j)e(k+1)¢ + (F - D) f (~ ,j)e(k-1)¢ n' n
2 q
sin "' . k¢ t-' Je n
= sin ~ (j-2) - 4 sin \3 (j-1) + 6 sin \3 j n n. n
- 4 sin P 0+1) + sin ~ (j+2) n n
Then, dividing Eq c.4 by e(k-l)¢, we obtain
135
(C.4 )
136
(C.S)
Substituting the following trigonometric identities.
sin P (j-2) = sin P , cos 2p - cos ~ , sin 2 P n nJ n nJ n
4 sin \3 (j-1) = 4(sin 13 ' cos 13 - cos I3n j sin P ) n nJ n n
4 sin 13 (j+ 1) = 4(sin i3 ' cos ~ + cos i3 • sin P ) n nJ n nJ n
sin \3 ('+2) = sin P , cos 213 + cos 13 ' sin 2 p n J nJ n nJ n
into Eq C.S and simplifying the equation, we obtain
(C.6)
We can simplify Eq C.6 by using the trigonometric identities
cos 2p = 2 cos2p -1 n n
(cos ¢ _ 1)2 n
= 2 cos P - 2 cos P + 1 ,
n n
and dividing by the coefficient of e 2¢
thus
2¢ + I 2i , ¢ e L - 2 2 Je
4 (F + D) (cos P - 1) + q n
137
1 4 (F - D) (co s \3 _ 1)2 + 2 q
+ L n 0 (C.7) J =
_ 1)2 2 4(F + D) (cos \3 + q
n
Let
2 B = 9 and ,- 2 2"""1
L4(F + D) (cos \3n - 1) + q J
4(F - D) (cos fl _ 1)2 + 2 q
C = n
4(F + D) (cos P _ 1)2 2 + q n
Hence Eq C.7 becomes
= o (C.S)
Equation C.S is the quadratic equation which can be used to evaluate the
stability criterion of the dynamic solutions of the beam. The criterion for
the solution of w in Eq C.l, which is to be bounded as time approaches j,k ~l ~
infinity, is that the roots of the quadratic, e and e 2 , must satisfy
the condition that
(C.9)
The two roots of Eq C.8 are
=
138
and
= - B - J B2 - C (C.lO)
The conditions of Eq C.9 are satisfied if the following i.nequality is
true
(C.ll)
Substituting the equalities of Band C into Eq C.ll yield~:
2 g
IL4 ( ) ( p. _ 1) 2 + q2 J' 2 F + D cos ""n
4(F - D)(cos ~ _ 1)2 + q2 ------=n---~2-~2 (C.12) 4(F + D)(cos ~ -,1) + q
n
The above inequality can be reduced to
- 1)4 + 8F (cos ~
n (C.13)
By omitting the positive term 2 (cos i3 - 1) , the criterion for the stability
n of the dynamic solution of a uniform beam with hinged ends and internal
damping coefficient becomes
Since
2 2 2 2 16 (F - D )(cos ~ - 1) + 8Fq ~ a n
is usually less than F. J
for a reasonable time increment
(C .14)
2 length h
t and the positive value of the term 8Fq is always greater than
l6(f2 _ D2)(cos Q _ 1)2 the nega tive va lue of the term "" when the time incre-n ment length is extremely small, the above inequality is true.
!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!
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DBC5 GUIDE FOR DATA INPUT -- Card Forms Page 1 of 13
IDENTIFICATION OF PROGRAM AND RUN (Two alphanumeric cards per run)
IDENTIFICATION OF PROBLEM (One card for each problem)
Prob. No. Description of problem
I 5 II
TABLE 1. PROGRAM CONTROL DATA (One card for each problem)
Enter "1" to keep data from prior tables Number of cards input for tables
2 3 4 5 6 7 8 9 2 3 4 5 6 7
10 15 20 30 35 40 45 50 55 60 70
TABLE 2. CONSTANTS (One, two, or three cards; none if preceding Table 2 is held)
NUM NUM NUM of NUM of of of monitor moni tor Plot-beam Beam time Time Print- STA Itera- STA ting incre- increment incre- increment ing (print- tion (itera- method ments length ments length Switch ing) switch tion) switch
M H MT HT IPS>'~ MONS ITSW,'d MONI MOP
I I I I I I I I 6 10 20 26 30 40 46 50 55 60 65 70
8
75
Option for line or point plots LOP
75
80
so
so
9
80
-'. If printing switch is -1, use the following card, otherwise ignore: Page 2 of 13
Comple te print out at
time s ta interval
KPS Monitor s ta tions
I I I I I I I I I I I 6 10 2f 2S 30 35 40 45 50 55 60 65 70
** If iteration switch is not zero, use the following card, otherwise ignore:
Max num of i tera tions for any time step
MAXIT
Maximum allowable deflection WMAX
20
Deflection closure tolerance WTOL
30
Monitor stations
TABLE 3. SPECIFIED DEFLECTIONS AND SLOPES (The number of cards as shown in Table 1, none if preceding Table 3 is held)
Case STA -Idd: Deflection
6 10 16 20 30
tdd, I f case ~ 1, specified deflection ~ 2, specified slope = 3, specified both
Slope
40
Initial or permanent
switch lOPS #
I I 46 50
'If If lOPS 1, initial deflection or slope (or both) is specified.
= 2, permanent deflection or slope (or both) is s fled.
Page 3 of 13
TABLE 4. STIFFNESS AND FIXED-LOAD DATA (The number of cards as shown in Table 1)
Enter "1" if can't Bending Transverse Static spring Static axial Rotationa 1
From To on next stiffness static load support thrust restraint STA STA card F QF S T R
I I I I I 6 10 15 20 JO 40 50 60
TABLE 5. MASS AND DAMPINGS (The number of cards as shown in Table 1)
Enter "1" if can't Internal External
From To on next Mass density damping damping STA STA card RHO DI DE
I I I I I I I 6 10 15 20 31 . 40 50 SO
TABLE 6. TIME DEPENDENT AXIAL THRUST (The number of cards as shown in Table 1)
Beam STA Time STA Axial thrust values (see page 7) from to from to TTl TT2 TT3
[ I I I I 6 10 15 21 25 30 36 45 55 S5
70
Transverse couple
CP
90
TABLE 7. TIME DEPENDENT LATERAL LOADING (The number of cards as shown in Table 1) Page 4 of 13
Beam STA Time STA Lateral Loads (see page 9) From To From To QTl QT2 QT3
4S 55 $5
TABLE 8. NONLINEAR SUPPORT CURVES (Three cards per curve, the number of cards as shown in Table 1)
Enter "1" if Symme-
From To cont'd Resistance Deflection Num try Deflection Resistance Beam Beam on next Multiplier Hultiplier Points Option Tolerance Tolerance STA STA card QM.P WMP NPOC KSYM WNTOL QNTOL
I I I l I ... I I 6 10 15 20 30 40 45 50 60 70
Resistance Values Q ~ __ ~~ ____ ~ ______ ~~ ____ ~ ____ ~ ____ ~~ ______ ~ ____ ~L-__ ~~ ____ ~
Deflection Values W ~ __ ~~ _____ ~ ____ ~~ ____ ~ ____ ~ ____ ~~ ____ ~ ____ ~L-__ ~~ ____ ~
TABLE 9. PLOTTING SWITCHES (One card for each plot, the number of cards as shown in Table 1)
&. e ~
Horizontal axis
Enter either "TIME " Enter either ''DEFL II Enter either a beam axis was specified.
Vertical axis
or "BEAM" or ''MOHr'' station number
Beam Hultiple
plot switch
if TIME horizontal axis was specified or a time station number if BEAM horizontal
5 of 13
GDThe consecutive plots, which are entered 1 for Multiple Plot Switch, will superimpose on common axes. The last plot of the group of plots which are superimposed on common axes must be entered 0 for Multiple Plot Switch.
For consecutive plots which are entered 0 for Multiple Plot Switch, each plot is plotted separately on different axes.
STOP QL~ (One blank card of the end of each run)
GENERAL PROGRAM NOTES Page 6 of 13
The data cards must be stacked in proper order for the program to run. A consistent system of units must be used for all input data, for example, lbs and inches. All words of 5-spaces or less are understood to be right justified integers, whole decimal
numbers, or alphanumeric words. ~ or ~ All words of 10-spaces are right justified floating-point decimal numbers. E4.231E+OU
Table 1. Program-Control Data
For each of Tables 2 and 3, a choice must be made between holding all the data from the preceding problem or entering entirely new data. If the hold option for any of these Tables is set equal to 1, the number of cards input for that Table must be zero.
For Tables 4 through 9, the data is accumulated in storage by adding to previously stored data. The number of cards input is independent of the hold option, except that the cumulative total
of cards cannot exceed 100 in Tables 4 through 7; 60 in Table 8; and 10 in Table 9. Card counts entered in Table 1 should be rechecked carefully after the coding of each problem is completed.
Table 2. Constants
Variables: Typical Input Units:
H in
HT sec
WMAX in
WTOL in
The maximum number of increments into which the beam-column may be divided is 200. Typical units for the value of beam increment length are inches. The maximum number of increments into which the time axis may be divided is 1000. Typical units for the value of time increment length are seconds. If the printing switch (IPS) is set equal to -1, only results at monitor beam stations will be printed, except at every time station interval specified (KPS) where results at all beam stations will be printed. If the printing switch (IPS) is set equal to +1, the complete results of all beam stations will be printed at every time station and the second card therefore is not necessary. If the printing switch is equal to zero or left blank, no intermediate results will be printed. MONS is the number of specified monitor beam stations which are entered on the second card from column 21 to column 70, for printing the monitor stations results at every time station. The maximum number of monitor beam stations which may be used is 10.
Page 7 of 13
The second input card should be omitted when the printing switch (IPS) is set equal to zero, blank, or +l.
For nonlinear support curves, which are stated in Table 8, the switch (ITSW) is set equal to either a negative or positive integer, for example, -lor +1. If ITSW is set equal to zero or left blank, no nonlinear support curves may be entered in Table 8 and the third input card must be omitted.
MONI is the number of monitor beam stations, which are entered on the third input card from column 41 to column 65, for printing monitor deflections during the iteration process. The maximum number of monitor beam stations which may be requested is 5.
MAXIT is the number of iterations to be allowed for this problem. The maximum number is 50. WMAX is the maximum allowable deflection for this problem. WTOL is the closure tolerance for the iteration process. MOP is a switch for selecting the method of plotting the results of beam or time stations which
are stated in Table 9. If MOP is set equal to -1, microfilm plots are made. If MOP is set equal to zero or left blank, printer plots are made. If MOP is set equal to 1, l2-inch standard ball-point paper plots are made. LOP is an optional switch for choosing point or line plots for the microfilm or paper plots. If LOP = -j, point plots are made at every jth point. If LOP = a or blank, line plots are made. If LOP = j, line plots with a point plot at every jth
point are made. If no cards are entered for Table 9, MOP and LOP may be left blank. If MOP is set equal to zero or left blank, LOP may be left blank.
Table 3. Specified Deflections and Slopes
The maximum number of beam stations at which deflections and slopes may be specified is 20. A slope may not be specified closer than 3 increments from another specified slope. A deflection may not be specified closer than 2 increments from a specified slope, except that
both a deflection and a slope may be specified at the same beam station. lOPS is a switch for specifying either an initial deflection (or slope) or a permanent deflection
(or slope) at the beam station stated on the same card. If lOPS is set equal to 1, initial deflection or slope (or both) are specified for the static
solutions only. If lOPS is set equal to 2, permanent deflection or slope (or both) are specified, both for the
static solutions and the dynamic solutions.
t-' lJ1 LV
Table 4. Stiffness and Fixed-Load Data
Variables: F
Typical Input Units: lb X in2 QF
lb
S
lb/in
T
lb
Page 8 of 13
R CP
ioxlb/rad ioxlb
Axial tension or compression values T must be stated at each beam station in the same manner as any other distributed data; there is no mechanism in the program to automatically distribute the internal effects of an externally applied axial force.
Data should not be entered in this table (nor held from the preceding problem) which would express effects at fictitious stations beyond the ends of the real beam-column.
The left end of the beam-column must be located at station O. For the interpolation and distribution process, there are four variations in the beam station
numbering and referencing for continuation to succeeding cards. These variations are explained and illustrated on page 11. There are no restrictions on the order of cards in Tables 4 and 5, except that within a distribution sequence the beam stations must be in regular order.
Table 5. Mass and Damping Data
Variables: RHO
Typical Input Units: lb 2/. x sec 1.n
See description in Table 4.
Table 6. Time Dependent Axial Thrust
Variables:
Typical Input Units:
TTl
lb
DI
lb X in2 x sec
TT2
lb
DE
lb X sec/in
TT3
lb
Time dependent axial thrust values "TTl" and "TT2" are the distributed values entered at beam stations "FROM" and "TO", respectively, of time station "FROM". The values "TT3" are the distributed values entered at beam stations "FROM" of time stations "TO".
All the values of time dependent axial thrusts at the beam stations within the two limits of "FROM" and "TO" of the time stations between the two limits of "FROM" and "TO" will be linearly
Page 9 of 13
interpolated by the program. There are no restrictions on the order of cards, except that beam stations which are entered as ''FROM" and "TO" in the same card must be in ascending order, or in other words, the same beam station is not allowed to be entered as lIFROM" and liTO II in the same card.
See page 12 for illustrated examples of interpolation and distribution.
Table 7. Time Dependent Lateral Loading
Variables:
Typical Input Units:
QT1
1b
QT2
lb
QT3
lb
All restrictions stated in Table 6 are also true in this Table, except that same beam station is allowed to be entered as ''FROM'' and liTO" in the same input card.
See page 13 for illustrated examples of interpolation and distribution.
Table 8. Nonlinear Support Curves
QMP is a constant which is multiplied by Q-VALUES to obtain the vertical resistance values of the corresponding points of the resistance-deflection curve.
WMP is a constant which is multiplied by W-VALUES to obtain the horizontal deflection values of the corresponding points of the resistance-deflection curve.
NPOC is the number of points input on the resistance-deflection curves. KSYM is the symmetry option of the input resistance-deflection curve. If KSYM is set equal to 1, then the total number of points on the curve will be twice the value
of NPOC. In this case, the point of origin of the curve should not be specified. If KSYM is set equal to ° (or left blank) or -1, the pOint of origin of the curve must be speci
fied. In this case, the deflections which are stated as the first pOint and the last pOint on the curve must be equal but opposite in sign.
If KSYM is set equal to 0, the support is a negative one-way support; that is the beam will lift off the support when it deflects upward.
If KSYM is set equal to -1, the support is a positive one-way support; that is the beam will lift off the support when it deflects downward.
There are no restrictions on the order of support curves, except that within any distribution sequence the beam stations must be in regular order. More than one curve may be placed at a beam station.
Page 10 of 13
The maximum number of points which may be input on any given curve is 10, although udditiona1 points up to a total of 20 may be created internally if the curve symmetry option is exercised.
WNTOL is the closure tolerance of deflections for matching up the unadjusted part of the resistance-deflection curve when the original curve needs to be adjusted.
QNTOL is the closure tolerance of resistances for matching up the unadjusted part of the resistancedeflection curve when the original curve needs to be adjusted.
For any particular resistance-deflection curve, the final deflections of the points in storage ~P times W-va1ues) must be increasing positively, while the final resistances (QMP times Q-va1ues) must be in descending or equaling order. In other words, the resistance-deflection curves must be continuously concave as viewed from the horizontal axis, except that horizontal lines of zero stiffness can be input for representing the plastic regions of the curve. Softening of the support is not permitted; that is, no reversal of slope sign of the segment on the support curve is permitted.
For both one-way (negative and positive) supports, the input order of the points on the curve is the same, e.g., the final deflections of the points in storage must be increasing positively; internally, the program reverses the order of the pOints on the positive one-way support curve after the necessary information for tracing the loading paths has been retained.
If continuing on next curves, the deflections must be equal for the corresponding points of each curve in the sequence of continuation. In this case, WNTOL, NPOC, and KSYM also must be equal.
The resistance-deflection curve at every nonlinearly supported beam station within the two limits of "FROM" and "TO" will be linearly interpolated by the program.
The maximum number of beam stations which can be nonlinearly supported is 100.
Table 9. Plotting Switches
Four kinds of plots, deflection or moment along time axis, and deflection or moment along beam axis may be requested.
If the horizontal axis is TIME, a beam station must be entered from column 26 to column 30. If the horizontal axis is BEAM, a time station must be entered from column 26 to column 30. As many as five plots may be superimposed by using the multiple plot switch. Superimposed plots must all be of the same kind. There are no restrictions on the order of cards except that beam or time stations must be in
regular order within a sequence of the same kind of plots.
IndividuOI- cord InptJt
Case 0.1 Data concentrated at one sta ..
Case 0.2. Data uniformly distri buted .
Multi pie - card Sequence
Case b. First -of - sequence
Case c. Interior-of-sequence ........ .
Case d. End - of - sequence ....
ReSUlting Distribution of Data
--STIFFNESS F I
I I I
-
-
Sto: 5 10 15 20
LOAD a -
frTll - -
Sic: 5 10 15 20
(;ONT'D FROM STA
TO TO NEXT STA CARD P
I 7 ..2...7 10:NO I r 5 ~15 I 0 -=1/0 I I 15 ----',---P 20 IO-NO I I 10~20 IO=NO I
~~ I I=Y£S I
1 1&0 ll=YES I I I 5 11 =Y£sl
I I 40 IO=NO I
4-
3
2 /
1 /
" /
25
3 -
2 - ,--, 1 -
25 30
F
I 2.0 I 4.0 I
I
0.0 I 4.0 I 2.0 I 2.0 I
35
Q
3.0
1.0
2.0
2.0
2.0
0·0
etc ... I I I I
J.
I I I I
40
-t
• ~
e o
Page 12 of 13 TABLE 6. TI~!E DEPENDENT AXIAL THRUST (CONTINUED)
The variable TT(J,K) is input at any bar-number and time station by specifying the value of axial thrust distributed at beam station "FROM" and time station "FROM" in the columns of inputting TTl, the value of axial thrust distributed at beam station "TO" and time station "FROM" in the columns of inputting T12 , and the value of axial thrust distributed at beam station "FROM" and time station "TO" in the columns of inputting TT3 .
Beam Sta Time S ta From To From To TTl T'1'2 TT3
0 5 0 5 4.0E+00 3.0E+OO 2.0E+0 10 15 2 2 2.0E+00 3.0E+00 2.0E+00
8 9 7 9 2.0E+00 2.0E+00 3.0E+OO 8 12 9 9 3.0E+00 1.OE+00 3.0E+00
15 20 6 7 2.0E+00 -2.0E+00 1.OE+OO
12 ----TT 1 r ----
10 ----
TTI=4
Beam Sta \J)
• 0
6
e 1/1
Page 13 of 13 TABLE 7. TI~!E DEPENDENT LATERAL LOADING (CONTINLTED)
The variable QT(J,K) is input at any beam station and time station by specifying the value of lateral load distributed at beam station "FROM" and time station "FROM" in the columns of inputting QTl , the value of lateral load distributed at beam station "TO" and time station "FROM" in the columns of inputting QT2 , and the value of lateral load distributed at beam station "FROM" and time station "TO" in the columns of inputting QT3 .
Beam Sta Time Sta rom To From To QTl QT2 QT3
0 5 0 5 4.0E+00 3.0E+00 2.0E+0 10 15 2 2 2.0E+00 3.0E+00 2.0E+0
8 8 7 9 2.0E+00 2.0E+00 3.0E+0
+-' 8 12 9 9 3.0E+00 1.OE+00 3.0E+00
15 20 6 7 2.0E+00 -2.0E+00 1.OE+00 \.. ~O
C:j QTI 4'l.
II ---- ~,~
OT
OTI=4
• 0
tJ.
• e
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IiL[~ ~A~ LINE6~ (0 O~ BLA~K), ~r;~lI- ,,1JUI NEA_ CR ROTHlnTHER THAN o} ~u.PORTS 07JUI
OOUTING SwITCH TO INDICATE THAT TH~ ~-. 07JUI C' AT ~TA ,EFns TO BE AOJUSTEO G7Jui
I~DEPE'nENT 'ARIA~LE IINDEXINGI 07JUI EXTEONAL TIME OR REA. STA INPUT I~ TARLE 007JUI INTE.NAl HEA~ STI 07JUI AROAy CON1AINI~G THE MONITOR STA FOR 07JUI
PRINTING THE TTERATIO' OITA 07JUI INTEON'l 'C',pOO qA NUH 07Jll INOEI OF NLM OF pnlNTS nF THE PRovIOes 07JLI
PLOTS TO RE ,CCUHUlATED nN ~~'T Ple. 07JLI A"NAY CONTII.ING THE MONIT~R STA FUR 07JU]
P~INTING T~E rnMPUTEO ~fSULTS ~7JUI INTE~NAL BEAM 5Tft ~nQ SPECrFIFO rO~0IT!O~C:C1Jul
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ALLU.ABlE nE~lErT!nN 07JUI
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NEXT CAIlD I TA8LES ~. 5. AND 8 I 07JUI SAVEO VALUE pF .SAVI ) Fell EAtH pLOT FRAMF07JUI TEMPORARY VALUE OF KASE I I 07JUI A~RAY CONTAINiNG f..TfR~AL aEA~ OR TIME STA07JvI
FOA THE TITLfS OF PLOTS 07JUI ROUTING SWITCH IN TARLE~ " 5 .N~ 07Jvl
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Q7JUI 07 JUI Q7JlIi 07JUI 07JIJI 07JUI 07Jl'1 07J"1 Q7JUI 07JUI 07 )UI Q7JUI 07JUI 07JLJj 07JUI ,,7JUI
NUM OF MONITO~ STaTIONS FOR PRINTING ITEkATION OATAlwA •• ~I
TWE 07JUI 07JUI
~U~ OF MONITOR STATIONS FOq P~I~TING SULTSIMU • in,
Rr- 07JUI
OPTIO.AL PLOTTI.~ uET·0~ SWITC" (-l:::l"'IC~OF'Il''', "I 1R f,':jLA .... ~:;>::'::.I ... TFQ, ·l·tlALL-POI~Tl
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PARTICULAR TARLE 07JUI ~U" OF C~RCS INPUT RFPEATEOLY IN TABLE 9 07JUl INCE' USED I. TME ITEIlATIoN CO LOOR 07JUI ITERATION ~U.PER AT WHICH THE LO.a ITERA- Q7JUI
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ITE~ATI"N~ 07JUI NU. C' SUPPORTING STAS W"ICH .AV, NO~_ 07JUI
LINeAR RESIST'NCE-DEFLECTION CURVES 07JUI ROUTING SWITCH WHiCH TELLS TMF PRO~RA~ 00 07JUI
O~E ~~~E CYCLE OF LOA~ ITERATIONS 07JUI W~EN THE SOLUTION DOES NOT cLOSE 07JUl
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NU~ ~F SLOPE_ ON THE NONLINEAR RESISTANCE_07JUI DEFLECTION CY AT A PARTICULAR ST. J 07JU!
INOEX 1'0" TOTAL "Uu of pLOTS 07JUI INDEx 1'0" ~u .. 0" "LOTS nF HENnINt; MO"E"TS 07JUI
'LO~G T"E REAM AXIs Q7JUI RouTING COUNTFR FoR THE NUH Of pnI~Ts ON 07JUj
THE NCNLINEAR RESISTANCE-OEFLECTION 07JUI CV .~TCH IS t~CREA~ING ITS ~PECIFIEO Q7JUI VALuE OF DEFLECTION FRO" NEr. TO ZERO 07JUI OR FIlCM NEG TO POSITIVE 07JUI
TEMP VALUE OF ,",UM OF POINTS ON suP CV 07JUI T"TAl 'UN OF POI~T~ ACTUALLY ON TME NQ~- 07JUI
LINEAR RESISTANcE-oEFLECTIO~ CV AT A 07JUl OAHTICULAR ST> J 07JUI
~U" OF PUI~T~ I~PUT I~ TABLE ~ Tn DEFINE A07JUI PIPIICULAR .nNLINEAR RESISTAN(E- 07JUI 'EF~E(TInN CIIRVE 01JUI
PHOSLE" NUY8ERIPROG STOPS IF • BLANK) 07JUI 1"0[ •• HIe .. INDICATES T"AT T~F 07JUI
OEF~ECTION AT A PARTICULAR ~TA IS 07JUI WITMI~ THE SFGMENT CO"NECTEn BETWEEN 07JUI POI~T NPS ANn POINT INPS-ll ON THE 07JUI ~ONLI~EAR CV OTJUI
INOEI FOM .U~ PLOTS 'lO~G TI~F AX!S 07JUI ~U~ OF P~INTs ON TME NONLINEAR RESISTANCE_07JUI
OEF~ECTION CV AT A PAQTICULAR STA J 07JUI POUTrNG (OUNTER WHICH DEFI~ES TM, 'UM,F 07JUI
~LOPES SPECIFIEn ON T~E PAP~ICU~AQ Q7JUI NON~I~EAP RF<I'iTANCE-nEFI cv 07JUI
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!NOE, '0" FOR SPEtIFIED CONOrTlo~S a7JUI ~UM OF O>CILl61ING CYCLES OURTNG T~E 07JUI
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(--
I I I I I I I + I I I I I I I I
r-I I I + I I I I I I I I
GENERAL FLOW CHART OF DBC5
.--------~------~ [READ & PRINT Problem I. D.I
I IREAD & PRINT Tables 1 thru 81
I Distribute and generate all the
I nonlinear curves of the sup sta
I Retain slopes, Qdrops, Wdrops ofl the segments on each sup curve
I IREAD & PRINT Table 9~
I I~nterpolate & distribut~1 data from Tables 4 and 5
I "k DO requested number of time sta
I Compute time variant axial thrusts and la tera 1 loads for the present time s ta k and time sta k-2
I - DO requested number of iterations)
I - DO for each nonlinear sup )
I ( 1
First cyclel Second cyc le 1
I Inter pola te to Sup. curve adjust-find resistance ment necessary'? stiffness & load
rYeS from sup curve Find resistance I stiffness & load from new curve
)
I ----·1 CONTINUE)
I
175
Terminate if Prob. No. 1.S blank
CALL S UBROUTINE 3 INTERP
''':/: See d flat.;r
pages
No
eta i led chart on
152 to 160
176
I I I I I t I I I I I I I I I I I I I I I I l
I I I I I I t I I I I I I l
I f
compute matrix coefficients, recursion or continuity coefficients,
.and deflection at each beam statiol
I Icount number of sta. not closed and number of sta exceed allowable deflection
maximum
Set up monitor data for this iteration
!Test closurance of deflection
Yes
------
Compute slopes, moments, shears, and ] sup reactions for the revious time sta
Retain deflections and moments for plot~
PRINT iterational monitor deflec tions and computed results
----------
Plot results if requested in table 9
Go back and READ new problem number
Go back for the second cycle of iterations
GENER.A,.L FLOW CHART OF TUlE DO LOOP
I Initialize KPC == 0 KK == 0 NPBT 0
I Clear storage for variables: IFUS ( ) , QI( ) , SS ( '; , , SSKMl ( \ QIN1( ) , & QLH2 ( ) } ,
I ,.------- DO 7000 K == 1 MTP4 ) f I I I I I I I I
, I
Initialize & clear storages
MOel == 0 ITeS 0 ITCSR == 0 lOCNT ;:: 0
I KK=KK+ 1
I I Compu te . dynamic axia 1 :hrus tsl & dynamlc lateral loads for !ti.me Sta. K & time Sta. K-2
I I I
______ 69_1_0 DO 6000 1 NIT == ,MAXIT r ~----~~--~
I I I
~\
Initialize for next iteration QI(J) Q(J) + QT(J,l) S S ) == S (J) - - J == 1, NP 7
YES
t I NIT) (linear case)
I I I f I I I I I I I I I I
(----
i I P® I I
DO 6056 NN t\NC
NP1S J NOS NPCT
ISTA(NN) NOSJ(NN)
== NPTTS(NN)
6504
177
If linear prob set MAXIT := 1 Q(J) == static load QT(J,I) dynamic
load at time Sta K S(J) linear spring
constant NCV8 == num of curves
in Table 8 NNe '" Total num of
nonlinear sup
178
I I I
I I I I I I I I I
~II I (NIT~
II ~ t t t I 1\
I I I I I \ I I I I I I l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
IS ABS (W (J ,KK)) "
ABS (W (J , KK - 1 ) )
669
YES
Initialize for the firs t load iteration
W(J,KK) = W(J,KK-l)
YES \First cycle)
YES
6086
6504
0910
IOSW \ J) Switch for load iteration
1,.JPT (NN, 1) deflection value of the firs t pt. on the sup curve
I 179
6910
I I
Yes
I ?
~ I Yes
I I CDI No
I I I I t
: FI I I I t I ~ No
I I I I I Yes
I I I I
Determine the location
I of the greatest slope
I (WPT (NN, NP)
I I
I Yes
I I one-way
I support)
I I
No
I I
Yes
I I I I I I
6086 6 6504
180
I
I I
I I
I I
t I I I b0l I
I + I FI I
~
~ I
I
Yes
Yes
605
IUS(J) IUS(J) + 1
Determine the resistant support stiffness at previous time station
IS the previous resistant stiff- Yes ness within elastic range?
No
Retain the deflections and the resistances of the points for the unadjusted part of the supcv
Revise the portion of the new path of the sup curve
Connect the new path to the unadjusted part and renumber the order of the aupport curve
60 6
6910
04
I US (J) = S wit c h to indicate tha t the sup cv is revised
I I I I I I I
I I t t I I
FI I I
I I I I I
~ I t I I
I Interpolate to find re-sistant stiffness and load value from sup cv
6086
IS NO r--...... IOSW (J) == 1 YES
Use spring-load (tangent modulus) iteration Md.
?
Iuse load it-I eration Md
\ 6056 '--- - - - -'1 CONTINUE )
INIT) YES ITCS?== 0 ~
6504IS
I I I I I I
I I I I I I I I I I
NO
IS there a sup sta NO where the Q-W curve }---...... has been revised?
Set ITCS ITCSR
6505
YES
== 0 == 1
Icompute matrix coeffi- I cients at each beam sta
Compute recursion or continuity coefficients at each beam station
Revise continuity coefficients for specified conditions
6175
181
182
I
I b50S
I I I
I I
~I I I i INIT)
I i
Compute deflection at each beam sta
YES
'2 (Linear case)
NO
Count number of stations exceeding max allowable deflection and number of stations not closed
YES
YES
IS there a sup sta where the deflection oscillates during the iteration process?
YES
Set switches for Load Iteration Md. IOSW(J) 1
YES
Check the deflections are within the ranges of adjusted parts of sup curves or not? If not, revise them
Retain monitor deflections for this iteration
6000
6175
6910
I
I I I I I I ~I I I
6505
YES
NO
I I NIT) IOeNT
YES
I I I I t t
IOeNT IOeNT + 1
YES
Use the largest resistant stiffnesses and iteration loads of new curves
YES
YES
Initialize for starting the second cycle of iterations
183
6910
184
I I I I I I f I I I
F I I I I I I I I I I I I
I I I I I I
I I I I INIT)
I I t I
YES
PRINT iterational monitor deflections
YES
l 6000 ----------------
YES Set IOSW (J) = 1
Set switches for printing error messages of not closed within the specified num of iterations
6175
YES
Compute slopes & shears oj
the bars and moments & sup reactions at the joints for the previous time station
Retain monitor deflections or moments for plots
PRINT monitor or complete results for the previous time station
PRINT monitor deflections during the iteration process
6910 Do one more cycle of load iterations for double check.
I I
~ I t I I I I I
YES
Reset W(J,I) W(J,2) KK == 2
NO
H(J,KK-I) h'(J,KK)
l 7000 '--------------
Plot deflections or moments along either time or beam axis for
I the reques ted Imonitor stations
PRINT ERROR message of not closed and STOP
GO back and READ new prob. no.
185
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GO TO ",q99 1:135 IF I .' "'Ck.
) 998~. 1431. 1400 1417 P!:"INT 903
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0(.051. J"151"). IN('51"1, ~Q?I~J. R .. ONt(~" DI"'2INI, UEN2(N) 4<SIo.~{'" J:: 1 • KI'2'1(NI ., 2 • KRt 1(~1 2 "172:- i"')
H ( t<S.13i"\l - 2 ) 1550, IS50 POI>'T '11' r~I'I'). 1~25INl.
~" TO 1<70
1560 INI, R~ON2(N)' OIN~(NI' DENtlhl
1~'5S PPINT 'P. ''15''', KR2SiN), RHO~2INI, OI~2(~). OE~2INI GO TI' lQO
156{) PPlr-.T 413. Tf\2~{r...). 1o(~25(N). P~Ci"'2U'J). Ol~2'N). OEN2tNl 1570 CO'TI'IJE
C-----I"'PUT T'~lE 6 l~(ln PPINT 6(""
I~ , KEF~6 J 9'190' I~OI' l~IU
I~OI
0~N04 2~sn 04JAS OSSEo 04JA5 04JAS 05sEO 04JA5 04JA5 OsSEO 04JAS O~SEO O~SE 0 O~SE 0 O<;Sf 0 OSSEO OSSEO U'5SEO O!iSEO Oo;SEO 05sEo 055[0 OSSEO O~SEO OssEO OSSEo 0l'>5F"0 OSSEO 055E 0 OS5~0 OSSEO O~sEo OSSEO 05SEo 055[0 0'55£0 O~sEO O!SSEO O!!SEO 055EO 055EO OSSEO 05sEO OSSEO 055EO OSSEO OSSEo OSSEO 055£0 OSSEO 05SEO 05SEO OSSEO O!s~EO
~CH 'CD6 GC' Tn 1~3'
1610 PPIU %S IF ( ~rTe - 1 ) It?~. 1"11. '~II
16Jl CO In?5 ,= 10 'CT~ p W 1 !'Ioi T 61!). ! T 1 ! I" 1, r 1 ? r N 1. I< r 1 i ~j l. ~ T 2" {'. 1, TTl (~), T T '2 ! f\', ~ 'f'? 1\:
1625 CO"!'UE 1620 C!,UINtJF
POI,T qn Nep ... ,..CT6 • 1 NCT~ ,..CT6. ~C06
1~10 '"' ~rll - 100 I 16]5, 1.]5, 163] I.]~ PRINT QC.
Gf'l Tr. QQCG
1~15 I~ I ·.C~f ) qQPO, 1611. l~'O 1.37 p\.Ir,T ·.0]
G(, TO I 100 16-0 If I ,rTf - -C16 ) 1100, 1'05, 1605 1.05 Cf 1'70 'r NClf, NCT.
PEfil' el. !,; I'), 112ff,}. I\tl (~,) t f(T2tN}, TTl (r.). TT2{~1. Tr3,,,,) PAT..,T ~l;)t TTl(",~, IT2{N}, KT1(NI, ,\l2(1\) , TTlH,j), TTZt"', r,"::(I\,
1'" 7 fl: (( r. T 1 ",-\'~ C-----TI'.PI-l r'~'_£ '1
1700 rr:!Tr-..T 1(-\) IF , 'rEP7 ) QQBO. 1701' 1710
1101 'C17 = 1
1710
1711
1725 1126
Nell C" !'>-CC7 GC rr. Inc.
P~T~T 905 n ( NC T7 - I , I 176, 1711. 171 I ~o 1725 •• I •• CT7
p~r~T ~lf'1t rrnu-). 1('!2fN), K~ll1'd. f'02Hdt QTl{N). QT211\), CT~(~) CChTINtJf CCt\.TT',lIf
P};INT Q10 f\I(,~1
C T 7 :::: t>(Y1 • 1 • r T 1 • r-.c 0..,
1730 IF ( _C.7 - IQ~ ) 17,5, 17'"' 1733 1711 PO I NT 90'
GO TO qQ99 17]5 IF ( ~cr7 ) ,g~O. 1737, 1740 17j7 PRINT
1740 1745
GO 1'0' IF ~rT7 - ~Cll ) lAOO. 11'~' 1145 no 177~ ,: NCll, NcT7
;:;>FAf) fSl~ 1(,)1P'). lQ21f1.1. KQ1(N~. KCC(f\j), (.;TIU"), I..:T2("")1 QT3;rd PRINT 61th rr.1P,}" IG2{N), Krnrl'lt •• I<Q?(.'''t OTIC"'}, QT2(N). W~3\t<.)
1770 rO'TI~~ c----- TNP{'T T ·'RlE P IPOO PQINT ~~~
IF I KEEP' 1 9Q80. 1"01' 1"10 1"~1 ~C'P = 1
~.(V9 ::: f\CT8 I
r.o TO 1 p?"
05SEO a-SED 055£0 O=SEO O~5EO 05SEr Q~<;EO
0<;5EO 05S,0 OSSEo O'S[O O-SEO O~SEO 0:5EO O~SEO C<SEO OSSEO OS5EO OSSEo O~<FO
C"EC 05~tO 05<,,0 O'Sfe 05SEO O'<EO O~<Eo 05<EO O:Sfo O~5EO C'SEO O~SEo 0.5Eo 05SEo OSSEO O~SE 0 OS'iEO O~SfO 05SEO 05SEO OSSEo OSSEO 05,[0 C!lSEO OSSEo O~SEO OSSEO 05SEO OSSlO 200EO 2.of 0 2.rEO 24nEO 2.Dfo
IFlIO PQIi\T q(,!1I;
TF' t 'r~p ... 1 ) lP?e. l A 11- 1811 1"11 UI"?'\ '=h,(vb
IF ( .5 .. ·PI,) - 2 ) 1013. 1017 • 1821 1~11 pl..rr"T ~ll, f!\P:tO.ltl!'·t1P("')~ K~?"\"-l' [JMP(N), \IIIMP(t~), f\p:JC{!\),
1 ~S)'~(~j, ",fl.TOl ir-,il, r.l\l(iL(N' r:f' T () 1 ~:?4
tA17 Pr.fT"T '31?, P·IAU). ~~2,g!t,). G;~P'{I'd, YlMP(Nl. I\p("IC{I>.n. KS'(/'Il!!\l. 1 ""p.,TCLtN)Y C~d(')L(~)
GI'" 1'1 1 F\?4 1$121 P~T"'1 811. !f\2PH.), 1(,j;?,g(N), QPlPH.:) , w""P'(I"<I). I\p~e(""lt K~,("(~)'
1 ",,,TOLrNl. Gf\T('lLir...) 1A24 ~~PCT e "POr:(p.,)
PPT"T A14. { t;PH .f\Pj f 1",:1' 1. ,PeT pRI",.T 8jf!;, (\'oP(~\.~,P), 1",:1' 1, fI.'FeT
IS?'\ (n~'I~UF 182& ce~TIplc!:
POINT 910 ~·,C 1 P NCV~
c~cv~·l
~ r Ii R • NCO,. I '? IP]O If ( ,evA - >0 1 1'3~. 183'. 1833 IAj] ."INT 900
Gn TO qQ,q l~~S IF ( ~,l~~ , Q~PO. lAl7, lA.O 18'7 P~I" 903
1840 GO '0 I A13
t(°1 • 0 IF I ~r.p - ~C18 I 1873. 1805. I~45 ~n JRln "'- = I",:cte. hr~~
DE.or Ph p'}!,:?O,i1 f"'?PI"'-I. KR;U:I/"'l, QMP(Nl. W;"PtNl. NPOCOo. 1 IlS'Y;" (~). wN1Cl ('d. t.iNT('J:L (N)
t:P(T :I ",P("IC("'-) QFlI F
. R2, ( c-Pt-",~PI' ",-p = " ... ,pc T ) Pt40 fl2~ ( W'~ f~,,~,P)' ~P • 1~ NPCl I
KSw2 I",,} • 1 • KRlfHN} • ? • KRl KRl • 1'i~2p(",-)
tF : y'S\>\P (f\l .. 2 ) lA5f.,. 1.'155, l~&O l~!O PRP;T 8\1, P'lq(~·), l"'-?RlN. t I<'P2tl(~). Q~Ptl'll" w~PH.,j), ~~OC(I\,j).
I KS'~"lo ""TOl(~', 0.,101..("1) GO Tr 1~11
1~5S PR1",T f:ll? tft,.19(",. t<P28iNl. Qt.'P(N), lifMP(h). ~POCH.j), f(~Y~(""l, 1 Yi'" TGL {N), Q" .. TfiL (1",:)
GC TO IP7l 11'1;":0 P~!"'T 813, tl.2PO'-) f K~7eUi), Q,.'1:HN}, lif~PP")' f\POCtr'Jl, j(~YIii'(Nl.
1 lI/",-TOLtN}t GNTflL("-) 1871 ~,P('TE"P('lC(",
P~INT fil"" r: QP(."t\Pl. IIJP ~ 1, Nf.'CT p~II",:T AI~, ( ~P (II; ,,,Pl, ~jp = 1. NrCT
1I:l10 r.;u" dNUf-1873 IF' NCV •• E~. 0 ) GO TO IQOU
C ... ----rn51~If'1.ITF ~L:P_Pf"lCT CL''::VE(;; (('\ 1 s;q~ f', c 1. !\CV8
NP(T II "pr;C(t-.i DO lARO f\P s 1. t\PCT
.oe<e 2.rEo 2·C~C .<C<~ 2'~EO (""EO 2OCEO Z4CEO ISFE I 2.rEO 2.r"~ 1~<E1 24!':£O 24f)EC 2<OEO 20D~O 20DEO 200EO 20DEO 'OCE 0 2"~~ 0 24rEo 24DEO 20DE 0 240£0 15<£1 20DEo 20Dcr 2_0EO 200EO 2 4 0£0 2.rlEO 20llEO 24DEO 2_0EO 2_0EO 2. e!:o 20DEO 240EO 240£0 240£0 I~HI 24DEO 241l£0 I~Fn 2_r,E 0 24DEO
!~~!~ '''H)t;.U
ISHI 20DEO 200"0 cOCEO 24DEO
QPV("' ..... ~) • CP("'JtNP) • l";",,PO~t WOVt~a~'~t = .P(~:."'D) 0 ~MPINI
1860 C0~T'~0E TF I "PO - 1 I le~5. IPo~. 1895
I BPS PQ TId qq~l
GO ,~ 9999 c-----c~EeK POI~'s 'r~ PoOFER 0R~.p
1891:, r(' p:1Qfl 1\P 7. f\PCT If ( ~P~c~.~~1 aLE. ~P\"~t~P-l) GO Tn IBe! IF l QPV{l'\.'Pl .('T. "';;\i(~ •• ~,p-n I GC H" iOA~
lAO" Cf'rTlf\Uf GO Tn 1"9\
1~90 If' ,sye(NI .'E. I ) r,n TO lABS 1891 IF I KS'~(~I .'F.. 1 ) (,0 TO IB92
C-----APPl y S'"'"FTOY OPTIO" TO en~~TQUcr UT~E~ MAL' C" SUPPO"! CuRVE NP1 • NPOCt~) + 1 ~P' • NPOCI'I • 2
00 1 ~Q:3 t P lit f\ Pl. NP2 t\f\ ~ "'JP ... ~FCT
QP~(~fNP) • CPVCN1~~)
~PVI"'1NP) : ~P~I"',,~N'
l~q3 !N~f 1~94 ",r. l' ~PCT
~~ % ~p, • 1 - ~p
PP\'l~tt!P) llI: .f'!PVCNfNNl _P\I (t .• r"oJ •• \lfPV(t·.N~l
IB90 C0'T1N~F NPCr~O,,\ :If ",P?
1';0 Tn }P7t5 IRQ2 ~JPC'5('jt , ~pnC'N'
1&15 c~~t[~U~ C- ____ ~Ff\EPAT~ ~LL T~F SLPPC~f'~G ~T~ hONlJ~EA~ CURVf~
, 'f . 0 1<"'1 : 0
DC 1'65 , = I. , eve IF i KOI ft;T", , ) (:(1 Te ~!""t.
,,) = " If i KP?F II\IJ fFO. I I (,0 T~ b070 6066 JI . IN181NII . •
J? . IN?SI"I . • Or"C" ~ JZ - ~l JE',C . J2
If OE"r". • r.T • 0 I «0 TO M"7 IF ~~<4 .Cf. JI , G~ 70 f;06e
G0 Tn 6075 6067 IF I MP4 .1 E. JI GO Te, tlO7~
IF I MD. .GE. Je an To ~O6~ JEf\D = .0,
GO 1 n 6t'16Q
Dru::", = 1.0 00 6080 J: JI. ~(Nn IF ':1 .fO. 1 I "r Tf> C,)."
IF ~~q .. to. 0' l GO TO «,("1 IF I ~":>S I>,-?I .fO. 0 ) en
ZO~EO <.n~o Z'~EO 20DEO 2.nEO 2"flF. 0 240Eo ZOOl':o IHEl l~'El 2411£0 240EO zom,o 2aOft} 2<"f.0 Z40ro 200EO ,<('EO '4rEO 24!lEO r4nFO •• r>Eo cI.GF:O 24nfO 240EO Z4DEO Z41'F 0 Z4nEo 24nEO 2<f'EO 2<OEO 15FEl 15Fli ?~OfO <FDEO ZPf'fO 2r~Ec 2pr" 0 <PDf 0 2PDfn 2pr>E 0 ,pOEO <PCEO 21<OEO Z80EO ZPOEo 2POfO ZOOEo ZPOEO 2pOfO ISHI I~ffl 2 .. ,.1IJt 2tJJ\!}
r~ ( 15T.A (~-,..;rl ,hE. ,J ) Gn TG fl082 !5\r. L.ti = I"l
f 10,,:( :: ,.." C + 1 IF ! ~SV~(~I' .~~. K~'M(N) ) GO TO lee! 1F "PCTSf',)l .',E. ,PCTS"! I GO TO IP85 1~ ( J .Fr.. "1 ,(\R. J .E~ • .;END ) lSlLltJ) c !f I Jl - A:' ('J ~ E:P). fI~2. 6fl
PR;~-T qQQ) r.r, Tn Q-:,qQ
I S ... Co') • () ~-Pr-T := ... (oJ("c:(~11
('C f.!" 1'l3 t r :: 1" "pc T !F' 1 "'P\'n,l.~'p) ."'~' tl,PVOouNF') } GO TO 1885
"PT (f~"'C.NP) • WPV(Nl,NP) 0.PT ["NC~NPl • OP\l(Nlf~p} • p~RT· ( QPVtN.NP}
I c ~'.", T : r ,1 'It"
n ( ~""! "-i} ,I'E. WNTOL("") 1 GO TO Ifl8S
26JUI I<'E\ IS"I IS'El l5'Ej ISHl 15'U 15'fl I~FEI 15'EI I~'EI 15FE I I~FEI I"FFI 1"'[1 15FEI 15fEI
- OPvl'l,~P'15'EI Is.f I I~HI 21Ac I <l~cI .... r..Tl"i.;'.?-.r:} 1: w~:TnL[""'l)
':t,·"'tL.;h • l..~.T(\llNl}
NPTT~U"C c ""FC;~(~l) I S l' ~ i ~ fir) • ,;
• PAQT • , Q~TOL IN! - QNT~L 'd I ) I~Fn ISHI 15'(1
KS¥~~I~~C} • ~S¥M{~l) [F ( A~C;{\Io:PT~~r.C.l}) ./\iF • .AF:St",Of(Np..C.t.JPCT,) ('C,,-T1MJf TF ~Q'P{~' .~E. 1 I ~n Tr 0070
If
p jC 'r J
hI :;. t~
,.1-1 ~ It! ~ t- 1 ¥~ 1 ~ n
GO Tr; ':l(l~':
K,01 c 1 CO~Tt"I;~
= If'\2RtNI
iF ( PC _t, ~ lce J '-L 'f() tP]4
Pj;;I~i 90. GO TO 9qG9
) GO TO IPS,! ISfE I ISFEI IS.EI ZPOEn 2PD'0 2FOEo 'POFO <PDFO ZPOEO ZPDfo
C-----~.VF SLOPF'. r.rpops. w~Rnp~ 'OQ EACH CURVE
ZPDF.O lC:fF:1 15F~ 1 ISfE I l~FE I 1~"1 IS" I !!'.EI ll:,rr:l 15'El l~'EI
IP74 D0 1876 " 1. "r
1877 1 A78
NP('T • ~PiTS\I\'l If [ NPrf - ;> } leAS. lPc11, lP7P TF ( KSVlJ,.IP''o!' .N~ .. 1 i Gr"') TC lflf:\S IF ( ~sy~~cr.) .~r. 1 ) on TO 1979
~,::;c;; • t~pCT I ? CO IH97 "P. 1. ~OS
,. (
NN : l\(jr::- ... ",-,P • Ql'c;;rp (~. ,"JP)
"(,POP (".NP, SLrPfIN.NPI
QPT(N'NN+l) - QPf(~.NNl ~ WPT(N,NN+l} - wPT(N.N~) • Qr>Hr,P{N.NPI WDROPlN"PI
",0 .EO. I ) Q[lAOP(N.NP} wi,Q(,,:P ! "" , .... IP J
('n TO lA97 a~QOP(N.,PI • 2.0
= ~pPnp(N.NP! • 7..0
I~'EI l~FEJ 15'E! lSFEl IOFEI 10. r.1 IOFEI 10FI'I
IM7
1~99
(n~T!"J, ",n <.: ... fl'>.} = ~('5
GC" Til 1 PPI'-~'p(I .• T ~ 0 ,")Pr-1 I :t! t'~CT - 1
to lQqp F\D::: I, ~PCTl
JF i t.FI«Io-PT!r---.t:'l~'.l) ... WPT{Ntl'-,P) .o.14~(,,,PT p'".,,'P)) 1 ) GO T(I 1898
I\;PIF~'(' :::I ,,-p
NDrl'\ T =- NPCt' • 1 r:r,t, T T\IJ~ If (~)pr,\T .~f.;; 1 (,0 T('I1Ailr.
~~C< ::: '.plF"~f'
\0<:,1 f r c:: ft 1
1 ~F ~ i I'on £lA PI IrFFI I~~tl Iurl 1 nf'!. 1 ISH I 10,[1 10,EI ll'r'fl lcr::r:l 1 OF'; 1 IS-U
OC) 1P99 ~P:= I' f\OSl l~~r::l f)('.Q("lP(-.."NP) • - ( OPT(NH.OS .. ""P·ll .. CPT(,"<I'~'1S-~~j;.n l!!oF"El wO~rplN.NP) = ~~TIN.NnS~NP.], - ~PT(~.NOS_~P) l!~EI SLnpE(~.NPJ • GnRnPIN,NPl I wDRnp(~.~PI l~~El
CONT l"l'E I oFF I N0S"jt~1 ~ NeSl l~F~l
Tr:: f SL0Pf'!N.NOS) .if. 1.1')£""'06) 60 To 1l!64 lSr-El 5Lr-.pr !t~.Nr,S) := thO l C fEl 'tJOP(,PH-J.NOSI :: WPTP.~.f\OS.l. - wPTfI\'.~05) l"f€l QOj:JnP(~.N()S) r I'}.O i5ffl "O~J('l • NCS l~~EI
G0 TO IP"t 21'''1 Ifl64 ;"QQ(,;Ph ... Nf')S)' • WDT(I'H!\CStll ""' wPT(/'It'NOS} • wc;;toPfNI~OS1115F"El
C-----~.fC. T.~ ppnpE~ 0"O£P D' SLOR,S ,r~ T"IS ~n'LINEAp 'UP cu~.r 211Pl IA~6 ,OSh • ~OSJI'l - I clAPI
00 lARl ~~. = If ~nS~1 2IAP! IF' t SL""PF'P".Nf\) .LE. 1.I)F"'Ob ) G'-' TC P~87 ZlADl IF ( SLr.PF'(N.N~,d .(,;T. C;LOCF" (""tNN+l) ) GO Tn l8Al 21AD1
..1..1 = f(;TA(N) ~. 4 (lAP}
PQ1~T <;;Q.QP. j.J 21A P l GO TO qqQ9 cIO"1
1""7 rOhTl'JF CIOPI C-----PFIIF:R5£:. THE ,"'Qflf.C (F P(",:hT$ r~ PLS1Tn,E 0/'.,E,- .. 4Y 5uPP"H;joT~ 2"',.1' 1
If ( KS "r't'",;(,\j, .N£ .. -1 r,(") TG 1811:0 C:I-..;'Il 00 }PAP ~p." f\PCT 21-JiJl
l\;DP ,. NOeT - ",p • 1 2"j!Jl QP\lT (~."J't:) ~ .. ~oT i"'''~) 21-..1 1 '1 ,,·pVT!,~ .. "PS:t ::: - WOT(~J."'~l 2Ajul
(O/'.,T p l';:' 2"JLJl r 0 lRRq ~o = 1. ~~CT ~~JlJl
(JPT i\""t,C:1 a C~~T (0, ...... 0) 2",Jll1 WPT (" ,>,0) :a ",f; IT {"',"JPt
1~89 (O~Tl'UF
"'Jell C hJ' ,I} IS'FI 3nut.Q 300fO lnnEO lnCO 0 )rli~ 0 3rnFO
187b CONTINI!F C--~--T~PUT TftALE 9
10nO PRI·'T 900
IQOI IF ( ~ffrq ) ~Q8r' lQ01. 191C
~C1C; = 1
NCT<; ~ ... cr~ Gil T"'" 1 Q30
IQl\ lr ( ~CTG - ~ ) l-;;::6, 1911t 1911 :: c, 1 v?~ " = 1. F\ (T q
pp'",r QOfl, li:,O_YS(NI r YAXTSI~) t I'fSWIN) t
r:cr-.TINUF (f't ... ~: t:F
Ct. '1'- T 41 I\-c;q !'-CT9' I ~CTq I'-CTQ· I\;r:09
193n I~ [ ~(TG - 10 ) 'Q3~t 191 c • ]933 lQ~l OwII'-T 91)Q.
G(" Tn 9~;":;
1<t11S IF"! r erg ) ~9POt 19'37, 1940 19'7 P~I'T 903
19 40 1945
G" Tn 2COO IF c :.(T9 ""' "C19 1 2('1001 lqo.~, 1~45 CO 1910 • = ~CI;, h(T9
3~rEC 3cnEO 30DEO I~FEI 3n~EO 3nDEO 3nDEO 3rDEO 30DEO 05JAI 20DEO 30~EO 30nEo 30~EO 30nEo 3n~EO 30DEO
p r to rJ q 1 l X All S (On, Y A X T S (N), t "r' S \If 'N) l MP S (""l DPT"'" '1e~. 'tA:t!5!Ni. YAXyS(N). TYSif"/'..H JoIPs[rd
10'[1 10,[1 300EO 055[0 300EO
IQ10 CONTINUE [-----r'TEAPCL"E ._D rIS'~IAUT[ .ALUES rAO~ TABLE.
2nOr; LSI<' IE 0 :-:.It 1""lE~r::" 1.\07. ~ CT4. (tlL r~Trpp~ ( ~o7t "-Cf4, CtlL I~TE~P~ V07, I'CT4, CAll If\TE:PPJ Ito'p7. F\(To.. r'Ll I~TERo~ 1 ~P1. ~C'4, r:Al( I"T(D~i i ,,",P7, "(TS, ('!t t P l£;:"P~ "'P7, ~ eT5, (All l"TEPP; ""'P7, ~CT5,
1".114, IN14, 1IIi14, tNl4. 1""14. ! r~ 1 '" • nde:.. Pd'"
I",,24 I
1r\'2'" 1N24. 1~"4. IN24, IN25. 1'25.
K~24,
Ko:t2 4 • KR24, KQ24. f(w24, KR2~,
K~2;.
K~25.
P',? r::. l S .... K5 W4) 1 noco aCN? Q. LSM, KS" I looco S"'2. S, lS"" KSW4 I 100CO R'2. R, LS'" <~.4 I lanco C"2. Cc. LS~, KSW4) loncr i=4r<CN2, ~~(1. lS",.J(SwS )lo(:L 01""?- nt, LSM, l(C;w5 l 1r.nr:o ~E~'. nEt L~~. K~~5 t lonco
L~'" • 1 CAll l/'.,Tf:PP, ,,",~7. "CT4_ TN} •• 1;','24.1'(;;(2'" n.,.. r. lor.:; .... K~W4 )
05S£0 InneD O~SEO O~SE 0
r- ____ <;Tt~JT I:<FAM""Ci I , ..... ~ tit'l LTlr.r. fOP FACH T[Mf S":ATlnN pp i ~ T 11 POINT I p.pr"T 1:' ( llol\.' (r-.l. " • l' 32 ) Pt::r;.' 1"'. ~.ot,;r:-" { ap.,t:fl',.l, ~ = I' 1 .. 1 PP!LT H?"I
Kpr: = fl I<~ • ;') ~~PRT .. r:
C---·-I~iTIJLrll~~
b997
rc ,Q91 I. I'l'< (j : nl (I) = ~S I ! I •
1, yP7
Q
0.0 0.0
c:.<;1"J.<1 q, • 0,0 'j I"" 1 III cO, 0 I,I"'( d j ;; \.i.(:
CO,,"T1"UF ,0 7rrn ~ = 1. ~TP4
/lJ('lf"t ::: ,.
~--;l T:3 IE 1 TOC" T c ()
O~SEO 0'5Eo 055"0 055£0 055£0 05sEO I~Ftl 05SEO 3cnEo I~FEl 2SJAI ISFtI 2!A"1 21A"1 " .. ,..",
'" "," .t
300£0 10FEI 15AP I 0~0"1 IC,.,U
f T(~ :::: " TTri:.~ -= I' r<tt(''5F'(ll • a KCloSEI?) C I<PC = !(oC 1 K K e: Kjr( ... 1
<M) < - } "t-!,' r: tr - 2 10(,.... .3 1<1;1,. :: )i( _ 4
r-----I'ITIAllll'G r0 h9QA 1. ~~7 IF ( ¥ ,CT, 7 ) ~C Tn ~QQ~
~dTtKK\ = ('.f .... In:::: 0,1') WWw (r I ; {l.
6q96 Ir ( K .If. , ) GC Tn ~Qq~
Wq,iqq :s: -'1,1<.1("')) y'W i 1) ~ ,,( 1, J< I()
f,Qg5 SSKp.i?(T} = C:SI(~' (f) <;~"q."l rp = '551]) 01"2<11 = O}~J!l) Q r"'1 iT 1 ~ (; T ( r) "OSw(II -= 0 f\,tU"('C \ ! 1 t: 0 fiJi:. ( T 1 ~ n
TF { It .E~, 1 ) r.c T0 /,'QCU'
"<:11) :I:: (11~) (Tl - GeTl - OTf1'1) !'9"P U,t 'I Tt 1'1'
~~ ( 1 "4 ... f< l 7('('12. 11'1111. l003 100? ~"l~"T SH~O
7003 ;~ ~ "< ,1"'. ,'TP.. Gil TI'': ~,;::: r.:c ,. '-I(JQ )< '1' ,. r C071'1()) J'l:1~"';"1
fTf""I'\T\ -: ~.o
(iT t .... ' I<; 1, = C. ~ 1'nnl COI\;Tl"'tlt
t----"'Cc.L('lilATE rtlVF V,"Ik"tAtl..l A'tlAL T~QL:ST F0~ EACH E!A~ AT H·I~ TIM,. SlA C-----K .akC K-~ 'IIV£ ~~ATlrtl..
P(: -= i( _ t i\" .. 1 ) • ? ~r 1iG J = ~, ~p~ r,: ( 1 .CT. ·CTo } ::'r lr 111 en 1"l7 '\C: 1. tCT6
)(1\'1 ;<;T 1 tt-,r) ..
Kl\? : l<T? (i'-.() •
JTl :T] P"C) • ~1T7 :: lT2P'.() • 4
TF ( JT? ... JTl ) '44. 74C;, 7"~ 144 pRH T 9~o
130 Hi 9<:'!t:;Q 145 H f I';L? ,(:L, 1'\1\ \ ) nO Tn p"S
Pin", T Q}7 G(l T:') 9Q9c:)
G! JA 1 2~J.Gl Of-AD) C~AOI
;J7C;rO OS~EO
O'~EO IIlHl Oc:.C:FC Ot;c:t:: f'I
3t",:i1'=" 0 300,[0 0'" .a CI1
3('\rr: I"t
2enFI' ! 3_ A I Ol-API O~AP I O~'P} 2po J I~Fq 2 J API 210"1 13JA} I~J'I 2"JA I 21L.P! 210"1 JpOFO }~fEI
Q~~fO
O!:~fO I ofE 1 O'JAj } o~C 0 loocr j "f'; C O~Sf 0 OSSE 0 Q55 F C lonco 1"""(':0 O~5EO
O~5'O looeo } C1{ 0 O!:ic;r 0 Oe.C;EO O'SF 0 C«E 0 O!";f 0 O~APl
OM"} OMP}
.. "-
7,1
If , Kj .. LT. .' I .('r'. ·1 • C-T • ~"'I? ) (·0 TO 70 7
]F , . .,:(: ,. . ' ; .. r'" • y 1 ","'0. n~? '0 146 j ,,0 T' 74"
PS f "'« 5 if \ .' .. LT .... T1 .. ~P • .J .r.T .... T2
OloA01 1 10 ';':'1 1~..JAl
I ~ " A 1 lL4J D;<;cO 1 1 "'( W'l'" 1<".1
rr t "q" .... ~ 1 ) 7,,0, 741. 1'+2 PP:I\ T G 1 !
[':, '';'';'' ... ;' GC:~~ c oe<€Q I~JAI O~qO
Tnt:'· r .... ~ I ..
I't:: T ~ 1,0
T(;I"~" = I<~:;; .. )(1"1 ;.'1>- C l 'l: J .. __ T I
Pj"'-C == PI' CT .. ",.e. CSSF::O RDPJ'" z ~T;' .. JTl • I O~C:;EIj TT(,,;.)(T\ TTi.J,~.Tl. i IT)ffl.,lC) • ( TT?I"'r) ... ITl[l\.tJ I llo.;Al
• p,P,C I Arfl\Olol .. ( TT3l,1\(> .. TTl ff'..!') 1 • Tl'\(lt-JAI I TCH'Jr:~ i • fST 11-JA1
7'11 C('1\1 ~~.'J~ 055[0 C- ... - ...... ('6~tl,1 r,TF TT""C \'-~k'4~_ i L~TfP~'
(-.---: A~r f<-' r{wF ~~A Lt..'H.:-ll'.~ f.Of.: F..ACH STI!! AT I;.-.qc !IJ04t: STAC:;[O
71 , IF ( 1 • ," C T 1 ) ~I"'> TO 710
.... 047
r;c: 11/1 I'.,t'" ill' 1. 1'.,(T1
, l' ~
l'f!' 1 It.f i • 3 1\"'::(- I'\.C) .. ) }f" 1 iNC I .. 4 1~? ;.t i .. 4
!P I ~~~ .~F .. w~l , ~o T~ ~4~ D:': fr, 1 911
Gr Tf" 9Qt;G
,r I .IT? .. 6F, w'" P~p.T 411
(.0 Ti"' qq(j<; If i i(} .i T. ,,~)
: f I I( 1 .. F f~. /1'1 I I=:ST :. l.~'
G0 Tn 71!Ci
I r.O T r .47
""2 ) GO TO 112 ,",i' ) 6<' TO 158
F:5 T ~ (", ,,'5 Tf ( ~' .. LT, .)Tl if \ J ~ n; ~ . T'
.c~. • ~.p.
.T2 GO T0 112
.T? I GO Tn H,~
-: r T ' 7 ~ ~
r<;~ ::. n.~
T T f\ r:: ;;: ~ 1 ... ;(~1
IF ~r?" I<t\] I 7C;O. 1S1. 752 7 C i' p~ i"- T '''j l'
(!(' ,. <'"1 g'1q(";
, .('
(r Tn
O~~EO !n~Co O!:C;'O 100)eO I ~~cn O-StO OBEO O~.·! O~ooJ
O"lI. P t OHDI O~O·l C~ AC 1 O"ACt II-.q l~~'l 1 ~ JAI 16J'1 Q5SEO 1 ~,;!ll 1~~'1 l~JAl l~JAl lrr:ca OSSEo Q~SFO OS5Eo OSSEO l~J'\ (; t;f;f I')
05'EO O~ScO
754
75S
IF I JT? - JTI 1 75~. PPI~T 917
(.~ T~ 9qq9 A[)f!\.{;~ ~ 1.0 ESP' 1.0
05SEO aS~EO 055EO O~sEC 16JAI OSSEO OSSEO 756
751 I
GO T(l 1~1
ROF.~O'" z (n IJ,!<Tl
jT2 - JT) QTiJ."T) • i aT! O"C) • I 'T?iNC) • YTllNC) I ~I"C . B()ENO~ • i QrJ("C) • OTIINC) ) • TINe I TOFNO~ ) • EST • Ese
016"AI IOJ"1 IfjAI 05SE 0 OSSEO looeo 2PDEO 2POEO 28DEO 280EO lADEC lADEo 2enEO 2POEO lPOEo 21AP 1 280EO 2POEo 2~OEO 1!!~EI IOFE! I!FEI ISFEI 15'EI C6A P I
CONTINUF eCNT l>.UE CO" T I NIIF IF ( ITS. j ''lao. ~905. ~900 IF I "AXIT .o.T. 0 I 00 To 6910
ORI'.T 99A;> C-.-.-ITS •• O. LI'EAR SPRI"GS. sO ~'T ",XIT • I
6905 MAXIT , I 6910 DC 6000 ,IT ~ I. ~AXIT
~OFFC(N'Tl • 0 C- •••• INITIALIZING FCP NEXT ITFOATIO'
6500
DO 6500 ~ I •• P7 aIL,,,) • QtJ) .. nTfJtll <is (..:) c 5 (Jl
ceNT H,uf IF [ Neve .Efl. a ) Gn TO 650~ CO ~D56 NN. I. 'Ne
~P1S • I')
J • IST/.(NN} NOS. NnSjiNNI NPCT • 'PTTSINN)
IF I InS.'J) .fO. a ) on TO 6S51 c--.--I'ITIALI/IN. 'OR TOE FI"~T LOAn ITERATICN
IF I NIT .GT. NIT5 I Gn T0 65~T W(J,KK) • Iln .. hKt<-p
6051 6085
IF [ IS;,iJl .EO, 0 ) Go T" 6057 ES~ • 0.5
GO TO 60PS ES~ • 1.0
IF I ITes .F~. 0 ) GO TO ~D86 IF I IFUSIJl .EO. I ) GO Tn ~081
C-----TOT TO SeE I~ NfEDS AnJlIST"E"T OF TI<E G-' CU~vE fOR Tot. FI~ST JF' i ARS(w(J.I<.;n) .LT" ARc:;n.'(,JtKK ... 1)) 1 GC TC !'IO] If ( ( ~p5IW(J.KK-l») • A~S(W{JtKK)) l .EQ. A~~{.\Jt~K-l' ...
W IJ,KK)) ) GO T0 tl03
GO TO 6~e~
ObAPI 060P I OU"I 15FF, I I!FEl 15~E1 lofEl I!lHI
Tl.E5 FI':I IOFEI I ~FE I 10F£! I OFE J
6091 IF I 'PTI".I) I 601. 9991 •• 02 C-----TFST TO SEE IF .FEOS AOJUST"fNT UF Q-' Cu"vE AFTE- T_E ~IRST TI~E
f,~l !" I i,','(,hN"K! r.;r; \ooIt_!~I(K ... !\ l ~n,.O 'd)Rh
IOFf! I~FEI InrP"',
602
603 503
Gr Tn ~q IF I WIJ.KKI .LE. "'J'~'-I J I GO TO 60P6 GO T~ Sf}3 IF' ( W!J,I<I< ... , i ) 669, f.OQ:~, ~{lJ IF ( WSY~~(N~') .Fe. 1 t n0 TU 604
I~FEi 10HI 15F£1 26JJ I 15FEI
IF' ( IF'US (J) .fa, 1 ) GO Tn tin. IF' ( v5Yjl!JtN"1 .fG .... 1 ; ,-..r t(l ~O_ GO Tn 6 h P6
C"'---"'''F'\tFR~[ TJ.oIF' rFfI;F'Q f':F' ~n"'j H:rAP ';L.PPD~T CU~VE ,,04 0<' f.(l" "p 1: 1, ~ peT .
~p~ • ~DCT _ NP • 1 W~VTt~N,NPl • WPT(~N,NPkl
QPVT{~~I,~PJ $ QPTiNN'~P~) ~06 CO,Tl NUF
CC 5~~ ~p z 1- "PeT ~PTC~~,~PI • wPVTI~~,'Pl QPT,~~ •• PI c QPVT(NN,~P)
506 CONTINUf TF { It"SY"'.J(~ ~.l .f::. 1 ) G" TL 60S
c--- ..... C~ECK Tn SEE TF' "fn;5 AeJuc;T""F"-. T OF T~E Q .. w CV f n~ O~E-WAY IF ( KSY"' ... (N~l .~E. -1 l GO TO 666 IF ( IFU'5(JI .ED. 0 ) GO TD 605
666 CO ~n1 ,P. 2. 'PCT STF"" • AAS { ( opT ,,"',"P I _ OPT ("'N.'P.II ) I
t .PT(N~'~P} - .PT(N~,~P~I) ) • JF I AH~(STE~P • SLOPE,NNolll .LF. I.OF,06 l Gn TO ~08
507 eCNTlNUF POINT 9M
GO To 999~ SOB IF I .PT 1"'.11 ( 509, 9991. 511 ~09 ~p 2 NP - \ 511 IF [ 'sr.,IN') .fe. 0 ) GO TO 667
)f ( wC • .:.k)(-11 .Lf. "PT',..N.NP} ) GO TO 6086 GO Hl 6"S
661 IF [ ~IJ."-'l .~E •• PTIN"'."'") I 10 TO 60@6 GO Tn U~
1\~9 IF' i I(S'( ... ~lM\) .EC. 1 ) (;0 JF I KS'.J IN' I .EO. 0 I GO GO Ta ~oe~
60S IV~(J) • rUSIJ) • 1 DC ~07 ,P.? ,PCT tF t .... PT {"I\., ~ I ) 608. 99(~11,
60~ JF ( w'J •• K-11 - .PTINN.NP) 609 IF' ~ ',HJ,"Of ... ,) - "PTINN.NPl 607 cO"T INUF
GO T'l 6'2 6Jl IF ( "IJ.~w-,) - ~PT{N~'l) 1\11 IF ( .... IJ.)()( .. 1) - iIIPTiNN.ll 61? K~FFCi~TTl
G0 T(l 61< 613 NPC; = NP •
GO T'l 61, 632 NPIS : ,
TO MS Tv 605
~O9 I 631. 613, 601 I 601. 613, 611
f.12. 632- 61< 614, 632. 612
1-.1 ok "'P~ t- H' C-:;--TF~T TO SFE .OJ ••• ·I) IS AT .HIC. SLnpE'N"'.NPI
~15 STf~P • APS I I QpT1N"'''PS) • OPT I"".NPS-II I I 1 ( WPT!NN.""PSi ...... PT{N'HN~5-1) ) )
JF I q<.P .J E. J .0E.0~ I GO TO HI QTfHP = STE", • ARSI~IJ"'-II - .pTINN.NPS-II)
I'API 2"'Jul ISFEI I roE I Ir<EI I~<FI 15FEI leFEI IOFEI lo<EI I"FE I 15FEl I~FEI
15Ai:ll SlJPPCRT26JUl
26JUI 2~JUI 26JUI clAPj 21API 15 AQI I"API 1~APl
15AP I ISAPI I~API UJ'JI 21!,Jl 2~J'JI 2~JUI 26JIJj UJ'JI 2~JU) 2fJvl ISFEI 10rEl loFEI 10~EI IOF51 IOFEI IOFEI I~~EI Iry~O IOFEI lOrE I loro 10FEl I OFF. '. lnHl IOFn 21API 2)API IOFF! IOFEI
530
617
(3{' T0 642 OTE"'P: APSI.IJ.~K-Il • WPTIIliIli.NPS-\ll
CO ~I~ ~IC. I. oOS IF ( IT(I'. - $I.MEINo'.NSCI ) 6H. 6J7. ;30 CONTINUE Gi' TO eJ7 IF INS, .EG. I 1 GO TO 611
SDHAL' • ( SV,P(!NN.NSC-ll - SLOPE (N'.NSCl I I 2.0 SPART = 5TE>'P - SLOPFINN.NSCI
IF [ SPART .r.T. SC .. OLF ) "'<C. Nse -aOPQPL • QOPOPINN.NSCI • QTE~P Nse 1 = ,-se • I
520
10FfI lurE! OUPI O~ool O~Aol
060PI 06A o l OH"I 060 P I 060PI 10FFI 10FEI
If I NSCI .GT. 0 I GO TO e-----NO >EED nF ArJU5TINa THE Q-~ ("-----1<; wITl-tT"1 '''f EL~STIC ~A"'IGE'
I!l$ (J) • 0
CURVE. SINCE TH~ ppEYI~US UEFL[CTICN \SFEI clAol 2lA"l
IF ( IFIlS I Jl .fO. I I GO TO 6096 I' I oPTIN~.ll .LT. 0.0 1 GO TO 6081
c-----~EVF05E THE OPOEP C,F POINTS nF O-W CURVE BAC" TO THE STAPTEO
S10
515
,-----56"F 520
hIS
DO 510 IliP,. 1. NPCT SPA • NPCT • NPA • 1 ~PVT(NN.~PA) • WPTINN,NPR) QPV'r~NtNPA' • QPT!NNtNPA)
CO"TINUE 00515 NPA = I. 'PCT
¥lPTt,....N.~PAJ :& "'PVTfNNtNPAl aPT{~N.~Pft) • apVT(NN.~PAI
CONTI'UE Tn Me~
I.r CPT '00' POINT I TO POINT NP-I fOP THIS c~~vE NN NP"" 1 :c "p - 1
00 118 .PVS. 1. NPMI WQYT'NN.NPVSl 2 WPT(NNt~PVS)
QPVl tt-.N.NPVS} -= OPT IN'" .~PVSl co.t..; l J tvl)F " I 'P15 .EC. 0 I Gn Tn ~33
NP • ~ •• I (.0 TO 6'4
C- •• ~-OFGI"fP~TE T~~ N~NLlNEA~ S\JPPOPT CU~Vf FOR T"IS CVRVf .. N ~3J ,.P\l1 (~J"' ... NP) a W(J.Kt<-ll
620
QPVT U",·f,..f\)p) a ( 'trip! iNI'\'f'«PS) .. "iJ,Kt<-l) i • STJ'MP • QPT {NN,NPS}
00 ~19 ,.ec' I, ",SCI ~I'S T :: r-• .'c:(C • NP
IF ( IroIPT ,"'/'I .. l} J 620, qqQl. 621 WPVTI~N.~STI wPvTIN .... ST·II •• OROP(NN.NSeC) QPVT ["'N.,ql • OPvT J"~.~ST-)) - QOROPJNN'NSCCI
r:.o Tn f, 1 q wPvT [NN.t-.tST) • wP\fTfNI\i.r-.ST"'l) - _OAOPO>JN.NS('C) QPVTJNN.NSTl • QPvTiNN."ST-ll • QOROPJNN •• SCCI
CONTINUF NSL c N<:C • ~p
IF J WPT/ ..... !l ) ~22. 9Q91. 023 If ( SLOPF '"'' _",se 1 .LE. 1. O~-06 ) GO To 639
ISHI 151'[1 15HI
ORH~ISHI 151'EI ISFEJ ISHI ISFF.I IS'[I ISFEI ISHI I~FE 1 ISHI \-FEI 210"1 ISHI 1 OFf.! I Off I InHI 10Ffl I nFF I IOrEI lonl 10FEI IOFf! 10Ff! IOFEi InFEI IOFfI IOFf! loHI 10FO IOFfI IOFEI 10FE! 10Ffl IOrEl I "HI 10FEI
W"VT(~N.NC:LI
npVT I>,N.NSLl GC T~ 62'
• WPVT(N~.~SL-lt • On~OPL / SLOPEiN~,~~Cl IOFf! IOHI IOFf! ISHI ISHI loFEI lorE!
wovT 1"".N~Ll
(lPVT "'.NSLl GC Tn 67'
• ~pVT(N~.~SL-ll • QOROPL
• ~pvTIN,.'SL-II • WDROP(NN'NSCI -AilSIOOOOPLI
• nPvTIN"NSL-!l
I' I SlC,PEIN' •• sCl .LE. I.Of-06 I GO To 640 WPVT('N.NSLI • WPVT!NI\."SL-ll - Qr>ROPl I SLnPEI"""Cl (lPVTINN.NSL) • QPVTINN.NSL-Il • QOPOPL
I~FEI IGHI 10FfI
MO 1
GO TO 674 WP\j'l(r· .. N.N~L •• wPvT(~N.~SL-U • WDJ;lOP(N''df,SCl •
~~SIQOPOPLI QPVTINN.NSL' • CPVTJN"'SL-I'
C--·--SAVE THE PFMbl'N'NG oPT oNe OPT IN wpvT ANn OPVT ARRA_S 6?4 DO 625 ,PC. 'Po NPC T
625 626
IF ABSIWPVTIt,N"SLl _ WPTIN(I,.NPCI •• GT. ~NT':)L,J(NN)
62-IF I ABSI;"VT(NN.NSLI _ OPTI'N,NPCII .GT. QNTOLJIN'!
~cS ,0 TO ~?t
CONT I NUf NP(lL c NPCT • ~PC
IF , NPeTL I 9'160' 677. 62" CO 662B 'A. I. ,peTL
WPvi(NN.NSL_~A) • wPT(N~tNPC_NAl OP\iTINN.NCiL .... Al .OPT(N"tNPC_NA)
CONT!NU' NSL ~ ""'c;L • ~t)C1L
r----.s.Vf T~E NEW .n~'STEC Q-. CUPVF IN WPT ! QPT ApPAYS 627 cr 6?Q ,P. I. 'SL
~PTl"N.~·P' c wPvTCNN.hP) OP1(~N,~P} • OPVTfNN.~P)
CC/vi I NUF NP1TS(Nt..} • "'SL I Fl\Cj (Jl • 1
~oe6 ~PCT c ~PTTS(~N)
IGFEI ISHI I~HI 10FEI I O"E I 10FEI
G- TC 15'EI 15"EI
Gt TC ISHI 15FEI 10""1 10FEI 10HI 10"£1 HFn IOHI loFEI lOrE! IOHI 21API \OHI \eHl IOHI 10F£1 lonl ISH! I~FEI
e----~INTEPPOLATE TO FINO p[SISTANT ST1'."FSS. AND LOAD 'ALLE~ FpO. SLP.ooT CV IOFfi IOF£1 I c~El loFEI IOFEI IOFfl loHI IOHI I.oE) 0"°1 O~ool
IeI'l'l 1O.El 15AP1 O~OPI
O.A'I
DO ~090 'P. 2. ,PCT IF J 'PTI".1) ) H910 99910 6092 IF I WI •••• , - WPTINN.NPI ) 6093. 6096, 60'0 IF I WIJ.KKI ~ ",pI (N",,,P) ) 6090. 6096. 6091 CONT '>,UF
t-"p • ~'P"'T<; 0"'\ 1 r.(' 'f Ii $-!'QP
IF { w {_,.:I<..:)
If ( wHJ.":'':. If' ! In~- (J)
,,!p C ,,"p
.. wPT (~"'.l) - •• "NN.II .EO. I I GO - I
r:o TO b(l:9l':Of"f"CfN1'Tl c
[,0 T ~ ~ I I r
) 609~. 6111. 6110 I ~IIO. 6111' 60'P T~ 01'6
o09~ IF ( fO~.IJl .EO. I I GO TO 6196 C-··--H'GE',T ~nn\'LUS ~ETHcr I LOAI).SP" ING
61Q8 I
6112 1
~113
IF
GO
IF
IF
GO
IF
IF
( NP .LT. NPCT I GO TO 6198 SLCPFZI • 0.0 Ill? 1 • 0.0
T0 6112 5LOPF?1 • MS ( I QPTIN~,NP'l) ~ OPTINlH,",P)
t ~PT(N~.NP.l) - WPT(N~'NPl
( 5L0P[cl ,LE. 1.0F-0~ I SLOPE21 • 0.0 OT?l Q tjPTCNI\,NPI • SLOPE21 • _PTtNNtNPt
( NP .GT. 1 ) On TO 6112 SLOPE?l • 5LCPE?1 • ?O Ill?1 • ~121 • 2.0 SLOPF'? • 0.0 0172 C #'1;.0
Tn ~1l' SLrPF2? • IRS ( I QPT(N~,NPI - GPTI~~.NP-li I
( WP1IN',NP) - WPT{NII...NP-li ) SL~P[ZZ .LE. J.Of-O~ I SLOPEZZ • 0.0 Ql?2 • ~PT!'~.NI>-ll • SLOP!':22 • WPTI'N.NI>_II I'll> .LT, NPCT I GO Tn 6113 SLOPE?? • SLCPE?? • ?O Ill?2 • 0122 • 2.0 ~SIJ) • S~(J) • 0.5 • I SLOPE21 • SLOPE22 ) • ~S~ C,IIJl = Q1IJ) • O.~ • I QI21 • 0lZ2 ) • ES.
GO Tn "(lS~ "110 IF I 10S.IJ) .EO. 0 ) M TO 6197
GO Tn 619~ C-----T"Gn,T ~nOI:LLS N[Y"OC ( LnAD-SPPING PERAnO" METHOD
6197 SLOPE? • ASS I I OPT('~'NP) - QPTIN",NP-lI I 1 ( wPTt'~t~Pl - ~PTCN~'NP·l) )
If ! SLOpF? .LF. I.OF-Of. I sLO"E2 • 0.0 SS(J) : S51Jl - SlOPF2 • ESM r.q i.,.:i ::: OffJ) • ( QPTtNh.NP' • SLOPEZ •• PTlft.t!l..tNP) ) •
FSM G(I TO ~nS6
C-----lDID ITERATION •• T~OD 61Q~ 00 ~2qb 'l.? ,"CT
STe:"'P;:; AF\$ ( i QPTi~JNt""Lt .. aPft~Nf""L-U ) I t WPT!Nf'",""Ll ... ~PTlM· .. ,f.,.L-l) } )
IF ( IRS ( STE." - SLO"E!N",11 ) .LE. ),0[-06 I GO !O 6cn 62q~ CONTINUF
PRINT QPD G~ Tn qQgq
6c07 IF ( ,S'.J(N') .EC. I ) GO TV ~2Qe c----- •• T IPppnll".TE llhEAP SPAIh. ~OhSTANT FOh ONE-"A' SUP ~~
IF ( ARs(QPT(N"oll) .LE. I.OE-12 ) 60 TO 6~10 51 • AAo ( OPT I,N,NL-ll I oPT (''''NL-11 )
GO Tn 6?9" 6310 51 • AR. I QPTINN,NLl I wPTINN,NLl )
e-----5Fl 629/\
uP 1 (1 6i'~; .PPPOII_AT. I I.EAF SPRING CO~STANT FOA S'W~ETNIC SuP C.
S I' ('PE 0". I ) IF I l.u. .EO, ~ I Gn TO 62<1q
~I •• ". I I IlPTIN".NL-ll - nPT 1.' .... L.p I i wPTfNN,NL-l) - wPT(Nt-.,NL·}) , )
ZI,OI 21AOI c·1 AP I clAOI 2lA"l 2)A P I 21A PI IOFn IOHI IOFfI IOFf! 10FE! InHI I ~FEI 21A"1 21API 21 AP 1 IOHI 21API clAP I clAPI I~FE I 1~~[1 I~~EI 06A"1 OUPI Ouol clA"1 21API 2U"1 15HI· ISHI 15HI 061"1 061P I cf<JUI 26JUI cf-JUI 26JUI C6JUI 26,/UI 26JUI 2fiJlJl c6JUI Z~JUI 26JUI c6JUI c'J'J!
~!~~i 26J'1\ 26Jdl 26Jul 2'Jul
5S( ... ) c 5S!..,.1 .. 51 • FS~
\.oC :: - \I. (""KK) • ",PTi"''','~,Pl ~} = ARc: (( QPT!N ........ 'I'I .. QPT("'~ .. I\P ... ll 1 I
( 'WPT(I"N,"/J;:) ... _PT(t-.,.t-.,.H,P-ll ) rF I <I .Lf. 1.01'-06 ) 51 0.0
QC = QPT(I\N.I\PI • we • 511 ns = - _(J .. ~~l • 51 Q I r J} .:: 0 r (J I • [ Qt:': ... ... S ) • E SM
60,6 CD~Tl~UE 6S04 IF ( ITC~ .[~. 0 ) Gn TO C-----TF~T TO SEE IF T~ERE 15 A ~To SUPPORT ~.LOADEO
6506
6503
6505
DC 6S~6 ~ c 4, WP4 IF ( !USIJ) .EG. I ) G~ TO 6503 co, n"JF GO TO 6175
ITCS • ~ lTCSR • 1 NS .& 1 AU} C 1'\
A tll ': (\ P j P • ,.. flPt C ( I I C I?) , 0
DO ~060 J • " MP~ C-----co."uTE ~oTAr' C~fFFS AT EACH ~TA ,.I FOR FIRST AND SECONO TJMF ST,
yA : F (J-l) ... ?5 • ~ • F!I(J-P
700~
I ?
'B • -?O· ( F(~_ll • fiJI I - "Ec' T(JI yc • F(J-l, .4.0. F(J) • F(J'l) • "'E'] • SStJI, •
.2" • l-< • ( O{J .. 11 • f,/eJ'U ) + HE2 • ( TIJ} +
TIJ'lI ) yo -2,0· t F1Jl + ~(J'll I - MEl· T(J+)) v[ F(J.l)" .?5 • ~ • F!I(J+l' 'F' HE" ~IIJ) - 0.'3' "'E2' I CPI,,-I) - ("It-Il
IF I K _ 3 i 700., 7005. TnOS AA • y/:.
PP • yR CC ~ 'C no • '0 H • 'I' FF' • YF
GO H) 7"O~ AA • Y/:' • OliJ-l) I HT FH::t • YA ... ?o • ( OIL! .. I) + OI(J} } I MT ... fl,S • ""E2 •
I TTIJ.Z) • TTIJ.lI ) CC II VC + ( CT(J"") + 4'0 • ol(J} + OTtJ+P • I "'r + 0.5
1 •
• ( TT(J.?) + TTIJI!) + TT(J+l.2) • TTIJ+l.l) C • H3T? • P"OI~1 • H3TI • OEIJ) to uC'''J; _ I _ C,Ct 1\ • CCIt'~?t 1\ )
DO. Vf'l ... 0·;"', 01(Jl :'01(..1.i)" )~/-~T - t'l.'S. HEZ· ! TTiJ.l .. 2i • TT(J+l.ll J
ff • Yf • u! f J" I I !-iT '?" • Yt • 4.0 • ~JT2 • RfooO(J) • W(J.KK-l! ?~ • Cl tJ ... t) I ... T ) • w t .. -2,KK-2)'· ( -Rrc ..... 0 •
FF I _AA
c~JUI Of API 21 AOl 210"1 2)AP) OU"I 26JUl 06API 15FF I 2~JAI c~JOl 2~Jol 15FE\ c~JAI 25JAI 15FEI 1'l.EI O'lS~O O~SEO OSSEO 05SEO O~SEO 0-<;[0 OSSEO 05SEO D'ISEO 100CO 100 04JAI 055[0 OSSEO loneo rooco 04JAr O~SEO 100CO I~OCO
l~ocO IOOCO Innco looeo looco IMeo 100CO 100CO loOCO 100CO 100CO "AI', 1 ~nco loneo IOOCD
'I~OCO {IOOCO
4 <; 6
C-----CO·P!;TE 1006
DI,J-n • DI 'JIll f<1 ) • wU-I,KK-2) • I -CC • 2.CIO~CO ., f ['>} \..,;-1) • 4 .. {\ 4t 01 {J' • 01 L"i.p ! I ... ,. • 2.0· 1 ('lOCO H3TI • CfIJ) ) •• 'J.,<_21 • I -DD - 4.0 • I DI''') .I~nco OJ(J'» ) I H1 ) • • 'J'I'KK-2) •• -Ef • Z.O • tn ... ·lonCo II I HT I • "J.2,'K-ZI • HE) • I - 01 (JI • QI"2(~1121'''1
ofc~~5In. 00 CO~1INUIT. COfFFS IT f'[~ 5'A O-SEO E • IA • R IJ-21 • RR O~SEO DE"'O" • E • BIJ_I) • AA • CIJ"21 • CC O~SEO
IF I DE~O~ I 6010. 6005. 6010 O!SEO C-----NC1[ IF DENa. TS ZERO. BFA" nnrs NOT "1ST, D = ° SfTS VE'l = O. OsSEO
oOOs n : 0.0 OSSEO GO TO 60'" O~SE~
n • - 1.0 I D,N"" O~SEO CUI' r • EE OSSEO RIJI : r • IE' [(J-n ,00 ) O~SEO AIJl 'C n • ( r • AfJ .. ll • 1.6 .. J{J-cl • Fr l Q~SEO
C-----CO~TPOL PESET Rm~INES Fn~ SPEcIFIED cO~OI'IO~S OSSEO KEYJ •• p'J) 05S[0
IF I K - 3 ) 601G. 6019. ~~Ib lonco 60lA IF I 10"IJI .NE. I 1 Gn TO 6Ul9 100CO
NS • ~s • I 100CO Gn TO 606~ 100CO
6019 GO TO ( 60,",0. 6020. ~OJO. ~020' M50 I. KEYJ O~SEo C-----"[5ET fOP ~PECIFTEO CEFLFCTION 05SEO
60Z0 CIJi : ~.O O~SEO ~lJi a ~.o OSSEO AIJ) •• SINSI 05S[0
If I ~EYJ - ~ ) ~0<;9. M3~, b060 O~SEO C-----RESET FOR SPfCIF,ED SLnp £ A1 NEll STA O~SEO
6030 rTF'p 0 05SEO CTf ••• CIJ) 055EO RTf'" • 81JI O~SEO ATr." •• 'J) 055EO CIJI • 1.n OSS~o ~(J) • n.O O'!SEo AIJ) • _ H12 • OW~IN~I O~~EO
GO TO 6n~0 OSSEO C-----I>ESET FOP SPECIFIED SLoPE AT PRECEOING STATlO~ 05SfO
6050 ORFV • 1.0 I ( 1'0 - I &TEMP • SIJ-ll • nE"" • 1.0 ). osUo I n I eTE!"" ) osno
CREv • nREV • CIJI OSSEO '1REV • OREV • I 81JI • I RTEMP • CIJ-P I • D I on .. ,,) 05~Eo ARfV .. r'iREV • ( AtJ! • t IoIT2 • O.S(~Si • ATf~P • QTF,",P o~sro
(IJI AU) A!J)
6059 NS • 60~O CONTINUE
• AIJ-ll 1 • 0 I DTEMP ) OSSEO • rREv 05SEO • aREV OSSF.O • AREV OSSEO ,S.I O!SEO
C-----C0vPUTf OFfLECTlrNS 00 ~I 00 J. 3.
05SEO OSSEO 05SEO 05<;EO 13JAI JaDEo
l*/o"·8-J " ... ILl = WWILl "'W(l) :=: W(L.K1(;
6100 W(L"I(t() = A(L} • FHl) • 'II/(L.ltfiod • C(L' •• dL*2,t(o{l
CO'" T 1 NUE O~SEo O~~EO OSSEO O'~FO
wt?)(Kl • 2.0 • W(3,KK} - Iti(4U(K) W(~P~.KK) c 2.0 • ~(VP~.~K) _ W(~P.,KKI
r-----tOU~1 ,u •• Ea OF STATIONS EXCEEnl'G MA.I~u~ OLLD~ABLE ~----- A~C ~U.RFR CF 510110.5 U01 CL0StD
DFFLEcT1C~5 2P~Eo
6165 61!O
K C.r ,~IT I • 0 Kt4AX • r,
1F { Neva .F~. a ) G~ TO ~noO CO bl~O v ~ 4, ~P4 IF , wMAl - b8~"IJ.K"'1 I 6155. 6160. OIO~
,<;MA): • K~A~ • 1 IF I IOC~T .r-T. ~ i (;0 T0 ~I!>O IF I AA<I.'J •• KI - •• IJ) I - .10t ) 6150. 6150. ~16S
1«".11;"'IT) • K(NT(t-.lTT) • 1
CO,,"T!NUE IF ( K .ll. ! I GC Tn 6~41 IF I NOCI .ra. I 1 Go TO A~41
C-----T£ST 10 SEE IF T~ERE IS • SUP STA WHERE THE DEFLECTION IS C-----DU~I.G THE ITEOATION "ROCEs~
6164 6~.0
00 6540 ~ •• 1. '"C J • TC;T,6.(~NJ
If ABSI .... CJ.I<IO .. WWir(J) 1 .GT. IdlE-OS) GO TO 61540 IF AaSI"J.KK)) .LE. "Tn!; I GO TO 6540 IF I A8SI~(J.KKI - "wIJII .LE. WTOL ) GO 10 6540
NU~OC(J' • ~~~QCrJ' • 1 IF 1 NU~CCIJ' - 2 ) 6540. "'164, 6164
NITSct.IT·l CO.T IMI£ IF I "IT~ .LT. 2 ) Gn 1060;41
CPDEO I~FEI 2PDr 0 ZSOEO ZPDF.o 2pero i1pnF:O lerFI IS"':I O~.JAI 2PDEO 21A P l 2IA P I
'1SclL.L.HING 2!API 06A"1 OUPI 2IA P I 210"1 21A"1 c]API O~'''I 2""1 06API
C-----SETTING S.ITC~ InSw!JI FOR LOAr 11ERA,ION ~ET~OO ~E'T I!E R01!C' 21A"1 214" I 210"1 CC ~~42 h~.}. kNC
J • !STAt"''' I 105',"J) • I "U"~C(JI • 0
6~42 (O~TINut
6541 IF' 10C" .FG. n I r,Q TO ~151 C-----TE~T TO SEE IF THE OEF( EC'IONS CO~PUTEO FRn~ THE INITIAL C-----ITERATION LeAns AOE ~I'HIN 1HF ADJUSTED RANGE OF THE 0-"
DO 61S2 •• I •• 'C J • ISTAINI
IF 1 IFU<IJ) .EO. ~ I GO TO 0152 IF 1 OSV"J") .Ea. I I GO 1'1 6149 fF ( KSy;.. •• Iff\) .(r.. 0 I Gt) TO b24q IF I WU,KO-;I .U. 0.0 ) r.o TO 6152 GO ,0 6149 1£ I WIJ,.O-)l .r.E. 0'0 ) ~o TO 615, IF I IoPT '~'II I ~1~3. 99'11, 6154 IF ( Iri(J.!tK) .GT, w(J,t(K~l) ) GQ TO 6152
;.I~Jtt(;() • W{ ..... K!(.jl • 1,Of-06 GO TO 61~? JF ( Irrt • .f.KI'() .LT, W(J,KK-}l ) GO TO' 6152
~(JtKK; • W(~.KK-l) - I·O!·O~ CO~TINU£
21A·I 21bOl 21'''1 21A"1 I~HI
',GS '21API 21A"1 1'5'" I ~Ff.1 I'5FEI I~FE I 26J'iI Z6JUI 2#-';lj1 26J'JI I~Ffl Is.q 15FEI ISFr.l I'5FEI 10;. F I 1<;"1
6151 IF I ~(" I .FIl. 0 ) Gn TO ,170 15FEI C-----SP L'
IF ~O~IIf("P :\f\'TA 2~J.61 I nrO" .,Q, 1 I r,O T('> -I)A 10FEI 'Mr,TI~ITI = KC~TINIT) ISFEI
CO .170 ,= 1, "eNI 2~OEO J~.JI~SIN).4 2~OEo W,..jf.,lT,1t.) • wfJMt;(IO 2~OEo
CO. T1NUF 2~DE 0 GO Tn 61 H I OH 1
6170
co 61N ,=], -eN I 10HI J,., • Jl~~ (t...J '" 4 10rEl \1110("1'\ .. ,,) \III (.JM,;Oq l{\,rl
b179 CCNTINliF I~FEl ,-----TF5T CLOS"P"CF 2eDEO
6174 IF i II. ..... /\.) .1,,1':", 0 } GO TO FtSOQ 2F10£0 IF I KO'Frl'.TTI .Eo. a 1 GO TO 627 4 15FEI IF i ncs" • .-0. 0 1 P~INT Q9~ •• NIT. '_3 IS'!I IF ( ITCo •• FQ. 1 1 PRINT 999., Nyr, K.3 15'£1
PGINT Q9A5 IS'PI GO Tn QQ9Q ISAP I IF i .,.TINln .'E. 0 ) "I' TC 6113 25JAI If i 10C, T .r.T. 0 ) <;0 TO ~J1s ISHI IF , ITCSP .FQ. 0 1 ,,0 TO .230 I5FE I
C-----T(5. TO SEE If T~E OFFLECTION~ CU_PUTEO FRO. THE NE~ .C~. Q-w CV 2IAPI C-----ARE CLOSED P'CK ~N '~E OLD ey O~ NOT. IF 'ES, USING THE GREAT,ST 26JU\ C-----STIFFNESS S ITER.TIO. LOAD ~F THE NEw G-w CV AND REenHPY'£ THE tEf26JUI
00 ~?35 • = I. ,.,,.,c ISHI J • IST.INI ISHI
IF , IUSUI .EO. 0 ) Gn Tn 6235 15HI NPeT = ,PTTSINl IsHI
IF IS.(JI .EO. 0 ) Go Tn 6236 I5FEI ES- = 0.5 I5FEI
GO 10 ~?31 15FEI fS- # 1.0 15HI
IF ( WPT1"oll ) H4G. 9991, 6"." I'HI IF ( KSV~J(N' .EC, 1 ) Gn TO 6241 2~JUI IF ( K~¥'~iI'l ,EC. 0 I Gn TO 62"1 2~JUI &0 Tn 623~ 26JUI IF ( '(J.n) .GE. W(J.<K-l) 1 GO TO 6235 26JUI DO ~2!O .P. I, ~PCT IS'£I If ( WiJ,"-li .EG. WPT(N.'P) ) GO TO 6260 15FEI CONTINUF IS'EI 00 Tn 6?3~ 15F[1 IF I "SV~J"1 ,g. 1 ) ,;(, TO ~245 ISH! iF ( ~SyI\l'JP~) .ro. -1 } r,r TO 624115 26JUl G0 Tn 6?3~ ISFEI IF i .(J,n) .LE. W(,;, •• -ll I GO TO 6235 15FEI CO ~255 ~p. 1f ~PCT ~:~~: ji=" i wi",., "-II .e .......... 1._.''"' nu 'v va.""'... • ... 1>'.\..,
CON.l~UF ISFEI GO .0 6235 ISfEI
55(",1 ~~(J)' SlnPF(N'!i • ES" 21API OIIJ! OIL.;)' GT!J,II • (5 •• i QPTIN,NP) • SLOPE(~,II 21APj
• WP'!'(N.f"lP) i 21APl
6260
IOCNT • 10eNT • I CONTINUE I' C 10CNT ,EQ, 0 ) GO TO .115 GO TO 6585
ITO • I TO 0I1l1 UH
I' CITeS ,EG. e ) CO I' ( K .L'. ] ) GO TO
c ••••• INITIALIZING ,OR STARTING TM! SfCONO CYCLE Q' IT£PATIONI 00 6111 N. I. NNe
J • ISU!N) 108W(J) • e NUMOC(J) • ~
CONTINUE NITT • NIT NIT~ • I Noe I • 0
GO TO 6910 I' ( K .0[, ] ) GO TO .eee I' ( MONI ,(G. 0 ) GO TO 5998 I' ( NIT .G!. 2 ) GO TO 5999
~RI"T QU. Klfl ~RINT 92]. ( JIHS(N). N • II MONI
!9" ~RINT 920. NIT, KCNT(NIT) PRINT 925. ( WH(NIT.N), N • I. HONI )
I' ( KCNT(NITl .fO, I ) GO TO 6115 CONTINUf I' ( K .LT, ] ) Gil Til I' ( NOe! ,fG, I ) GO I' ( NCye ,£Q. 0 )
U99 TO b!n
C ••••• SINC[ TANQf .. T MOIlULUS M!TMOO NOC I • I
.0 TO 6'599 11111 "'OT CLOUt
NITS. I 110 66e. N. I, NNC
J • IITA (N) 10U(J) • I 'WHIlC (J) • II
CONTI"UE GO TO .9111 I' ( UCSR .[0,
"'ITT. NIT 1 GO TO UU
I' ( IT8W ) 6!95, "ge. 6!9! I' ( NCVI .N!, II ) Gil TO .601
I'RINT 9919 GO TO 9999 H ( 'ICYI .U, II
~RINT 9918· GO TO 9999
GO TO U7!
00 lACK UI[ LOAD IT •
e=~e=:=~: 'MITCH~! ~~~ '~!~1!NQ TH! !~~a~ W!!!!~! ~~ ~~, CLO!!~ ~!!M!~ C ••••• TM! .,ICI'I!O NUMBfA 0' IT!RATIONI
6611 I' ( ITe~R ,!O. e ) KCLOI'(I) • I' ( ITeSR .!a. I ) KCLOI!(ll •
C ••••• CALCULATf ~lO~!S,.wIA.S 0' TH! IARI ANII ~O.INT.,eu~,RIACTIONe AT C ••••• TH[ JOINTS '0. TM! ~A!V!OUS TIMf ITA .1'5 I' ( K .LT, ] ) CO TO 11111
00 8108 J.], ."
15'fl 15~!I UHI I5HI 15'£1 ZSJAI U,f1 2UPI I!UI'I I!AI'I I5A~1 I!'(t UJAI U'!I 2UIII 15 All I !e~lI UIlU 2\AIII 15H! 280[11 lIllU 2111'" ulln UHI non 2,,1'1 I,A~I
UNOZ • Il,A~!
I!A'I I!U>I I!AI'l I!A~I IU~!
IU'! HUI'! I!u~! I!A~I II.A~l '6A~1 nO!0 nou Z91lU 29IlU 290111 2911[11 !! ~D! 11A~1 1l1tA'! .U,I I!HI I!HI I!HI II!UII
8006 I 2
HOOO
8105 1
BI06 I 2 3
BlOII BI20
"121
1 2 3 • <, 6
BIn I 2 3
R <:I
1 ? 3 4
= ( ... \idJ-ltKK""'ll • w(J.K~-l) I'" FfJ) • i wtJ-l,KI'(-l) ... 2.0 ... \Io(J'KK-ll •
WIJd.KL-1l I I HE2
OS~EO
0"5EO OS~EO 21API IF I I< .. , ~OGO. BOOO. BOOb
AM(J} ,. OtL,n • ( .. W1J .. l,KI'(-2) • ... W(J.l,Kt(-t!) • W(J-l.I'(KJ ... 2." • WIJ'l.KKI ) I I HE2 • 2.0 • "T )
0.0 Rfr.4t"'P~) *' 0.0
CQ Pl00 j.). ~P~
2.C •• (J,P(K-Z)Z)6P] 1IiI1. .. ""~) • 211l P l
21AI>I OSHO OSSEO O,!\ Sf: 0 OSSEO OSSEO IF I • - 3 ) 8105. 810~. Aln6
DR"ljl • I -ew{J-Il • e~IJl * w(Jtt<K"'l) ) I H
I H - TIJI • I -'IJ-I"'-II05~EO O'!l~EO
GO TO ~! 07 055(0 DR"IJl • I -P-IJ-J) 0 8"tJI I I H - I 0.5 • I ITIJ.lI 0 10nco
T!lj.?l 1 0 TIJI I • I -.IJ-l,KK-ll 100CO o .lj.I<I<_11 I I H- 0IIJ-P' 1-.lj-2'KK_cI· 100CO 2.0 • wIJ"1.KK~21 .. WlJ.KK-21 • w(J-2,KM, • 2.0 100CO • wIJ-l,!(K) • I;aI(J,Kld ) I ( ~f3 • ",T ~ 2.0' 100CO .. CHJl • ( -1R(J-1,t<K-2) • 2,e • wtJtwK-2} - .( 1fl:OCO J.lt~t<.21 • IIiIfJ .. ltKK} .. 2 .. 0 • lIf!j.t<K} .. w(J.1,KM100CO I I I I HEJ • "T • 2.0 I 100CO
I<EYJ • -EYI"I 05SEO IF 1< .. f[;;. IF lOP IJl GO TO Pl40
, ) GO TO 810R 2)AP I .NE. I ) GO TO SlOB 2jA P I
GO 1'1 I "140. Al20. A140. Plco, 81 4 C l. KfYJ 2lA P 1 2IAPI OSSEO 04jAI O'!lSEO OSSEO 055£0 O!5EO 055EO O!5EO Q5'!\EO OS5EO
REAC II ( RM!J-11 - 2.0 ., BM{J) - RM!J.11 ) I '" .. a'J) • ( CP(J-1) - CP(,J+1) } I ( 2.0 • '" ) - ( Q(J-l) • W(J-2.!(K-1) - tltJ"'l) • W(J,!(l<-lJ - R(J-1) .. w(Jt!(K-1} .. Rt~.l) • w(J e 2.I<K_l} )
t 4!O • ~E£ ) - ( T{J) .. WtJ-1.KK-li .. T(~l (t W(J.KK_l) - TIJ.l) • WLJ,;t!(-U .. TtJ.11 • '(Jolt""-!! ) I ...
IF ( K _ 3 I R121. 8121. AI2Z "E'CT IJI • PEAC
GC Tn AIOO REAC1 (Jl • ~EoC - I I}n.;.Jl 0 QTIJ.2) I • 0.5 • p"O!.' • 100CO
( wtjtKI(-;:q - 2.0 • wiJ,KP("'l) • WIJtKK) 1 I 100CQ "Tf2 • OEIJI • ( _W(J.KK_c) •• IJ.K.I I I 2.100CO fit "T1_ ( f"'l1 tJ ... n • ( .... fJ-l.f.J( ... ;q • 2.('1 • 10n(0 'lIdj-l'KIC-;?) .... fJ,KK-2l • \rIi\J""2,KKl - ;(,(1" lorco \l.C"''''lt!(Ki • w(Jtl<.d ) ... 2.0" OItJl • r l(!OCO "'w(J .. ltKK'-?l .. 2.0 • tIICJt KK-2J - w(~·11I<K-21 ·loOCO "'<J-ltI(K) .. l.O •• (Jtl<l() • III(J.l*KKl ). lQ(')CO Cl (J.l; • ( "W(JtKK ... l) .. Z,o • ~d,J ••• KK·'} - 100cO irdJ*2tKK-;:q • WIJ.KK) ... 2.0 .. IiIfJ.l.I(I<!. lQOCO WIJ02.~K) iii I HE3 • ~T • 2.0 i 100CO
PEAC!,JI • PEACT'J) - ( 0.5' ( It,J'11 • ~Tr •• 21 I. loor~ wIJ-I.~M-I) - o.S • ( TIIJ.II 0 TTI •• 21 I' looco \IdJtKI<-11 ... 0.5. TT(J+l.U. T1(,J·1.2) ) • looeo w(J.j(K.ll • 0.5" , TTC .... 1.P .. TT(J.l..Z) l • looeo WIJ·l .. K-ll I I H IODeo
Gf' Ttl 8100 ~140 t:1fAC'T(J) ;t .. <;St'("', (..,1) .. W(jiKK-U .. QC;:JI
~lno (O~TI"UF (-----5A'F OEFLECTICNS FCq PLOTS AlO'G TH~ TI~E AxIS
AltH ,,:PT C n C~ ~l~O J = 4. ~P4
NFpr: II: I"l
l Sf /. ~ I ... '"
If NCT9 j ~qA~t bIRO. blA5 IF • - 2 I R317. 6)90. 6190 DO flq<, ,C9. I. ~CT9 iF XAxT<t~rQ) .~f. TESTT IF YAXT5lM'9) .... f. TE~Tn if TYS .. nCQ) ..... f.. lSTA )
NPT 'C ,..oT ,. 1
KS.V I~PTI • LSTA KEYPINPTI • 1 .... Pr: tt', PT) II: JroIPS (~Ccn I(Y~ {"'PT 1 ;: )
I GO To b19~ ) GO TO 019' GO To 6195
tF { K .FQ, ~TP. ) GO TO ~195 Yf~PT.I(""U • WLh~l'()
NEQR cr: ~fPR .. 1 C~~ Tl ',U< IF I N[OO .l<. I I Gil TO 'l~O
pOr~T 99Q~. "'FP~. l$TA GO T~ 9q_.
bIRO CONTINUE C-----~AVE ~CMENTS F0p PLOTS AI.O"G THE TI"'~ AXIS
00 6?PO ~ ••• ~P4
~285
NERP • ~
LSTA • J - 4 IF I NCT9 \ Q9aO. b2aO. 62~5 co b290 '(9. I. NCT9 IF I XAxl<l~(9) .'E. TESTT If I YA_ISINr91 .'E. TESTH IF ( r'f~"fNCQ) .... r. LSTA 1
~,;P'T 1:1 1\0'T • 1
'SA\d~PT) • L~TA P(FYPlf'llPf) • 1 Mpr~~p'T; s ~F~iNCQi
JC;YSH .. PT} - ? IF I K • LT. , I EO TI) 6?"~
yt~PT,~~2f ; ~~{JI
NEQA :' "FR~ + 1 CN,TINUf
GO TO 6290 , GO TO 6290 GO TO 0290
IF ( NEPQ .Lt. 1 ) on TO b?~O Prilf\'T Q9Q'5. "FPP. lSTA
GO T0 9QqQ ~?PO CO~TlhUE
C-----~.VE OEFlECTIO" F~R PLOTS ALONG THE ~EA~ AtlS
6205 6210 6215
IF I NCT9 I <:19M. 620". '205 IF I K _ 3 \ 6?Or. o?in. b?15
NPR • NPT NfQP III 1'\
O~~EO
21'''1 Oc:.'EO l~fEI 2"('£0 290£0 15Ft! 290EO 290EO OU"I 290FO 290EO I~F£I 290£0 290EO 290£0 290EO 290[0 IOHI 15FF I 10FEI 10'EI 29DEO I~FEI 290En 29r.EO 29r.EO 15FEI ISFEI 15HI 15FEI 15FEI ISFEI 15FE I 15FEI 15FEI !!iFf I I~FE I I~HI I~FE I 10FEI 15HI 10HI ISFEI ISFf! ISHI 15FEI 15HI I~FF:I 15HI 290EO 290£0 29Cfo 29('\F:0
DO 6220 'Cq. I. NCTq IF I XAX!SINr9) .'f. TEST~ IF YA'15INr9) .'E. TfSTO ) IF IVS~{Nce) .~E, ~~3 ) ~0
~JPFl = ~~pR • 1 t<S{lV(~!pP) ;;; ~""'3
I<.£YP(~PPl So 2 MP(,,)(""PRj II: fo'PC;(NCQ) I'\:YC;O,PR) • 1
00 6??S ~P~. I. ~P3 NPP2 s: f- PP • ?
GO TO bUO GO T0 bUD TO 6?20
Y(""PfI, ""PPI :r. \Ij(NPP? KI'\:)
6225 CO'TINUE NEPR II: ~,fRR •
6220 CO'TINUE IF ( ~EQR .IF. 1 ) Gn TO ~?95
P~I""T Q996, ""EPR. K~3 GO TO 900,
C-----~AVE ~O"FNTS ALr,G T"E BfAM AXIS FO~ pLOTS 6ZQ, NFRR • ~
CO ~3~n ""Cq. 1, NCT9 IF I XA<T51"r9) .'E. TFST~ ) GO TO b300 IF I YAxISINr9) .'E. TESTM ) GO TO b300 IF I TYS.I'CQ) .'E. KM4 ) GO TO 6300
NPFIT • ~'PRT • 1 ~SAVT(NPBT) • K~4
KfvPT INPAT) • 2 MPOTINPpT) • MP~INC91
I(V'5T (NPpT) • 2 CO ~30S NPP. 1, MP1
NPPl="PP,? YT n·PRT,NPP) • R~(NPPJ)
6305 CO~TIN"F NFnp II: ~ERR •
6300 CONTINUE IF I NEOO .LF. I ) Gn TO 6~OU
P~TNT 9996, ~F~~. KM4 GO Tn 9999
6200 If ( K .LT. 1 ) GC Tn A317 IF I IP< ) ~lO?' 832" AIO?
C-----pOINT ~ESULTS BIO? IF I K .r_T. 1 r.r Tn ~?oo
PPINT 902 (';0 Tn A?OC;
8700 IF ( I<. .GT. 4 GC Tn 8201 P~I"'T QI,
8i?01 P~INT ql~, 1<~4
820S pOINT 901 !F ( IP~ } H~OO' ~J~~' MJ?O
e3~0 IF' •• LF •• ) GC TO 83?D IF I KPr .[Q. K054 ) Gn Tn 8)20 IF ( Mn~1t .fr'. 0 ) Gn TO A1j?5
C-----p"I'T ~nNIT"O 5ToTION5 RFSULTS ~o P340 J = I. ~c~S
2qOEO 2qOEO 10HI IC,FEl ISFEI 2QOEo 2QnfO 2QOEO 10FE! 2QOEo 2QOEO I~F E I 2QOE 0 2QOEO 2QOEO ISHI 2QnEO ISHI ISFEI ISHI ISFEI I~FEI ISFEI I~HI I~FEI IsHI I~FEI I~HI IsHI IsHI ISHI ISHI I~HI I~FEI ISHI I~HI ISHI I~HI 06API ISHI 0~5fO 290fO 2Qnf 0 OSSEO OS~H O~~F 0 OS SE 0 O~SEO
O~SEO OSSEO O~API
055[0 OS~EO
J5TA = JP5 ( ... il • 4 Z 1 ;;; ",,'Pc:. ( ... Jl X II: Z 1 ....
PRY,..,T Q??, r~(JC;TA), CRM(JSTAl PPTI'oIT Q21. JP~{JI, x, \Ij(JSTA,KK-ll, A~(JSTA). REACT(..;STA)
ij3.0 CO'TINUE GO Tn 832~
C- ____ POI'T cn-PLfTE R.SWLT5 B3?n JSTA • _I
71 • J5TA J: 'II Z I • ~
P~II'oIT Q21, J5TA, X, .... (J'ltt< ... l) , ~~(3), qEACT(3)
CO e210 J. 4.MP!
J5Tf, & J .. " ZI = JSTA X • 71 ....
PI=lII'oIT 9i?i?, D\Ij (J). DB~ \J) P~INT 9i?1, JSTA, x •• IJ,I(I'I: .. l', ~"'(Jl, REACT(J)
B210 CO'TINUE KPC • "
IF , IT~. ) O)?6. 8315. 8326 IF ( ~ .EO. ~TP4 ) Gn Tn A315 IF I MO'I .En, 0 ) GO TO 031 7
C-----PPI'T ITEPATIO' .O'ITOO nATA PRI'T 916 ••• 3
B316
B330 ~)17
Pj;jJNT 9i?~, ( JIP-4C.(~), N • I. 1Io10Nl DO P31~ ~T ~ 1, ~lTT
PPI~T 9i?4, "I. K ... C~T(~T) PkT~'T 9i?~. ( .... "'(~hN), N • 1. ~O~I )
CO'TINUf IF I ITrSP .FO. 0 ) GO TO P317
P~I"~T 9977 rC" 1=1331) ~I:z 1, ~TT
P~I~T 9c4. ~I' ~rNTI~t) P~[I'oIT 9i?S. ( .U(~ItN), N. 1. ~O~I )
CO'TINUF. IF ( KCLCSF(11 .F~.
IF ( ~CL()SFI11 ,EC. IF I KCLr~fl?) .EO.
POIU 998, GO TO 000 0
.ANn. 'CLOSE(2) .EG. n ) Go TO e31~ ) PRINI 90~6 ) POINl 9987
8318 IF' •• LT •• J r_c Tn 7000 C"---"~f5FT .... tJe!l :I \Ij(J,I'I:K"'l) AND "(J'i?) •• (J,1<1<1 AND I<.K I: ,
8315 ro P21' ~ ~ I. ~~1
B21S
"(J,ll II "I.J.I<I'I:-ll -,J(J.i?) 2 "(j.KI'I:)
CO~TINUF.'
~~ = ?
r-----PlCTTI~r_ CuoVfS IF I "peT .EO. 0 ) Gn TO ~500 C(1 P":'OI'l ~ = 1, ~PRT
I<SAV(~P~.~l 2 KSAVT(N) KFVP (~PC.~l = I<FVPT (N)
O~SEO O~ SF. 0 OSSEO OS~EO OSSEO O~SEO 290EO O~SEO OSSEO O~SEO OSSE 0 O~SEO OSSEO OSSEO O~SfO O~SEO OSSEO OSSE 0 O~SEO OSSE 0 2QnEo O~JAI 2~ JA I 06JAI 06JAI 06JAI 10FF.I I SF' 2QnEO 2QOEO I OFE I InFU 10FU InFEI 10FEI 10FEI O~API 06API OOPI ISAPI ISAPI 06API OSSF.O 100CO OSSEO OSSEO OSSEO 05SE 0 Ol;l;.Fn
O~SEO 15FEl I~Ffl ISFEI ISFEI
N o o
"POI'PR.Nl • MPOTiNI ~VS!~PR.N) & K¥ST(Nl
00 e-o' ~~« I. ".3 Yt~P~.N.N~) • YTIN,NN'
8605 CCNTINUE 86no CO'TI~UE
~p~ • NPB • ~P8T
850n IF I K"" .E"Q. 0 ) Go TO ~Sol PRT~T Q9S3, K~Ax. KM3 PQl"T 9985
Gr ,0 9Q<;Q eo;nl IF I IP~ ) .~02. 8503. HSn2 esn3 penn 9.0 .~02 IF I NCT9 ) Q98D. 1010. 7050 7050 IF I MOP) P112. n14' ~113 8312 PRINT qll
G(' TO .n40 R314 PRI,T qn
GO In 11'140 R3l3 PRI~T 972 '0'0 v'"11I 0.0
"1",../2') • 0.0 NAC & 0 fib. • 1 lOS • 0
00 7065 ,p. 1. ~PB NAC = NAC • I
IF "lAC .EQ. I ) JNI • 0 IF KEYP (>,P, .Eo. I I GO TO '051 I~ ( ~~yP(~Pl .EC. 2 ) lnx(~AC' • ~P3
~PS = ? • ~YS(NP) K~(~Ar} • KSAV(NPl
GO T0 7r.~? 1()51 F~Y(--A('\ #. w'fPZ
~S<~AC' : K$AvtNP, KPS = K .... StNP)
10~2 ~A)I'1 = rOX(f\ACi If I MOP) 77S0. 72SS. 1:>SO
7250 IF ( ~P .'E. NA ) GO TQ 1?'5S C----.SAVE TI~E OR REo. STAS FOR TITLE$ ON T~E PLOT
1158 IA • ~A • NP • I ISIIAI • KSAV,NA)
11SQ
IF I MPC(~A) .EO. 0 ) GO Tn 7159 ~A • Nil • 1
GO TO 7ise NA • NA . I I ~S (Il II: ~PRCR
IF KE¥P(t1P} .EO. I I GO 10 10S9 10121 . I O~T 1"[ 5TA a
IF ( KPS .EO. 3 1 IS~(21 . 5" ovp IF ( KPS .~a" 4 1 T55111 . S;' ~VP
FNrt"Cr I 10. 4. l~!l ) I ISS GO TO 7"lQ
1n;9 Ifl( 21 . laHBEA" SU e
I!lFE! I!lF£1 !!lFfI ISFE! I!HI 1 !IFf I I!lFfl 2'10EO l!fEl 2'10EO 29DEO l!lfEl I!lFEI O!lSEO USEo 17SEO 17!~0 17SEo 17~EO 11!£0 I!lHI 1 !I FE 1 O!l~EO O!lSEO 0'55£0 O!lSEO O!lSEO 2!1JAI 17SEO O!lSEO IOFEl O!lSE a O!l!! 0 O!lSE 0 OMEO 10HI O!lSEo ISffl ISHI l!\fEl 17!1EO 175£0 17n'0 175£0 l7UO 17SfO 26Jul I75EO USEO 26JUI 26JUI 26JUI 17!EO U5Eo
'079
T O'R
7077
'oel '0,"
IF KPS .Ea. 1 ) 15s171 = IF f KP~ .fa. 2 1 TS~(7' :
f,(Orr I 10 •• J I~II) IAre • IA I 2 II • 2 • lAD,
51< DVT 51< ~VT ) ISS
IF I II .FO. II ) GO Tr 'nTA IA 11: Ii • 1 I(Jc:. I:: 1 I S'I t. j 111 5H ! " 'S t A I ?
on 7076 ~. 1. lA2 NP? • N • ? F\!N • 2 • I'. ... 1 15<::! i} lIJ IS It..N)
iS~\?1 ::J' tSp.N.n ~ .~[. tA? I GO Tn '077 1 F
IF I~S .EO. 1 ) GO TO 70AI F~ccr,f ( 10. 5. IQINP71 ) ISS
GO 10 .nT~ ENCorE I 10. 6. IOlNP?)
CO~TI"'UE ISS
2~J'J 1 26JIJI 26JUI 17SEO USEO 17SEO IBEo 17SEO USEO 15Hl 115EO 175(0 175EO 115[0 175[0 175(0 115EO 11~EO 17SEo I1SEO 17SEO
IQS a 0 Me • r 162 • 2 , •
C-----SAVE T;'E VE~lICAI VALLES OF C--.--ARRA~ Qt Yp
175£0 In 17~EO TNE ACCUMULATtO PLOTS I~ A' O~E-DIME~SIO'AL
2lA P I '2S~ or .060 'I. I. ",xI
10~O
YDc";l\l.",T) • V~NPtNrl
CO'TI'UE Ir { ~Pn(I\Pl .fO. 1 , GO Tn 7090 If I .Pnl'P) .~E. a ) ao TO 9990
15H'1 O~~fO 055£0 O!lSEO O!l5( a USEO If I MOp ) 'CS3. '054. 7053
c-----Sf' SCALE FOP THIS FRA"E OF ACCU~ULATED C-----PAPER PLnTTl'~ MFT~OD
.~oT5 RV MICROFILM OP BALL-POINT ~lAPI
7053 NPP • NAC • ~AXT 1 !lFE 1 00 7260 •• I. ,Pp IF ( '(tIIIt)) .(:1. ,(peN} V~(ll c yp(p.,.) IF ( "'''(2) .1 T. YP(N) 1 'r'~(21 )"p(,,)
CONTtNUE
15Ff I 15Ff 1 15Hl I !I FE I
XM(P 1: "'1. tjt,ljf2j • MAXI - ? tROLL « I LOP" Jl tOP LOP. -,
CALL lOT 1 ( )ffolt VfoI .. 2,. TO LOP • L~PT
C~ 1o~P J. 1. ~A.I
:rF" (",,) :II: J - 2 CONTI NLJE
IF IpotAXY.p Z' """31 Xr(ftoIAxy.2) • XMi4l
c~.---PLCT T~IS Fe.~E rF ACClJ"uLAT(~
C-----PLOTTI~G ,,[T.rr no 726! • a 1 • • ,c
N~ ~ ! t - I J • ~AXT
I!\Ffl I!\FfI ISFEI ISFU IlIFE I I!lHI I~FEI 17SE 0 O!lSEO OS5EO ISfEl 15,,1
PLOTS RT "lc~orIL" OR BA~L-P01" P'PEP 21A P I ISHI 15F(1
CO 7270 ~I. I. _AXT VPI,1l • VPl''''·NI.
7270 CONTI~UE V"III • yPI"AXI.11 Yl"121 c YPl~AXI.i>1 YPI~Aq'I' c YMI31 VPI~AXI.2} • YMI41 IRnLL = MPOI'P - NAC • ~I
IF l N~ .~". NOS 1 Go TO 1?75 If I N .EO. NAC ) IFU"lLL • -I
7275 CALL ZOT I I xr, yp. "AXI. TO 1 VPI.HI.I} • YM(\) YPI.AXI.21 • YMI2,
T26S CC"'TINI~ VMlll 0.0 nuz) " 0.0 NAC • 0
GO Til 7~65 C-____ PlOT T~IS rR."r nr ACCUMULATED PLOTS 8Y PRINTER PLIlTTIN8 ~[THOO 7n5~ 00 1080 J. I •• AXI
IXIJ1"J-2 70@0 CONTINUE
PRI,'T II PRINT I PRINT 13. ( ANI IN). N • I, 12 ) PRINT 16. NPRnR. ( AN2IN" N • I. 14 )
IF I KpS .EO. I ) PRINT 9il IF I KOS .EO. 2 1 PRTNT 91q IF I KOS .EO. 3 ) PRINT 9?9 IF ( KP5 .Eo. 4 ) PRINT 939 IF I NAC .Lt:. 5 ) GO TO 1~84
PRINT 9976 GO TO 9999
10S4 00 1085 "I. I. ~AC PRINT 912. KI' SVMalKII. K5(KI)
70es CONTINUE
70<10 7065
9980 9999
C
PRINT 'Ill CALL SPLOT 9 I NAC. yp, lOX. IX
NAC • 0 GO TO 1065
JNI • JNI • IDXINAC) CONTINUE
PRINT 11 PRINT 16. NP~06. ( A,'IN). N • I' 14 ) CALL TIC TOC (41
GO Til I~IO PRINT 99'e
GO TO 9999 Pt-lfNI ';""1
GO Tn 9999 pol"T 9P.O
CQNTINlif END OF CPC5
1~'E1 I~'EI I~FEI 15'0 15FE 1 15'E! 15'E 1 15FEI 15FE! I'\FE I 15HI 15HI 15HI 15HI 15HI ISHI 15Hl 175[0 Z\ApI 15'EI ·055EO 055[0 O5SEO OliSEO Oll5[0 055[0 055[0 055[0 IO'El 10'[1 I!lHI 1!I'El I!lHI O!lSEO O!lS[o 05S[0 O!l5E 0 O!lSEO O!lS[o 055[0 O!ISU O!l5EO I!lHI I!lHI 055'0 055EO 055[0 O!lSEO l~rEl 10'0 055£0 O!l5EO 05JAI O!lSEO
N o N
16\3 16,5 16 17 )618 16}9
1600 16~1 1652 Ih5
SUqROUTINE INTERP3 ( MP7. f\lCT, ,)"'1. IN?t 1(J:f2. ZN, 1. LS~, I('S'-I l OI'IEIIISTO~ JNlilOOI. JN211ool. Ko'tlOol, 1>'0001> 112Q71, "5_11001
C'S~FO
'OF~6 14~Y5 FORMAT (114;]" ERROl> qe ••• STAT!ON~ Nr>T IN "IIDE. 1
FOR'IAT I 114j" UNnESIANATFD rRRO" _Tnp IN SUBAOUTTN. ) !4 V v5 , ..... yl)
"5~~O
~6MQ5
'~"Q~ 12A='5 "96"5 I'\. 5c; J"(]
IlJ6S 7SSP '6SC'1 lOr-r-6 'Z J65 12A05 ,OFF6 ?4 Fr6 IOF~6
~1'E6 lOFn tO FE6
FORMAT (,,4bH ER~OA __ ~'~N_?ERo nATA R~VONO ~N~ ~F ~~AM DO 1~Q3 J; 1, YP7
l(J) • 0.0 CONTINU[
AS" AS"
= LS"t ::t AS,,", , 2. "P7 • 7 M •
J(~I • 0 IF( NCT • I ) 167.'l~'4"60. DO 1~75 NC. I. NCT IF J(RI 1 16Q~. 1605. 1'0'
If IF
NCI a NC JV a JNIIN(1) 'LSM . J(RZI~C) I le'8, IIOT, I~TO JN2INC) _ M ) HO'. 1"'0<>' 1~li JSZ • IN;>{NC) lNS • zt.UNC)
GO TQ 1619 JSZ • lNS •
.. ZN(NC1) • ( Z"INCI - 1NINCII I ( JN2(NC) • JV • lS~ )
KAlOS • KSWINCI GO TO 1 1613, 1615, 1617. 1~17 ). ~6SS If I JNIINC, _ .. 1 1619. 161 Q , Iq~
IF I JNIINCl' ... 1 1619; 16j'l, 1~'O IF I ~S.INC\l • Z I I~I~, 161 5 • 16jS If JIII21NCII _ .. I 1~19. 1~j9, 1610
If
JI • Jv , 4 J2 • JS2 + 4 DENU" • JZ - JI • LSI< JINCR • 1 E5" • 1.0 IS- • I DE NO" ) 169S. I '2~, 16H DENOM • 1.0 ISW • 0
IF ! JZ - JI 1 1651. I~l;;. 1~30 00 1650 J. Jl. JZ, J!NrO
F • J F! • JI
• 65M nE1\10,",
1 • I JS2 - J~ , l5" "O~~b 10~"6 'I~E66 '15~bb 1 QFfb 10'[6 o91oC1t6 (19"'°6 10 F [6 10F!6 l2AD5 1 ZJAS l!JAS llJA~ , lJ6~ 12J6<; , ZJA5 I'\BIo4RA
lZJ6S 1'\ 96 C:l'5 09A"5 "9605 12JA5
DI'I' ••• '1 PART. OT" I llJI • ZIJI , ( ZN{'JCU .. PhRT • r 1NS .. IN!~C1I II • ES"IOF~t>
CO .. TlNuE If I l51>1 1 I' I 15~ I
JINCR ES" • 15~ •
GO TO 1630
Ib9R. Ib52. 1'60 1698. 16"'0. 1.55 • JZ • Jl ~O,5
o
Il.IAS (,8j"dl'~
o9APS \l'JAS IZ.1AS lZJAS llJAS
161'00 iF 1 KR2u;CI I l~qq:. }#"7r, '#.05 16~5 J'iJ ::: IN? (J"~C) • I,-;U 1610 wt:'1 !(q?(NI":)
~.jC 1 ~JC
Ib~~ CU~T1NL[ 16~' CC~TJ~~l
,"1p~ II: MP1 - ? ~f>b :: ~P1 - 1
(--.--TES' F~~ nATA £R~n~EouSLY ~TnRF~ BEvnNn F~nS OF AFAl AFAV lCfIo. l(l)" Zf?l .. 71') .. 11""1''') .. ZfMPfo!l .. ?iMP")
IF ( ~C~ ) lOQQ, Ib7~, l~Qq
1616 I=\£TV~ti
16,5 PRINT qn~ GO Tt) 17jQ
IIj~8 PAl'NT Q'.).4
GU Tf') 17"'9 161':19 PRTNT QQY 11,9 CONT1'vf
[N0
'?JAS 12Ap5
1?JA5
1 'JAC; 12,IAS
;]1~F'f,"
"SS-" 0 r'5~~O nAIt,I~A
f'dSC;C' 0
\ 4,.yl'j
14,I.;IY5
'4"""5 ''\J661 14 .... ",C;
''\ !h67 14~¥5
Ib~ 1
tV o w
SU.ROUTINE TIC Toe lJI e----- TIC Toc (II • eO~PILE TTMF C TIC TllC (2) • ELAPSEn Co" TT~E C rye Toe (31 • liM, FOR T'il~ PRO"LE"
TIC Inc (41 • TI"E rOR T.<l5 PRORLEM AN(\ FLA o 5EO cPU Tt"~ 10 fORMAT(IIt30'lQHELAPSED roll TI"''' = I~.c", ~lN"TESFQ,),A'" ~rcOND5 11 F'O~t4AT(11130:"1-:'HC~,",P!LE TT~F' u .T5.~ .... ~t""'ITE<;.F'9.3'Ro1 ~F'r:I"I"IOS 12 FORMATtII130~2.HT1ME F'O~ THt~ PRA8LEv •• t~.A~ ~rNUTE~,F'q.~.
1 A'i SECONDS I t • J - ,
iFf T-1 1 ItO,~(I.3l)
'0 fl~ = f (~o Cb.LL S[CI"JNO (F)
III • f 11 c 111 I ~o Fy2 c f' ... ll.~o
IF( T ) 5ry.10,~O "0 PRINT I!' Il'F!2
GO F' Q90 ~o FIJ s F _ F14
12 • fT3 I M fI3 = Fl' _ 12*~o
PRINT 12' 12' fl3 1Ft I-I • 9QO.Q90,70
10 PRINT 10. 11."12 9QO CONTINUE
RETURN EN"
,4nCb6 "8~OB r"R~QiR
nAfoiQA
i"At.fPA
::>SC::ff.6 ~S~f66 ;SSF66 ,S5E66 ,one66 ?'''f66 ;SSf66 ::':t'f'f.6 7Sj'F6~ ;S5f~6
'SSE66 >55f66 ~5sr66 ?SSt66 ?5SE66 >5SE66 ?SSE66 ,5"H,6
~~~~~~ >5 5E66 '5"E66 ?5SF66
Sljq~OUTINE SPLDT q ( NPLT, 'II, I'1't't II( 1 1 I J[O C •••• THE LATEST REVISION nATA FOQ THIC; ,oOIITIN[ IS
CO..,MON I SPLOT I WIDTH, S"-4ALi. ~TG, C;yuR II JUN 70 cEVISE~
1 1 JrO II JrO "l4Jt"O
DIIrotIENSYON IOXI}). X(]l. IX(;1t C;PACrl",21, SY"-4j:\f!o' INTEGEQ "10TH DATA SPACE I 6Z-lH I • SY"-4A I IIo4T I • qLA~K I IH I
c •••• •••• THIS ROUTINE ~UPERI ... pnSE~ UP TO 10 PLOTS PER FR~~~ C C C C C C C C C C C C C C C C C C C C C
....
....
THE flQST PLOT SHOULn QE TNf LON"F~T 'NOST POINTS, TH~ PAPEN SHOULD BE pnSITIO~fD PCO"EcLY AND ALL HEAnlNGS PNINTED BEfnp~ rALLI~G. INPUT -NPLT, THE ~UMqER of PLOTS FnR TIo4I~ FRA~E
InX, AQRAY CONTAINI,," LE"GTH nf THE CESPECTIVf PLoTS IDX(! 1 = NU~RE. Of DOTNTS IN nQsT PLOT, nCo
x, STARTINr, AnnRESc; OF TuE ~,RST PLoT - NOTE • TH~ PL",TS IoIUCjT qE STOREO CO~TTGOUSLY
IX, INDE,oEN~rNT vARfABLF' ( T~OExING ) WIUTH, WIDTH OF PLnT( '_ES5 T~A~ ~3 ) S~'LL, LOWEP LINIT or vALUfS Tn RE CONSIDERfD
OUTPUT-diG, ~PPER LI~IT Of vALUf~ To RE CONSIDERFD
NO ACTUAL VALUfS A"E RfT"QNEO TO THE CALLING RO"TI~f • THE VAl IJ~S AQF PRINTEO AND PLOTTEo VEDTICALLY.
T~E VALUES PQINTEO AQ~ T~OSf Of THf flQST PLOT. IF "10TH IS GQEATER THAN 0 TNE 1<· .. 15 , .... ) IS PLOTTED WIT" THE APPROPRIATE AOJ"STMfNTS I" 'CALE MADE' IF NECCESARY IF WIDTH IS LrSS THA" 0 THE ,.AXTS I~ "OT PLOTTED A"n T~E ROUTINE fUNCTIONS JUST LIKE ~PLOT i
•••• FRAN~ L E~ORES PAX l~q2 •••• i2 fOQMiT I 41<,14,)X EI0.),21< 6pAI 1
)0
~O
NENO • 0 IrotI""A,( I: 0 I_SK • I
00)0 I. I, NPLT IF ( ""P<lAX.LT.IDX(I), "-4ra.!A'II = Tn'llITl
NENU • NfND • IOx(I) CONTINUE IF I NEND.LE.o 1
IF IF
loWE. WIDTH lAX • 0 1"IOE.LT.O IWIDE,LT.O , IOTA. IWIDE D""EGA • XII) TMET A :I 'II (1)
IF I NENU.EG.I ) DO So I. 2. NEND If I O_EGA.LT.'II, IF I THEIA,GT.XII' CONTINUE IF , S_ALL,GE.AIG ) IF , O_EGA.GT.RIG 1
2
IF TMETA.LT.SIrotlALL' If IAX.EG.I) IF TMETA.GT.O. l
r,0 TO
lA'll • IWI~E • -IWlnE
r;0 TO
n""'Fr;A = 'II(tl THETA. 'II It)
nO TO Ora.!Fr,A THF'TA 1;0 TO TIo4FTA
• ~rG • S ... ALL ~~
• n.
1 LifO t'14JrO o5JfO 10 nCq 10 n Cq "4.1E' 0 f14JrO r'\4Jrn r4J~0 n4JEO II J~O 14nCq II J~O IIJfO loncq , OOrq lonrq filS. lEO "'4JFO o4JEO 04JfO n4JfO !1JrQ 11Jro 1"'4J"O n4J r o 11JrO 04JEO '1 4 JF'O r4JrQ "4J~O 11~~q
(I. 4 "IrQ i14J~O r4JEO o4JEO IIJEO 10nCq 100Cq 11"nq 100Cq 10nCq 10 0 Cq 11,IEO 'OJfO o8JEO o BJEO II JfO "4J~O
IF owE~A.LT.o.} O""F~A = ~. ~5 IF O~EuA.EQ.THETA 1 GO TO ~O
SIG 'o1A = ( IWIDE _ 1 ) I I "'''''FGA - THE'TA IF ( IAX.E'Q.l .OQ. I'IJE~n.FO.l 1 Gn TO 60
RETA = 1. - SI~"-4A • THfTA IOTA = APA
IF I (BETA - IOTA) .r,~ • • 5 ) IOTA. IOTA - 1 ISK;C. K TnTA I~S'" = ISKX
~O 00 Ino I. I, M"-4AX IF ( OP<lEuA.NE.THfTA r,o TO ~~
I""'S~ • IOTA SPACE(lOTA) • "'Y"'~(C:;I
GO TO 8" ~5 III<. = 0
IF ( IAX.EO.l) 1;0 Tn '7n SPACE (ISIf''II) • !;Y,,",A
10 uO 85 NP ~ I, N~LT IF NP.uT.l) IfX •• TI"'))'(N~-P - IIXX IF I.GT.InX(NP)) r,0 TO 8~
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31 0 _4.924E.04 lEAM DEI"L 280 0 I 31 0 3,21OE'io 1,9IU·0. lEAM MOMT 60 1 0 0 0 3.210E·01 B[AM ..oMT 'iIIO 0 1 11 I 10 -3.090E.02 O. -,.0'0[ •• 3 • TII"E[-,"A" 8EA'" wIT .. ONE-WA. A "'0 S·I4"ETIIIC NONLIN[AII su" L.OAD£~ ... T." 1 II 10 20 -"990[.01 q. -6.110[.oS • 0 ° 0 0 0 a 0 3 2 2 I 0 5 9 In 1 11 20 '0 _6. 1 1l0E.0] O. -8.9S8['0) '0 4.100E.01 40 4.000[-n2 -I 4 -I 4 0 I II 30 40 -8.~6IE.0l O. -1,111['.4 4 II 15 16 IT I II 40 50 _I,118E.04 O. -1.t60Eoq4 50 4.000£·00 I,OOOE-06 A 15 l6 11
I II 50 60 -1.UOE.04 O. -1.30'1['04 0 I O. 2 I 11 60 10 -1.'09E.04 O. -1.2S0E.04 10 I O. 2 I II 70 80 _1.UOh04 O. -I,IIIEo04 0 0 0 3.000['10 I Ii 80 90 -I. i18£.04 O· -8.,6eE·n 0 30 0 2.400E·IO -1.000f.05 1 1\ 90 100 .8.968[.0' O. -6.1I0E.OS 0 30 0 5.184[-01 1 11 100 110 .6. i 101.03 ~, -3.0901.03 17 iT 0 4 q,q 0.0 -1.000['04 1 II 110 11. .'.0901.03 /I, -3.090["2 17 17 4 8 -1.000£.04.1.000[.04 1.000[,03 1 11 121 130 2.1'81[.02 O. 2. 11" • ., 11 IT a I~ 1,/1001.03 1.000['03-1,000['04 I 11 130 140 2.781"03 O. 11.0001." 11 17 Il 16 -I,OOOE.04·1.000[.04 1.000£,03
11 140 150 '5.000E.03 O. 6. 421£.°' 17 11 16 20 I.OOOE.03 1.000[.0'-1.000['04 11 150 160 6.42\E.03 O. 6.910E.03 8 8 O-\.OOOE·03 1.000E-OI 7 1.000E.02 1.000['00 1\ 160 lTo 6.910E·03 Q' 6.42\['03 .40 -40 _39 -35 .20 0 ~
11 \70 leo 6.421E.03 o. 5.000E.03 .40 -10 -20 -;0 -5 0 40 11 180 190 5.~00E.03 O. 2.111I1E.03 15 1-1.000['02 1.000E-~1 4 I 1,000[_04 1.000[·00 11 190 199 2.78IE·03 O' 2.7a~E·02 ~ 0 0 0 11 201 210 -2.212[,02 O. -2.212['03 3 10 20 40 11 210 220 -2,212E'03 O. -3.633E·03 17 0-1.000E·02 1.000E-Ol 4 I 1.000£_04 1.000['00 11 220 230 _3,4'3[.03 n. -4,122('0' 30 R6 136 136 11 230 240 .4.12eE.03 o. .3.1033E.03 3 10 20 40 II 240 250 .3.633E.03 o. -2.'ltE.03 TII4E OEFL. 8 1 11 250 259 -2.U2E.03 ~. -2.112[.02 TI!4E O[FL. 11 0 II 261 270 1,421E.02 6. 1.421['03 TI"1E "01011 ~ 1 11 210 2ao 1,421E.03 ~. 1.910E.03 T I .. E "O"r 11 n
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7 PARTfALLy'EMBEOOEO PILE ElCI~I!:D RY [ARTHQUA'I[ fNO~e[D "'ORCES!I/IO 81 0 0 0 0 0 0 0 0 3 0 I 2 0 19 6 10
50 l.lOOE'OI 160 S.000E-03 -I 5 ·1 0 I 1 10 10 !O 30 40 50 SO 1.0OOEoOI 1.000E-06
0 50 0 1.090E·10 ·1.000E.04 0 40 a 50184£-01
41 50 0 Z.592[000 0 40 I 1 1.~231!:001 , • 2231!: 001 I,Z2U o OI 0 40 2 l 2.417£001 2.4171!:001 2,41T[001 0 40 3 3 3.550E.OI 3.550[001 3,550E oOI 0 40 4 4 4.~9ThOI 4.591[001 4,59T!'01 0 40 5 5 5.53U.OI 5.,]0[001 5.UOhal a 40 6 6 6.3Z"f[001 6.]2T[001 6.32T£001 0 40 1 T 6.961[.01 6.968[001 6.'68£'01 0 40 6 8 7 • .,1[.01 1,.37[001 1,437£.01 0 40 9 9 7.724[001 7,724[001 7.724£'01 0 40 10 10 7.'I20EoOI 7.e20£001 7.820['01 0 40 11 11 7,124£001 7.n41.01 T.7hE oOI 0 40 12 12 7,437['01 T •• 37EoOI 7.un.0\ 0 40 13 13 6,'6U'01 6,968[001 6,'68[001 0 40 14 14 6,127EoOI 6.327!·0l 6,32TE'01 0 40 15 15 5.530[001 5.Sl0EoOI 5,5J O£'01 0 40 16 16 4,5,7EoOI 4.5'7[001 4.5'71001 0 40 17 17 3.5$OEoOI 3.550[·01 3.5'0['01 0 40 1. 18 2,417hOI l!.;ln'OI f.41T1!:oOI 0 40 I' I. 1,223E.Ol 1,,23£'01 1.223[.01 0 40 0 1.000Eo OO-I.000[-ft2 4 0 I,OOO[.OJ 1.000['00
100 100 0 0 1000 5 0-1000
0 40 0 1.000E o OO-l.000[-02 4 .1 1.000f-&) 1.0001'00 100 100 0 0
1000 5 0-1000 TIME DE'!. 10 1 n ... £ Dl':'!. 30 Q TiME DE"!. 40 I n ... E DE'!. 50 0 BfAM O£F'L 10 1 8EA'" O["'!' 20 0 au ... DE'!, 50 i 8!AM O['!, 150 0 8EA ... MOMr 30 1 8EAM MOMT 100 0
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PRor;RAM nBC:S - MASTER - JACK C~d"l _ M"TLOCIt' - I")EC,I(l-~EVyC;ll~! nt,.T[ • ?6 ~I'I"'I 71 EXAMPLE PPU~LF"~S F"O~ m~Ct; po.,GRA,j ~y JACK Co.jA~ APRIL 197 1 lJYNAt.1IC ttEAM-inL,I~N PQI1r;O,6!'t 'ISINti c;,-l-t; T~PLICTT I"IPFQATnR
PRO~
I
HOLn FROo.t PRErEDINlJ PI"H")~LE~ (l_l-in[l1) NUM CARDS INPlJT THIS PRn~LEM
TAALE 2 _ CONSTAOTS
~UM OF REAM INCREMFNT5 BEAM INCREMENI LENr,TH NUM OF TIME INCRE~FNTS Tl~c INC~EHENT LEN~TH OPTTONAL PRINTING SWITCH NUM OF "IONITOR STA I P~INTT~r, iTERATION S'!TCH (O=LINEAR) NUM OF .... ONyTOH STA ( ITER4Tyr'')''1 , PLOTTING METHOD(-I=MIC.,O.PPT~'TE~,l-PAP~~) LINE OR POINTS PLOT OPTION
--- __ I1ERA!ION DATA FOR NO~LIN[AR ~UP. CV. ___ --MA~TMUM ITERATION NUMRER MAXIMUM ALLO'.SLE "EFLECTION DEFLECTION CLOSURE TOLERANCE MONITOR STAIIONS(ITERATION)
I 2 3 • 5 STA J -0
TABLE 3 - SPEC IF lEU DEFLECTIONS ANn SLOPES
STA OErLEC T I ""J
FRO~ TO CONTO
_0. I.OOOF.O? _0. -no
TAa[ F" ~UM9[R
5 ~ 7
1.000E+o(\ aOo
6.283[-0) n
-0 .1
o I I
~O 1.000E·01 I.OOOE-O,
lOPS (IF.I,INITIAL Tr.?,PERJIIIANfNTl
R CP
-0. -0.
q
F~O,~ TO r-:Ol'lTrl ""0 nr DE
1.00 'JE. ')0 -u. -n.
T ABI E b - T I ~F nEPE~OENT AxJAL TI-iQII~T
BE A~ STA TIMF STA FROI-l TO F~Ou Tn TTl
.I'IjOI~C: 0
TARLE 7 - T I ~E D[Pt...NDENT L(')ADYN"
BE A .... STA TIMF STA FROM TO FRO'" TO QTl
-5.I"!OOf+Ql
0 BOO -1.(lOOE+OZ Ron 800 -5. n O'lE"+01
TAaLE 8 - NONLINEAW SUPPORT CURVF~
I 20
-1.000F+QO 3~ ,0
e
1. "nnr-O 1 I?O pO
?o Ina
1040~lZONTAL Vt~TIC~L rjElI"" nR TT"'F- '-'IILTTPLE PlOT
TTZ TT3
QT? QT3
-5.000[.01 _~.OOOE+Ol
-1.000F+0~ _1.000E·02 _5.000[+01 _S.OOOE·OI
1.onnF"'_OZ
AXIS AXIS 5T~TY"N ~~ITC~ rTrEltSIJPE~T""pn~F ~ITH ~~XT TF-O,PLrT dLi SAvED PLOTC;~
N >-' Vl
PRO.RA~ 1SC! - MASTE~ - JACK CHAN _ MA'LOCW _ ~EC~I-REVJSInN nATE - Zb JU~ 71 EXA~PLE ~QO~Lf.S Fnp ry~(K P~"GRAU '" JAC~ CMAu AP~JL 1971 il'NA"lC ~EAM·cnL" .. N P'IOGPA'" .. S IN" ~-1-5 J~pLIC I T OPFIiATOI>
PROR ICONT!)} lONE MASS SUPPn"TE:) "y • "()"Ll~EAO <PilING U"IOER ~RFE VTR'IATlI)N
PROq ICO~TI.)I
lONE "ASS SUPP"~Tf.D 0, A "ONLTNEAP ~PQJNG UNDER ~IIFE VT~"ATION
TABLE lQ- CALCULAT~D RESULTS J-SEAM AXiS. ~'TI~E "15
n "'JNUn'S 'PRINTJ~r, RE5ULTS IS NOT REQUESTFn'
PROGRA~ nBCS - MASTER - JACK C'"1AN _ "4ATLOCIC'4O nECI(I4OREVISI"'~ I"'lAT!: - ~b ...III'" 11 £XA~PLE fo'ROBlP''''S rOR PPI"l~PAUI O,",Cc; R'f JACK C~A"t JLI~E lq71 DY"'IJA~IC t=tEAM4OrIjLIJ"'~ PQnr;cAI'4 I 5-, ... c; yuolP"IT I"'lPrRaT,.,Q I
PROR b
TABLE 1 - PROG~A"-CO~TROL OATA
HOLO F_O. PRECEDING PRORlEM (l="OLO' NU~ CARaS INPuT T~lS PRORl,M
TARLE 2 - CO~STANT'
NvM OF ~E'~ INCREMENTS 8EA~ INCREMENT LEN~TH NU~ OF TI~E INCREMENTS TIME INCREMENT lENr,TH OPTTONAl PRINTING ~'ITCH NUM Of MO'ITOR STA ( PRINTI~G ITERATION S~ITCH (O.LINEAOI NUM Of ~ONITOR STA ( ITER'TIOM I PLOTTING METHOD(-IB~IC.,O.PRI~ITEo.l.PAP,~1 LINE UR POINTS PLOT OPTION
o 2
TA8l.f NLI"'RER ~ ~ 7
o 5
3~
•• eOOE'Ol 40
4.000E-02 -1
4 -i
-- __ -COMPLET[ PRINTING TIM, INTfRV.L A~D
TIME STA INTERVAL Of COMPlrT[ PRI~T ~ONTTOR STATIONSIP~INTING)
Mn'TTO~ STATIONS fOR PRINTING-----4
1 2 3 4 ~ ~ STA B 15 16 17
_____ lTr'ATlnN OATA FOR NONLINEAR ~UP. CU.- ___ _ MAXTMUM ITERATION ~,Ii_I~RER MA~IMUM ALLO.'BLE DEFLECTION DEFLECTION CLOSURE TOLERANCE MONITOR STATIONSIITERATIONI
I 2 345 STA B 15 16 11
TABLE 3 - SPEClfltU DEfLECTIONS ANn SLOPES
STA CASE DEHE~TTON ~I O~F
0 o. "'O~F 30 O. .. ONE
TABLE 4 ,Tlf.NESS A~ln F p,En"ll"lfln naTA
fROM TO CONTo QF S
9 10
rops
5~
•• DOOE'OO 1.000E- 06
rfF_I,iNITIAl Tr_.:?PEAMANENT)
CP
10
-0. -0. _ n. on. 1. f) I') 0", I 0 -0. 30 <.40uE·10 -0. -n. -1.00()f·05 -n. -no
TABLE 5 - MASS AND DAMPINGS DATA
F"RO"4 TO I"'ONTn RHO 01 DE
30 5.184E-Ol -0. -0.
TABLE b - TlME DEPENDENT A~IAL TI-fQ,.S T
BEh STA TIME STA FRO ... TO FRO'" Tn TT, TT2 TT3
-,.,.ON£.-
TABLE 7 - TI~E DEPENDENT Lf')AOlloJr;
BEA .. STA TIME STA fRO~ TO fROM TO QT; QT2 QTJ
11 11 O. O. _1.000E.04 11 11 4 ~ -1.000E'04 -1.00OF"·04 I.I)OOE'Ol 11 11 ~ 12 l.oOOE.03 1.000F+Ol _1.OOOE,Ott.
11 11 1~ 16 -1.nOnf',04 _1.000[,04 I.OOOE·O)
11 11 16 ~o 1.('101"1['03 l,OOOF.Ol _I,OOOE·04
TABL[ 8 - NO"-/L I N£.A~ SUPpnRT ClleVfC;
F"Rn'" Tn CONTD Q-~UL TIPL lEO ~-MULTIPL'ER POINTS S'~ OPT W-TnLfOANCE a-T(')lf!'gA~rr
-1.OO"F:'·"~ 1,1')00E-01 -0 1.O .... 0~-O~ 1. 001')f!'. On Q _40 _4" _l9 -35 _?O 0
-.0 _3U _?O -10 _s n 40 15 -1.OOQE"Q2 1 • "(H)jC' .. 0 I 4 1 I, "OO~ .. OI.j. 1.00nf!'."t"I
\I u 0 0 10 20 4Q
17 -1.000E·02 1.000E-01 1.000·-04 1.000"·0('1 II 30 bO Db ll6
1 u ~O 40
TABLE 9 _ PLoTTING S"ITCHE,
HORIZONTAL VEHTICAL BEA~ 00 Tr .. r wULTTPL[ PLOT
AXIS AXIS q A TION ~~!TC~ 11'.l.SUPEOI~PO~E ~ITH NEXT If-O'PLOT ALL SAVED PLOTS'
T I ME lJ[fL 8 1 T I ",E I)EfL 17 0 TI"E .... O..-T 8 1 T I"'E ~Op.tT 11 0
9EA~ 0EFL 4 BEAM OEFL ~
8EAM DEFL ~O
8£UI ~OMT 12 8El~ ~DMT 16 8U~ "OIOT ~O
PROARA~ ~aC5 - NA~r£~ - JAC~ C~b~ _ ~~TLOC~ _ nEC~l-REVISI'~ 'ATE : 2~ JIJ~ 71 ElI.AtAPL[ t>'ROHLF"o.AS ~0R PQ'i~PA'" O~Cc; ~'( JACK C~A~! JUt.ff 1911 O'f~~i\"'"IC 'JfAM-rf'LI ~N cQnr,QIlM t 5-' ... " T"PLIf'Il' OP£::\IATr.Q t
PRO .. (CO~TU) 6 THClEE-SPA~ REA'" wITH rH>JF ... wAY AN.0 C;V""MF"T~TC NO~I t J~6c C:;'.)P UiADr:'l q ... T.p
TA8LE 10_ CALCULAT~O RESULTS j·8~AM Axrs. ~'TIMF •• rs
TIME STA k. .,
'iT 4T rONS j • I ~ 17
FOR ITERATION, TMERE ARF. 0 'iTATlnN~ NnT CLOSED -1,lllE-16 -I.I?~E-l~ -1,~36E-IS -1.317~·15
TIME STA ¥. -,
STlTlONS J • I'" ]7
FOR ITERATION, TMERE AR£ 0 'iTATlnN~ ~~T CLOSEn ·1,lllE-I~ -1.j2.E-i~ -1,']6F·l~ -1.317~.j~
•.•.. $T,T1C RESULTS . .... STA J OIST nEFL ~LOOF "0" SHE.R
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•• tlOOE.OI 5.]46£-18 '1.0111'-11 ~.11.,~ .. 19 -1.1~",~",1? , o.onOE·ol I.SSI£-11 • 1,936"-12 1.Qt;Af'.19 -1,189'-12
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4 l.970E.02 2.708<-11 _1.2141'-1 0
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-1.' I 0 .. _ 18 -1.1119.-12 3.3~~E.02 -.,.1\29 .. -11 -2.M3~-1 0
_1 •• ~3F-18 -1.1~9"-1? 3.840£·02 -1.313E-16 -3,3~~<-10
,,'.'1:i4~_lA 1.436 .. -12 0 •• ]'OE·02 -2.30SE-I~ _2.5 .. 0"._1 0
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2 ITERATION, TMERE APE .9 5TATln .. ~ NOT CLOSED -1.613,-02 -Z.9RAE-ni -3.404[-01 -3.140'-01
rOR 3 ITERATION, T"'ERE ARf "STATIoNS NOT CLOSED -1.613'-02 -~ •• 9.E-~; ·,.404~~01 -3.744'-nl
TI ... E sa " ..
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T,446£+04
I. 22?. 05
~.6C;?E·n5 -2,547£+02
3.~IOf·05 .2.7q3E.~2
.1.522 •• 05 _3.073~.n?
3.3"7(+05 -3.37~E.02
3.Z04 •• 05 -3.b9.~.n2
2.97f\E+05 -3.982~.n2
2.0~OE·"~
1.830E·03
O.
O.
0,
O.
4.578f+03
0,
O.
O.
O.
O.
O.
i .b7"'I':.O~ O.
FOR
1.3q2E.03
1. ".OE+ 03
1, 4R~E. 03
J a
-3. 743E-0 I
-1 •• q2E-0 I
~ •• 27f_03
'. ,83._03
3.~SS~_03
'.942~ .. 03
3 .. Q42r .. OJ
TINE STA I( •
~r)NTTO~
I~
-4, __ 65F:+02 1 .. 27AE+05
-4.q~4F.'.n2
~.b2If·0' .. 5,062[.02
4.341[+04 -S.IOIE.Ol
O.
STAT TONS I' 17
TTE~ATION. T~E~E ARE ,q ~TATInN< NOT CLOSED -4.600E-02 -8.57.E-o' -Q.q3nE-01 -1.125f.OO
FOR ITE~ATION, THERE ARE '~TATInN< N0T CL~SEO -4.600E-O~ -~.'i7A[-ni -q.q311~-OI -le125F"+iil'l
• USED ,o"nJUSTt:o LOAn"'r'!FfL C1JQVF' T"l CAl C'IL6.TE nEr:-LECTIt)N~ •
O.
O.
O.
STA J
I~
FOR
FO_
FO.
FOR
FOR
FOR
FOR
FOR
TT~RArION, T~~Rr ARE ,9 ~TATI~~~ ~OT ~LOSEn -~.410F-02 -Q.40,E-nl -q.74~F-Ol -1.106F+nl1
ITERAT!ON, THEQ[ 4RE ,q ~TATI~~~ N~T CLnSEn -4.67Qf-02 -A.~A4f-~l -1.~04~.OO -1.lJsr.n0
-4.73Ar-02 -Q.A~~f-ni ·1.~24f.OO -1.15~r.~~
ITERATION. T~ERE A.f ,q ~TATIn~< ~OT CLOSEO -4.7RSr-0? -~.q'AE-nl -1.~33E.00 -1.1~7r.~0
ITERATION, THE.E AR< ,q ~TATIn~< ~0T CLOSED -'.7~ar-02 -A.q~Jf-~l -1.n3~f.OO -1.170r.o~
8 ITERATION, THERE ARE ,q ~TATInN< NOT CLOSEO -4.901[-02 -A.Q~~E-nl -1.~3~[.00 -1.171r.OO
FOR 9 ITERATION. THERE ARE ,7 ~TATIn~s NOT (LOSED -4.802(-02 -~.q~AE-nl -1.~3~~.OO -1.171~.~0
rOR 10 TTE~ATIO~. THE~E A~r ,6 ~TATI~N~ NOT CLOSED -~.80Zr-02 -~.Q~Ar-~l -1.~3~£.00 -1.171r.oe
FOR II ITERATION, THERE ARE ,ii ~TATIO~S "'OT CLOSE!) - •• 802f-02 -A.q~AF-OI -1.n3~F'00 -1.171,'00
FOR lZ ITERATION, T~ERE ARr 15 5TATInN. NOT CLOSED -4.802E.02 -A.9b~f-Ol -1.~3~~.00 -1.171~.~0
FOR 13 ITERAIION. T~ERE AR~ n .TATIn ... NOT CLOSED
FOR
OIST
7.21)0[.02
J •
TI"'E 5TA I(' •
-3 •• 74E-02
-ti.~3·E-01
-Q.q3 0E-01
-t'. ~ i 3~-03
_2,7 47 r ... 03
TI"'£ 'ITA I(' •
-2.115f·05 -2. 77RE ·iI'!
STATIONS I' I7
ITE~ATION, T~EQ£ AR[ ,Q ~TATlnN~ ~nT rLOS£n -1.b98r-Ol -~.nnnE-nl -~.QR7r-01 -7,00~r-Ol
1.390E·OJ
O.
24
31
1.200E.03
l,h8E.03
1.344E.03
5.885E·02
1.361'.';2
O.
o.
o.
1.3~2E.03 9.9]2E.02 .~.2~~E'04 O • • 2.~~9,.03 8.892~.~?
FOR
1.440E.Ol O. _4.6~6F~09 .A.892E·02 _2.069~.0] ~.~31~_il
1.468E.03 -9,932[-02 O. O.
TI~E ~TA • • Q
J •
ITERATION, THERE ARE 19 ~TATlnN~ NOT CLOSED -1.559(-02 ~.5A4r-n? 1.~39~-OI 1.656r-OI
FOR ITERATION, THERE ARE 0 ~TATlnN~ NOT CLOSED -1.559E-02 5.~~.E-0' l,ry19E-01 1.~56£-OI
• U5ELl ADJUSTE!) LOAO-nEFL CURvE T" CALCIILHE rlEFI.ECTIONS •
FOR
FOR
ITERATION, THERE AOf ,~ ~TATlnNS NOT r.LOS[D -1.430E-02 1.7IaE-n, 1.772£-01 1.8~6F-nl
Z ITERATION. THERE AaE ,~ STATrnN~ NOT CLOSED -1.4191'-02 6.64~£-n2 l.isOF-OI 1.764E-OI
FOR 3 TTERATION. T~EaE ARE ,9 ~TATInN~ NOT CLOSEO -1,430£-02 1.1i~E-0' 1.,1,r-OI 1.896£-01
FOR 4 rTEII.TION. T~ERE ARE ,9 ~TATInN~ NnT CLOSED -1.4'~E-02 6.~4'E-n, 1.1501'-01 1.1~4F·nl
FOR 5 ITEIIATION, T~ERE ARE ,9 ~TATI"N~ NOT CLOSEO -1,4901'-02 ~.llqE-O' 1.162£-01 1.183'-01
FOR c ITERATION, THER~ ARE ,8 STATIONq NOT CLOSEO -I,485E-02 6,79~E-O? 1,171£-01 1.792F.-61
~OR ITERATiON, THERE Aat; ,~ ~TATI"N" 'lOT CLOSEO -I,48~£-02 ~,7Q1F-O' 1.11"r-OI 1.791'-01
FOR d ITERATION, THERE ARE ;5 ST'TI"Nq NoT CLoSEn -I,485F-02 6,1qi~-o' 1.'70f-01 1.?91~-~1
FOR TTERATION, THERE AOE ~ ~TATlnN~ N0T CLOSEn -1.485[-02 ~,1qiE-n' 1,170'-01 1.19IF-~1
STA J DIST
STA J
II
STA J
I"
11
fOR
TI~E ST. •• 10
J • 8
ITERATION, THERE ARE ?Q STATlnNS NOT CLOSEn -1.738[-02 -l,A44E-OI -'.Q7~F-OI -2.00R~-~1
FOR 2 ITE~.TION, THERE AQ[ n ~TATlnN~ NOT CLoSEn
FOR
OYST
7.200E.02 -I,A44E-OI
7.6110E.02 -I,970E-OI
8.I~OE.02 -2.008~-OI
_?,6Z 0J'_04
TI~E STA .. • II
J • 8
Q.IOor+ 04
i.S4flF+1'I5
4.540~.nz
1.9"5F:+~J
STATIONS It. !T
ITERATION, THERE ARE ?9 ~TATION~ NOT CLOSED -i.115£-02 -t.,4~iE-~1 -T.~90f-OI -8."6I1F-hl
l ITERATION, THERE ARE ?9 5TATlnN~ NOT CLoSEn -2,036(-02 -",JJ?E-oi .7,464,-01 -8.~40F-ol
1 ITERATION, THERE AilE "5TATInN~ NOT CLoSED -<,036._02 -~,~3?E-?i -7,464,-01 ·8.~40'-nl
.7.505E'O?
_I,OOlE·01
SIJP REACT
0,
8.195E·02
8,928[.02
N N N
20
21
2~
27
2.
3n
31
FOR
fOR
1.OO~E.03 -1.75)~.O"
1.056E.03 -1.736E.00
1.1~2E.03 -1.S46E.00
1.2~OE.OJ -1.375E.OO
1.2.8E.03 -1.160E.OO
O.
1.4QaE.nJ 3.1~2E-OI
,.c;121='.0",
l,4 4 0F .. Ol ':;.407F.:.05
-2.R4'5F:.';1 C,.14QE·/lc;
3.c;52~_OJ -6.231,::.';2
n.
Tl"-'E ~TA I< • 11
~TATIONS J • e I~ I7
ITERATION, THEQE ARE ~9 STAT InNS NnT CLOSED -5.080~-02 -q.~17E-O' -1.130E'OO -l.?q~~·oO
-5.080F.:-02 -q.~7~F-Ol -1.13nE.on -1.29~~.;'O • U5EU ADJUSTED LOAD-OEFL CURVE Tn CALCI~ATE nEFLEcTION5 •
FOR
fOR
ITERATION, THERE APE ,Q ~TATtn~c:; ~nT CL~SEO
-5.~99~-OZ -1.037~.on -l.?O~F'OO -1.374F·OO
ITERATION, THERF ARE ,Q ~TATlnN~ NOT CLoSEn -5.080F-02 ~Q.~71E-oi -1.13nF.OO -1.?q6F.OO
FOR 3 ITERATION, THERE AR, ?9 STATlnN< ~nT CLaSEn -5.49QF-02 -1."3~E.on -'.10~E.OO -1.~7.~·nO
FOQ 4 TTERATION. THERE A~E ,Q ~TATIn~~ ~nT iLOSE~
-~.3Q5E-02 -1.n2r.E.nn -,.,q~~.o~ -1.~S4~.no
FOR ITERATION, THEQE AO, ,9 ,TATln~< ~nT CLoSEn -5.403r-02 -l.n~lE.nn -l.'~~r.on -1.3S6~·~n
FOR b TT~RATION, THEQE AqE ,R ~TATYn~~ ~OT i.LOSEO -5.404E_02 -l.n~lE.O~ -1.l~~F.on -l.~~~F.nn
FOR 1 IT~QATION, THERE AQF i7 ~TATrn~~ ~nT i.LnSEn -5.404r.nz -l."~ir.ry~ -l.iq~E.OO -l.~~~,.nO
FO~ H yTEWATION, THE~E A~~ ~ ~TATIn~t~ ~nT iLosrn -J.QQ6r-02 -~.'~~E-n1 -qt~lQ~-Ol -l,lbS~.nn
u. 51 A J
o.
O.
n.
O. 17
O.
c.
STA J
l~
17
STA J
FOR
:)y S T
..3.~07f_03 6.1561='.n3 -1.1~5E'OO 1.S47F.nc,
TJ~E C:;TA II' •
STATIONS J • I~ 17
ITEpATION, THEpE 4PF ?Q ~TATlnN~ NOT rLOSE~
-j.344~-02 -~.~7~'-~i -~.C,21r-Ol -7.2971='_nl
ITERATION, THERE AR~ 'Q ~TATloN5 NOT CLoSEn -3,422[-02 -~.~4"~-nl -~.706E-Ol -7.497~-nl
FOR 3 ITERAlloN, THERE ARr n 5TATlnN~ NOT CLOSEn -J.422r-02 -S."4nE-O! -~.706f-OI -7.497F-nl
FOR
FOR
TIME STO •• 14
DIST II::LOP.e"
.. 7.';16, .. 04
STATIONS J • I~ 17
1 ITERATION. THEPE APf ~Q ~TATIn~~ NnT CLnSEn -1.0'9F-O? -1.44~E-o, -1.1BS'-01 -1.102'-nl
ITERATION, THE~E A~~ ,Q ~TATInN~ ~OT CLOSEO -1.095F-01 -1.43~E-n, -1.'7A~-Ol -l.OQb~-nl
3 ITERATION, THEO' AR~ 0 <TATloN< NnT CLOSED -1.OQSf-02 -1.4]QE-nl -1.17~r-OI -1.096°_01
DIST
~IJP RJ:.ACT
0.
SUP ~EACT
'-...l N +'-
7.0441:" ... (11 1,,044[-(11 ,.t::Q't"_01 3.384'-01 .. 2.j.74~ ... 04 -2.7l)6F".i'l2
Fe" 17 !THIAT 10"1, T"E'IE AQ~ '9 C;TATlf'Hd(; .,.:"IT cLoSEn A 3,840E.02 -1.095E-O' ... ,.9"F'+()4 4.37AE+O?
7.160'-01 t .'Iuf.-n1 ,.C;Q'F"-Ol 3.:3~4J:''''''1 _1.A"IF_04 1.?~5F".~3 I, 1.<00E·OZ -1 •• 39,,-01 1. 5'iAF.:+ f'l5 n.
FOR lti I TE'<4 r '0"'. T"ERE ARE ,II ~TATIt)"~ "0T CLOSED I. '«;6E-04 1.585F.03
7.252f-01 t.94t;E-n1 ,.c:~e-F-Nl 3.3R4F"-nl I~ 7.I>~OE.Q2 -1.,\78E-OI ~.'l'lF".OC; _1.435[·01
5,AIIU_04 -4,396f.02 FCR I~ ITERA [tON, THE"E AQf ,~ C;TATlt"!"Jt:: "'H CLOS[n
11 8.II>0E.O, -1.0'l6E-01 2.074F"+"5 _4,OIlE'0? 7,l26r-01 1.04'1E-Ol ".C::Q2F' ... r, 3.384F"~tl1
T!"E 5rA ~ . I~ FOR 20 ITERA fl ON, THERE A"E 'A ~TAUn". "'(>T CLoSEn 7.3~;f-03 1.'I4"E-01 ~.~92E-OI 3.3A4F-Ol
"Cl"Ti'n~ 5TA T10"'5 fOR 21 ITERATION. T"ERE .. IE '7 HAT!"". "OT CLOSED
j . 8 I~ I" !1 7.4)2E-03 1.94",,-01 2.c;'I~f-OI 3.lA4F-01
FOR lTERAT !o~, THERE ARE ''1 S"ATTI"IN~ \lIn CLoSEn FOR 22 ITER> T 10". T"ERE AgE '7 .TAU"",· "nT CLoSEn
;'.534f-0. 1. ''';'F'-O,l ,."\9°f'-Ol l.lqlF-iil 1.4b9F_03 1.94"E-~1 ~.ti{Q'1F-Ol 3.3A4F"-rq
FOR .: ITERATION. TyEflE AflF ,. ~TATln~~ ~nT CLOSEt) FOR 2J ITEIUT!ON. TwEIIE A"E '7 t;TATr,.v."c:;; NOT CLOSEr)
-2.lhE-02 I.~q~~-~l >.)511'-01 3.11>4F-ol 7.4q9F-03 1.94"'E-Ol ~.~91E-OI 3.384F"-ril
FOR PERA TlON. T~EflE ARE ,9 5TA TIn~~ ~nT CLOSED fOR 24 ITERATION. THE"E Allf ,5 qAT!m.~ ,,"T CLOSED
;'.534f-04 1.16"E-01 '.l99f-01 3.lql"-~1 7.522E-0] l.q4"E-01 ,.~q'n"-Ol 3.JA4F'-nl
FOR 4 !TE~ATION, T~ERE ARE ,9 S"ATtnl'l,lt:; ~"T CLOSED FOR 25 ITERA TlON, THE~E A~[ ,3 ~TAT!"". NOT CLoSEn
-2.174E-0~ 1.!>fl"'E-ni :>.l'ilF-OI 3.1b4r-~1 7.S0IF-03 1.'1."'£-01 ,."'93E-OI 3.3841'-';1
FOR 5 !TERATlOt .. THE"E ARE '9 5TATlnN~ ~OT CLOSEn FOR ~I> HERA T ION, THERE ARf >2 5TATIO~~ "OT CLOSED
7.264f-04 1.19nE-nl 2.432E-OI 3,?24"-01 7.556E-03 1.941-tF.-Ol 2 ... ·llE-01 '.'ME-ill
FOR 6 ITERATION. To;ERE ARE ,9 5TATInN5 NOT CLOSEt) FOR 27 ITERATION, THERE A fir >" 5TATln~~ NOT CLOSEn
2.0951'-0] l.q~3E-01 ,.';'1';F-OI 3. 31Z'-n I 7.568(-03 1.9.""E-nl ,.';q1F-Ol 3.3RO"-nl
fOR ITERATION, THERE ARE ,'1 ~TATInN5 NOT CLOSEt) FOR 28 ITERATION. T~EIIE AilE ;1> 5TATln"'~ NflT CLOSEO
3.214r-03 1.'1'\ j E-O 1 ~.~AlF-OI 3.379"-iil 1. S7Br-03 1.941-1£-0\ '.~93E-Ol 3.380"-01
fOR ITERATION, THERE A"f >9 STATln,,~ ~nT CLoSEn FO~ Z9 !TERAIION, THERE UIE 15 .TATln~~ NtH CLOSEn
4.107F-03 1.934E-OI ,,~8~f.-Ol 3.380~-~1 7.S85f-03 1.94"'F-Ol ~.o;9'f.-0l 3.385'-01
rOR y ITERATION. THERE ARE ,q 5TAT!~,,~ NnT CLOSED FOR 3U ! TERA TlON. THERE A~E ;4 C;TATYntoJc:: NnT CLoSEn .,81 0E-03 '.93 7 E"'('I1 ,."An-Ol l.J81~-nl
1.!>92f.-01 1.Q4t'-E-'P 7.C;QJF' ... Ol 3.385'-01
~OR 10 ITERATION, THEilE ACIE ,9 5TAT!n~~ NOT CLOSH! FOR 31 ITERA TlON, THERE ARI' 14 ~TAT!nN~ "OT CLOSEn
5,387(-03 I, 930E-0 I ?,~8·E-Ol 3.3R2F-~1 7,5'161'-01 1.q4~E ... Ol ' .... ql~-Ol 3.3RS'-01
fOR 11 ITERA T ION, THERE AR' ;q ~TATln"~ ~~T CLaSEn FOR 3Z !TERATION, THERE ARE '3 ~TAT!nN~ NOT CI OSEn
5.83QF-OJ 1. 94f!E-O 'I ".~AqF'-Ol 3.382"-~1 7.6001'-01 I. 94~.-n f 2 .... Q'u··O\ 3.385<-0\
FOR IZ ITERATION, THERE AQf '0 'STAT In"lC:; 'JOT CLoSEn FOR 33 ITERATION. THERE AR' iz ~TATI~N~ NOT CLI)SEn ".ZOOf-OJ l.q4!E-~1 2,0;901'-01 3.3831'-01
7.603F-OJ 1. q4"F:-~·ti 2.~q'f-Ol 3.3851'-01
'OR , 3 ,.FRATION. T '"IF R'F ARF ,'I 5TATTn"~ ~nT CLoSECl fOR 34 !TE~ATIO~, THERE A~f II C;TtlTlnN«:: NtH CLOSEO
b.48U:-01 1.Q4,E-Ol ,."qOF-OJ l.383r-OI 7.60I>r-03 1.Q41.F.:-f)' ,.C.Q)F'-Ol l.3A<;F-OI
FOR 14 ITERATION, THERE AR, ~9 STATI"',. NOT CLI)SED FOR 35 ITERAT ION. THERf. ARE 9 .TAn",,· "I)T CLOSE!">
,>.716F-OJ 1,941f.-01 '.'1'111'-01 3.3A3F-Ol 7.608f-03 1.Q4",E"fll ,.,,911'-01 3.3851'-01
FOR IS ITER A liON, TH"RE AilE '9 ~TAT!fI". NnT CLOSEO FO" 36 PERl TlON. THERE A"E 7 STATln"" "OT CLOSED
b.8QAE-03 1.944E-01 ,,';911'-01 1.384F-nl 7.609E-03 1.Q4",F-ot '.~9'\F-OI J, 3aSF- 0 I
FOR 16 ITERATION, T~ERE A'lF ,'I .TATlfl~. NnT CLOSED
STA J
-I
6
11
I'
I.
I~
11
10
20
ZI
2~
Z.
FOR 37 yTERATION, T~E~E ARF 5 ~TATTn~~ ~"T CLOSEO 7,611F'-OJ 1. Cl 4"F'-r.., ~.lI;q"F-Ol 3.3Asr·nl
fOR 38 TTERATION, THERE eQF ~ ~TATI"~~ ~OT CLoSEn
DIST
-_,a"OE.Ol
O.
3,:3~OE.02
3.840E.02
6.7?0E.,02
9.600E.Ol
7.&12E-01 1.Q4"E- rll i".II;Q'F-Ol :\,:\R5':·r'll
TTME STA It • I~
i'lFFI.
-5,?78E-04 o. 1. 1 00E-05
O. -~.6"BF·03 _1.175F._05 -7.4JOr.on
1.~60E-OZ
Z.9. ZE-QZ
1.4Z8E-OI
1.9·~E-OI
_1.~t;4"_OS
4,~4l!:_06
'.'56.: .. 05
7.~t6,,_05
~.~ZQ~_O.
" •• 7~F_O.
1.476£:.04
i.QRo r. 04
q.71~F·I"I"
i.I~9f.n5
l.lOO,:.O?
1.n]q,:.n2'
1.ft8 4 !:.02
1.218,:.02
1.474F.;'2
t.e74F.';'?
2.433~. ~2
.. ,382~.n2
4,76 3r. ft i'
".eS1r. n2
SUP REACT
O.
O.
o.
O.
O.
o.
O.
O.
O.
o.
O.
o •
O.
O.
7.474E.Q4 _2.19]E.03 l,7Qqr=_OJ -l.7Q3F'~'
4.~~8£-Ol _1.QQ4E.r)4 O. 1.7C;QF:'.03 -1.651~.~,]
o. O.
1. '~"'F"_04 -4.450,:.n2 _'].1"17Qr-.05
_4.QQ1F" ... 04 -a.370F.nl n.
_~.OQ5E·n5
27
2-
Z9
STa J
I~
17
o.
J •
n.
1.?3flF'.n3
9."'31""11
FOR ITERATION. THERE aRF ~Q ~TaTI"~s N~T ~LnSED
-1.726F-OZ 3.9~~E-nZ ~.717F-0~ 1.005r-ol
Fa~ l ITERaTION. THERE aRE ~Q STArIoN~ NOT CLoSEn -b.020r-03 •• 33~E-0' ~.Q6~F-0~ 1.019F-nl
FOH ITE~aTION. TME~E A~E n STaTION~ NOT CLOSED -6.0Z0r-03 •• 33~E-0' ~,Q6~~-0?' 1.019F-nl
• u~~u AOJUSTEO LOaO-OEFL CUc!VI; TO caLCIlLaTE OULECTIO""S •
FOR rrERArION, THERE aC!E ;9 ~TATION5 ""aT CLOSED -3.Q77F_03 7.70Qf-n~ 1.064~-Ol 1.3Q~F-nl
FOR 2 ITERArlON, THERF aPF 0 STaT InN' ""aT CLOSED -3.977F_03 1.70QE-n~ l.n64,: .. nl 1,~QR':-nl
OIST
).840E·02 -3.Q17E-03
1.2nn[.02 1.70~E.-02
8.1"'~E.02
~.i47F_0.
"'. 1,01=:_04
~.()4~F_04
TIME ~TA
J = 1<4()~1 T T(')~
1<
I-
1.015F.oZ 1.22?E·04
-1, SA4,:. 0 1 4.s1nF.n4
-7.083,,01
STATlO""S I~ i1
ITERATION, THERE .RE ~Q ~TATIoN~ NOT CLaSEn -3.73QF-02 -'.~QQf-nl -4.~ll~-Ol -5.06~~-nl
~ TTERA1JJN. THE~E .R' 'Q STATION~ NoT CLOSEO -J"S3F-OZ -,.'"4~-oi -'.2.QF-OI - •• AOZ'-Ol
n.
o.
1.591E·OZ
.. A.)AIiE·02
10 10 V1
I~
17
rOR
FOR
STA J
A
~OR
fOr<
FOR
EOR
fOR
fOR
1!bR(lE.02
8.160[.02
J = ITERATION, THERE ARE ,Q 5TATI~~' ~nT ~L05En
1.2~O~-02 '.~4~E-~l 4.e4~'·Ol 5.12lr-nl
TTERATION, THERE ARE ?Q srATInN. ~Or CLOSED 8.56IE-0~ 3,"31E-01 4.<3QE-OI 5.rI8E-nl
ITER4TION, THERE AOt , 5TATln~s NOT CLoSEn d.561~-03 ~.~'iE-nl 4.~3QF-OJ S.11S~-~1
DIST
a.I~OE.02 s. "lA(-Ot " 147r:. nS .".127E·03
TIME ~TA ~. 24
J •
TTERATION, T~EPE AP' ,q ~T'TloN~ ~OT CLOSEn -1.OISf-OO 1.150[-01 I.Q43[-01 2.981£-01
TTERATION, THERE ARE ,9 ,TATlnN' ~OT CLOSEO 1.96Sf-Q3 1.1,,£-0\ I.04A,-OI 2.9Alr-nl
ITERATION. T~ERf ~Pr. ?Q ~TATI"N, ~nT CLoSEn -1.Ol51'=''''04 1.15::,nr-OI J.Q'+'~"'Ol 21()Sl~-nl
• ITERAIION. THERE A.' ,q ~TATln~. ~OT ~LnSEn 1"q"'G~ .. 111 L,r:;.,r-nl 1.Q4~':-.Ol 2.9B)':--"1
I ITERATION. T~EAE AR' ,q 'TATln~. NOT CLOSEn ~.429~_01 1.?A~E-01 ?_1AE_01 3.lllf-~1
6 IT~RATION, T~ERE ARE ,q ~TATlnNS NOT CLOSEO ~.511~-01 1.1R.E-ot '.QA~E-Ol 3.01IE_nl
TTERAT10N. T~ERE APr ,q ~TATI"N~ ~"T CLn5EO 9.664.-03 1,19,E-'\ 1,019E-OI 3.00Af'~1
FDA ~ TTiQAT10~. T~ERE AqF ,Q ~TATlnN~ ~~T cLnSEn ~.737F-O' 1.1Q~E-~' 1.Q1qr-Ol 3.00SF'-nl
F'OIi '-J TTEPATln~t r ... EQE Aq,t' .,,:; (;TATln",Jc Jll"'IT CLOSEn ~.195f"'O' \ .,Q"E-rl1 1.Q1Qt::'-Ol 3. o OS5r:'"" !"I 1
,OR 10 ITERATION, T~ERE ARF 07 ~TATln~. NnT ~LOSEO ~.~41F-OJ 1.1q£E-~1 1.o1Qr-Ol 3.QO~V-~1
'O~ II TTERATIJN. T""EQE AQr ,7 <TATrn~c ~nT CLnSEn ~.S18r-03 1.1q~f-nl ,.q1Q~-Dl J.OO~r:'-Ql
,OR 12 ITERATION, THERE ARE ,7 ~TATln~. NOT CLoSEn ~.QOTF-03 1.1q~E-Ol 1.q1Q(-Ol J.oo~~-nl
EOR 13 ITEQATION, r~ERE ARE 05 STATIONS NOT CLOSEO "1.931('-03 1.1Q",F'-/)' 1.Q1Qr-Ol 3.n!'J~j:"-Ol
FOR I_ ITERATION, THEoE AOr ,4 5TATln~. NOT CLOSEn
rOR Ib ITERA'ION, THE~E AQE ? 'TATlnN~ NoT CLOSEn ~.q76r-03 l,lq~E-~' I.QrQE-Ol 3.~o~~·nl
FOR 17 ITERATION. THERE ARE i6 STATI"~S NnT CLOSEn ~,9B6E-03 l.lq~E-Ol I.Q70,-01 1.008;-01
fOR I~ ITERATION, THERE AQE is ~TATION' N~T CLOSEn ~,993r-0) 1,lq~E-nl 1.9r9'-01 3,OOA;-ol
fOR j¥ ITERATlON, T~ER' AR' ,. 5TATI~N' NnT rLnSEn ~.~~9f.03 1.1q~F-~l I.Q7 Qr-Ol 3.008~·nl
EO" 20 !TE"ATjO~. THERE ARE i. ~TATI~'J~ ~OT CLOSr.O l,OOOE-02 l,lq~E-Ol 1.979,-01 3.008E-OI
FOR 21 'TERATION, T~ERE APr i3 ~TATlnw. NOT eLOSrn 1.001[-0~ 1.lq~~·O; 1.979'-01 3.00R.·nl
FOR ZZ ITERATION, T~ERE ARE i2 ~TATI"N' NnT CLOSEO 1.001'-02 1,19~E-Ql 1.919'.01 3.~OA.·nl
FOR 23 TTERATION. T~ERE A~~ il ~T'TI"~~ NnT ~LOSEn l.aOI~-02 1,lq~E-nl 1.979'-01 3.00R·-"1
FOR 2~ ITERAT:ON. T~EQF AO~ Q ~T.TJ"~~ ~~T CLnSEn 1.OO?r.oz 1.lQ~r-n' 1.Q1QF-Ol l.On~F-nl
FO~ 25 TTE~AT10N, TWE~r APr 7 ~TATI"~~ ~"T ~Ln~E~ 1.002~-02 1,lq~E-nl 1.Q7Q~-Ol 3.00~~-~1
EOR 20 IT~RATION. THERE 1. Q02f:-OZ
A.' 5 'TATI~N' ~nT CLOSEn l.l0~E-nl I.Qr9~-OI 3.008E-OI
fOR 21 ITERATION, T~ERE AOf 0 ~TAT!n~. NOT CLOSEO 1.002E-02 1,lq~r-O\ 1.9r9>.-01 3,OOBf-nl
• U~EU AD lUSH.!) LOAD-OrFL CII1"IVf Tn CA' C"LAT[ n~ELEcTIO~S •
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ITERATION. TMERE A~F ,9 ~TATI~N~ ~OT ~LOSEO -1,518.-02 -2.3Q'F-OI -'.430F-01 -2,520f-OI
2 ITERATION. THERE ARE ?q ~TATION~ ~OT CLOSED .~,e3TE-OI -1.9q~E-ai -'.~5AE-OI -2.181f-i!
3 ITERATION. T~EqE AQ, n ~TATI~N~ NOT CLoSED -Z.837f-02 -I.QA~E-Ol -,."SOF.-OI -2.187'-01
TI~E ~T. •• 2~
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STATIONS J • I~ j1
ITERATION, T~ERE '~E ~<j ~TATI~N~ NOT rLOS[O .J.ObQr-02 -'.?5~(-~' - •• '1o~-Ol -S.5?1~-nl
ITERATION. THER. ARE 0 ~TATI~N~ NOT CLOSEn -3.06Qr-O? -1.'~~E-ni -4.?1~r-nJ -5.;~1~-nl
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TIlE AUTIlORS
Jack Hsiao-Chieh Chan was a Research Engineer Assis
tant with the Center for Highway Research, The University
of Texas at Austin. His experience includes work as a
teaching assistant in Taiwan and work as a highway engineer
with Howard, Needle, Tammon, and Bergendoff, Consulting
Engineers, Kansas City, Missouri. His research interest is
primarily in the area of discrete-element analysis in structural dynamics.
Hudson Matlock is a Professor of Civil Engineering at
The University of Texas at Austin. He has a broad base of
experience, including research at the Center for Highway
Research and as a consultant for other organizations such
as the Alaska Department of Highways, Shell Development
Company, Humble Oil and Refining Company, Penrod Drilling
Company, and Esso Production Research Company. He is the author of over 50
technical papers and 25 research reports and received the 1967 ASCE J. James
R. Croes Medal. His primary areas of interest in research include (1) soil
structure interaction, (2) experimental mechanics, and (3) development of
computer methods for simulation of civil engineering problems.
229