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.

v

vi

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

10

~-l

/ s N \ + IS. + S. k+ 1 ) J \ J J,

= 3

4h p. J

= - bk+1 -

( .\ 2 D~ )j+l = -ek+1 + h

t

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.

· o

t ~

3 OVT 8EAM STA. 20

Fig 31. Example Problem 3. Deflection vs time for the center of beam.

0

~ ~ .

• 0

3~MVT BEAM STA = 20

Fig 32. Example Problem 3. Moment vs time for the center of beam.

-- -------.----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.

APPENDIX A

DERIVATION OF THE DYNAMIC IMPLICIT OPERATOR BASED ON THE CRANK-NICOLSON IMPLICIT FORMULA

!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+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

f. k J, .

123

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44!5"6!7$1*'*0!8$($.$9'.$/-!")':!

APPENDIX B

DERIVATION FOR RECURSIVE SOLUTION OF EQUATIONS (Extracted from Appendix 2 of Ref 4)

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44!5"6!7$1*'*0!8$($.$9'.$/-!")':!

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

!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!

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.

APPENDIX D

GUIDE FOR DATA INPUT FOR DBCS

!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!

44!5"6!7$1*'*0!8$($.$9'.$/-!")':!

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

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-'. 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.

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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

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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

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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)

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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.

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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

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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

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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.

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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.

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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

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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

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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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APPENDIX E

GLOSSARY OF NOTATION FOR DBCS

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C \Jj ( I L: 1'15 (.J::')

C" -=.~ ./ ( J

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t ,,~ .. t ~ ~ ( ) t F ( J)

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(ONTT~UIf'l' C~~F"rIrT£NT

8~A~ T~C~E~ENT3 nF"N0~1~6T04 f-lEA~ n,CHE"'E",TC; FnQ TNTr;,q~-:LATI~'\ ~\

TAHLE5 f, 6"n 1 ~ENDtNG MO~E""T CO~TINUIT' CC"FI~TFNT

I? 1 Jill 07Jlil ,'11_'lI1 " 1 Jl' 1 C 7 JL' 1 1')1 JvI \17 Jill 07 )( I \

07":Ul STATTr T"~~SVF~S': TnQ\1L'F: tI"-P-'; 1."-_ ~ULTTPLIl~ I'" CC~TTN~IT'I' COEFF" r~5 SHEAQ BEI~EE. 'nJ,rFNT ~TATIONS rrNO~I~A}OO

r0TAL107JUl 07JUI 07 ,)U]

C7JUI Ll,ro'PED fXTE~",6L OAt.tPTNG COE"FICI~f'IIT

fl".lD TCT 6t ) o l"EOfNCE

! INPUT07JUI 07 JLI, 01 III 1

LU~P~~ rNTfP~AL ns~PING COEF"F1:Ir;\1 ,~"L TCTALI

: ~··;::lJT07J~·: 07JLJl C 7 ,,,,I n 1 Jlll

FI"'~T OE .... rVATIVF" nr O,MCOL ,)EFl (C:"~p~\ 5~ECIFIEv \I"L'IE OF" SLOP~ AT 5;A IS Tf··p~l:::Jr.R'I' \lAl U~ rF" n .. C;f 1

TP'~ IN CO'lIN'JITY cnEf', US F~CTrI:::JS fO~ cIc;r~IQdTl",r; HALF ~'!, l.'t.S

['0 S TA~ uLJLTTPLIEW FnQ ~ALr; VALUES AT E~r STAS FLt:XI'~AL STIF"P.JF.:S~ IFl) (I"-JPLIT 6"12' i(,TALI REAM rF\C~F"'E~ T L~~JI'lT~

'"' SOl"·~E..J I- CIJi-lE r-

T I ""E P,C~t: "'E~ T LE"J~ TH ... TI~F"S C I-4T <:~II-'1;H.[1

~ Cur~: ~llJlrFn ~'I' ~T

1-4 CUREC u!VICFO ~'I' HT S~U6~Er

~Ut-' OF PLOTS TO RF PIOTTEO 0'" C'ljF F ~AI.IE TA DI\llO[(J By 2 IT" TFGEQ)

IV,kIARLES "n~ ST~CTN~ T~E ~LrT rTrLt~ f'..L'·' rF- I-lUIF\Tc: 1~, JTf-I J..'LtiT . swITCH f,-,'p..( FIQC;T II~LOAL11NG Cllc:t: BEAM C;TA ~I-ICH IS C:PECIF"IE') r;("~ r.1::'';':1 A~£-

"'E~l I~ TA.RLF" 1 INITIAL £XTEpNAL I=IFA~ STA(T481 ~c: .. A"r 5' I~ITrAL ANC FIN4L FXrfQ~'AL ST6 I~A~Lf ~)

FI~AL FlTE~N6L ~EA"" ~TA(TA4lF~ ~ 6\~ ~\

TE~P~~AR'I' \lAII}!=: Or I"i13f ) I\ll~ I"':F ~~L.INLI~fAI:::J ~lIPPO~TS. '''~E~E ,u..,P,-,TfO

I)EI=L. A~E r')C;CTLl6TI',r, !ol.tT\oIEE".. ""J:;r('~

A"'O ACJI STFC ~-\Jj C"~\I~'·, ,,"I"'I~' LI\.:..,"'I6('1_ P,G C~C:;: """"D~

"'(IUTp.iG :::'''ITCH :-OQ INITTAlLY A"-i "'t,.";"6-Nf~TL'I' C:P~cl~TFn cn~OITl~~S

(\1 ~Ill 1 0'" JlJl 1"17 Jl' 1 07 )1.' 1 07 JU I 07 JU 1 O?JUI n 7 J" J C 7 JU) 07.1L11 07 ..1l' I 07 1\' 1 01.1\11 07 JL,! 07J~1 "7 Jl'l 07 JL' 1 D7JlJl ;j 1 JU 1 (11 ~llJ 1 07 Jll I 1'l1Ju1 01 J(!!

07 Jll) ('I1JLll n1J U l 07 Jl'l 01 Jt,' I 07 Jl' I o 1Jl' 1 07JlJI

1 C!= 5 ' 1

I o~ 5 A"

IG5 \If ( 1 IPS

C C If~ 5 C I C;' (

" IQ?! 1

IHC lL I <; I, I 1

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" I ..J? < K t.SE (

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KC~T( l I'<.EEP2 THRoJ ,[E"9 KEYr 1

I'<.E:YP( 1

I'<.E'I'PTt I. KYS T (

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I( ~~ x

I

SPECIFIEu SWTTCH INPUT ,N T'.Ll , TU "DI_07JUI (~TE T1-46T T~F ~oECIFlfr) (ON~IIIn~JS Q£:'01JUI ~ISPL~(EME~Tc: 6I:::JE PERJANF"~T O~ tNITt601JUl

TfIo(Pr~~.I::r \lAI UE OF" t.,P5( ) 07JUI ROUTING SwITCHllOAC ITEOATION MET,OD\ 07JU' CPTInNAl SWITCH Fnp PRINTI~G THE RlSIJLTS 07JU]

OF MO~ITO.Q 'STA5 (-1, ~LANK. A .... )) "7JUI INOICATO_ THAT II, I~ EvEN OR 000 07JUI INITI6L ANC F"IN6L F~TERNAL 5T6 (TAtjLF 71 07JUI O~TION Tv ~OVE TO 6 NEw PLr')T F"~A~E 07JUI INTE~Nt.L AEA~ C;T6 ~HE~E A NO~t INFAIoI: 07JUI

<"PPOOT EXIST< 07JUI ~~UTlNG 5~ITCI-4 ~nQ HALF" ~Al_UE~ ~7JUI

.uUTING SwlTCHIjST CYCLF O. TTE"":IO'Sl 07JUI ROUTING S~ITCHI?n CYCLE O. !TE""Tl0~SI 07JUI INITIAL ANO FINIl FXTERNAl STA ITAblE 6) 07JUI OOUTI~G ,0ITCH TO INDICATE THAT TME P00- O?JUI

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

I' IAelE ] C7J 1c] INTEq~AL 8EA~ 5TA ~nQ p~tNrI~~ T~E ~~5'jLT~07J~1 INITIAL rNTE~~AL STA(T6RLES b A~~ 7) :7Jl.l FINAL INTll:::JNAL ~T6fTAAL~S ~ AND 7· 07JUI I~ITIAL INTE~NAL ~~AM STA t;SEn r~ TAqLE p ,7Jul FINAL INTEONIl q'A. STA USfO r' TAdlE ~ 07JUI I""TEq~AL lIME ST4 1)1Jl)1 Cf.SE NllM OF t:;PF:CIF"fED cnNOITTnNC;, S1Jll

\ ~ I'EFL. ? ~ SLOPF" , 1 = hOT~ 07J~~J SwITCf-I TLJ INOTCIITF" THAT OEj:L[l"ry"".:: ,:;lE :1J'v'1

NOT CLOSED!IF". 0) J7Jl'i COUNTER FO~ CLO!UQ.N~E 0F ~EFlECT!ONS 07JUI IF~lt KE£P PI:::JIOQ n6TA, TARLES 2 _ ~ 07JUl QOUTprG SWITCH F"OQ InE'~TIFYI"'~ C4SE.: ~II~ <")£:,01Jul

SPECIFIEr (ONnITIONS 0F nlsplACE.E'T~07JLJI S~ITC" Tu INOICATF THAT Pl0T ,XI< IS ?7JU]

TIM[IIF=1I OR BfAMIIF=2l ?7JIJ] TFMP~~ARY \lALUf OF KEYP{ ) S KYS( ) Fn~ 07JLl

PlOT CF MO_ENT ALONG THE BE •• A'IS 07JUI COUNTFR FD~ THREE CONSEcUTIVE TI.E STA 07J"1 C()LINTER FOR NUM oF- STAC; EXCEE~PJ~ /"lAX G1JUl

ALLU.ABlE nE~lErT!nN 07JUI

c c c C c c r C C C C C C C C C C C C c C

C t C (

c C (

C

c c C

r r C (

c

c (

<~CNTI

""1 I ) 1(1\'2 \ )

<OFFC (

KPC

f'i"SI. <0 I I jt;Ql{

<~ 1 KRH I I. ~R25 I ), KlllB I ) '5 ( I KS.V ,SAlIl

KSil4 ( j, KSW5 ( ).

"s.~ I ) ,SYlH ) KSYMJ!J) KTl ( ) KT"I ) ~Y5 ( I

LOF

L5~

LSlA ~

,..rltI h',tIT _r _C~I

~PC ( ) ~PCT I I

~PS I )

"PI II-"U "P1 "T "TF2 .T •• ~A

1\,\(

vALU. OF KCNT( I (nRST rTE4AT10Nl C7J;;[ INITIAL INTEpNAL ilME 51A,TA!lES b A~D 7, 07J"1 FINAl INTERNAL TI~F. STAITA:lLE~ 6 "0 7' 07J'l SwiTCH TO INDICATF TMAT DULfCTloNS EtCEEon7JUl

LIMITS 0' SUPPORT CURvES OTJul ROUTING CO~NTER FnQ (ouPLETE PRI~T-O~T 07JUI

OF ~ESUL TS 07 JU 1 CO~PLETE PPINTOUT OF TMf RESLLTS OF EVERy 07JUI

STA AT EVERY KPI TIME ST. 07JUI INIERNAL TI"E STA I'F ~PI I <PI'4 I OTJUI INITIAL EXTERNAL TI"r STA IjSEn I'! TA~LE 7 07JUI FINAL ExTER'IAL Tly' <TA ,'c<n I" YA~lE 1 07JlI !'!'IOR VALUE OF "R?4 I )0 KR25 I " K';ZA I) Q7JUI CONTINUE SWITCH Fn~ rNPliT COUI~III~G ON Q7JUI

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

SYMMETRY OPTION U~FD I~ TA~Lf SPECIFIED VALUE O~ KSYM( I AI EACH ST. J INITIAL EXTE~N'l Tr"F STA IISEe IN 'A8U 6 FINAL ExTERNAL TluE STA USED IN ,ASlE 6 S.ITCH TO INOIC_TE THAT VE_' PLOI AAIS I~

DEFlECTION.IF"I) OR M""FNTI,F"Z' OPTIONAL LINE OR POINT PLOT S,ITcH .Hf~

USING MIcRoriLM OR RALL-POINT PLOTS SWITCH W~ICH CAUSFS THE AXIAL LOADS TO BE

DISTRI"uTEn TO HALF-STAS INUE. FOR EXTfR~AL BFA- STA 'IJM OF BEAY rNCREwEN'~ NUM OF PUINTS fOR EACH PLOT FRAME 'liM OF JTERATTO~S ~PFCIFlEI) IMAI • S'l ~UM OF CHA_ACTERS !N PLoT TITLES

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

~1;LT IPLE PLOT 5_1 TeM SAVED FOR E4C~ PL',T TEMPOIlARY VAL vF OF "pO I I ,OR PU)TS OF

MOMENTS ALONa T~E .EAM AXIS MULTrPLE PLOT S.ITCH I~PUT IN lA~LE qlI'

). PLOT IS SI)PERI,"POstu WITH NEH) Jr,I; PLUS ONE f,..RU W !)Lll~ ~E.VEt.t

~UM OF TIME INCpEuFNTS uT PL US T~C ~T PIIJS fOLP ~~~ Of PLOtS ~UPE~t~POS£O lIN r~~ FRAME INDEx OF NLM OF Pl~TS

07JUI 07 JU 1 07 )U 1 07JUj 07JvI o7JlJI Q 7Jl'J

~07JUI 07JUI 'J7Jlil 07 J!'l '11Jl!1 07JUI 07JU) 07JUI

C C C C C (

(

C C C

c C

C C C C C C C C C C (

C C C

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HTT

~PCC I

M" OF CAkeS INPUY IN TABLES ? T .. ~~ 4 <OP 01Jl!j TMIS PRc8LFu a7JUI

TOTAL N0~ OF CARD5 I~ TME PAQTICllLAR TABLr07JU! T~TAL .UM OF cURVES IN TABLE' 07JUj INITIAL I~CEx VALUF FOR THE INPUT To TME 07JUI

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

TIO~ .ETHO~ ~TAIlTS 07JUI SAVEr VALeE OF NIT FnR THE FIRST CYCLE OF 07JUI

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

I.OE. FOH COUNTING THE NU~ O' SLoPES OF 07JUI NONLI~EAR CV Q7JUI

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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This page replaces an intentionally blank page in the original --- CTR Library Digitization Team

APPENDIX F

FLOW CHARTS AND LISTING OF DECK OF PROGRAH DBCS

!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!

44!5"6!7$1*'*0!8$($.$9'.$/-!")':!

(--­

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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19tC4 04J·S 04JAS 2~SE7 2f5F 7 ;:6~n

04".S O~SEO 0'J'5 04J~~

OSSe, C 04';·5 04.AS Oc. rE' 0 04J A5 215£66 04JAS 210C4 IQOC4 04JAS 04JA5 05SEo 04JAS 04JAS 04JA~ 04J"5 usn 26SE7 1'Ir.C4 O~"EO 231")(4

Kk1 s: f(Q24 it<,) IF( ~'W4d"" ., (1 ) 14C:;0.14C;~.1.60

1450 PQINT 4l1. 1"-1",(11. Hi). KR240~)~ rr...2(,..). Cf)".~l~l. !)N2t"l. 1 T'?U:}. !N} t CN?{"")

i,v Tn 1470 J4C::~ P~P,T 41? p.]41";. ~f:;24tN), F'r-.?tt-Ji, QO~2(~lt t;N2ttn. TN2(",,.

J QI\?I~l. C".:2fN) 30 Tr'i 147~

1460 P~INT 413. TP\r4(r...)t ~~?'J(Nl. F",,';:H~'i. QD~2("')' S~2fN). T~2("'), 1 p,,~ H!l. Cr t? (/'.ol

1470 ceq!r". C----~I,..Pl'T 1'"lf !

lC;nO P~i"'T 50("' IF , 'FFP· J 9QSQ. 1~01. ISIO

} 501 ! C15 :: 1

NCT" '" t-·Cr,; (~O TO 1530

1510 t=:n:~:'T Qr.r;, rF( NfT!:) .. 1 ; iC;cfn' lC:;llt 1511

1~1l nO ,r;?~ t,,; 1, pol 1'~1"''5(''} _ if 3. 1511. 1521

,C,11 pr .. t",T 4)1, po; {r...}. (N). )(P2S{~h RfoiO"2(1'o!), OI"J2(I'I;). tEN?t"'} r-C' Jr- l52~

1511 Pf.lTfo.T 41;.'t Th1c:.:p". f();:?5(N,. R ... O""?tN1. OI"?(I".}t DE-N2t N )

c:~- T'" 1~?5 152] P~TI\T 41"1. 1:t-.?'50,) , 1{~?5(NI. P"'O"2(t·n. OINcH.l, DEN2{Nl 15?S '\;'f"

15?~ ll~

~.C]S It ,,(T5 • 1 NCT~ & P\CT5 • ~C05

153(1 If ( ~'Ci'!' ... lOC 1 '535. IS15t 1533 IS33 PRl~T 90.

GC" 1'1 15J~ fF ( ) ()9AO. 1537. 15-0 1~"7 DDt",T Q~3

r;!"' T (\ 1" n C ..... R 1 • 0

IF , ~CT5 - ,CIS ) 1~00. 1,,5, 1545 IS4S DO 1570 "E ~CI5. ~rT5

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..

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)(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

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3t",:i1'=" 0 300,[0 0'" .a CI1

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Q~~fO

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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

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I ~ " A 1 lL4J D;<;cO 1 1 "'( W'l'" 1<".1

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[':, '';'';'' ... ;' 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~

IF x(I'IIxXI.LT.THFTA} GO Tn .q5 IF xd·I1XX).GT.O""~r,A I GO Tn QIi

RET~ • ST~MA • ( XIT_IT'IIX) - THE'TA • 1. leTA. BFTA

IF IBETA - TOTo.) .r,~.n.c;, tOTA. IOTA -IF IOTA,GT,IMSK) IM~ •• TOTA

SPACEII0TA) • Sy"'~p,,0)

q5 CO~TI"UE .. 8 PRINT 12- lXIII, XI!), ( c;~ArE(L)' l.l~tt.4!;K I

00 qo J. I, 62 qO SPACEIJ) I: qLA"'I1(

100 CONTI~uE qqO RETURN

END SPLOT q END

1'"14 Iro ll~oq

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11 JfO 11 J~O lIJ~O

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N o V1

SU~ROUTI'IE lOT I ! XF, yr. "I', TO I C •••• Tt'tF LATEST UEVISIQN nATA FO~ Tlfl~ "OIiTIN, IS 11 JUN 10

CO ... "ON I lOI I LOP, MC, TROll.' .. 01' DIMENSIO~ XFfll, YFIlI, Inl') OATA I~C. 10, x. Y. XL. Vj. yo. yO I 1.12.0 •• 0.,Q •• 8 •• 0.,QO. DATA ITI' IT2 I -1"0 I

C XF ARRAY CONTAT~I'Ir. THr x _ rnnRol'lATES C vF ARRAY CONTAININ~ TH' Y _ conpnl'lATES C Np NUM8FR OF P~TNT~ Tn 8E pLnTTFO C LOP LINE OR POINT PLoT npTln" cO, LINf PlOT C • -J • pot"T PLOT AT EVFPy JoTIf POiNT C •• J , LINE oLOT WITH A POI"T PLOT AT EVFRy J_TIf PT. C 10 VARIAAL~ OR ARR,y co"TAT"T"G TITLE OF PLOT C MC NUMarR nF CIfARArTEP~ tN TTTLE I 0 If NO TITL' , C IROLL - OPTION TO MOVF TO A NEW 'PA"', - AfTER T~TS PLOT C • I , SAME fRAM, C EOUAL TO 0 • NEW FRANE C LESS THAN 0 , TFRMI"AT£ C MOP MiCROFILM OR PAPER 0LOT DOT 10" C • I • PAPER PLOTS C • -I , ... ICROflLM C •••• FRA~K E~ORES PAX lA92 ••••

C

C

IF IF IF IF

NC " MC ITI"ITl.1 ITl.NE.O , MOp'EO.1 ) MOP.EO, .... 1

I\IC·"'4£.O ) NC " 10 10 " 101f

CO"'INUE POSITION aRIGI" IF ! I12.EO.O, If ! IT2.EO.l ) SCALE x - AXI~

r,n Tn 20 CALL 'l6""LT CALL AG,,"LT GO To "iii

CALL PU 0.0. I.e;. -J) 110 T" i nO

CALL SCALE I Xf. XL. NP. INt" C - - - • SCALE y-AXIS

CALL SCALE ! Yl'. YL. NP. IN" C - - - ~ SET UP X-AXIS

y~ • - 'FC~p.l) I Y~(NP.Z'

c CALL AXIS (x. YMtlH , -1, XL. XO. xF(NP+l), XF(NP.~l •• SET UP Y-AXIS

x~ • - XFrHP.l) I X~(NP·2J CALL AXIS C XM, Y, IH , 1, 'l. YO. YF(NP.lt, YF(NP'~}

C _ • - • PRrNT TITLE CALL SY~ROL C .1, -,~, -I'. TO_ yO. ~c

C - ~ - • PLvT TIfE rU"CT!ON lr.O CALL LfN£ ()O", .,r-, !"W,:;., 1. L\}j:r, li~ I

If I !T2.ECl.1 I IF I IROLL.Ea. o IF I IROLL.L T.o If I !ROLL.GT.O IF I IROLL.LI.O 1

rT? • 0 CALL PLT l ,1\, COLI. FNn"LT tT? • 1 TTl • _1

.0, 999 )

11 JFO DEVISEo 11.1rO ° 9J<0 o9JrO r.9JFO n9J~O

09J~O

19JrO "9J,O M~J'O o\lJ,O n9J'0 "9JEO °9J,O '9JEO ,9J'O "JEO o'J!,O n9J~O 09JEO n9JEO O'J~O n9JEO o'Jr O n9JrO n9J!,O 09JEO o9JFO ,,9J[0 o9JrO .,9J[0 "\lJ~O '9J,O n9JrO n9JrO '9J[0 o9JeO o9JfO n9J,0 n\lJ[O "'JEO "JEO o9J ro "JEO ,,9JrO r"J£O o'J[O -:'}-.jE:~

09JEO O.IlJFO r9JrO ollJFO ! I J<O

N o 0'

APPENDIX G

LISTING OF DATA FOR EXAMPLE PROBLEMS

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0 41 4 5 -8.Z50[.02.A,ZI0[·02-7,500[·OZ 1070 1'13l 2465 30ZZ 3380 3135 4Z40 4750 5800 5800

0 41 I 6 _7.~00[.02_7.500[·02.6.250[.02 34 2"2 418 169 1072 14.4 2119 2963 !l42620000

0 41 6 7 .6.'50[.02.6.250[·OZ-5. 15 OE·02 13 1-I,OOOE-OO 1.000E-OJ 10 1 1.000E_04 1.000£'00 0 41 7 8 _~,150E.02_5,150E.Ol_3.150[·02 2650 47nO 6100 7490 836!1 4240105001175014380143RO 0 41 ~ 9 _3.R50E.02.3.850[·OZ-2. 75OE.02 34 20Z 4111 7,,9 107Z 1444 2119 2963 ~4Z620000 0 41 9 10 _~.750E.02.Z,710[·02·1.600E.02 16 1-1.000E·00 1.000E-03 10 1 I.OOOf-04 1,000['00 0 41 10 11 _1.600E.02.1.600£.OZ.7.0 00£.01 102011840156961 952221 72223652z5340263052R23528235 0 41 11 12 _7,nOOE.01.1.000E·01 0.0 34 2"2 41R 769 1072 1444 Z1\9 2963 5-2620000

S PARTIALLY ~M~EDO[O STEEL plpr LOAOED BY WAYE ~ODCE! 31 0-1.000E·00 1.000E-03 10 I 1.000~_04 I.~OOE·OO

0 0 0 0 0 0 0 2 0 7 2 0 32 12 10 10Z01184015"9619~222172223652253402630S2B23S28235

31 4.100E.Ol 400 1.000[-02 0 _1 0 1 1 34 202 .\8 169 1072 14.4 2119 2963 5'2620~00 50 2.000[,01 1.000[-06 TIM[ DEFL 10 1

\ I 0 3.0001"01 2.UO£·10 TINE OE'L 21 0 31 31 0 1.000t·1D TIM! _T 10 1

0 3\ 0 2.877['11 TI141 "'O"T 21 0 1 1 -2.000E·04 8EAM DE'I- 60 !

II 0 -I.OOOE.~. BUM OE'I- 160 0 I -1.220E·04 I£AM OE'I- 230 I

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

N I-' N

flUM DE"'!. 4 1 flUM OEF'L " j fiE AM DE"'!. lO 0 SEAM MO"r 12 1 IIEAM MOMT 16 1 SEA" MOMT 30 0

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

APPENDIX H

PARTIAL SAMPLE COMPUTER OUTPUT FOR EXAMPLE PROBLEMS

!!!!!!!!!!!!!!!!!!!"#$%!&'()!*)&+',)%!'-!$-.)-.$/-'++0!1+'-2!&'()!$-!.#)!/*$($-'+3!

44!5"6!7$1*'*0!8$($.$9'.$/-!")':!

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

-, .4.ttnOE·Ol S."OIE-I~ ti. -1.,.?~-19 1 ,Ii!ltr-I?

O. 0, I. i14F-19

~. 300£-11 -6.6'iH, .. j6

•• 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

1.440E.Oc ?.~1'-11 ... ~.t;Qf,F'-ll ~,'9? .. _20 -l.lQQF'-l?

4 l.970E.02 2.708<-11 _1.2141'-1 0

_1 •• 21'< •• 19 -1.1R9r::-it» ~ c.4nOE.02 l.q20~-11 _1.H~E·l0

_1\ •• 19 •• 19 -1.19""·12 ~ 2. 6~O(. 02 -6. ~I O,,_IA _'!.341r_ 1O

-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

_?76~F_la 1.430"-P

SUP REACT

0,

_I,U9E-12

0,

O •

O •

O.

O.

o.

o.

2.625E-12

o.

10

II

I>.

I'

14

I~

I·

10

21

22

23

2"

29

30

3'

7. b~OE -02

1.Ol)bE+03

1.104£-03

I.1S2E.03

1.200E.03

-3. n2~-16

-S.217E-I,

-1.236E-15

-1.315E-15

-I. ?42E-15

-1.144E-15

-7.:?71E-16

_?Jt18E_18

_?~::'6f_18

?042f_ 18

?~07[-18

_15.2 .. "'f-12

~ •• bO'-IO

3.261[-1~

3.50 <;'-10

3.'~4f-'0

1. 4 161='-1::'

1.43""1=" .. ,-::,

.. 3.50"PI:"_13

-3.507[_13

3. 0 .4r:- 1 t)

-3.~07F.-13 2.8HE-IO

-3.!;01F" .. IJ ?S"8(-10

-3.507~_IJ 2.3?2E-IO

~.OJJE-IO -3.S07E-;3

I.'?~(_IO -3.507~-i3

1. 4 0IE-10 -3.507E_13

n.

O.

n.

o.

O.

_1.09.E-12

O.

O.

c.

O.

O.

O.

O.

0,

O.

1.29bE.03 -5.~73'-1' I.OIIU-IC O.

1.:44E.03 -3.772E-16

1.30lE.03 -1.on3E-16

'. I '9E-' I O. -3.S07E-13

3.5""-11 O.

1.44IJE+03 o. n. O.

1.qOJE-16 0.

frM[ c:TA ~ •

~TATIONS

J = 8 14'1 17

TTE~ATION. THE~E AQF n ~T4TIn~lC; ~n! CI.05£0

"b.bOIF-17 -,.104F-l~ -).?ZJ~-15 -1.l10F"-I~

0,

••••• DY~4MIr ~ESULTS •••••

TIJ.4[ C;TA jo( =

5TA J DIST SUP REACT

-,

•

9

10

II

I~

13

14

1~

16

17

I"

10

21

22

2,

24

O.

3.a4nE.n2

4.320E.02

4.t:itlOE.02

6!120E.02

7.200E.02

O.

-1.104E-15

?,oC'SI=" .. lq -1.41"1='-17 -I. I, 2E- I 0

~.~n6F'_20 -1.414F' .. 1? -1.8t'oo3f-1 0

.. 3,1""6,, .. 19 "l •• ~OI="-J?

_~.;:t.5F'_18

_3.4601=" .. 18

_~.OO;8r_IO

_2.01~E-10

_1.4?6r-1 1 ..J.4 Q Q",_I8

"'.,33E .. le

_t',QQ8F: .. 18

7.noE-11

1.679E-IO

2.~~5E-IO

-1.4Q'-F'-12'

1.6J~F-i2

1.600~_1?

1.511'-1?

O.

O.

O.

O.

O.

n.

O.

O.

O.

O.

O.

O.

8.b'OE'02 -1.36IE-15 3.514'-10 O.

9.1~OE.02 -1.379E-l~ 3.J44E-IO o. ~.o.0<_19 -3.qO~<-j1

9.bf')O(.Oc -1.1t'ooSE-15 '.14:?~-t" O.

1.00~E.03 -1.3?IE-I~

1.104F.Ol -I.I"IF-IS

1.1"2E.03 -1.030E-IS

1.~06F_18

?03 7f_ 18

;t.e;12 ... 18

O.

2.6~~f.-IO O.

?3'<;F-IO O. _3.74~F'_'1

:::!.07I:SE-l" O. -3.705F _13

25 1.2nnE.03 -~.A98r-l~ 1.7~~f-10 0, 1.?"P8F'_lB -3.66't~_'1

\,4~JE-I 0 O. -3,629~-;1

o.

O.

20 n.

Jl

STA J

1'\

I ..

~TATIONS

J • I~ !1

ITERATION, T"ERE ARE ,. ~TATloNS NOT ~LOSEn

-0,665'-03 -1.190f-ol -1.10 7[-01 ·1.062'·01

'OR Z ITERATION, T~E.E A., 0 STATlnN. NOT CLOSED -o,665E-03 -1.19nE-nj -1.147E-OI -1.462'-01

'OR

fOR

OIST

8.160E'02 -1.462(-01

T1"''' STA " •

J • 8

SHEAR

-1.3S2r·02

STATIONS 1" 17

lTERATIO", T"ERE ARE .9 5TATloN'I NOT CLOSED -,.690'-02 -1.'2oE-01 -'.~31'-OI -3.eeor-.1

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 " ..

STA J OIST

15

I"

11

7.Z00E.OZ • 2 •• 98E-Ol

-3.404[-01

8.16~E'OZ -3.744E-0!

TI"'E STA " s

SUP IIEACT

SUP R[ACT

STA J

IS

11

~TAT1ONS

J = ,. , , ITERATION, T"ERE AO< ,Q ~TATI~~~ ~nT CLOSED

-c.qqlE-O~ ·~.~~~~·~1 -~.4bA~-Ol ·7.1AO~-~1

'OR Z ITERATION, T~ERE ARf ~ ~TATlou. ~nT (LOSE" -2,99IE-~Z -S.6~'E-,i - ••• 6""-01 -7,180'-01

J • e

T11'1[ ~TA t( •

"'ONITaR 1'1

2.727E·05

5TATIONS 1~ 17

IT(IIATI0N. T~ERE ARE '9 STATlnNS NOT CLoSEn ~ •• 613~-02 -A.Aq'E-nl -l.nlQE·OO ·1.137~·OO

rOR 2 ITERATION, T ... ERE .RE .9 STATIDN~ NOT CLOSED --.136'-02 -".3~.E-ij, -Q,,,,"r-Ol -I,07eE'OO

~OR J ITER.TION, TH[~E ARE .Q STATloN~ ~OT CLOSEO ~4,576E.02 ~~.~Ro(·n; ~Q.Q~7~-01 -1.lll~·n~

'OR ITERATION, THERE ARE n ~TATI"~~ NDT CLoSEO ·'.576f-02 - •• ~.oE-O' -Q.951~-OI -1.lllr·n~

T!~E sa ••

STA J DIST ... ~ .. SMEAQ

-I -.,SOOE.OI

0,

4.600E.Ol

1,71111':-0?

J.1UE-02

" PlE-02

O.

_'.50'5E·00 2.i35~-04 -1.0~5~.OJ

_2.:>63,,_00

-5.,,12[_04

SUP IIEACT

o.

-1.028E+01

o •

0,

N N o

q

I~

II

I~

I~

I~

I~

Iq

21

24

2<;

27

Z.

31

8.I,OE.02 -1.lIIE'OO

8.b40E.02 -1.lq2E'00

Q.120E.Oc -1.238~'00

_1.,7Br_03

_q.<bOf_ O•

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

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TIME ~TA

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-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

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17

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FOR

STA J

A

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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

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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

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A.' 5 'TATI~N' ~nT CLOSEn l.l0~E-nl I.Qr9~-OI 3.008E-OI

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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

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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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This page replaces an intentionally blank page in the original --- CTR Library Digitization Team

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